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704
Sparsity-certifying Graph Decompositions
We describe a new algorithm, the $(k,\ell)$-pebble game with colors, and use it obtain a characterization of the family of $(k,\ell)$-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the $(k,\ell)$-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and Westermann and Hendrickson.
Sparsity-certifying Graph Decompositions Ileana Streinu1∗, Louis Theran2 1 Department of Computer Science, Smith College, Northampton, MA. e-mail: streinu@cs.smith.edu 2 Department of Computer Science, University of Massachusetts Amherst. e-mail: theran@cs.umass.edu Abstract. We describe a new algorithm, the (k, `)-pebble game with colors, and use it to obtain a charac- terization of the family of (k, `)-sparse graphs and algorithmic solutions to a family of problems concern- ing tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characteri- zation of arboricity. We also present a new decomposition that certifies sparsity based on the (k, `)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9]. 1. Introduction and preliminaries The focus of this paper is decompositions of (k, `)-sparse graphs into edge-disjoint subgraphs that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a graph is (k, `)-sparse if no subset of n′ vertices spans more than kn′− ` edges in the graph; a (k, `)-sparse graph with kn′− ` edges is (k, `)-tight. We call the range k ≤ `≤ 2k−1 the upper range of sparse graphs and 0≤ `≤ k the lower range. In this paper, we present efficient algorithms for finding decompositions that certify sparsity in the upper range of `. Our algorithms also apply in the lower range, which was already ad- dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs and graphs admitting the decomposition coincide. Our algorithms are based on a new characterization of sparse graphs, which we call the pebble game with colors. The pebble game with colors is a simple graph construction rule that produces a sparse graph along with a sparsity-certifying decomposition. We define and study a canonical class of pebble game constructions, which correspond to previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide a unifying framework for all the previously known special cases, including Nash-Williams- Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the properties of the augmenting paths used in matroid union and intersection algorithms[5, 6]. Since the sparse graphs in the upper range are not known to be unions or intersections of the matroids for which there are efficient augmenting path algorithms, these do not easily apply in ∗ Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author. 2 Ileana Streinu, Louis Theran Term Meaning Sparse graph G Every non-empty subgraph on n′ vertices has ≤ kn′− ` edges Tight graph G G = (V,E) is sparse and |V |= n, |E|= kn− ` Block H in G G is sparse, and H is a tight subgraph Component H of G G is sparse and H is a maximal block Map-graph Graph that admits an out-degree-exactly-one orientation (k, `)-maps-and-trees Edge-disjoint union of ` trees and (k− `) map-grpahs `Tk Union of ` trees, each vertex is in exactly k of them Set of tree-pieces of an `Tk induced on V ′ ⊂V Pieces of trees in the `Tk spanned by E(V ′) Proper `Tk Every V ′ ⊂V contains ≥ ` pieces of trees from the `Tk Table 1. Sparse graph and decomposition terminology used in this paper. the upper range. Pebble game with colors constructions may thus be considered a strengthening of augmenting paths to the upper range of matroidal sparse graphs. 1.1. Sparse graphs A graph is (k, `)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤ kn′− `. We observe that this condition implies that 0 ≤ ` ≤ 2k− 1, and from now on in this paper we will make this assumption. A sparse graph that has n vertices and exactly kn−` edges is called tight. For a graph G = (V,E), and V ′ ⊂ V , we use the notation span(V ′) for the number of edges in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge. There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of a sparse graph. A component is a maximal block. Table 1 summarizes the sparse graph terminology used in this paper. 1.2. Sparsity-certifying decompositions A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees. Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight graphs. A map-graph is a graph that admits an orientation such that the out-degree of each vertex is exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map- graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible configuration certifying that each color forms a map-graph. Map-graphs may be equivalently defined (see, e.g., [18]) as having exactly one cycle per connected component.1 A (k, `)-maps-and-trees is a graph that admits a decomposition into k− ` edge-disjoint map-graphs and ` spanning trees. Another characterization of map-graphs, which we will use extensively in this paper, is as the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that the converse holds as well. 1 Our terminology follows Lovász in [16]. In the matroid literature map-graphs are sometimes known as bases of the bicycle matroid or spanning pseudoforests. Sparsity-certifying Graph Decompositions 3 Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a (2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is shown with a certifying orientation. A `Tk is a decomposition into ` edge-disjoint (not necessarily spanning) trees such that each vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2. Given a subgraph G′ of a `Tk graph G, the set of tree-pieces in G′ is the collection of the components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come from the same tree or be single-vertex “empty trees.” It is also helpful to note that the definition of a tree-piece is relative to a specific subgraph. An `Tk decomposition is proper if the set of tree-pieces in any subgraph G′ has size at least `. Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree- pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges. These count as three tree-pieces, even though they come from the same back tree when the whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three gray tree-pieces and one black one. Table 1 contains the decomposition terminology used in this paper. The decomposition problem. We define the decomposition problem for sparse graphs as tak- ing a graph as its input and producing as output, a decomposition that can be used to certify spar- sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper `Tk decompositions; and the pebble-game-with-colors decomposition, which is defined in the next section. 2. Historical background The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint, 4 Ileana Streinu, Louis Theran Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a single vertex) and one black tree-piece. Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps- and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19]. In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman) graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay [21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a direct proof of Laman’s theorem and generalized the 3T2 condition to all `Tk for k≤ `≤ 2k−1. Haas [7] studied `Tk decompositions in detail and proved the equivalence of tight graphs and proper `Tk graphs for the general upper range. We observe that aside from our new pebble- game-with-colors decomposition, all the combinatorial characterizations of the upper range of sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24]. A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick- son’s Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and Streinu [12] generalized the pebble game to the entire range of parameters 0≤ `≤ 2k−1, and left as an open problem using the pebble game to find sparsity certifying decompositions. 3. The pebble game with colors Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative integers k and `. We will use the pebble game with colors as the basis of an efficient algorithm for the decomposition problem later in this paper. Since the phrase “with colors” is necessary only for comparison to [12], we will omit it in the rest of the paper when the context is clear. Sparsity-certifying Graph Decompositions 5 We now present the pebble game with colors. The game is played by a single player on a fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the addition and/or orientation of an edge. At any moment of time, the state of the game is captured by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored by the pebbles on them. While playing the pebble game all edges are directed, and we use the notation vw to indicate a directed edge from v to w. We describe the pebble game with colors in terms of its initial configuration and the allowed moves. Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices are shown as black or gray dots. Edges are colored with the color of the pebble on them. Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2, . . . ,k. Add-edge-with-colors: Let v and w be vertices with at least `+1 pebbles on them. Assume (w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw to E(H) and put the pebble picked up from v on the new edge. Figure 3(a) shows examples of the add-edge move. Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace vw with wv in E(H); put the pebble that was on vw on v; and put p on wv. Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows examples. The convention in these figures, and throughout this paper, is that pebbles on vertices are represented as colored dots, and that edges are shown in the color of the pebble on them. From the definition of the pebble-slide move, it is easy to see that a particular pebble is always either on the vertex where it started or on an edge that has this vertex as the tail. However, when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is sometimes convenient to think of this path reversal sequence as bringing a pebble from the end of the path to the beginning. The output of playing the pebble game is its complete configuration. Output: At the end of the game, we obtain the directed graph H, along with the location and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble game configuration colors the edges. We say that the underlying undirected graph G of H is constructed by the (k, `)-pebble game or that H is a pebble-game graph. Since each edge of H has exactly one pebble on it, the pebble game’s configuration partitions the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble- game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a pebble-game decomposition. Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges, and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con- 6 Ileana Streinu, Louis Theran (a) (b) (c) Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph. (c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges contain a cycle and do not contribute a piece of tree to the subgraph. Notation Meaning span(V ′) Number of edges spanned in H by V ′ ⊂V ; i.e. |EH(V ′)| peb(V ′) Number of pebbles on V ′ ⊂V out(V ′) Number of edges vw in H with v ∈V ′ and w ∈V −V ′ pebi(v) Number of pebbles of color ci on v ∈V outi(v) Number of edges vw colored ci for v ∈V Table 2. Pebble game notation used in this paper. nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′ otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with the corresponding definition for `Tk s, the set of tree-pieces is defined relative to a specific sub- graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned by G′. The properties of pebble-game decompositions are studied in Section 6, and Theorem 2 shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows this. For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom- position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees. In the following discussion, we use the notation peb(v) for the number of pebbles on v and pebi(v) to indicate the number of pebbles of colors i on v. Table 2 lists the pebble game notation used in this paper. 4. Our Results We describe our results in this section. The rest of the paper provides the proofs. Sparsity-certifying Graph Decompositions 7 Our first result is a strengthening of the pebble games of [12] to include colors. It says that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games discussed in this paper are our pebble game with colors unless noted explicitly. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. Next we consider pebble-game decompositions, showing that they are a generalization of proper `Tk decompositions that extend to the entire matroidal range of sparse graphs. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained by playing the pebble game defined in the previous section. Notice the similarity between the requirement that the set of tree-pieces have size at least ` in Theorem 2 and the definition of a proper `Tk . Our next results show that for any pebble-game graph, we can specialize its pebble game construction to generate a decomposition that is a maps-and-trees or proper `Tk . We call these specialized pebble game constructions canonical, and using canonical pebble game construc- tions, we obtain new direct proofs of existing arboricity results. We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo- sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning trees contributes at least one piece of tree to every subgraph. The case of proper `Tk graphs is more subtle; if each color in a pebble-game decomposition is a forest, then we have found a proper `Tk , but this class is a subset of all possible proper `Tk decompositions of a tight graph. We show that this class of proper `Tk decompositions is sufficient to certify sparsity. We now state the main theorem for the upper and lower range. Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Theorem 4 (Main Theorem (Upper Range): Proper `Tk graphs coincide with pebble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. As corollaries, we obtain the existing decomposition results for sparse graphs. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . Efficiently finding canonical pebble game constructions. The proofs of Theorem 3 and Theo- rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem. Our last result improves on this, showing that a canonical pebble game construction, and thus 8 Ileana Streinu, Louis Theran a maps-and-trees or proper `Tk decomposition can be found using a pebble game algorithm in O(n2) time and space. These time and space bounds mean that our algorithm can be combined with those of [12] without any change in complexity. 5. Pebble game graphs In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game with colors. Since many of the relevant properties of the pebble game with colors carry over directly from the pebble games of [12], we refer the reader there for the proofs. We begin by establishing some invariants that hold during the execution of the pebble game. Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following invariants are maintained in H: (I1) There are at least ` pebbles on V . [12] (I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12] (I3) For each V ′ ⊂V , span(V ′)+out(V ′)+peb(V ′) = kn′. [12] (I4) For every vertex v ∈V , outi(v)+pebi(v) = 1. (I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with a pebble of color ci or a cycle. Proof. (I1), (I2), and (I3) come directly from [12]. (I4) This invariant clearly holds at the initialization phase of the pebble game with colors. That add-edge and pebble-slide moves preserve (I4) is clear from inspection. (I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of the same color on it. If there is no pebble of that color reachable, then the path must eventually visit some vertex twice. From these invariants, we can show that the pebble game constructible graphs are sparse. Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the pebble game. Then H is sparse. If there are exactly ` pebbles on V (H), then H is tight. The main step in proving that every sparse graph is a pebble-game graph is the following. Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce the out degree of v by one. Lemma 9 (The `+1 pebble condition [12]). Let vw be an edge such that H + vw is sparse. If peb({v,w}) < `+1, then a pebble not on {v,w} can be brought to either v or w. It follows that any sparse graph has a pebble game construction. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. 6. The pebble-game-with-colors decomposition In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We start with the following lemmas about the structure of monochromatic connected components in H, the directed graph maintained during the pebble game. Sparsity-certifying Graph Decompositions 9 Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub- graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for i = 1, . . . ,k. Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex. Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H in a pebble game construction contains at least ` monochromatic tree-pieces, and each of these is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge. Recall that an out-edge from a subgraph H ′ = (V ′,E ′) is an edge vw with v∈V ′ and vw /∈ E ′. Proof. Let H ′ = (V ′,E ′) be a non-empty subgraph of H, and assume without loss of generality that H ′ is induced by V ′. By (I3), out(V ′)+ peb(V ′) ≥ `. We will show that each pebble and out-edge tail is the root of a tree-piece. Consider a vertex v ∈ V ′ and a color ci. By (I4) there is a unique monochromatic directed path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle. Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the monochromatic path from v leaves V ′), then the path cannot have a cycle in H ′. Since this argument works for any vertex in any color, for each color there is a partitioning of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each pebble and out-edge tail is the root of a monochromatic tree, as desired. Applied to the whole graph Lemma 11 gives us the following. Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of color ci is the root of a (possibly empty) monochromatic tree-piece of color ci. Remark: Haas showed in [7] that in a `Tk , a subgraph induced by n′ ≥ 2 vertices with m′ edges has exactly kn′−m′ tree-pieces in it. Lemma 11 strengthens Haas’ result by extending it to the lower range and giving a construction that finds the tree-pieces, showing the connection between the `+1 pebble condition and the hereditary condition on proper `Tk . We conclude our investigation of arbitrary pebble game constructions with a description of the decomposition induced by the pebble game with colors. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub- graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs. For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can span at most n− ti edges; summing over all the colors shows that a graph with a pebble-game decomposition must be sparse. Apply Theorem 1 to complete the proof. Remark: We observe that a pebble-game decomposition for a Laman graph may be read out of the bipartite matching used in Hendrickson’s Laman graph extraction algorithm [9]. Indeed, pebble game orientations have a natural correspondence with the bipartite matchings used in 10 Ileana Streinu, Louis Theran Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there are no cycles in ` of the colors, then the trees rooted at the corresponding ` pebbles must be spanning, since they have n− 1 edges. Also, if each color forms a forest in an upper range pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de- composition is a proper `Tk . In the next section, we show that the pebble game can be specialized to correspond to maps- and-trees and proper `Tk decompositions. 7. Canonical Pebble Game Constructions In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves- tigation of decompositions induced by pebble game constructions by studying the case where a minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15 and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that this is always possible, implying that monochromatic map-graphs are created only when we add more than k(n′−1) edges to some set of n′ vertices. For the lower range, this implies that every color is a forest. Every decomposition characterization of tight graphs discussed above follows immediately from the main theorem, giving new proofs of the previous results in a unified framework. In the proof, we will use two specializations of the pebble game moves. The first is a modi- fication of the add-edge move. Canonical add-edge: When performing an add-edge move, cover the new edge with a color that is on both vertices if possible. If not, then take the highest numbered color present. The second is a restriction on which pebble-slide moves we allow. Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a monochromatic cycle. We call a pebble game construction that uses only these moves canonical. In this section we will show that every pebble-game graph has a canonical pebble game construction (Lemma 14 and Lemma 15) and that canonical pebble game constructions correspond to proper `Tk and maps-and-trees decompositions (Theorem 3 and Theorem 4). We begin with a technical lemma that motivates the definition of canonical pebble game constructions. It shows that the situations disallowed by the canonical moves are all the ways for cycles to form in the lowest ` colors. Lemma 13 (Monochromatic cycle creation). Let v ∈ V have a pebble p of color ci on it and let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created in exactly one of the following ways: (M1) The edge vw is added with an add-edge move. (M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse edge vw. Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7. By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble game construction, since the color of an edge only changes when it is inserted the first time or a new pebble is put on it by a pebble-slide move. Sparsity-certifying Graph Decompositions 11 vw vw Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are labeled according to their role in the definition of the moves. Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves, respectively, in a (2,0)-pebble game construction. We next show that if a graph has a pebble game construction, then it has a canonical peb- ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa- rately. The proof gives two constructions that implement the canonical add-edge and canonical pebble-slide moves. Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc- tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤ i ≤ `′, where `′ = min{k, `}. Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If this is not possible, then there are `+1 distinct colors present. Use the highest numbered color to cover the new edge. Remark: We note that in the upper range, there is always a repeated color, so no canonical add-edge moves create cycles in the upper range. The canonical pebble-slide move is defined by a global condition. To prove that we obtain the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma 9 to only canonical moves. The main step is to show that if there is any sequence of moves that reorients a path from v to w, then there is a sequence of canonical moves that does the same thing. Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading to an add-edge move can be replaced with one that has no (M2) steps and allows the same add-edge move. In other words, if it is possible to collect `+ 1 pebbles on the ends of an edge to be added, then it is possible to do this without creating any monochromatic cycles. 12 Ileana Streinu, Louis Theran Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this the shortcut construction by analogy to matroid union and intersection augmenting paths used in previous work on the lower range. Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one application of the shortcut construction reorients a simple path from a vertex w′ to w, and a path from v to w′ is preserved, the shortcut construction can be applied inductively to find the sequence of moves we want. Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle, shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is simple, and the shortcut construction can be applied inductively to it. Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v and w are contained in a maximal monochromatic tree of color ci. Call this tree H ′i , and observe that it is rooted at w. Now consider the edges reversed in our sequence of moves. As noted above, before we make any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on this path in H ′i . We modify our sequence of moves as follows: delete, from the beginning, every move before the one that reverses some edge yz; prepend onto what is left a sequence of moves that moves the pebble on w to z in H ′i . Sparsity-certifying Graph Decompositions 13 Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path. The path where the pebbles move is indicated by doubled lines. Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is (M2); (b) avoiding the (M2) and simplifying the path. Since no edges change color in the beginning of the new sequence, we have eliminated the (M2) move. Because our construction does not change any of the edges involved in the remaining tail of the original sequence, the part of the original path that is left in the new sequence will still be a simple path in H, meeting our initial hypothesis. The rest of the lemma follows by induction. Together Lemma 14 and Lemma 15 prove the following. Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction. Using canonical pebble game constructions, we can identify the tight pebble-game graphs with maps-and-trees and `Tk graphs. 14 Ileana Streinu, Louis Theran Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game graph. For the reverse direction, consider a canonical pebble game construction of a tight graph. From Lemma 8, we see that there are ` pebbles left on G at the end of the construction. The definition of the canonical add-edge move implies that there must be at least one pebble of each ci for i = 1,2, . . . , `. It follows that there is exactly one of each of these colors. By Lemma 12, each of these pebbles is the root of a monochromatic tree-piece with n− 1 edges, yielding the required ` edge-disjoint spanning trees. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. We next consider the decompositions induced by canonical pebble game constructions when `≥ k +1. Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb- ble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. Proof. As observed above, a proper `Tk decomposition must be sparse. What we need to show is that a canonical pebble game construction of a tight graph produces a proper `Tk . By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom- position into ` edge-disjoint trees. Finally, an application of (I4), shows that every vertex must in in exactly k of the trees, as required. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . 8. Pebble game algorithms for finding decompositions A naı̈ve implementation of the constructions in the previous section leads to an algorithm re- quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n) applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running time of Θ(n3) for the decomposition problem. In this section, we describe algorithms for the decomposition problem that run in time O(n2). We begin with the overall structure of the algorithm. Algorithm 17 (The canonical pebble game with colors). Input: A graph G. Output: A pebble-game graph H. Method: – Set V (H) = V (G) and place one pebble of each color on the vertices of H. – For each edge vw ∈ E(G) try to collect at least `+1 pebbles on v and w using pebble-slide moves as described by Lemma 15. Sparsity-certifying Graph Decompositions 15 – If at least `+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma 14, otherwise discard vw. – Finally, return H, and the locations of the pebbles. Correctness. Theorem 1 and the result from [24] that the sparse graphs are the independent sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction found is canonical, the main theorem shows that the coloring of the edges in H gives a maps- and-trees or proper `Tk decomposition. Complexity. We start by observing that the running time of Algorithm 17 is the time taken to process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an edge of G that is added to H. Each of the pebble game moves can be implemented in constant time. What remains is to describe an efficient way to find and move the pebbles. We use the following algorithm as a subroutine of Algorithm 17 to do this. Algorithm 18 (Finding a canonical path to a pebble.). Input: Vertices v and w, and a pebble game configuration on a directed graph H. Output: If a pebble was found, ‘yes’, and ‘no’ otherwise. The configuration of H is updated. Method: – Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and return ‘no.’ – Otherwise a pebble was found. We now have a path v = v1,e1, . . . ,ep−1,vp = u, where the vi are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use the array c[] to keep track of the colors of pebbles on vertices and edges after we move them and the array s[] to sketch out a canonical path from v to u by finding a successor for each edge. – Set s[u] = ‘end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in reverse order: vp,ep−1,ep−2, . . . ,e1,v1. For each i, check to see if c[vi] is set; if so, go on to the next i. Otherwise, check to see if c[vi+1] = c[ei]. – If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge. – Otherwise c[vi+1] 6= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2, . . . , fq−1,xq = x that is monochromatic in the color of the edges; set c[xi] = c[ fi] and s[xi] = fi for i = 1,2, . . . ,q−1. If c[x] = c[ fq−1], stop. Otherwise, recursively check that there is not a monochro- matic c[x] path from xq−1 to x using this same procedure. – Finally, slide pebbles along the path from the original endpoints v to u specified by the successor array s[v], s[s[v]], . . . The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut construction. Efficiency comes from the fact that instead of potentially moving the pebble back and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three times: once in the initial depth-first search, and twice while converting the initial path to a canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time spent processing edges in H. Although we have not discussed this explicity, for the algorithm to be efficient we need to maintain components as in [12]. After each accepted edge, the components of H can be updated in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1) time each. 16 Ileana Streinu, Louis Theran Summarizing, we have shown that the canonical pebble game with colors solves the decom- position problem in time O(n2). 9. An important special case: Rigidity in dimension 2 and slider-pinning In this short section we present a new application for the special case of practical importance, k = 2, ` = 3. As discussed in the introduction, Laman’s theorem [11] characterizes minimally rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com- binatorially, we model the bar-slider frameworks as simple graphs together with some loops placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each color. We characterize the minimally rigid bar-slider graphs [20] as graphs that are: 1. (2,3)-sparse for subgraphs containing no loops. 2. (2,0)-tight when loops are included. We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse graphs studied in our paper [14]. The connection with the pebble games in this paper is the following. Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph. Proof. Follows from invariant (I3) of Lemma 7. In [15], we study a special case of slider pinning where every slider is either vertical or horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction. For this axis parallel slider case, the minimally rigid graphs are characterized by: 1. (2,3)-sparse for subgraphs containing no loops. 2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each monochromatic tree spans exactly one loop of its color. This also has an interpretation in terms of colored pebble games. Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)- pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the graph of a minimally pinned axis-parallel bar-slider framework. Proof. Follows from Theorem 4, and Lemma 12. 10. Conclusions and open problems We presented a new characterization of (k, `)-sparse graphs, the pebble game with colors, and used it to give an efficient algorithm for finding decompositions of sparse graphs into edge- disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the upper range from [12]. We also used the pebble game with colors to describe a new sparsity-certifying decomposi- tion that applies to the entire matroidal range of sparse graphs. Sparsity-certifying Graph Decompositions 17 We defined and studied a class of canonical pebble game constructions that correspond to either a maps-and-trees or proper `Tk decomposition. This gives a new proof of the Tutte-Nash- Williams arboricity theorem and a unified proof of the previously studied decomposition cer- tificates of sparsity. Canonical pebble game constructions also show the relationship between the `+1 pebble condition, which applies to the upper range of `, to matroid union augmenting paths, which do not apply in the upper range. Algorithmic consequences and open problems. In [6], Gabow and Westermann give an O(n3/2) algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from dense ones. Their technique is based on efficiently finding matroid union augmenting paths, which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch scanning, which finds groups of disjoint augmenting paths. We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester- mann’s algorithm without changing the running time. The data structures used in the implemen- tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those used to support cyclic scanning. The two major open algorithmic problems related to the pebble game are then: Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain an implementable O(n3/2) algorithm for the lower range. Problem 2. Extend batch scanning to the `+1 pebble condition and derive an O(n3/2) pebble game algorithm for the upper range. In particular, it would be of practical importance to find an implementable O(n3/2) algorithm for decompositions into edge-disjoint spanning trees. References 1. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th European Symposium on Algorithms (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003) 2. Crapo, H.: On the generic rigidity of plane frameworks. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988) 3. Edmonds, J.: Minimum partition of a matroid into independent sets. J. Res. Nat. Bur. Standards Sect. B 69B, 67–72 (1965) 4. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Optimization—Eureka, You Shrink!, no. 2570 in LNCS, pp. 11–26. Springer (2003) 5. Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. Journal of Computer and System Sciences 50, 259–273 (1995) 6. Gabow, H.N., Westermann, H.H.: Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica 7(1), 465–497 (1992) 7. Haas, R.: Characterizations of arboricity of graphs. Ars Combinatorica 63, 129–137 (2002) 8. Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompo- sitions. Journal of Combinatorial Mathematics and Combinatorial Computing 62, 3–11 (2007) 9. Hendrickson, B.: Conditions for unique graph realizations. SIAM Journal on Computing 21(1), 65–84 (1992) 18 Ileana Streinu, Louis Theran 10. Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the pebble game. Journal of Computational Physics 137, 346–365 (1997) 11. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 331–340 (1970) 12. Lee, A., Streinu, I.: Pebble game algorihms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008) 13. Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proc. Cana- dian Conference of Computational Geometry. Windsor, Ontario (2005). http://cccg. cs.uwindsor.ca/papers/72.pdf 14. Lee, A., Streinu, I., Theran, L.: Graded sparse graphs and matroids. Journal of Universal Computer Science 13(10) (2007) 15. Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG’07) (2007) 16. Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North-Holland, Amsterdam (1979) 17. Nash-Williams, C.S.A.: Decomposition of finite graphs into forests. Journal of the London Mathematical Society 39, 12 (1964) 18. Oxley, J.G.: Matroid theory. The Clarendon Press, Oxford University Press, New York (1992) 19. Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees. Mathematics of Operations Research 10(4), 701–708 (1985) 20. Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: SCG ’08: Pro- ceedings of the twenty-fourth annual Symposium on Computational Geometry, pp. 365– 374. ACM, New York, NY, USA (2008). 21. Tay, T.S.: Rigidity of multigraphs I: linking rigid bodies in n-space. Journal of Combinato- rial Theory, Series B 26, 95–112 (1984) 22. Tay, T.S.: A new proof of Laman’s theorem. Graphs and Combinatorics 9, 365–370 (1993) 23. Tutte, W.T.: On the problem of decomposing a graph into n connected factors. Journal of the London Mathematical Society 142, 221–230 (1961) 24. Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM Journal on Discrete Mathematics 1(2), 237–255 (1988) http://cccg.cs.uwindsor.ca/papers/72.pdf http://cccg.cs.uwindsor.ca/papers/72.pdf Introduction and preliminaries Historical background The pebble game with colors Our Results Pebble game graphs The pebble-game-with-colors decomposition Canonical Pebble Game Constructions Pebble game algorithms for finding decompositions An important special case: Rigidity in dimension 2 and slider-pinning Conclusions and open problems
Introduction and preliminaries The focus of this paper is decompositions of (k, `)-sparse graphs into edge-disjoint subgraphs that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a graph is (k, `)-sparse if no subset of n′ vertices spans more than kn′− ` edges in the graph; a (k, `)-sparse graph with kn′− ` edges is (k, `)-tight. We call the range k ≤ `≤ 2k−1 the upper range of sparse graphs and 0≤ `≤ k the lower range. In this paper, we present efficient algorithms for finding decompositions that certify sparsity in the upper range of `. Our algorithms also apply in the lower range, which was already ad- dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs and graphs admitting the decomposition coincide. Our algorithms are based on a new characterization of sparse graphs, which we call the pebble game with colors. The pebble game with colors is a simple graph construction rule that produces a sparse graph along with a sparsity-certifying decomposition. We define and study a canonical class of pebble game constructions, which correspond to previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide a unifying framework for all the previously known special cases, including Nash-Williams- Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the properties of the augmenting paths used in matroid union and intersection algorithms[5, 6]. Since the sparse graphs in the upper range are not known to be unions or intersections of the matroids for which there are efficient augmenting path algorithms, these do not easily apply in ∗ Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author. 2 Ileana Streinu, Louis Theran Term Meaning Sparse graph G Every non-empty subgraph on n′ vertices has ≤ kn′− ` edges Tight graph G G = (V,E) is sparse and |V |= n, |E|= kn− ` Block H in G G is sparse, and H is a tight subgraph Component H of G G is sparse and H is a maximal block Map-graph Graph that admits an out-degree-exactly-one orientation (k, `)-maps-and-trees Edge-disjoint union of ` trees and (k− `) map-grpahs `Tk Union of ` trees, each vertex is in exactly k of them Set of tree-pieces of an `Tk induced on V ′ ⊂V Pieces of trees in the `Tk spanned by E(V ′) Proper `Tk Every V ′ ⊂V contains ≥ ` pieces of trees from the `Tk Table 1. Sparse graph and decomposition terminology used in this paper. the upper range. Pebble game with colors constructions may thus be considered a strengthening of augmenting paths to the upper range of matroidal sparse graphs. 1.1. Sparse graphs A graph is (k, `)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤ kn′− `. We observe that this condition implies that 0 ≤ ` ≤ 2k− 1, and from now on in this paper we will make this assumption. A sparse graph that has n vertices and exactly kn−` edges is called tight. For a graph G = (V,E), and V ′ ⊂ V , we use the notation span(V ′) for the number of edges in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge. There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of a sparse graph. A component is a maximal block. Table 1 summarizes the sparse graph terminology used in this paper. 1.2. Sparsity-certifying decompositions A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees. Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight graphs. A map-graph is a graph that admits an orientation such that the out-degree of each vertex is exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map- graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible configuration certifying that each color forms a map-graph. Map-graphs may be equivalently defined (see, e.g., [18]) as having exactly one cycle per connected component.1 A (k, `)-maps-and-trees is a graph that admits a decomposition into k− ` edge-disjoint map-graphs and ` spanning trees. Another characterization of map-graphs, which we will use extensively in this paper, is as the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that the converse holds as well. 1 Our terminology follows Lovász in [16]. In the matroid literature map-graphs are sometimes known as bases of the bicycle matroid or spanning pseudoforests. Sparsity-certifying Graph Decompositions 3 Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a (2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is shown with a certifying orientation. A `Tk is a decomposition into ` edge-disjoint (not necessarily spanning) trees such that each vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2. Given a subgraph G′ of a `Tk graph G, the set of tree-pieces in G′ is the collection of the components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come from the same tree or be single-vertex “empty trees.” It is also helpful to note that the definition of a tree-piece is relative to a specific subgraph. An `Tk decomposition is proper if the set of tree-pieces in any subgraph G′ has size at least `. Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree- pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges. These count as three tree-pieces, even though they come from the same back tree when the whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three gray tree-pieces and one black one. Table 1 contains the decomposition terminology used in this paper. The decomposition problem. We define the decomposition problem for sparse graphs as tak- ing a graph as its input and producing as output, a decomposition that can be used to certify spar- sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper `Tk decompositions; and the pebble-game-with-colors decomposition, which is defined in the next section. 2. Historical background The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint, 4 Ileana Streinu, Louis Theran Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a single vertex) and one black tree-piece. Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps- and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19]. In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman) graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay [21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a direct proof of Laman’s theorem and generalized the 3T2 condition to all `Tk for k≤ `≤ 2k−1. Haas [7] studied `Tk decompositions in detail and proved the equivalence of tight graphs and proper `Tk graphs for the general upper range. We observe that aside from our new pebble- game-with-colors decomposition, all the combinatorial characterizations of the upper range of sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24]. A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick- son’s Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and Streinu [12] generalized the pebble game to the entire range of parameters 0≤ `≤ 2k−1, and left as an open problem using the pebble game to find sparsity certifying decompositions. 3. The pebble game with colors Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative integers k and `. We will use the pebble game with colors as the basis of an efficient algorithm for the decomposition problem later in this paper. Since the phrase “with colors” is necessary only for comparison to [12], we will omit it in the rest of the paper when the context is clear. Sparsity-certifying Graph Decompositions 5 We now present the pebble game with colors. The game is played by a single player on a fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the addition and/or orientation of an edge. At any moment of time, the state of the game is captured by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored by the pebbles on them. While playing the pebble game all edges are directed, and we use the notation vw to indicate a directed edge from v to w. We describe the pebble game with colors in terms of its initial configuration and the allowed moves. Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices are shown as black or gray dots. Edges are colored with the color of the pebble on them. Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2, . . . ,k. Add-edge-with-colors: Let v and w be vertices with at least `+1 pebbles on them. Assume (w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw to E(H) and put the pebble picked up from v on the new edge. Figure 3(a) shows examples of the add-edge move. Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace vw with wv in E(H); put the pebble that was on vw on v; and put p on wv. Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows examples. The convention in these figures, and throughout this paper, is that pebbles on vertices are represented as colored dots, and that edges are shown in the color of the pebble on them. From the definition of the pebble-slide move, it is easy to see that a particular pebble is always either on the vertex where it started or on an edge that has this vertex as the tail. However, when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is sometimes convenient to think of this path reversal sequence as bringing a pebble from the end of the path to the beginning. The output of playing the pebble game is its complete configuration. Output: At the end of the game, we obtain the directed graph H, along with the location and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble game configuration colors the edges. We say that the underlying undirected graph G of H is constructed by the (k, `)-pebble game or that H is a pebble-game graph. Since each edge of H has exactly one pebble on it, the pebble game’s configuration partitions the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble- game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a pebble-game decomposition. Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges, and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con- 6 Ileana Streinu, Louis Theran (a) (b) (c) Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph. (c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges contain a cycle and do not contribute a piece of tree to the subgraph. Notation Meaning span(V ′) Number of edges spanned in H by V ′ ⊂V ; i.e. |EH(V ′)| peb(V ′) Number of pebbles on V ′ ⊂V out(V ′) Number of edges vw in H with v ∈V ′ and w ∈V −V ′ pebi(v) Number of pebbles of color ci on v ∈V outi(v) Number of edges vw colored ci for v ∈V Table 2. Pebble game notation used in this paper. nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′ otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with the corresponding definition for `Tk s, the set of tree-pieces is defined relative to a specific sub- graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned by G′. The properties of pebble-game decompositions are studied in Section 6, and Theorem 2 shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows this. For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom- position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees. In the following discussion, we use the notation peb(v) for the number of pebbles on v and pebi(v) to indicate the number of pebbles of colors i on v. Table 2 lists the pebble game notation used in this paper. 4. Our Results We describe our results in this section. The rest of the paper provides the proofs. Sparsity-certifying Graph Decompositions 7 Our first result is a strengthening of the pebble games of [12] to include colors. It says that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games discussed in this paper are our pebble game with colors unless noted explicitly. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. Next we consider pebble-game decompositions, showing that they are a generalization of proper `Tk decompositions that extend to the entire matroidal range of sparse graphs. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained by playing the pebble game defined in the previous section. Notice the similarity between the requirement that the set of tree-pieces have size at least ` in Theorem 2 and the definition of a proper `Tk . Our next results show that for any pebble-game graph, we can specialize its pebble game construction to generate a decomposition that is a maps-and-trees or proper `Tk . We call these specialized pebble game constructions canonical, and using canonical pebble game construc- tions, we obtain new direct proofs of existing arboricity results. We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo- sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning trees contributes at least one piece of tree to every subgraph. The case of proper `Tk graphs is more subtle; if each color in a pebble-game decomposition is a forest, then we have found a proper `Tk , but this class is a subset of all possible proper `Tk decompositions of a tight graph. We show that this class of proper `Tk decompositions is sufficient to certify sparsity. We now state the main theorem for the upper and lower range. Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Theorem 4 (Main Theorem (Upper Range): Proper `Tk graphs coincide with pebble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. As corollaries, we obtain the existing decomposition results for sparse graphs. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . Efficiently finding canonical pebble game constructions. The proofs of Theorem 3 and Theo- rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem. Our last result improves on this, showing that a canonical pebble game construction, and thus 8 Ileana Streinu, Louis Theran a maps-and-trees or proper `Tk decomposition can be found using a pebble game algorithm in O(n2) time and space. These time and space bounds mean that our algorithm can be combined with those of [12] without any change in complexity. 5. Pebble game graphs In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game with colors. Since many of the relevant properties of the pebble game with colors carry over directly from the pebble games of [12], we refer the reader there for the proofs. We begin by establishing some invariants that hold during the execution of the pebble game. Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following invariants are maintained in H: (I1) There are at least ` pebbles on V . [12] (I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12] (I3) For each V ′ ⊂V , span(V ′)+out(V ′)+peb(V ′) = kn′. [12] (I4) For every vertex v ∈V , outi(v)+pebi(v) = 1. (I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with a pebble of color ci or a cycle. Proof. (I1), (I2), and (I3) come directly from [12]. (I4) This invariant clearly holds at the initialization phase of the pebble game with colors. That add-edge and pebble-slide moves preserve (I4) is clear from inspection. (I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of the same color on it. If there is no pebble of that color reachable, then the path must eventually visit some vertex twice. From these invariants, we can show that the pebble game constructible graphs are sparse. Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the pebble game. Then H is sparse. If there are exactly ` pebbles on V (H), then H is tight. The main step in proving that every sparse graph is a pebble-game graph is the following. Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce the out degree of v by one. Lemma 9 (The `+1 pebble condition [12]). Let vw be an edge such that H + vw is sparse. If peb({v,w}) < `+1, then a pebble not on {v,w} can be brought to either v or w. It follows that any sparse graph has a pebble game construction. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. 6. The pebble-game-with-colors decomposition In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We start with the following lemmas about the structure of monochromatic connected components in H, the directed graph maintained during the pebble game. Sparsity-certifying Graph Decompositions 9 Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub- graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for i = 1, . . . ,k. Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex. Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H in a pebble game construction contains at least ` monochromatic tree-pieces, and each of these is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge. Recall that an out-edge from a subgraph H ′ = (V ′,E ′) is an edge vw with v∈V ′ and vw /∈ E ′. Proof. Let H ′ = (V ′,E ′) be a non-empty subgraph of H, and assume without loss of generality that H ′ is induced by V ′. By (I3), out(V ′)+ peb(V ′) ≥ `. We will show that each pebble and out-edge tail is the root of a tree-piece. Consider a vertex v ∈ V ′ and a color ci. By (I4) there is a unique monochromatic directed path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle. Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the monochromatic path from v leaves V ′), then the path cannot have a cycle in H ′. Since this argument works for any vertex in any color, for each color there is a partitioning of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each pebble and out-edge tail is the root of a monochromatic tree, as desired. Applied to the whole graph Lemma 11 gives us the following. Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of color ci is the root of a (possibly empty) monochromatic tree-piece of color ci. Remark: Haas showed in [7] that in a `Tk , a subgraph induced by n′ ≥ 2 vertices with m′ edges has exactly kn′−m′ tree-pieces in it. Lemma 11 strengthens Haas’ result by extending it to the lower range and giving a construction that finds the tree-pieces, showing the connection between the `+1 pebble condition and the hereditary condition on proper `Tk . We conclude our investigation of arbitrary pebble game constructions with a description of the decomposition induced by the pebble game with colors. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub- graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs. For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can span at most n− ti edges; summing over all the colors shows that a graph with a pebble-game decomposition must be sparse. Apply Theorem 1 to complete the proof. Remark: We observe that a pebble-game decomposition for a Laman graph may be read out of the bipartite matching used in Hendrickson’s Laman graph extraction algorithm [9]. Indeed, pebble game orientations have a natural correspondence with the bipartite matchings used in 10 Ileana Streinu, Louis Theran Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there are no cycles in ` of the colors, then the trees rooted at the corresponding ` pebbles must be spanning, since they have n− 1 edges. Also, if each color forms a forest in an upper range pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de- composition is a proper `Tk . In the next section, we show that the pebble game can be specialized to correspond to maps- and-trees and proper `Tk decompositions. 7. Canonical Pebble Game Constructions In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves- tigation of decompositions induced by pebble game constructions by studying the case where a minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15 and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that this is always possible, implying that monochromatic map-graphs are created only when we add more than k(n′−1) edges to some set of n′ vertices. For the lower range, this implies that every color is a forest. Every decomposition characterization of tight graphs discussed above follows immediately from the main theorem, giving new proofs of the previous results in a unified framework. In the proof, we will use two specializations of the pebble game moves. The first is a modi- fication of the add-edge move. Canonical add-edge: When performing an add-edge move, cover the new edge with a color that is on both vertices if possible. If not, then take the highest numbered color present. The second is a restriction on which pebble-slide moves we allow. Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a monochromatic cycle. We call a pebble game construction that uses only these moves canonical. In this section we will show that every pebble-game graph has a canonical pebble game construction (Lemma 14 and Lemma 15) and that canonical pebble game constructions correspond to proper `Tk and maps-and-trees decompositions (Theorem 3 and Theorem 4). We begin with a technical lemma that motivates the definition of canonical pebble game constructions. It shows that the situations disallowed by the canonical moves are all the ways for cycles to form in the lowest ` colors. Lemma 13 (Monochromatic cycle creation). Let v ∈ V have a pebble p of color ci on it and let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created in exactly one of the following ways: (M1) The edge vw is added with an add-edge move. (M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse edge vw. Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7. By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble game construction, since the color of an edge only changes when it is inserted the first time or a new pebble is put on it by a pebble-slide move. Sparsity-certifying Graph Decompositions 11 vw vw Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are labeled according to their role in the definition of the moves. Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves, respectively, in a (2,0)-pebble game construction. We next show that if a graph has a pebble game construction, then it has a canonical peb- ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa- rately. The proof gives two constructions that implement the canonical add-edge and canonical pebble-slide moves. Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc- tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤ i ≤ `′, where `′ = min{k, `}. Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If this is not possible, then there are `+1 distinct colors present. Use the highest numbered color to cover the new edge. Remark: We note that in the upper range, there is always a repeated color, so no canonical add-edge moves create cycles in the upper range. The canonical pebble-slide move is defined by a global condition. To prove that we obtain the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma 9 to only canonical moves. The main step is to show that if there is any sequence of moves that reorients a path from v to w, then there is a sequence of canonical moves that does the same thing. Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading to an add-edge move can be replaced with one that has no (M2) steps and allows the same add-edge move. In other words, if it is possible to collect `+ 1 pebbles on the ends of an edge to be added, then it is possible to do this without creating any monochromatic cycles. 12 Ileana Streinu, Louis Theran Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this the shortcut construction by analogy to matroid union and intersection augmenting paths used in previous work on the lower range. Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one application of the shortcut construction reorients a simple path from a vertex w′ to w, and a path from v to w′ is preserved, the shortcut construction can be applied inductively to find the sequence of moves we want. Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle, shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is simple, and the shortcut construction can be applied inductively to it. Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v and w are contained in a maximal monochromatic tree of color ci. Call this tree H ′i , and observe that it is rooted at w. Now consider the edges reversed in our sequence of moves. As noted above, before we make any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on this path in H ′i . We modify our sequence of moves as follows: delete, from the beginning, every move before the one that reverses some edge yz; prepend onto what is left a sequence of moves that moves the pebble on w to z in H ′i . Sparsity-certifying Graph Decompositions 13 Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path. The path where the pebbles move is indicated by doubled lines. Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is (M2); (b) avoiding the (M2) and simplifying the path. Since no edges change color in the beginning of the new sequence, we have eliminated the (M2) move. Because our construction does not change any of the edges involved in the remaining tail of the original sequence, the part of the original path that is left in the new sequence will still be a simple path in H, meeting our initial hypothesis. The rest of the lemma follows by induction. Together Lemma 14 and Lemma 15 prove the following. Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction. Using canonical pebble game constructions, we can identify the tight pebble-game graphs with maps-and-trees and `Tk graphs. 14 Ileana Streinu, Louis Theran Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game graph. For the reverse direction, consider a canonical pebble game construction of a tight graph. From Lemma 8, we see that there are ` pebbles left on G at the end of the construction. The definition of the canonical add-edge move implies that there must be at least one pebble of each ci for i = 1,2, . . . , `. It follows that there is exactly one of each of these colors. By Lemma 12, each of these pebbles is the root of a monochromatic tree-piece with n− 1 edges, yielding the required ` edge-disjoint spanning trees. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. We next consider the decompositions induced by canonical pebble game constructions when `≥ k +1. Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb- ble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. Proof. As observed above, a proper `Tk decomposition must be sparse. What we need to show is that a canonical pebble game construction of a tight graph produces a proper `Tk . By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom- position into ` edge-disjoint trees. Finally, an application of (I4), shows that every vertex must in in exactly k of the trees, as required. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . 8. Pebble game algorithms for finding decompositions A naı̈ve implementation of the constructions in the previous section leads to an algorithm re- quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n) applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running time of Θ(n3) for the decomposition problem. In this section, we describe algorithms for the decomposition problem that run in time O(n2). We begin with the overall structure of the algorithm. Algorithm 17 (The canonical pebble game with colors). Input: A graph G. Output: A pebble-game graph H. Method: – Set V (H) = V (G) and place one pebble of each color on the vertices of H. – For each edge vw ∈ E(G) try to collect at least `+1 pebbles on v and w using pebble-slide moves as described by Lemma 15. Sparsity-certifying Graph Decompositions 15 – If at least `+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma 14, otherwise discard vw. – Finally, return H, and the locations of the pebbles. Correctness. Theorem 1 and the result from [24] that the sparse graphs are the independent sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction found is canonical, the main theorem shows that the coloring of the edges in H gives a maps- and-trees or proper `Tk decomposition. Complexity. We start by observing that the running time of Algorithm 17 is the time taken to process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an edge of G that is added to H. Each of the pebble game moves can be implemented in constant time. What remains is to describe an efficient way to find and move the pebbles. We use the following algorithm as a subroutine of Algorithm 17 to do this. Algorithm 18 (Finding a canonical path to a pebble.). Input: Vertices v and w, and a pebble game configuration on a directed graph H. Output: If a pebble was found, ‘yes’, and ‘no’ otherwise. The configuration of H is updated. Method: – Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and return ‘no.’ – Otherwise a pebble was found. We now have a path v = v1,e1, . . . ,ep−1,vp = u, where the vi are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use the array c[] to keep track of the colors of pebbles on vertices and edges after we move them and the array s[] to sketch out a canonical path from v to u by finding a successor for each edge. – Set s[u] = ‘end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in reverse order: vp,ep−1,ep−2, . . . ,e1,v1. For each i, check to see if c[vi] is set; if so, go on to the next i. Otherwise, check to see if c[vi+1] = c[ei]. – If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge. – Otherwise c[vi+1] 6= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2, . . . , fq−1,xq = x that is monochromatic in the color of the edges; set c[xi] = c[ fi] and s[xi] = fi for i = 1,2, . . . ,q−1. If c[x] = c[ fq−1], stop. Otherwise, recursively check that there is not a monochro- matic c[x] path from xq−1 to x using this same procedure. – Finally, slide pebbles along the path from the original endpoints v to u specified by the successor array s[v], s[s[v]], . . . The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut construction. Efficiency comes from the fact that instead of potentially moving the pebble back and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three times: once in the initial depth-first search, and twice while converting the initial path to a canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time spent processing edges in H. Although we have not discussed this explicity, for the algorithm to be efficient we need to maintain components as in [12]. After each accepted edge, the components of H can be updated in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1) time each. 16 Ileana Streinu, Louis Theran Summarizing, we have shown that the canonical pebble game with colors solves the decom- position problem in time O(n2). 9. An important special case: Rigidity in dimension 2 and slider-pinning In this short section we present a new application for the special case of practical importance, k = 2, ` = 3. As discussed in the introduction, Laman’s theorem [11] characterizes minimally rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com- binatorially, we model the bar-slider frameworks as simple graphs together with some loops placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each color. We characterize the minimally rigid bar-slider graphs [20] as graphs that are: 1. (2,3)-sparse for subgraphs containing no loops. 2. (2,0)-tight when loops are included. We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse graphs studied in our paper [14]. The connection with the pebble games in this paper is the following. Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph. Proof. Follows from invariant (I3) of Lemma 7. In [15], we study a special case of slider pinning where every slider is either vertical or horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction. For this axis parallel slider case, the minimally rigid graphs are characterized by: 1. (2,3)-sparse for subgraphs containing no loops. 2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each monochromatic tree spans exactly one loop of its color. This also has an interpretation in terms of colored pebble games. Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)- pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the graph of a minimally pinned axis-parallel bar-slider framework. Proof. Follows from Theorem 4, and Lemma 12. 10. Conclusions and open problems We presented a new characterization of (k, `)-sparse graphs, the pebble game with colors, and used it to give an efficient algorithm for finding decompositions of sparse graphs into edge- disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the upper range from [12]. We also used the pebble game with colors to describe a new sparsity-certifying decomposi- tion that applies to the entire matroidal range of sparse graphs. Sparsity-certifying Graph Decompositions 17 We defined and studied a class of canonical pebble game constructions that correspond to either a maps-and-trees or proper `Tk decomposition. This gives a new proof of the Tutte-Nash- Williams arboricity theorem and a unified proof of the previously studied decomposition cer- tificates of sparsity. Canonical pebble game constructions also show the relationship between the `+1 pebble condition, which applies to the upper range of `, to matroid union augmenting paths, which do not apply in the upper range. Algorithmic consequences and open problems. In [6], Gabow and Westermann give an O(n3/2) algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from dense ones. Their technique is based on efficiently finding matroid union augmenting paths, which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch scanning, which finds groups of disjoint augmenting paths. We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester- mann’s algorithm without changing the running time. The data structures used in the implemen- tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those used to support cyclic scanning. The two major open algorithmic problems related to the pebble game are then: Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain an implementable O(n3/2) algorithm for the lower range. Problem 2. Extend batch scanning to the `+1 pebble condition and derive an O(n3/2) pebble game algorithm for the upper range. In particular, it would be of practical importance to find an implementable O(n3/2) algorithm for decompositions into edge-disjoint spanning trees. References 1. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th European Symposium on Algorithms (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003) 2. Crapo, H.: On the generic rigidity of plane frameworks. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988) 3. Edmonds, J.: Minimum partition of a matroid into independent sets. J. Res. Nat. Bur. Standards Sect. B 69B, 67–72 (1965) 4. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Optimization—Eureka, You Shrink!, no. 2570 in LNCS, pp. 11–26. Springer (2003) 5. Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. Journal of Computer and System Sciences 50, 259–273 (1995) 6. Gabow, H.N., Westermann, H.H.: Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica 7(1), 465–497 (1992) 7. Haas, R.: Characterizations of arboricity of graphs. Ars Combinatorica 63, 129–137 (2002) 8. Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompo- sitions. Journal of Combinatorial Mathematics and Combinatorial Computing 62, 3–11 (2007) 9. Hendrickson, B.: Conditions for unique graph realizations. SIAM Journal on Computing 21(1), 65–84 (1992) 18 Ileana Streinu, Louis Theran 10. Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the pebble game. Journal of Computational Physics 137, 346–365 (1997) 11. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 331–340 (1970) 12. Lee, A., Streinu, I.: Pebble game algorihms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008) 13. Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proc. Cana- dian Conference of Computational Geometry. Windsor, Ontario (2005). http://cccg. cs.uwindsor.ca/papers/72.pdf 14. Lee, A., Streinu, I., Theran, L.: Graded sparse graphs and matroids. Journal of Universal Computer Science 13(10) (2007) 15. Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG’07) (2007) 16. Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North-Holland, Amsterdam (1979) 17. Nash-Williams, C.S.A.: Decomposition of finite graphs into forests. Journal of the London Mathematical Society 39, 12 (1964) 18. Oxley, J.G.: Matroid theory. The Clarendon Press, Oxford University Press, New York (1992) 19. Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees. Mathematics of Operations Research 10(4), 701–708 (1985) 20. Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: SCG ’08: Pro- ceedings of the twenty-fourth annual Symposium on Computational Geometry, pp. 365– 374. ACM, New York, NY, USA (2008). 21. Tay, T.S.: Rigidity of multigraphs I: linking rigid bodies in n-space. Journal of Combinato- rial Theory, Series B 26, 95–112 (1984) 22. Tay, T.S.: A new proof of Laman’s theorem. Graphs and Combinatorics 9, 365–370 (1993) 23. Tutte, W.T.: On the problem of decomposing a graph into n connected factors. Journal of the London Mathematical Society 142, 221–230 (1961) 24. Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM Journal on Discrete Mathematics 1(2), 237–255 (1988) http://cccg.cs.uwindsor.ca/papers/72.pdf http://cccg.cs.uwindsor.ca/papers/72.pdf Introduction and preliminaries Historical background The pebble game with colors Our Results Pebble game graphs The pebble-game-with-colors decomposition Canonical Pebble Game Constructions Pebble game algorithms for finding decompositions An important special case: Rigidity in dimension 2 and slider-pinning Conclusions and open problems
Descomposiciones del gráfico de certificación de la sparsity Ileana Streinu1*, Louis Theran2 1 Departamento de Ciencias de la Computación, Smith College, Northampton, MA. Correo electrónico: streinu@cs.smith.edu 2 Departamento de Ciencias de la Computación, Universidad de Massachusetts Amherst. Correo electrónico: theran@cs.umass.edu Resumen. Describimos un nuevo algoritmo, el (k, `)-pebble juego con colores, y usarlo para obtener un charac- la terización de la familia de gráficos (k, `)-sparse y soluciones algorítmicas a una familia de problemas ing árbol descomposicións de gráficos. Casos especiales de gráficos escasos aparecen en la teoría de la rigidez y tienen ha recibido una mayor atención en los últimos años. En particular, nuestros guijarros de colores generalizan y fortalecen los resultados anteriores de Lee y Streinu [12] y dar una nueva prueba de la Tutte-Nash-Williams carácteri- Zación de arboricidad. También presentamos una nueva descomposición que certifica la esparcidad basada en la (k, `)-pebble juego con colores. Nuestro trabajo también expone conexiones entre los algoritmos de juego de guijarros y anteriores algoritmos gráficos escasos de Gabow [5], Gabow y Westermann [6] y Hendrickson [9]. 1. Introducción y preliminares El foco de este documento son las descomposicións de (k, `)-sparse gráficos en bordes-disjunto subgraphs que certifique la escasez. Usamos el gráfico para significar un múltiplo, posiblemente con bucles. Nosotros decimos que un grafo es (k, `)-sparse si ningún subconjunto de n′ vértices abarca más de kn ` bordes en el gráfico; a (k, `)-sparse gráfico con kn ` bordes es (k, `)-estrechado. Llamamos al rango k ≤ 2k−1 el superior rango de gráficos escasos y 0≤ k el rango inferior. En este artículo, presentamos algoritmos eficientes para encontrar descomposicións que certifiquen la escasez en el rango superior de `. Nuestros algoritmos también se aplican en el rango inferior, que ya era ad- vestido por [3, 4, 5, 6, 19]. Una descomposición certifica la escasez de un gráfico si los gráficos dispersos y los gráficos que admiten la descomposición coinciden. Nuestros algoritmos se basan en una nueva caracterización de gráficos escasos, que llamamos el juego de guijarros con colores. El juego de guijarros con colores es una regla de construcción de gráficos simples que produce un gráfico escaso junto con una descomposición certificadora de la escasez. Definimos y estudiamos una clase canónica de construcciones de juego de guijarros, que corresponden a previamente estudiado las descomposiciones de los gráficos escasos en los árboles disjuntos del borde. Nuestros resultados proporcionan un marco unificador para todos los casos especiales conocidos anteriormente, incluidos Nash-Williams- Tutte y [7, 24]. De hecho, en el rango inferior, las construcciones canónicas de juego de guijarros capturan la propiedades de las rutas de aumento utilizadas en los algoritmos de unión de matroides y de intersección[5, 6]. Dado que los gráficos escasos en el rango superior no se sabe que son uniones o intersecciones de la matroides para los que hay algoritmos de ruta de aumento eficiente, estos no se aplican fácilmente en * Investigación de ambos autores financiada por la NSF bajo subvenciones NSF CCF-0430990 y NSF-DARPA CARGO CCR-0310661 al primer autor. 2 Ileana Streinu, Louis Theran Significado del término Gráfico escaso G Cada subgrafo no vacío en n′ vértices tiene ≤ kn ` bordes El gráfico ajustado G G = (V,E) es escaso y V = n, E= kn− ` El bloque H en G G es escaso, y H es un subgrafo apretado El componente H de G G es escaso y H es un bloqueo máximo Gráfico cartográfico que admite una orientación de grado-exactamente-uno (k, `)-maps-and-trees Edge-disjunt union de ` árboles y (k- `) map-grpahs `Tk Unión de ` árboles, cada vértice está exactamente en k de ellos Conjunto de piezas arbóreas de un `Tk inducido en V ′ ́V Piezas de árboles en el `Tk extendido por E(V ′) `Tk Apropiado Cada V ′ V contiene ≥ ` pedazos de árboles de la `Tk Cuadro 1 Gráfico escaso y terminología de descomposición utilizada en este artículo. el rango superior. Pebble juego con construcciones de colores por lo tanto puede ser considerado un fortalecimiento de caminos de aumento a la gama superior de gráficos de la escasez matroidal. 1.1. Gráficos escasos Un gráfico es (k, `)-sparse si para cualquier subgrafo no vacío con bordes m′ y n′ vértices, m′ ≤ kn `. Observamos que esta condición implica que 0 ≤ ` ≤ 2k− 1, y a partir de ahora en este Haremos esta suposición. Un gráfico escaso que tiene n vértices y exactamente bordes kn se llama apretado. Para un gráfico G = (V,E), y V ′ V, utilizamos el intervalo de notación (V ′) para el número de bordes en el subgráfico inducido por V ′. En un gráfico dirigido, out(V ′) es el número de bordes con la cola en V ′ y la cabeza en V −V ′; para un subgráfico inducido por V ′, llamamos a tal borde un borde superior. Hay dos tipos importantes de subgrafías de gráficos escasos. Un bloque es un subgrafo apretado de un gráfico escaso. Un componente es un bloque máximo. La Tabla 1 resume la escasa terminología gráfica utilizada en este artículo. 1.2. Descomposiciónes de certificación de la sparsidad Un k-arborescencia es un gráfico que admite una descomposición en k borde-desjunto que abarca los árboles. La Figura 1(a) muestra un ejemplo de una 3-arborescencia. Se describen los gráficos k-arborescentes por los conocidos teoremas de Tutte [23] y Nash-Williams [17] como exactamente el (k,k) apretado gráficos. Un map-graph es un gráfico que admite una orientación tal que el grado de cada vértice es Exactamente uno. Un k-map-graph es un gráfico que admite una descomposición en k borde-disjunto mapa- gráficos. La Figura 1(b) muestra un ejemplo de un 2-map-graphs; los bordes están orientados en uno posible configuración que certifica que cada color forma un mapa gráfico. Los mapas pueden ser equivalentes definido (véase, por ejemplo, [18]) como tener exactamente un ciclo por componente conectado.1 A (k, `)-maps-and-trees es un gráfico que admite una descomposición en k− ` borde-disjunta - mapas y árboles que se extienden por los árboles. Otra caracterización de los mapas, que utilizaremos ampliamente en este artículo, es la siguiente: los gráficos (1,0) ajustados [8, 24]. Los k-map-graphs son evidentemente (k,0)-stight, y [8, 24] muestran que lo contrario se sostiene también. 1 Nuestra terminología sigue a Lovász en [16]. En la literatura matroide los mapas a veces se conocen como bases del matroide de la bicicleta o pseudobosques que se extienden. Descomposiciones del gráfico de certificación de la Sparsity 3 Fig. 1. Ejemplos de descomposiciones certificadoras de la escasez: a) una 3-arborescencia; b) una 2-map-graph; c) una (2,1)-maps-y-árboles. Los bordes con el mismo estilo de línea pertenecen al mismo subgrafo. El 2-map-graph es se muestra con una orientación certificadora. Un `Tk es una descomposición en `árboles disjuntos de borde (que no necesariamente abarcan) de tal manera que cada uno vértice está en exactamente k de ellos. La figura 2 a) muestra un ejemplo de un 3T2. Dado un subgrafo G′ de un gráfico `Tk G, el conjunto de piezas arbóreas en G′ es la colección del componentes de los árboles en G inducidos por G′ (dado que G′ es un subgrafo cada árbol puede contribuir piezas múltiples en el conjunto de piezas de árbol en G′). Observamos que estas piezas de árboles pueden venir del mismo árbol o ser un solo vertex “árboles vacíos.” También es útil tener en cuenta que la definición de un árbol-pieza es relativo a un subgrafo específico. Una descomposición `Tk es apropiada si el conjunto de las piezas arbóreas de cualquier subpárrafo G′ tienen un tamaño mínimo `. La Figura 2(a) muestra un gráfico con una descomposición 3T2; observamos que uno de los árboles es un vértice aislado en la esquina inferior derecha. El subgrafo de la Figura 2(b) tiene tres árboles negros- piezas y un árbol-pieza gris: un vértice aislado en la esquina superior derecha, y dos bordes individuales. Estos cuentan como tres árboles-piezas, a pesar de que vienen del mismo árbol trasero cuando el Gráfico completo considerado. La figura 2 c) muestra otro subgráfico; en este caso hay tres piezas de árboles grises y una negra. En el cuadro 1 figura la terminología de descomposición utilizada en este documento. El problema de descomposición. Definimos el problema de descomposición para gráficos escasos como tak- • un gráfico como su entrada y producción como salida, una descomposición que se puede utilizar para certificar sity. En el presente documento se estudiarán tres tipos de productos: mapas y árboles; descomposiciones adecuadas de `Tk; y la descomposición de guijarros-juego-con-colores, que se define en la siguiente sección. 2. Antecedentes históricos Los conocidos teoremas de Tutte [23] y Nash-Williams [17] relacionan los gráficos (k,k) ajustados a la existencia de descomposicións en los árboles que se extienden por los bordes. Tomando un punto de vista matroidal, 4 Ileana Streinu, Louis Theran Fig. 2. (a) Un gráfico con una descomposición 3T2; uno de los tres árboles es un único vértice en la parte inferior derecha esquina. (b) El subgrafo resaltado dentro del conteo rayado tiene tres piezas de árbol negro y una gris pieza de árbol. (c) El subgrafo resaltado dentro del conteo rayado tiene tres piezas de árbol grises (uno es un solo vértice) y una pieza de árbol negro. Edmonds [3, 4] dio otra prueba de este resultado usando uniones de matroide. La equivalencia de los mapas- los gráficos y árboles y los gráficos ajustados en el rango inferior se muestran utilizando uniones de los matroides en [24], y rutas de aumento matroide son la base de los algoritmos para el rango inferior de [5, 6, 19]. En la teoría de la rigidez un teorema fundacional de Laman [11] muestra que (2,3)-ajustado (Laman) los gráficos corresponden a marcos de barras y conjuntos genéricamente mínimamente rígidos en el plano. Tay [21] ha demostrado ser un resultado análogo para los marcos de la barra del cuerpo en cualquier dimensión utilizando (k,k) gráficos. Rigidez por conteos de interés motivado en el rango superior, y Crapo [2] probó la equivalencia de gráficos Laman y gráficos 3T2 apropiados. Tay [22] utilizó esta condición para dar un prueba directa del teorema de Laman y generalizada la condición 3T2 a todos `Tk para k≤ 2k−1. Haas [7] estudió detalladamente las descomposicións de `Tk y demostró la equivalencia de gráficos ajustados y gráficos `Tk apropiados para el rango superior general. Observamos que aparte de nuestro nuevo guijarro... game-with-colors descomposición, todas las caracterizaciones combinatoria de la gama superior de Los gráficos escasos, incluidos los conteos, tienen una interpretación geométrica [11, 21, 22, 24]. Un algoritmo de juego de guijarros fue propuesto por primera vez en [10] como una alternativa elegante a Hendrick- algoritmos de gráfico Laman de hijo [9]. Berg y Jordania [1], facilitaron el análisis formal de la juego de guijarros de [10] e introdujo la idea de jugar el juego en un gráfico dirigido. Lee y Streinu [12] generalizó el juego de guijarros a toda la gama de parámetros 0≤ 2k−1, y izquierda como un problema abierto utilizando el juego de guijarros para encontrar la escasez certificando las descomposicións. 3. El juego de guijarros con colores Nuestro juego de guijarros con colores es un conjunto de reglas para la construcción de gráficos indexados por no negativos enteros k y `. Usaremos el juego de guijarros con colores como la base de un algoritmo eficiente para el problema de descomposición más adelante en este documento. Puesto que la frase “con colores” es necesaria Sólo en comparación con [12], lo omitiremos en el resto del documento cuando el contexto sea claro. Descomposiciones del gráfico de certificación de la sparsity 5 Ahora presentamos el juego de guijarros con colores. El juego es jugado por un solo jugador en un conjunto finito fijo de vértices. El jugador hace una secuencia finita de movimientos; un movimiento consiste en el adición y/o orientación de un borde. En cualquier momento, el estado del juego es capturado por un gráfico dirigido H, con guijarros de colores sobre vértices y bordes. Los bordes de H son de color por los guijarros en ellos. Mientras que jugando el juego de guijarros todos los bordes están dirigidos, y utilizamos el notación vw para indicar un borde dirigido de v a w. Describimos el juego de guijarros con colores en términos de su configuración inicial y el permitido se mueve. Fig. 3. Ejemplos de juego de guijarros con movimientos de colores: (a) add-edge. b) Deslizamiento de guijarros. Guijarros sobre vértices se muestran como puntos negros o grises. Los bordes están coloreados con el color de la rocalla en ellos. Inicialización: Al principio del juego de guijarros, H tiene n vértices y no tiene bordes. Comenzamos colocando k guijarros en cada vértice de H, uno de cada color ci, para i = 1,2,...,k. Add-edge-with-colors: Dejar v y w ser vértices con al menos â € 1 guijarros en ellos. Asumir (w.l.o.g.) que v tiene al menos un guijarro en él. Recoger un guijarro de v, añadir el borde orientado vw a E(H) y poner el guijarro recogido de v en el nuevo borde. La Figura 3(a) muestra ejemplos del movimiento de add-edge. Pebble-slide: Dejar w ser un vértice con un guijarro p en él, y dejar vw ser un borde en H. Reemplazar vw con wv en E(H); poner el guijarro que estaba en vw en v; y poner p en wv. Tenga en cuenta que el color de un borde puede cambiar con un movimiento de guijarros. La figura 3 b) muestra ejemplos. La convención en estas figuras, y a lo largo de este documento, es que los guijarros sobre los vértices se representan como puntos de color, y que los bordes se muestran en el color de la rocalla en ellos. A partir de la definición del movimiento de guijarros-deslizamiento, es fácil ver que un guijarro en particular es siempre en el vértice donde empezó o en un borde que tiene este vértice como la cola. Sin embargo, al hacer una secuencia de movimientos de guijarros que invierten la orientación de un camino en H, es a veces es conveniente pensar en esta secuencia de inversión del camino como trayendo un guijarro desde el final del camino al principio. La salida de jugar el juego de guijarros es su configuración completa. Salida: Al final del juego, obtenemos el gráfico dirigido H, junto con la ubicación y los colores de los guijarros. Observe que ya que cada borde tiene exactamente un guijarro en él, el guijarro la configuración del juego colorea los bordes. Decimos que el gráfico G de H subyacente no dirigido es construido por el juego (k, `)-pebble o que H es un gráfico de juego de guijarros. Puesto que cada borde de H tiene exactamente un guijarro, las particiones de configuración del juego de guijarro los bordes de H, y así G, en k diferentes colores. Llamamos a esta descomposición de H un guijarro... juego-con-colores descomposición. La Figura 4(a) muestra un ejemplo de un gráfico ajustado (2,2) con un Descomposición de juego de guijarros. Que G = (V,E) sea gráfico de juego de guijarros con la coloración inducida por los guijarros en los bordes, y dejar que G′ sea un subgrafo de G. Entonces la coloración de G induce un conjunto de con- 6 Ileana Streinu, Louis Theran a) b) c) Fig. 4. A (2,2)-término gráfico con una posible descomposición del juego de guijarros. Los bordes están orientados a mostrar (1,0)-esparsidad para cada color. a) El gráfico K4 con una descomposición del juego de guijarros. Hay un árbol negro vacío en el vértice central y un árbol gris que se extiende. b) El subgráfico resaltado consta de dos: árboles negros y un árbol gris; los bordes negros son parte de un ciclo más grande pero aportan un árbol al subgrafo. c) El subgrafo resaltado (con fondo gris claro) tiene tres árboles grises vacíos; los bordes negros contienen un ciclo y no aportan un pedazo de árbol al subgrafo. Significado de la notación longitud (V ′) Número de bordes que se extienden en H por V ′ V ; es decir, EH(V ′) Peb(V ′) Número de guijarros en V ′ ́V fuera (V ′) Número de bordes vw en H con v ́V ′ y w ́V −V ′ pebi(v) Número de guijarros de color ci en v • V outi(v) Número de bordes vw coloreados ci para v â € € TM V Cuadro 2 Pebble notación de juego utilizado en este papel. Subgrafías de G′ (puede haber más de uno del mismo color). Tan monocromático subgraph se llama un mapa-foto-pieza de G′ si contiene un ciclo (en G′) y un árbol-pieza de G′ De lo contrario. El conjunto de piezas arbóreas de G′ es la colección de piezas arbóreas inducidas por G′. Al igual que con la definición correspondiente para `Tk s, el conjunto de piezas arbóreas se define en relación con un sub- grafo; en particular, una pieza de árbol puede formar parte de un ciclo más grande que incluye bordes que no se extienden por G′. Las propiedades de las descomposicións del juego de guijarros se estudian en la Sección 6 y en el Teorema 2 muestra que cada color debe ser (1,0)-sparse. La orientación de los bordes en la Figura 4(a) muestra Esto. Por ejemplo, la Figura 4(a) muestra un gráfico ajustado (2,2) con un posible decom de juego de guijarro- posición. El gráfico completo contiene una pieza de árbol gris y una pieza de árbol negro que es un aislado vértice. El subgrafo de la Figura 4(b) tiene un árbol negro y un árbol gris, con los bordes del negro árbol procedente de un ciclo en el gráfico más grande. En la Figura 4(c), sin embargo, el ciclo negro no contribuir con una pieza de árbol. Las tres piezas de árbol en este subgrafo son árboles grises de un solo vértex. En la siguiente discusión, utilizamos la notación peb(v) para el número de guijarros en v y pebi(v) para indicar el número de guijarros de colores i en v. La Tabla 2 enumera la notación de juego de guijarros utilizada en este artículo. 4. Nuestros resultados Describimos nuestros resultados en esta sección. El resto del periódico proporciona las pruebas. Descomposiciones del gráfico de certificación de la sparsity 7 Nuestro primer resultado es un fortalecimiento de los juegos de guijarros de [12] para incluir los colores. Dice que los gráficos escasos son exactamente gráficos de juego de guijarros. Recuerde que a partir de ahora, todos los juegos de guijarros discutidos en este artículo son nuestro juego de guijarros con colores a menos que se anote explícitamente. Teorema 1 (Los gráficos Sparse y los gráficos de juego de guijarros coinciden). Un gráfico G es (k, `)-sparse con 0≤ 2k−1 si y sólo si G es un gráfico de juego de guijarros. A continuación consideramos las descomposiciones de juego de guijarros, mostrando que son una generalización de las descomposiciones adecuadas de `Tk que se extienden a toda la gama matroidal de gráficos dispersos. Teorema 2 (La descomposición de guijarros-juego-con-colores). Un gráfico G es un juego de guijarros gráfico si y sólo si admite una descomposición en k borde-discoint subgraphs tales que cada uno es (1,0)-sparse y cada subgrafo de G contiene al menos ` piezas de árbol de la (1,0)-sparse gráficos en la descomposición. Las subgrafías de (1,0)-parse en la declaración de Teorema 2 son los colores de los guijarros; por lo tanto Teorema 2 da una caracterización de las descomposicións de guijarros-juego-con-colores obtenidos jugando el juego de guijarros definido en la sección anterior. Nótese la similitud entre el requisito de que el conjunto de piezas arbóreas tenga por lo menos un tamaño ` en el Teorema 2 y la definición de un propiamente dicho `Tk. Nuestros siguientes resultados muestran que para cualquier gráfico de juego de guijarros, podemos especializar su juego de guijarros construcción para generar una descomposición que es un mapa-y-árboles o `Tk. Nosotros llamamos a estos especializada construcción de juegos de guijarros canónicos, y el uso canónico juego de guijarros construc- ciones, obtenemos nuevas pruebas directas de los resultados de arboricidad existentes. Observamos Teorema 2 que los mapas-y-árboles son casos especiales del juego de guijarros decompo- Situación: tanto los árboles que se extienden y los mapas que se extienden son (1.0)-parse, y cada uno de la extensión los árboles aportan al menos un pedazo de árbol a cada subgrafo. El caso de los gráficos `Tk apropiados es más sutil; si cada color en una descomposición del juego de guijarros es un bosque, entonces hemos encontrado un adecuado `Tk, pero esta clase es un subconjunto de todos los posibles apropiados `Tk descomposiciones de un gráfico apretado. Demostramos que esta clase de descomposiciones apropiadas `Tk es suficiente para certificar la escasez. Ahora declaramos el teorema principal para el rango superior e inferior. Teorema 3 (Teorema Principal): Mapas y árboles coinciden con el juego de guijarros grafos). Que 0 ≤ ` ≤ k. Un gráfico G es un gráfico de juego de guijarro apretado si y sólo si G es un (k, `)- mapas y árboles. Teorema 4 (Teorema principal): Los gráficos `Tk adecuados coinciden con el juego de guijarros grafos). Deje k≤ 2k−1. Un gráfico G es un gráfico de juego de guijarros apretado si y sólo si es un adecuado `Tk con kn− ` bordes. Como corolarios, obtenemos los resultados de descomposición existentes para gráficos escasos. Corollario 5 (Nash-Williams [17], Tutte [23], White y Whiteley [24]). Deja k. Un gráfico G es estrecho si y sólo si tiene una descomposición (k, `)-maps-and-trees. Corollario 6 (Crapo [2], Haas [7]). Dejar k ≤ 2k−1. Un gráfico G es estrecho si y sólo si es un propiamente dicho `Tk. Encontrar eficientemente construcciones canónicas de juego de guijarros. Las pruebas de Teorema 3 y Theo- rem 4 conduce a un algoritmo obvio con O(n3) tiempo de ejecución para el problema de descomposición. Nuestro último resultado mejora en esto, mostrando que una construcción canónica juego de guijarros, y por lo tanto 8 Ileana Streinu, Louis Theran un mapa-y-árboles o `Tk descomposición apropiada se puede encontrar usando un algoritmo de juego de guijarros en O(n2) tiempo y espacio. Estos límites de tiempo y espacio significan que nuestro algoritmo puede combinarse con los de [12] sin ningún cambio en la complejidad. 5. Gráficos de juego de pebble En esta sección demostramos Teorema 1, un fortalecimiento de los resultados de [12] al juego de guijarros con colores. Dado que muchas de las propiedades relevantes del juego de guijarros con colores directamente de los juegos de guijarros de [12], nos referimos al lector allí para las pruebas. Comenzamos estableciendo algunas invariantes que se mantienen durante la ejecución del juego de guijarros. Lemma 7 (invariantes de juego de pebble). Durante la ejecución del juego de guijarros, lo siguiente los invariantes se mantienen en H: (I1) Hay por lo menos ` guijarros en V. [12] (I2) Para cada vértice v, span(v)+out(v)+peb(v) = k. [12] (I3) Para cada V ′ ́V, span(V ′)+out(V ′)+peb(V ′) = kn′. [12] (I4) Por cada vértice v V, outi(v)+pebi(v) = 1. (I5) Cada ruta máxima que consiste sólo de bordes con ci de color termina en el primer vértice con un guijarro de color ci o un ciclo. Prueba. (I1), (I2), y (I3) vienen directamente de [12]. (I4) Este invariante se mantiene claramente en la fase de inicialización del juego de guijarros con colores. Esa reserva de movimientos de bordes añadidos y guijarros (I4) está clara de la inspección. (I5) Por (I4), un camino monocromático de los bordes se ve obligado a terminar sólo en un vértice con un guijarro de el mismo color en ella. Si no hay guijarros de ese color alcanzable, entonces el camino debe eventualmente Visita un vértice dos veces. De estos invariantes, podemos mostrar que los gráficos constructibles del juego de guijarros son escasos. Lemma 8 (Los gráficos de los juegos de pelota son escasos [12]). Dejar H ser un gráfico construido con el Juego de guijarros. Entonces H es escasa. Si hay exactamente ` guijarros en V (H), entonces H es apretado. El paso principal para probar que cada gráfico escaso es un gráfico de juego de guijarros es el siguiente. Recordemos que al traer un guijarro a v nos referimos a reorientar H con movimientos de guijarro-deslizamiento para reducir el grado de v por uno. Lemma 9 (La condición de guijarro â € 1 [12]). Dejar vw ser un borde tal que H + vw es escaso. Si peb({v,w}) < â € 1, entonces un guijarro no en {v,w} se puede llevar a v o w. Se deduce que cualquier gráfico escaso tiene una construcción de juego de guijarros. Teorema 1 (Los gráficos Sparse y los gráficos de juego de guijarros coinciden). Un gráfico G es (k, `)-sparse con 0≤ 2k−1 si y sólo si G es un gráfico de juego de guijarros. 6. La descomposición de guijarros-juego-con-colores En esta sección demostramos Teorema 2, que caracteriza todas las descomposicións de juego de guijarros. Nosotros empezar con los siguientes lemas sobre la estructura de los componentes monocromáticos conectados en H, el gráfico dirigido mantenido durante el juego de guijarros. Descomposiciones del gráfico de certificación de la sparsity 9 Lemma 10 (los subgrafos monocromáticos del juego de guijarros son (1,0)-sparse). Deja que Hi sea el sub- gráfico de H inducido por los bordes con guijarros de color ci en ellos. Entonces Hi es (1,0)-parso, para i = 1,...,k. Prueba. Por (I4) Hi es un conjunto de bordes con grado a lo sumo uno para cada vértice. Lemma 11 (Piezas de árbol en un gráfico de juego de guijarros). Cada subgrafo del gráfico dirigido H en una construcción de juego de guijarros contiene por lo menos ` piezas monocromáticas de árboles, y cada uno de estos tiene sus raíces en un vértice con un guijarro en él o un vértice que es la cola de un borde. Recordemos que un borde superior a un subpárrafo H ′ = (V ′,E ′) es un borde vw con v′ V y vw /′ E. Prueba. Dejar que H ′ = (V ′,E ′) sea un subgrafo no vacío de H, y asumir sin pérdida de generalidad que H ′ es inducida por V ′. Por (I3), fuera (V ′)+ peb(V ′) ≥ `. Mostraremos que cada guijarro y cola de borde es la raíz de una pieza de árbol. Considerar un vértice v V ′ y un color ci. Por (I4) hay un único monocromático dirigido ruta de color ci a partir de v. Por (I5), si este camino termina en una rocalla, no tiene un ciclo. Del mismo modo, si este camino alcanza un vértice que es la cola de un borde también en color ci (es decir, si el trayectoria monocromática desde v hojas V ′), entonces la trayectoria no puede tener un ciclo en H ′. Dado que este argumento funciona para cualquier vértice en cualquier color, para cada color hay una partición de los vértices en aquellos que pueden alcanzar cada guijarro, cola de borde superior, o ciclo. De ello se deduce que cada uno de guijarros y cola de borde superior es la raíz de un árbol monocromático, como se desee. Aplicado a todo el gráfico Lemma 11 nos da lo siguiente. Lemma 12 (Los pebbles son las raíces de los árboles). En cualquier configuración de juego de guijarros, cada guijarros de color ci es la raíz de un (posiblemente vacío) monocromático árbol-pieza de color ci. Nota: Haas mostró en [7] que en un `Tk, un subgráfico inducido por n′ ≥ 2 vértices con m′ los bordes tienen exactamente piezas de árbol knm′ en él. Lemma 11 refuerza el resultado de Haas al ampliarlo a la gama inferior y dando una construcción que encuentra las piezas de árbol, mostrando la conexión entre la condición de guijarro â € 1 y la condición hereditaria en la adecuada `Tk. Concluimos nuestra investigación de construcciones arbitrarias de juego de guijarros con una descripción de la descomposición inducida por el juego de guijarros con colores. Teorema 2 (La descomposición de guijarros-juego-con-colores). Un gráfico G es un juego de guijarros gráfico si y sólo si admite una descomposición en k borde-discoint subgraphs tales que cada uno es (1,0)-sparse y cada subgrafo de G contiene al menos ` piezas de árbol de la (1,0)-sparse gráficos en la descomposición. Prueba. Deja que G sea un gráfico de juego de guijarros. La existencia de la k borde-disjunta (1,0)-sparse sub- Los gráficos fueron mostrados en Lemma 10, y Lemma 11 prueba la condición en subgrafías. Para la otra dirección, observamos que un ci de color con piezas de árbol ti en un subgrafo dado puede espacio a lo sumo n- ti bordes; sumando sobre todos los colores muestra que un gráfico con un guijarro-juego la descomposición debe ser escasa. Aplique el Teorema 1 para completar la prueba. Observación: Observamos que una descomposición del juego de guijarros para un gráfico de Laman puede ser leída de la coincidencia bipartita utilizada en el algoritmo de extracción de gráficos Laman de Hendrickson [9]. De hecho, las orientaciones de juego de guijarros tienen una correspondencia natural con los emparejamientos bipartitos utilizados en 10 Ileana Streinu, Louis Theran Mapas y árboles son un caso especial de descomposición de juegos de guijarros para gráficos apretados: si hay no son ciclos en ` de los colores, entonces los árboles enraizados en los ` guijarros correspondientes deben ser que se extienden, ya que tienen n - 1 bordes. Además, si cada color forma un bosque en un rango superior la descomposición del juego de guijarros, entonces la condición de piezas de árbol asegura que el juego de guijarros de- la composición es un `Tk. En la siguiente sección, mostramos que el juego de guijarros puede ser especializado para corresponder a los mapas- y árboles y las correspondientes descomposicións `Tk. 7. Construcciones Canónicas de Juego de Pebble En esta sección demostramos los principales teoremas (Teorema 3 y Teorema 4), continuando las inves- de las descomposiciones inducidas por las construcciones de juego de guijarros mediante el estudio del caso en el que un Se crea un número mínimo de ciclos monocromáticos. La idea principal, capturada en Lemma 15 e ilustrado en la Figura 6, es evitar la creación de ciclos al recoger piedras. Demostramos que esto es siempre posible, lo que implica que los mapas monocromáticos se crean sólo cuando añadir más de k(n1) bordes a algún conjunto de n′ vértices. Para el rango inferior, esto implica que Cada color es un bosque. Cada caracterización de descomposición de gráficos ajustados discutidos arriba sigue inmediatamente del teorema principal, dando nuevas pruebas de los resultados anteriores en un un marco unificado. En la prueba, vamos a utilizar dos especializaciones de los movimientos de juego de guijarros. El primero es un modi- ficación del movimiento de add-edge. Add-edge canónico: Al realizar un movimiento de add-edge, cubra el nuevo borde con un color que está en ambos vértices si es posible. Si no, entonces tome el color numerado más alto presente. La segunda es una restricción en la que los movimientos de guijarros-deslizamiento que permitimos. Deslizamiento canónico de guijarros: Un movimiento de guijarros se permite sólo cuando no crea un ciclo monocromático. Llamamos a una construcción de juego de guijarros que utiliza sólo estos movimientos canónicos. En esta sección vamos a mostrar que cada gráfico de juego de guijarros tiene una construcción canónica de juego de guijarros (Lemma 14 y Lemma 15) y que las construcciones canónicas de juego de guijarros corresponden a `Tk y las descomposicións de mapas y árboles (Teorema 3 y Teorema 4). Comenzamos con un lema técnico que motiva la definición de juego canónico de guijarros construcciones. Muestra que las situaciones desaprobadas por los movimientos canónicos son todas las maneras para que los ciclos se formen en los colores más bajos. Lemma 13 (creación del ciclo monocromático). Let v â € ¢ V tener un guijarro p de color ci en él y dejar w ser un vértice en el mismo árbol de color ci como v. Un ciclo monocromático de color ci se crea exactamente de una de las siguientes maneras: (M1) El borde vw se añade con un movimiento de add-edge. (M2) El borde wv es invertido por un movimiento de guijarro-deslizamiento y el guijarro p se utiliza para cubrir el reverso edge vw. Prueba. Observe que las condiciones previas en la declaración del lema están implícitas en Lemma 7. Por Lemma 12 ciclos monocromáticos se forman cuando el último guijarro de color ci se elimina de un Subgrafía monocromática conectada. (M1) y (M2) son las únicas maneras de hacer esto en un guijarro construcción del juego, ya que el color de un borde sólo cambia cuando se inserta la primera vez o un guijarro nuevo es puesto en él por un movimiento de guijarro-deslizamiento. Descomposiciones del gráfico de certificación de la sparsity 11 vw vw Fig. 5. Crear ciclos monocromáticos en un juego (2.0)-pebble. a) Un movimiento de tipo (M1) crea un ciclo por añadir un borde negro. (b) Un movimiento de tipo (M2) crea un ciclo con un movimiento de guijarros-deslizamiento. Los vértices son etiquetado de acuerdo a su papel en la definición de los movimientos. La figura 5 a) y la figura 5 b) muestran ejemplos de movimientos de creación de mapas (M1) y (M2), respectivamente, en una construcción de juego (2.0)-pebble. A continuación mostramos que si un gráfico tiene una construcción de juego de guijarros, entonces tiene un peb canónico- ble construcción de juegos. Esto se hace en dos pasos, considerando los casos (M1) y (M2) sepa- - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. La prueba da dos construcciones que implementan el add-edge canónico y canónico movimiento de guijarros-deslizamiento. Lemma 14 (El movimiento canónico de add-edge). Let G ser un gráfico con un juego de guijarros construc- tion. Los pasos de creación de ciclo de tipo (M1) se pueden eliminar en colores ci para 1 ≤ i ≤, donde = min{k,. Prueba. Para los movimientos de add-edge, cubra el borde con un color presente en v y w si es posible. Si esto no es posible, entonces hay â € 1 colores distintos presentes. Usar el color numerado más alto para cubrir el nuevo borde. Observación: Observamos que en el rango superior, siempre hay un color repetido, por lo que no canónico los movimientos de add-edge crean ciclos en el rango superior. El movimiento canónico de guijarros se define por una condición global. Para demostrar que obtenemos la misma clase de gráficos usando sólo movimientos canónicos de rocalla-deslizamiento, tenemos que extender Lemma 9 a sólo movimientos canónicos. El paso principal es mostrar que si hay alguna secuencia de movimientos que reorienta un camino de v a w, entonces hay una secuencia de movimientos canónicos que hace lo mismo Cosa. Lemma 15 (El movimiento canónico de guijarros). Cualquier secuencia de deslizamiento de guijarros se mueve llevando a un movimiento de add-edge se puede reemplazar por uno que no tiene pasos (M2) y permite el mismo add-edge move. En otras palabras, si es posible recoger 1 guijarros en los extremos de un borde a añadir, entonces es posible hacer esto sin crear ningún ciclo monocromático. 12 Ileana Streinu, Louis Theran La Figura 7 y la Figura 8 ilustran la construcción utilizada en la prueba de Lemma 15. Nosotros llamamos a esto la construcción de atajos por analogía a la unión matroide y caminos de aumento de intersección utilizados en trabajos anteriores en el rango inferior. La Figura 6 muestra la estructura de la prueba. La construcción de acceso directo elimina un paso (M2) al principio de una secuencia que reorienta un camino de v a w con deslizamientos de guijarros. Desde uno la aplicación de la construcción abreviada reorienta un camino simple de un vértice w′ a w, y un ruta de v a w′ se conserva, la construcción de acceso directo se puede aplicar inductivamente para encontrar la secuencia de movimientos que queremos. Fig. 6. Esquema de la construcción del atajo: (a) Un camino sencillo arbitrario de v a w con líneas curvas indicando caminos simples. b) Una etapa (M2). El borde negro, a punto de ser volteado, crearía un ciclo, se muestra en gris rayado y sólido, del (único) árbol gris enraizado en w. Los bordes grises sólidos eran parte de la ruta original de (a). (c) El camino acortado a la rocalla gris; el nuevo camino sigue el gris árbol todo el camino desde la primera vez que el camino original tocó el árbol gris en w′. La ruta de v a w′ es simple, y la construcción del atajo se puede aplicar inductivamente a él. Prueba. Sin pérdida de generalidad, podemos asumir que nuestra secuencia de movimientos reorienta un simple camino en H, y que el primer movimiento (el final del camino) es (M2). El paso (M2) mueve un guijarro de color ci de un vértice w en el borde vw, que se invierte. Porque el movimiento es (M2), v y w están contenidos en un árbol monocromático máximo de color ci. Llame a este árbol H ′i, y observar que está arraigado en w. Ahora considere los bordes invertidos en nuestra secuencia de movimientos. Como se ha señalado anteriormente, antes de hacer cualquiera de los movimientos, estos bosquejan un camino simple en H que termina en w. Que z sea el primer vértice en este camino en H ′i. Modificamos nuestra secuencia de movimientos de la siguiente manera: eliminar, desde el principio, cada mover antes de la que invierte algunos yz borde; prepend en lo que queda una secuencia de movimientos que mueve el guijarro en w a z en H ′i. Descomposiciones del gráfico de certificación de la sparsity 13 Fig. 7. Eliminando movimientos (M2): (a) un movimiento (M2); (b) evitando el (M2) moviéndose por otro camino. El camino donde se mueven los guijarros está indicado por líneas duplicadas. Fig. 8. Eliminación (M2) movimientos: (a) el primer paso para mover el guijarro negro a lo largo del camino doble es (M2); (b) evitando el (M2) y simplificando el camino. Puesto que ningún borde cambia de color en el comienzo de la nueva secuencia, hemos eliminado el movimiento (M2). Porque nuestra construcción no cambia ninguno de los bordes involucrados en el cola restante de la secuencia original, la parte de la ruta original que queda en el nuevo secuencia seguirá siendo un camino simple en H, cumpliendo con nuestra hipótesis inicial. El resto del lema sigue por inducción. Juntos Lemma 14 y Lemma 15 prueban lo siguiente. Lemma 16. Si G es un gráfico de juego de guijarros, entonces G tiene una construcción canónica de juego de guijarros. Usando construcciones canónicas de juego de guijarros, podemos identificar los gráficos apretados de juego de guijarros con mapas y árboles y gráficos `Tk. 14 Ileana Streinu, Louis Theran Teorema 3 (Teorema Principal): Mapas y árboles coinciden con el juego de guijarros grafos). Que 0 ≤ ` ≤ k. Un gráfico G es un gráfico de juego de guijarro apretado si y sólo si G es un (k, `)- mapas y árboles. Prueba. Como se observó anteriormente, una descomposición de mapas y árboles es un caso especial del juego de guijarros descomposición. Aplicando el Teorema 2, vemos que cualquier mapa y árbol debe ser un juego de guijarros gráfico. Para la dirección inversa, considere la construcción canónica de un juego de guijarros de un gráfico apretado. Desde Lemma 8, vemos que quedan piedras en G al final de la construcción. Los definición del movimiento canónico de add-edge implica que debe haber al menos un guijarro de cada ci para i = 1,2,........................................................................................................... Se deduce que hay exactamente uno de cada uno de estos colores. Por Lemma 12, cada uno de estos guijarros es la raíz de una pieza arbórea monocromática con n - 1 bordes, dando los árboles de separación de bordes necesarios. Corollario 5 (Nash-Williams [17], Tutte [23], White y Whiteley [24]). Deja k. Un gráfico G es estrecho si y sólo si tiene una descomposición (k, `)-maps-and-trees. A continuación consideramos las descomposicións inducidas por las construcciones canónicas de juego de guijarros cuando k +1. Teorema 4 (Teorema Principal): Árboles y árboles adecuados coinciden con el ble-game graphs). Deje k≤ 2k−1. Un gráfico G es un gráfico de juego de guijarro apretado si y sólo si es un `Tk con bordes kn− ` adecuado. Prueba. Como se ha señalado anteriormente, una descomposición adecuada de `Tk debe ser escasa. Lo que tenemos que mostrar es que una construcción canónica de un juego de guijarros de un gráfico apretado produce una adecuada `Tk. Por Teorema 2 y Lemma 16, ya tenemos la condición en los árboles-piezas y el decom- posición en `árboles de borde-desconectado. Por último, una aplicación de (I4), muestra que cada vértice debe en exactamente k de los árboles, según sea necesario. Corollario 6 (Crapo [2], Haas [7]). Dejar k ≤ 2k−1. Un gráfico G es estrecho si y sólo si es un propiamente dicho `Tk. 8. Algoritmos de juego de pebble para encontrar descomposicións Una ejecución naïve de las construcciones en la sección anterior conduce a un algoritmo re- tiempo para recoger cada guijarro en una construcción canónica: en el peor de los casos aplicaciones de la construcción en Lemma 15 requiriendo tiempo cada uno, dando un total de ejecución tiempo de فارسى(n3) para el problema de descomposición. En esta sección, describimos algoritmos para el problema de descomposición que se ejecutan en el tiempo O(n2). Comenzamos con la estructura general del algoritmo. Algoritmo 17 (El juego canónico de guijarros con colores). Entrada: Un gráfico G. Salida: Un gráfico de juego de guijarros H. Método: – Conjunto V (H) = V (G) y colocar un guijarro de cada color en los vértices de H. – Para cada borde vw E(G) tratar de recoger al menos 1 guijarros en v y w utilizando guijarros deslizante movimientos según lo descrito por Lemma 15. Descomposiciones del gráfico de certificación de la Sparsity 15 – Si al menos 1 guijarros se puede recoger, añadir vw a H utilizando un movimiento de borde añadido como en Lemma 14, por lo demás descarte vw. – Finalmente, devolver H, y las ubicaciones de los guijarros. Correcto. Teorema 1 y el resultado de [24] que los gráficos escasos son los independientes conjuntos de un matroide muestran que H es un subgrafo de tamaño máximo escaso de G. Desde la construcción encontrado es canónico, el teorema principal muestra que el color de los bordes en H da un mapa- y-árboles o descomposición adecuada `Tk. Complejidad. Comenzamos observando que el tiempo de ejecución del Algoritmo 17 es el tiempo necesario para proceso O(n) bordes añadidos a H y O(m) bordes no añadidos a H. Primero consideramos el costo de un borde de G que se añade a H. Cada uno de los movimientos de juego de guijarros se puede implementar en tiempo constante. Lo que queda es a describir una manera eficiente de encontrar y mover los guijarros. Utilizamos el siguiente algoritmo como un Subrutina de Algoritmo 17 para hacer esto. Algoritmo 18 (Encontrar un camino canónico a una rocalla.). Entrada: Vertices v y w, y una configuración de juego de guijarros en un gráfico dirigido H. Salida: Si se encontró un guijarro, ‘sí’ y ‘no’ de otra manera. Se actualiza la configuración de H. Método: – Comience por hacer una búsqueda de profundidad desde v en H. Si no se encuentra ningún guijarro en w, detener y devolver «no.» – De lo contrario se encontró un guijarro. Ahora tenemos una ruta v = v1,e1,. ..,ep−1,vp = u, donde el vi son vértices y ei es el borde vivi+1. Que c[ei] sea el color del guijarro en ei. Usaremos la matriz c[] para hacer un seguimiento de los colores de los guijarros en los vértices y los bordes después de moverlos y el array s[] para dibujar un camino canónico de v a u encontrando un sucesor para cada uno borde. – Establecer s[u] = «end′ y establecer c[u] al color de una piedra arbitraria en u. Caminamos en el camino en orden inverso: vp,ep−1,ep−2,. ..,e1,v1. Para cada i, verifique si c[vi] está configurado; si es así, vaya a la siguiente i. De lo contrario, compruebe si c[vi+1] = c[ei]. – Si lo es, establece s[vi] = ei y establece c[vi] = c[ei], y pasa al siguiente borde. – De lo contrario c[vi+1] 6= c[ei], tratar de encontrar un camino monocromático en color c[vi+1] de vi a vi+1. Si un vértice x se encuentra para el cual c[x] se establece, tenemos una ruta vi = x1, f1,x2,. .., fq−1,xq = x que es monocromático en el color de los bordes; establecer c[xi] = c[fi] y s[xi] = fi para i = 1,2,...,q−1. Si c[x] = c[ fq−1], pare. De lo contrario, comprobar recursivamente que no hay un monocro- c[x] ruta mática de xq−1 a x usando este mismo procedimiento. – Finalmente, deslizar guijarros a lo largo del camino desde los puntos finales originales v a u especificado por el array sucesor s[v], s[s[v],... La corrección de Algoritmo 18 viene del hecho de que está implementando el atajo construcción. La eficiencia viene del hecho de que en lugar de potencialmente mover el guijarro hacia atrás y adelante, Algoritmo 18 pre-computa un camino canónico que cruza cada borde de H a lo sumo tres times: una vez en la primera búsqueda de profundidad inicial, y dos veces al convertir la ruta inicial a una Canónico. De ello se deduce que cada borde aceptado toma O(n) tiempo, para un total de O(n2) tiempo los bordes de procesamiento gastados en H. Aunque no hemos discutido esta explicitación, para que el algoritmo sea eficiente necesitamos mantener los componentes como en [12]. Después de cada borde aceptado, los componentes de H se pueden actualizar en el tiempo O(n). Por último, los resultados de [12, 13] muestran que los bordes rechazados toman un O(1) amortizado tiempo cada uno. 16 Ileana Streinu, Louis Theran Resumiendo, hemos demostrado que el juego canónico de guijarros con colores resuelve la decom- problema de posición en el tiempo O(n2). 9. Un caso especial importante: Rigidez en la dimensión 2 y slider-pinning En esta breve sección presentamos una nueva solicitud para el caso especial de importancia práctica, k = 2, ` = 3. Como se explica en la introducción, el teorema de Laman [11] caracteriza mínimamente gráficos rígidos como los gráficos ajustados (2,3). En el trabajo reciente sobre el slider pinning, desarrollado después de la El documento actual fue presentado, introdujimos el modelo de slider-pinning de rigidez [15, 20]. Com- binatoriamente, modelamos los marcos bar-slider como gráficos simples junto con algunos bucles colocados en sus vértices de tal manera que no haya más de 2 bucles por vértice, uno de cada uno color. Caracterizamos los gráficos de deslizadores de barras mínimamente rígidos [20] como gráficos que son: 1. (2,3)-parse para subgrafías que no contengan bucles. 2. (2,0)-ajustado cuando se incluyen los bucles. Llamamos a estos gráficos (2,0,3)-clasificados-ajustados, y son un caso especial de la clasificación-parse gráficos estudiados en nuestro artículo [14]. La conexión con los juegos de guijarros en este artículo es la siguiente. Corollary 19 (juegos de pebble y slider-pinning). En cualquier gráfico de juego (2,3)-pebble, si Reemplazar los guijarros por los bucles, obtenemos un gráfico ajustado (2.0,3)-calificado. Prueba. Seguidos de invariantes (I3) de Lemma 7. En [15], estudiamos un caso especial de slider pinning donde cada slider es vertical o horizontal. Modelamos los deslizadores como bucles precoloreados, con el color que indica la dirección x o y. Para este caso de deslizador paralelo eje, los gráficos mínimamente rígidos se caracterizan por: 1. (2,3)-parse para subgrafías que no contengan bucles. 2. Admitir un 2-coloración de los bordes para que cada color sea un bosque (es decir, no tiene ciclos), y cada uno árbol monocromático abarca exactamente un bucle de su color. Esto también tiene una interpretación en términos de juegos de guijarros de colores. Corollary 20 (El juego de guijarros con colores y slider-pinning). En cualquier canónico (2,3)- Guijarro-juego-con-colores gráfico, si reemplazamos los guijarros por bucles del mismo color, obtenemos el gráfico de un marco de eje-paralelo de barra-slider mínimamente fijado. Prueba. Sigue desde el Teorema 4, y Lemma 12. 10. Conclusiones y problemas pendientes Presentamos una nueva caracterización de (k, `)-sparse gráficos, el juego de guijarros con colores, y lo utilizó para dar un algoritmo eficiente para encontrar descomposicións de gráficos escasos en el borde- árboles desarticulados. Nuestro algoritmo encuentra tales descomposiciones certificadoras de esparcimiento en el rango superior y se ejecuta en el tiempo O(n2), que es tan rápido como los algoritmos para reconocer gráficos escasos en el rango superior a partir de [12]. También usamos el juego de guijarros con colores para describir una nueva descomposición de la sparsity-certificating- ciones que se aplican a toda la gama matroidal de gráficos dispersos. Descomposiciones del gráfico de certificación de la sparsity 17 Definimos y estudiamos una clase de construcciones canónicas de juego de guijarros que corresponden a o bien una descomposición de mapas y árboles o bien una descomposición adecuada de `Tk. Esto da una nueva prueba de la Tutte-Nash- Teorema de arboricidad Williams y una prueba unificada de la descomposición previamente estudiada cer- tificates de la esparzidad. Las construcciones canónicas de juego de guijarros también muestran la relación entre la condición de guijarro â 1, que se aplica a la gama superior de â, para aumentar la unión de los matroides rutas, que no se aplican en el rango superior. Consecuencias algorítmicas y problemas abiertos. En [6], Gabow y Westermann dan un O(n3/2) algoritmo para reconocer gráficos escasos en el rango inferior y extraer subtítulos escasos de Densos. Su técnica se basa en la búsqueda eficiente de caminos de aumento de unión de matroides, que extienden una descomposición de mapas y árboles. El algoritmo O(n3/2) utiliza dos subrutinas para encontrar rutas de aumento: exploración cíclica, que encuentra rutas de aumento uno a la vez, y lote escaneado, que encuentra grupos de caminos de aumento disjuntos. Observamos que Algoritmo 17 se puede utilizar para reemplazar el escaneo cíclico en Gabow y Wester- algoritmo de mann sin cambiar el tiempo de ejecución. Las estructuras de datos utilizadas en la aplicación de guijarros, detallado en [12, 13] son más simples y más fáciles de implementar que los utilizado para apoyar el escaneo cíclico. Los dos principales problemas algorítmicos abiertos relacionados con el juego de guijarros son entonces: Problema 1. Desarrollar un algoritmo de juego de guijarros con las propiedades de escaneado por lotes y obtener un algoritmo O(n3/2) implementable para el rango inferior. Problema 2. Extender la exploración por lotes a la condición de guijarro â € 1 y derivar un guijarro O(n3/2) algoritmo de juego para el rango superior. En particular, sería de importancia práctica encontrar un algoritmo O(n3/2) implementable para las descomposiciones en los árboles que se extienden por los bordes. Bibliografía 1. Berg, A.R., Jordán, T.: Algoritmos para la rigidez gráfica y el análisis de la escena. In: Proc. 11a Simposio Europeo sobre Algoritmos (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003) 2. Crapo, H.: Sobre la rigidez genérica de los marcos planos. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988) 3. Edmonds, J.: Partición mínima de un matroide en conjuntos independientes. J. Res. Nat. Bur. Normas Secc. B 69B, 67–72 (1965) 4. Edmonds, J.: Funciones submodulares, matroides y ciertos poliedros. En: Combinatoria Optimización: ¡Eureka, encogerte!, no. 2570 in LNCS, pp. 11–26. Springer (2003) 5. Gabow, H.N.: Un enfoque matroide para encontrar conectividad de borde y arborescencias de embalaje. Journal of Computer and System Sciences 50, 259–273 (1995) 6. Gabow, H.N., Westermann, H.H.: Bosques, marcos y juegos: Algoritmos para sumas de matroide y aplicaciones. Algoritmica 7(1), 465–497 (1992) 7. Haas, R.: Caracterizaciones de la arboricidad de los gráficos. Ars Combinatoria 63, 129–137 (2002) 8. Haas, R., Lee, A., Streinu, I., Theran, L.: Caracterizando gráficos escasos por mapa decompo- Situaciones. Revista de Matemáticas Combinatoria y Computación Combinatoria 62, 3-11 (2007) 9. Hendrickson, B.: Condiciones para realizaciones gráficas únicas. SIAM Journal on Computing 21(1), 65–84 (1992) 18 Ileana Streinu, Louis Theran 10. Jacobs, D.J., Hendrickson, B.: Un algoritmo para la percolación de rigidez bidimensional: la Juego de guijarros. Revista de Física Computacional 137, 346-365 (1997) 11. Laman, G.: En gráficos y rigidez de las estructuras esqueléticas planas. Revista de Ingeniería Matemáticas 4, 331-340 (1970) 12. Lee, A., Streinu, I.: Algorihms de juego de pebble y gráficos escasos. Matemáticas discretas 308(8), 1425-1437 (2008) 13. Lee, A., Streinu, I., Theran, L.: Encontrar y mantener componentes rígidos. In: Proc. Cana... Conferencia de Geometría Computacional. Windsor, Ontario (2005). http://cccg. cs.uwindsor.ca/papers/72.pdf 14. Lee, A., Streinu, I., Theran, L.: Gráficos bajos y matroides. Diario de Universal Ciencias de la computación 13(10) (2007) 15. Lee, A., Streinu, I., Theran, L.: El problema del slider-pinning. En: Actas del 19 Conferencia Canadiense sobre Geometría Computacional (CCCG’07) (2007) 16. Lovász, L.: Problemas y ejercicios combinatorios. Akademiai Kiado y North-Holland, Amsterdam (1979) 17. Nash-Williams, C.S.A.: Descomposición de gráficos finitos en los bosques. Diario de Londres Sociedad Matemática 39, 12 (1964) 18. Oxley, J.G.: Teoría de los matroides. The Clarendon Press, Oxford University Press, Nueva York (1992) 19. Roskind, J., Tarjan, R.E.: Una nota sobre la búsqueda de un coste mínimo borde de árboles disjuntos que se extienden. Matemáticas de la investigación de operaciones 10(4), 701-708 (1985) 20. Streinu, I., Theran, L.: Genericidad combinatoria y rigidez mínima. En: SCG ’08: Pro- cedidas del 24o Simposio anual sobre Geometría Computacional, pp. 365– 374. ACM, Nueva York, NY, USA (2008). 21. Tay, T.S.: Rigidez de los multógrafos I: uniendo cuerpos rígidos en n-espacio. Diario de Combinato- rial Theory, Serie B 26, 95–112 (1984) 22. Tay, T.S.Una nueva prueba del teorema de Laman. Gráficos y combinatorios 9, 365–370 (1993) 23. Tutte, W.T.: Sobre el problema de la descomposición de un gráfico en n factores conectados. Diario de Sociedad Matemática de Londres 142, 221–230 (1961) 24. Whiteley, W.: La unión de los matroides y la rigidez de los marcos. SIAM Journal on Matemáticas discretas 1(2), 237–255 (1988) http://cccg.cs.uwindsor.ca/papers/72.pdf http://cccg.cs.uwindsor.ca/papers/72.pdf Introducción y preliminares Antecedentes históricos El juego de guijarros con colores Nuestros resultados Gráficos de juego de pebble La descomposición de guijarros-juego-con-colores Construcciones Canónicas de Juego de Pebble Algoritmos de juego de pebble para encontrar descomposicións Un caso especial importante: Rigidez en la dimensión 2 y slider-pinning Conclusiones y problemas pendientes
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The evolution of the Earth-Moon system based on the dark matter field fluid model
The evolution of Earth-Moon system is described by the dark matter field fluid model proposed in the Meeting of Division of Particle and Field 2004, American Physical Society. The current behavior of the Earth-Moon system agrees with this model very well and the general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The closest distance of the Moon to Earth was about 259000 km at 4.5 billion years ago, which is far beyond the Roche's limit. The result suggests that the tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The average dark matter field fluid constant derived from Earth-Moon system data is 4.39 x 10^(-22) s^(-1)m^(-1). This model predicts that the Mars's rotation is also slowing with the angular acceleration rate about -4.38 x 10^(-22) rad s^(-2).
The evolution of the Earth-Moon system based on the dark fluid model The evolution of the Earth-Moon system based on the dark matter field fluid model Hongjun Pan Department of Chemistry University of North Texas, Denton, Texas 76203, U. S. A. Abstract The evolution of Earth-Moon system is described by the dark matter field fluid model with a non-Newtonian approach proposed in the Meeting of Division of Particle and Field 2004, American Physical Society. The current behavior of the Earth-Moon system agrees with this model very well and the general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The closest distance of the Moon to Earth was about 259000 km at 4.5 billion years ago, which is far beyond the Roche’s limit. The result suggests that the tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The average dark matter field fluid constant derived from Earth-Moon system data is 4.39 × 10-22 s-1m-1. This model predicts that the Mars’s rotation is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2. Key Words. dark matter, fluid, evolution, Earth, Moon, Mars 1. Introduction The popularly accepted theory for the formation of the Earth-Moon system is that the Moon was formed from debris of a strong impact by a giant planetesimal with the Earth at the close of the planet-forming period (Hartmann and Davis 1975). Since the formation of the Earth-Moon system, it has been evolving at all time scale. It is well known that the Moon is receding from us and both the Earth’s rotation and Moon’s rotation are slowing. The popular theory is that the tidal friction causes all those changes based on the conservation of the angular momentum of the Earth-Moon system. The situation becomes complicated in describing the past evolution of the Earth-Moon system. Because the Moon is moving away from us and the Earth rotation is slowing, this means that the Moon was closer and the Earth rotation was faster in the past. Creationists argue that based on the tidal friction theory, the tidal friction should be stronger and the recessional rate of the Moon should be greater in the past, the distance of the Moon would quickly fall inside the Roche's limit (for earth, 15500 km) in which the Moon would be torn apart by gravity in 1 to 2 billion years ago. However, geological evidence indicates that the recession of the Moon in the past was slower than the present rate, i. e., the recession has been accelerating with time. Therefore, it must be concluded that tidal friction was very much less in the remote past than we would deduce on the basis of present-day observations (Stacey 1977). This was called “geological time scale difficulty” or “Lunar crisis” and is one of the main arguments by creationists against the tidal friction theory (Brush 1983). But we have to consider the case carefully in various aspects. One possible scenario is that the Earth has been undergoing dynamic evolution at all time scale since its inception, the geological and physical conditions (such as the continent positions and drifting, the crust, surface temperature fluctuation like the glacial/snowball effect, etc) at remote past could be substantially different from currently, in which the tidal friction could be much less; therefore, the receding rate of the Moon could be slower. Various tidal friction models were proposed in the past to describe the evolution of the Earth- Moon system to avoid such difficulty or crisis and put the Moon at quite a comfortable distance from Earth at 4.5 billion years ago (Hansen 1982, Kagan and Maslova 1994, Ray et al. 1999, Finch 1981, Slichter 1963). The tidal friction theories explain that the present rate of tidal dissipation is anomalously high because the tidal force is close to a resonance in the response function of ocean (Brush 1983). Kagan gave a detailed review about those tidal friction models (Kagan 1997). Those models are based on many assumptions about geological (continental position and drifting) and physical conditions in the past, and many parameters (such as phase lag angle, multi-mode approximation with time dependent frequencies of the resonance modes, etc.) have to be introduced and carefully adjusted to make their predictions close to the geological evidence. However, those assumptions and parameters are still challenged, to certain extent, as concoction. The second possible scenario is that another mechanism could dominate the evolution of the Earth-Moon system and the role of the tidal friction is not significant. In the Meeting of Division of Particle and Field 2004, American Physical Society, University of California at Riverside, the author proposed a dark matter field fluid model (Pan 2005) with a non-Newtonian approach, the current Moon and Earth data agree with this model very well. This paper will demonstrate that the past evolution of Moon-Earth system can be described by the dark matter field fluid model without any assumptions about past geological and physical conditions. Although the subject of the evolution of the Earth-Moon system has been extensively studied analytically or numerically, to the author’s knowledge, there are no theories similar or equivalent to this model. 2. Invisible matter In modern cosmology, it was proposed that the visible matter in the universe is about 2 ~ 10 % of the total matter and about 90 ~ 98% of total matter is currently invisible which is called dark matter and dark energy, such invisible matter has an anti- gravity property to make the universe expanding faster and faster. If the ratio of the matter components of the universe is close to this hypothesis, then, the evolution of the universe should be dominated by the physical mechanism of such invisible matter, such physical mechanism could be far beyond the current Newtonian physics and Einsteinian physics, and the Newtonian physics and Einsteinian physics could reflect only a corner of the iceberg of the greater physics. If the ratio of the matter components of the universe is close to this hypothesis, then, it should be more reasonable to think that such dominant invisible matter spreads in everywhere of the universe (the density of the invisible matter may vary from place to place); in other words, all visible matter objects should be surrounded by such invisible matter and the motion of the visible matter objects should be affected by the invisible matter if there are interactions between the visible matter and the invisible matter. If the ratio of the matter components of the universe is close to this hypothesis, then, the size of the particles of the invisible matter should be very small and below the detection limit of the current technology; otherwise, it would be detected long time ago with such dominant amount. With such invisible matter in mind, we move to the next section to develop the dark matter field fluid model with non-Newtonian approach. For simplicity, all invisible matter (dark matter, dark energy and possible other terms) is called dark matter here. 3. The dark matter field fluid model In this proposed model, it is assumed that: 1. A celestial body rotates and moves in the space, which, for simplicity, is uniformly filled with the dark matter which is in quiescent state relative to the motion of the celestial body. The dark matter possesses a field property and a fluid property; it can interact with the celestial body with its fluid and field properties; therefore, it can have energy exchange with the celestial body, and affect the motion of the celestial body. 2. The fluid property follows the general principle of fluid mechanics. The dark matter field fluid particles may be so small that they can easily permeate into ordinary “baryonic” matter; i. e., ordinary matter objects could be saturated with such dark matter field fluid. Thus, the whole celestial body interacts with the dark matter field fluid, in the manner of a sponge moving thru water. The nature of the field property of the dark matter field fluid is unknown. It is here assumed that the interaction of the field associated with the dark matter field fluid with the celestial body is proportional to the mass of the celestial body. The dark matter field fluid is assumed to have a repulsive force against the gravitational force towards baryonic matter. The nature and mechanism of such repulsive force is unknown. With the assumptions above, one can study how the dark matter field fluid may influence the motion of a celestial body and compare the results with observations. The common shape of celestial bodies is spherical. According to Stokes's law, a rigid non- permeable sphere moving through a quiescent fluid with a sufficiently low Reynolds number experiences a resistance force F rvF πμ6−= (1) where v is the moving velocity, r is the radius of the sphere, and μ is the fluid viscosity constant. The direction of the resistance force F in Eq. 1 is opposite to the direction of the velocity v. For a rigid sphere moving through the dark matter field fluid, due to the dual properties of the dark matter field fluid and its permeation into the sphere, the force F may not be proportional to the radius of the sphere. Also, F may be proportional to the mass of the sphere due to the field interaction. Therefore, with the combined effects of both fluid and field, the force exerted on the sphere by the dark matter field fluid is assumed to be of the scaled form (2) mvrF n−−= 16πη where n is a parameter arising from saturation by dark matter field fluid, the r1-n can be viewed as the effective radius with the same unit as r, m is the mass of the sphere, and η is the dark matter field fluid constant, which is equivalent to μ. The direction of the resistance force F in Eq. 2 is opposite to the direction of the velocity v. The force described by Eq. 2 is velocity-dependent and causes negative acceleration. According to Newton's second law of motion, the equation of motion for the sphere is mvr m n−−= 16πη (3) Then (4) )6exp( 10 vtrvv n−−= πη where v0 is the initial velocity (t = 0) of the sphere. If the sphere revolves around a massive gravitational center, there are three forces in the line between the sphere and the gravitational center: (1) the gravitational force, (2) the centripetal acceleration force; and (3) the repulsive force of the dark matter field fluid. The drag force in Eq. 3 reduces the orbital velocity and causes the sphere to move inward to the gravitational center. However, if the sum of the centripetal acceleration force and the repulsive force is stronger than the gravitational force, then, the sphere will move outward and recede from the gravitational center. This is the case of interest here. If the velocity change in Eq. 3 is sufficiently slow and the repulsive force is small compared to the gravitational force and centripetal acceleration force, then the rate of receding will be accordingly relatively slow. Therefore, the gravitational force and the centripetal acceleration force can be approximately treated in equilibrium at any time. The pseudo equilibrium equation is GMm 2 2 = (5) where G is the gravitational constant, M is the mass of the gravitational center, and R is the radius of the orbit. Inserting v of Eq. 4 into Eq. 5 yields )12exp( 1 R n−= πη (6) (7) )12exp( 10 trRR n−= πη where R = (8) R0 is the initial distance to the gravitational center. Note that R exponentially increases with time. The increase of orbital energy with the receding comes from the repulsive force of dark matter field fluid. The recessional rate of the sphere is dR n−= 112πη (9) The acceleration of the recession is ( Rr Rd n 21 12 −= πη ) . (10) The recessional acceleration is positive and proportional to its distance to the gravitational center, so the recession is faster and faster. According to the mechanics of fluids, for a rigid non-permeable sphere rotating about its central axis in the quiescent fluid, the torque T exerted by the fluid on the sphere ωπμ 38 rT −= (11) where ω is the angular velocity of the sphere. The direction of the torque in Eq. 11 is opposite to the direction of the rotation. In the case of a sphere rotating in the quiescent dark matter field fluid with angular velocity ω, similar to Eq. 2, the proposed T exerted on the sphere is ( ) ωπη mrT n 318 −−= (12) The direction of the torque in Eq. 12 is opposite to the direction of the rotation. The torque causes the negative angular acceleration = (13) where I is the moment of inertia of the sphere in the dark matter field fluid ( )21 2 nrmI −= (14) Therefore, the equation of rotation for the sphere in the dark matter field fluid is ωπη d −−= 120 (15) Solving this equation yields (16) )20exp( 10 tr n−−= πηωω where ω0 is the initial angular velocity. One can see that the angular velocity of the sphere exponentially decreases with time and the angular deceleration is proportional to its angular velocity. For the same celestial sphere, combining Eq. 9 and Eq. 15 yields (17) The significance of Eq. 17 is that it contains only observed data without assumptions and undetermined parameters; therefore, it is a critical test for this model. For two different celestial spheres in the same system, combining Eq. 9 and Eq. 15 yields 67.1 1 −=−=⎟⎟ (18) This is another critical test for this model. 4. The current behavior of the Earth-Moon system agrees with the model The Moon-Earth system is the simplest gravitational system. The solar system is complex, the Earth and the Moon experience not only the interaction of the Sun but also interactions of other planets. Let us consider the local Earth-Moon gravitational system as an isolated local gravitational system, i.e., the influence from the Sun and other planets on the rotation and orbital motion of the Moon and on the rotation of the Earth is assumed negligible compared to the forces exerted by the moon and earth on each other. In addition, the eccentricity of the Moon's orbit is small enough to be ignored. The data about the Moon and the Earth from references (Dickey et .al., 1994, and Lang, 1992) are listed below for the readers' convenience to verify the calculation because the data may vary slightly with different data sources. Moon: Mean radius: r = 1738.0 km Mass: m = 7.3483 × 1025 gram Rotation period = 27.321661 days Angular velocity of Moon = 2.6617 × 10-6 rad s-1 Mean distance to Earth Rm= 384400 km Mean orbital velocity v = 1.023 km s-1 Orbit eccentricity e = 0.0549 Angular rotation acceleration rate = -25.88 ± 0.5 arcsec century-2 = (-1.255 ± 0.024) × 10-4 rad century-2 = (-1.260 ± 0.024) × 10-23 rad s-2 Receding rate from Earth = 3.82 ± 0.07 cm year-1 = (1.21 ± 0.02) × 10-9 m s-1 Earth: Mean radius: r = 6371.0 km Mass: m = 5.9742 × 1027 gram Rotation period = 23 h 56m 04.098904s = 86164.098904s Angular velocity of rotation = 7.292115 × 10-5 rad s-1 Mean distance to the Sun Rm= 149,597,870.61 km Mean orbital velocity v = 29.78 km s-1 Angular acceleration of Earth = (-5.5 ± 0.5) × 10-22 rad s-2 The Moon's angular rotation acceleration rate and increase in mean distance to the Earth (receding rate) were obtained from the lunar laser ranging of the Apollo Program (Dickey et .al., 1994). By inserting the data of the Moon's rotation and recession into Eq. 17, the result is 039.054.1 10662.21021.1 1092509.31026.1 (19) The distance R in Eq. 19 is from the Moon's center to the Earth's center and the number 384400 km is assumed to be the distance from the Moon's surface to the Earth's surface. Eq. 19 is in good agreement with the theoretical value of -1.67. The result is in accord with the model used here. The difference (about 7.8%) between the values of -1.54 and - 1.67 may come from several sources: 1. Moon's orbital is not a perfect circle 2. Moon is not a perfect rigid sphere. 3. The effect from Sun and other planets. 4. Errors in data. 5. Possible other unknown reasons. The two parameters n and η in Eq. 9 and Eq. 15 can be determined with two data sets. The third data set can be used to further test the model. If this model correctly describes the situation at hand, it should give consistent results for different motions. The values of n and η calculated from three different data sets are listed below (Note, the mean distance of the Moon to the Earth and mean radii of the Moon and the Earth are used in the calculation). The value of n: n = 0.64 From the Moon's rotation: η = 4.27 × 10-22 s-1 m-1 From the Earth's rotation: η = 4.26 × 10-22 s-1 m-1 From the Moon's recession: η = 4.64 × 10-22 s-1 m-1 One can see that the three values of η are consistent within the range of error in the data. The average value of η: η = (4.39 ± 0.22) × 10-22 s-1 m-1 By inserting the data of the Earth's rotation, the Moon’s recession and the value of n into Eq. 18, the result is 14.053.1 6371000 1738000 1021.11029.7 1092509.3105.5 )64.01( (20) This is also in accord with the model used here. The dragging force exerted on the Moon's orbital motion by the dark matter field fluid is -1.11 × 108 N, this is negligibly small compared to the gravitational force between the Moon and the Earth ~ 1.90 × 1020 N; and the torque exerted by the dark matter field fluid on the Earth’s and the Moon's rotations is T = -5.49 × 1016 Nm and -1.15 × 1012 Nm, respectively. 5. The evolution of Earth-Moon system Sonett et al. found that the length of the terrestrial day 900 million years ago was about 19.2 hours based on the laminated tidal sediments on the Earth (Sonett et al., 1996). According to the model presented here, back in that time, the length of the day was about 19.2 hours, this agrees very well with Sonett et al.'s result. Another critical aspect of modeling the evolution of the Earth-Moon system is to give a reasonable estimate of the closest distance of the Moon to the Earth when the system was established at 4.5 billion years ago. Based on the dark matter field fluid model, and the above result, the closest distance of the Moon to the Earth was about 259000 km (center to center) or 250900 km (surface to surface) at 4.5 billion years ago, this is far beyond the Roche's limit. In the modern astronomy textbook by Chaisson and McMillan (Chaisson and McMillan, 1993, p.173), the estimated distance at 4.5 billion years ago was 250000 km, this is probably the most reasonable number that most astronomers believe and it agrees excellently with the result of this model. The closest distance of the Moon to the Earth by Hansen’s models was about 38 Earth radii or 242000 km (Hansen, 1982). According to this model, the length of day of the Earth was about 8 hours at 4.5 billion years ago. Fig. 1 shows the evolution of the distance of Moon to the Earth and the length of day of the Earth with the age of the Earth-Moon system described by this model along with data from Kvale et al. (1999), Sonett et al. (1996) and Scrutton (1978). One can see that those data fit this model very well in their time range. Fig. 2 shows the geological data of solar days year-1 from Wells (1963) and from Sonett et al. (1996) and the description (solid line) by this dark matter field fluid model for past 900 million years. One can see that the model agrees with the geological and fossil data beautifully. The important difference of this model with early models in describing the early evolution of the Earth-Moon system is that this model is based only on current data of the Moon-Earth system and there are no assumptions about the conditions of earlier Earth rotation and continental drifting. Based on this model, the Earth-Moon system has been smoothly evolving to the current position since it was established and the recessional rate of the Moon has been gradually increasing, however, this description does not take it into account that there might be special events happened in the past to cause the suddenly significant changes in the motions of the Earth and the Moon, such as strong impacts by giant asteroids and comets, etc, because those impacts are very common in the universe. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological evidence. Based on Eq. 9, the recessional rate exponentially increases with time. One may then imagine that the recessional rate will quickly become very large. The increase is in fact extremely slow. The Moon's recessional rate will be 3.04 × 10-9 m s-1 after 10 billion years and 7.64 × 10-9 m s-1 after 20 billion years. However, whether the Moon's recession will continue or at some time later another mechanism will take over is not known. It should be understood that the tidal friction does affect the evolution of the Earth itself such as the surface crust structure, continental drifting and evolution of bio-system, etc; it may also play a role in slowing the Earth’s rotation, however, such role is not a dominant mechanism. Unfortunately, there is no data available for the changes of the Earth's orbital motion and all other members of solar system. According to this model and above results, the recessional rate of the Earth should be 6.86 × 10-7 m s-1 = 21.6 m year-1 = 2.16 km century-1, the length of a year increases about 6.8 ms and the change of the temperature is -1.8 × 10-8 K year-1 with constant radiation level of the Sun and the stable environment on the Earth. The length of a year at 1 billion years ago would be 80% of the current length of the year. However, much evidence (growth-bands of corals and shellfish as well as some other evidences) suggest that there has been no apparent change in the length of the year over the billion years and the Earth's orbital motion is more stable than its rotation. This suggests that dark matter field fluid is circulating around Sun with the same direction and similar speed of Earth (at least in the Earth's orbital range). Therefore, the Earth's orbital motion experiences very little or no dragging force from the dark matter field fluid. However, this is a conjecture, extensive research has to be conducted to verify if this is the case. 6. Skeptical description of the evolution of the Mars The Moon does not have liquid fluid on its surface, even there is no air, therefore, there is no ocean-like tidal friction force to slow its rotation; however, the rotation of the Moon is still slowing at significant rate of (-1.260 ± 0.024) × 10-23 rad s-2, which agrees with the model very well. Based on this, one may reasonably think that the Mars’s rotation should be slowing also. The Mars is our nearest neighbor which has attracted human’s great attention since ancient time. The exploration of the Mars has been heating up in recent decades. NASA, Russian and Europe Space Agency sent many space crafts to the Mars to collect data and study this mysterious planet. So far there is still not enough data about the history of this planet to describe its evolution. Same as the Earth, the Mars rotates about its central axis and revolves around the Sun, however, the Mars does not have a massive moon circulating it (Mars has two small satellites: Phobos and Deimos) and there is no liquid fluid on its surface, therefore, there is no apparent ocean-like tidal friction force to slow its rotation by tidal friction theories. Based on the above result and current the Mars's data, this model predicts that the angular acceleration of the Mars should be about -4.38 × 10-22 rad s-2. Figure 3 describes the possible evolution of the length of day and the solar days/Mars year, the vertical dash line marks the current age of the Mars with assumption that the Mars was formed in a similar time period of the Earth formation. Such description was not given before according to the author’s knowledge and is completely skeptical due to lack of reliable data. However, with further expansion of the research and exploration on the Mars, we shall feel confident that the reliable data about the angular rotation acceleration of the Mars will be available in the near future which will provide a vital test for the prediction of this model. There are also other factors which may affect the Mars’s rotation rate such as mass redistribution due to season change, winds, possible volcano eruptions and Mars quakes. Therefore, the data has to be carefully analyzed. 7. Discussion about the model From the above results, one can see that the current Earth-Moon data and the geological and fossil data agree with the model very well and the past evolution of the Earth-Moon system can be described by the model without introducing any additional parameters; this model reveals the interesting relationship between the rotation and receding (Eq. 17 and Eq. 18) of the same celestial body or different celestial bodies in the same gravitational system, such relationship is not known before. Such success can not be explained by “coincidence” or “luck” because of so many data involved (current Earth’s and Moon’s data and geological and fossil data) if one thinks that this is just a “ad hoc” or a wrong model, although the chance for the natural happening of such “coincidence” or “luck” could be greater than wining a jackpot lottery; the future Mars’s data will clarify this; otherwise, a new theory from different approach can be developed to give the same or better description as this model does. It is certain that this model is not perfect and may have defects, further development may be conducted. James Clark Maxwell said in the 1873 “ The vast interplanetary and interstellar regions will no longer be regarded as waste places in the universe, which the Creator has not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find them to be already full of this wonderful medium; so full, that no human power can remove it from the smallest portion of space, or produce the slightest flaw in its infinite continuity. It extends unbroken from star to star ….” The medium that Maxwell talked about is the aether which was proposed as the carrier of light wave propagation. The Michelson-Morley experiment only proved that the light wave propagation does not depend on such medium and did not reject the existence of the medium in the interstellar space. In fact, the concept of the interstellar medium has been developed dramatically recently such as the dark matter, dark energy, cosmic fluid, etc. The dark matter field fluid is just a part of such wonderful medium and “precisely” described by Maxwell. 7. Conclusion The evolution of the Earth-Moon system can be described by the dark matter field fluid model with non-Newtonian approach and the current data of the Earth and the Moon fits this model very well. At 4.5 billion years ago, the closest distance of the Moon to the Earth could be about 259000 km, which is far beyond the Roche’s limit and the length of day was about 8 hours. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The Mars’s rotation is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2. References S. G. Brush, 1983. L. R. Godfrey (editor), Ghost from the Nineteenth century: Creationist Arguments for a young Earth. Scientists confront creationism. W. W. Norton & Company, New York, London, pp 49. E. Chaisson and S. McMillan. 1993. Astronomy Today, Prentice Hall, Englewood Cliffs, NJ 07632. J. O. Dickey, et al., 1994. Science, 265, 482. D. G. Finch, 1981. Earth, Moon, and Planets, 26(1), 109. K. S. Hansen, 1982. Rev. Geophys. and Space Phys. 20(3), 457. W. K. Hartmann, D. R. Davis, 1975. Icarus, 24, 504. B. A. Kagan, N. B. Maslova, 1994. Earth, Moon and Planets 66, 173. B. A. Kagan, 1997. Prog. Oceanog. 40, 109. E. P. Kvale, H. W. Johnson, C. O. Sonett, A. W. Archer, and A. Zawistoski, 1999, J. Sediment. Res. 69(6), 1154. K. Lang, 1992. Astrophysical Data: Planets and Stars, Springer-Verlag, New York. H. Pan, 2005. Internat. J. Modern Phys. A, 20(14), 3135. R. D. Ray, B. G. Bills, B. F. Chao, 1999. J. Geophys. Res. 104(B8), 17653. C. T. Scrutton, 1978. P. Brosche, J. Sundermann, (Editors.), Tidal Friction and the Earth’s Rotation. Springer-Verlag, Berlin, pp. 154. L. B. Slichter, 1963. J. Geophys. Res. 68, 14. C. P. Sonett, E. P. Kvale, M. A. Chan, T. M. Demko, 1996. Science, 273, 100. F. D. Stacey, 1977. Physics of the Earth, second edition. John Willey & Sons. J. W. Wells, 1963. Nature, 197, 948. Caption Figure 1, the evolution of Moon’s distance and the length of day of the earth with the age of the Earth-Moon system. Solid lines are calculated according to the dark matter field fluid model. Data sources: the Moon distances are from Kvale and et al. and for the length of day: (a and b) are from Scrutton ( page 186, fig. 8), c is from Sonett and et al. The dash line marks the current age of the Earth-Moon system. Figure 2, the evolution of Solar days of year with the age of the Earth-Moon system. The solid line is calculated according to dark matter field fluid model. The data are from Wells (3.9 ~ 4.435 billion years range), Sonett (3.6 billion years) and current age (4.5 billion years). Figure 3, the skeptical description of the evolution of Mars’s length of day and the solar days/Mars year with the age of the Mars (assuming that the Mars’s age is about 4.5 billion years). The vertical dash line marks the current age of Mars. Figure 1, Moon's distance and the length of day of Earth change with the age of Earth-Moon system The age of Earth-Moon system (109 years) 0 1 2 3 4 5 Distance Length of day Roche's limit Hansen's result Figure 2, the solar days / year vs. the age of the Earth The age of the Earth (109 years) 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6
Introduction The popularly accepted theory for the formation of the Earth-Moon system is that the Moon was formed from debris of a strong impact by a giant planetesimal with the Earth at the close of the planet-forming period (Hartmann and Davis 1975). Since the formation of the Earth-Moon system, it has been evolving at all time scale. It is well known that the Moon is receding from us and both the Earth’s rotation and Moon’s rotation are slowing. The popular theory is that the tidal friction causes all those changes based on the conservation of the angular momentum of the Earth-Moon system. The situation becomes complicated in describing the past evolution of the Earth-Moon system. Because the Moon is moving away from us and the Earth rotation is slowing, this means that the Moon was closer and the Earth rotation was faster in the past. Creationists argue that based on the tidal friction theory, the tidal friction should be stronger and the recessional rate of the Moon should be greater in the past, the distance of the Moon would quickly fall inside the Roche's limit (for earth, 15500 km) in which the Moon would be torn apart by gravity in 1 to 2 billion years ago. However, geological evidence indicates that the recession of the Moon in the past was slower than the present rate, i. e., the recession has been accelerating with time. Therefore, it must be concluded that tidal friction was very much less in the remote past than we would deduce on the basis of present-day observations (Stacey 1977). This was called “geological time scale difficulty” or “Lunar crisis” and is one of the main arguments by creationists against the tidal friction theory (Brush 1983). But we have to consider the case carefully in various aspects. One possible scenario is that the Earth has been undergoing dynamic evolution at all time scale since its inception, the geological and physical conditions (such as the continent positions and drifting, the crust, surface temperature fluctuation like the glacial/snowball effect, etc) at remote past could be substantially different from currently, in which the tidal friction could be much less; therefore, the receding rate of the Moon could be slower. Various tidal friction models were proposed in the past to describe the evolution of the Earth- Moon system to avoid such difficulty or crisis and put the Moon at quite a comfortable distance from Earth at 4.5 billion years ago (Hansen 1982, Kagan and Maslova 1994, Ray et al. 1999, Finch 1981, Slichter 1963). The tidal friction theories explain that the present rate of tidal dissipation is anomalously high because the tidal force is close to a resonance in the response function of ocean (Brush 1983). Kagan gave a detailed review about those tidal friction models (Kagan 1997). Those models are based on many assumptions about geological (continental position and drifting) and physical conditions in the past, and many parameters (such as phase lag angle, multi-mode approximation with time dependent frequencies of the resonance modes, etc.) have to be introduced and carefully adjusted to make their predictions close to the geological evidence. However, those assumptions and parameters are still challenged, to certain extent, as concoction. The second possible scenario is that another mechanism could dominate the evolution of the Earth-Moon system and the role of the tidal friction is not significant. In the Meeting of Division of Particle and Field 2004, American Physical Society, University of California at Riverside, the author proposed a dark matter field fluid model (Pan 2005) with a non-Newtonian approach, the current Moon and Earth data agree with this model very well. This paper will demonstrate that the past evolution of Moon-Earth system can be described by the dark matter field fluid model without any assumptions about past geological and physical conditions. Although the subject of the evolution of the Earth-Moon system has been extensively studied analytically or numerically, to the author’s knowledge, there are no theories similar or equivalent to this model. 2. Invisible matter In modern cosmology, it was proposed that the visible matter in the universe is about 2 ~ 10 % of the total matter and about 90 ~ 98% of total matter is currently invisible which is called dark matter and dark energy, such invisible matter has an anti- gravity property to make the universe expanding faster and faster. If the ratio of the matter components of the universe is close to this hypothesis, then, the evolution of the universe should be dominated by the physical mechanism of such invisible matter, such physical mechanism could be far beyond the current Newtonian physics and Einsteinian physics, and the Newtonian physics and Einsteinian physics could reflect only a corner of the iceberg of the greater physics. If the ratio of the matter components of the universe is close to this hypothesis, then, it should be more reasonable to think that such dominant invisible matter spreads in everywhere of the universe (the density of the invisible matter may vary from place to place); in other words, all visible matter objects should be surrounded by such invisible matter and the motion of the visible matter objects should be affected by the invisible matter if there are interactions between the visible matter and the invisible matter. If the ratio of the matter components of the universe is close to this hypothesis, then, the size of the particles of the invisible matter should be very small and below the detection limit of the current technology; otherwise, it would be detected long time ago with such dominant amount. With such invisible matter in mind, we move to the next section to develop the dark matter field fluid model with non-Newtonian approach. For simplicity, all invisible matter (dark matter, dark energy and possible other terms) is called dark matter here. 3. The dark matter field fluid model In this proposed model, it is assumed that: 1. A celestial body rotates and moves in the space, which, for simplicity, is uniformly filled with the dark matter which is in quiescent state relative to the motion of the celestial body. The dark matter possesses a field property and a fluid property; it can interact with the celestial body with its fluid and field properties; therefore, it can have energy exchange with the celestial body, and affect the motion of the celestial body. 2. The fluid property follows the general principle of fluid mechanics. The dark matter field fluid particles may be so small that they can easily permeate into ordinary “baryonic” matter; i. e., ordinary matter objects could be saturated with such dark matter field fluid. Thus, the whole celestial body interacts with the dark matter field fluid, in the manner of a sponge moving thru water. The nature of the field property of the dark matter field fluid is unknown. It is here assumed that the interaction of the field associated with the dark matter field fluid with the celestial body is proportional to the mass of the celestial body. The dark matter field fluid is assumed to have a repulsive force against the gravitational force towards baryonic matter. The nature and mechanism of such repulsive force is unknown. With the assumptions above, one can study how the dark matter field fluid may influence the motion of a celestial body and compare the results with observations. The common shape of celestial bodies is spherical. According to Stokes's law, a rigid non- permeable sphere moving through a quiescent fluid with a sufficiently low Reynolds number experiences a resistance force F rvF πμ6−= (1) where v is the moving velocity, r is the radius of the sphere, and μ is the fluid viscosity constant. The direction of the resistance force F in Eq. 1 is opposite to the direction of the velocity v. For a rigid sphere moving through the dark matter field fluid, due to the dual properties of the dark matter field fluid and its permeation into the sphere, the force F may not be proportional to the radius of the sphere. Also, F may be proportional to the mass of the sphere due to the field interaction. Therefore, with the combined effects of both fluid and field, the force exerted on the sphere by the dark matter field fluid is assumed to be of the scaled form (2) mvrF n−−= 16πη where n is a parameter arising from saturation by dark matter field fluid, the r1-n can be viewed as the effective radius with the same unit as r, m is the mass of the sphere, and η is the dark matter field fluid constant, which is equivalent to μ. The direction of the resistance force F in Eq. 2 is opposite to the direction of the velocity v. The force described by Eq. 2 is velocity-dependent and causes negative acceleration. According to Newton's second law of motion, the equation of motion for the sphere is mvr m n−−= 16πη (3) Then (4) )6exp( 10 vtrvv n−−= πη where v0 is the initial velocity (t = 0) of the sphere. If the sphere revolves around a massive gravitational center, there are three forces in the line between the sphere and the gravitational center: (1) the gravitational force, (2) the centripetal acceleration force; and (3) the repulsive force of the dark matter field fluid. The drag force in Eq. 3 reduces the orbital velocity and causes the sphere to move inward to the gravitational center. However, if the sum of the centripetal acceleration force and the repulsive force is stronger than the gravitational force, then, the sphere will move outward and recede from the gravitational center. This is the case of interest here. If the velocity change in Eq. 3 is sufficiently slow and the repulsive force is small compared to the gravitational force and centripetal acceleration force, then the rate of receding will be accordingly relatively slow. Therefore, the gravitational force and the centripetal acceleration force can be approximately treated in equilibrium at any time. The pseudo equilibrium equation is GMm 2 2 = (5) where G is the gravitational constant, M is the mass of the gravitational center, and R is the radius of the orbit. Inserting v of Eq. 4 into Eq. 5 yields )12exp( 1 R n−= πη (6) (7) )12exp( 10 trRR n−= πη where R = (8) R0 is the initial distance to the gravitational center. Note that R exponentially increases with time. The increase of orbital energy with the receding comes from the repulsive force of dark matter field fluid. The recessional rate of the sphere is dR n−= 112πη (9) The acceleration of the recession is ( Rr Rd n 21 12 −= πη ) . (10) The recessional acceleration is positive and proportional to its distance to the gravitational center, so the recession is faster and faster. According to the mechanics of fluids, for a rigid non-permeable sphere rotating about its central axis in the quiescent fluid, the torque T exerted by the fluid on the sphere ωπμ 38 rT −= (11) where ω is the angular velocity of the sphere. The direction of the torque in Eq. 11 is opposite to the direction of the rotation. In the case of a sphere rotating in the quiescent dark matter field fluid with angular velocity ω, similar to Eq. 2, the proposed T exerted on the sphere is ( ) ωπη mrT n 318 −−= (12) The direction of the torque in Eq. 12 is opposite to the direction of the rotation. The torque causes the negative angular acceleration = (13) where I is the moment of inertia of the sphere in the dark matter field fluid ( )21 2 nrmI −= (14) Therefore, the equation of rotation for the sphere in the dark matter field fluid is ωπη d −−= 120 (15) Solving this equation yields (16) )20exp( 10 tr n−−= πηωω where ω0 is the initial angular velocity. One can see that the angular velocity of the sphere exponentially decreases with time and the angular deceleration is proportional to its angular velocity. For the same celestial sphere, combining Eq. 9 and Eq. 15 yields (17) The significance of Eq. 17 is that it contains only observed data without assumptions and undetermined parameters; therefore, it is a critical test for this model. For two different celestial spheres in the same system, combining Eq. 9 and Eq. 15 yields 67.1 1 −=−=⎟⎟ (18) This is another critical test for this model. 4. The current behavior of the Earth-Moon system agrees with the model The Moon-Earth system is the simplest gravitational system. The solar system is complex, the Earth and the Moon experience not only the interaction of the Sun but also interactions of other planets. Let us consider the local Earth-Moon gravitational system as an isolated local gravitational system, i.e., the influence from the Sun and other planets on the rotation and orbital motion of the Moon and on the rotation of the Earth is assumed negligible compared to the forces exerted by the moon and earth on each other. In addition, the eccentricity of the Moon's orbit is small enough to be ignored. The data about the Moon and the Earth from references (Dickey et .al., 1994, and Lang, 1992) are listed below for the readers' convenience to verify the calculation because the data may vary slightly with different data sources. Moon: Mean radius: r = 1738.0 km Mass: m = 7.3483 × 1025 gram Rotation period = 27.321661 days Angular velocity of Moon = 2.6617 × 10-6 rad s-1 Mean distance to Earth Rm= 384400 km Mean orbital velocity v = 1.023 km s-1 Orbit eccentricity e = 0.0549 Angular rotation acceleration rate = -25.88 ± 0.5 arcsec century-2 = (-1.255 ± 0.024) × 10-4 rad century-2 = (-1.260 ± 0.024) × 10-23 rad s-2 Receding rate from Earth = 3.82 ± 0.07 cm year-1 = (1.21 ± 0.02) × 10-9 m s-1 Earth: Mean radius: r = 6371.0 km Mass: m = 5.9742 × 1027 gram Rotation period = 23 h 56m 04.098904s = 86164.098904s Angular velocity of rotation = 7.292115 × 10-5 rad s-1 Mean distance to the Sun Rm= 149,597,870.61 km Mean orbital velocity v = 29.78 km s-1 Angular acceleration of Earth = (-5.5 ± 0.5) × 10-22 rad s-2 The Moon's angular rotation acceleration rate and increase in mean distance to the Earth (receding rate) were obtained from the lunar laser ranging of the Apollo Program (Dickey et .al., 1994). By inserting the data of the Moon's rotation and recession into Eq. 17, the result is 039.054.1 10662.21021.1 1092509.31026.1 (19) The distance R in Eq. 19 is from the Moon's center to the Earth's center and the number 384400 km is assumed to be the distance from the Moon's surface to the Earth's surface. Eq. 19 is in good agreement with the theoretical value of -1.67. The result is in accord with the model used here. The difference (about 7.8%) between the values of -1.54 and - 1.67 may come from several sources: 1. Moon's orbital is not a perfect circle 2. Moon is not a perfect rigid sphere. 3. The effect from Sun and other planets. 4. Errors in data. 5. Possible other unknown reasons. The two parameters n and η in Eq. 9 and Eq. 15 can be determined with two data sets. The third data set can be used to further test the model. If this model correctly describes the situation at hand, it should give consistent results for different motions. The values of n and η calculated from three different data sets are listed below (Note, the mean distance of the Moon to the Earth and mean radii of the Moon and the Earth are used in the calculation). The value of n: n = 0.64 From the Moon's rotation: η = 4.27 × 10-22 s-1 m-1 From the Earth's rotation: η = 4.26 × 10-22 s-1 m-1 From the Moon's recession: η = 4.64 × 10-22 s-1 m-1 One can see that the three values of η are consistent within the range of error in the data. The average value of η: η = (4.39 ± 0.22) × 10-22 s-1 m-1 By inserting the data of the Earth's rotation, the Moon’s recession and the value of n into Eq. 18, the result is 14.053.1 6371000 1738000 1021.11029.7 1092509.3105.5 )64.01( (20) This is also in accord with the model used here. The dragging force exerted on the Moon's orbital motion by the dark matter field fluid is -1.11 × 108 N, this is negligibly small compared to the gravitational force between the Moon and the Earth ~ 1.90 × 1020 N; and the torque exerted by the dark matter field fluid on the Earth’s and the Moon's rotations is T = -5.49 × 1016 Nm and -1.15 × 1012 Nm, respectively. 5. The evolution of Earth-Moon system Sonett et al. found that the length of the terrestrial day 900 million years ago was about 19.2 hours based on the laminated tidal sediments on the Earth (Sonett et al., 1996). According to the model presented here, back in that time, the length of the day was about 19.2 hours, this agrees very well with Sonett et al.'s result. Another critical aspect of modeling the evolution of the Earth-Moon system is to give a reasonable estimate of the closest distance of the Moon to the Earth when the system was established at 4.5 billion years ago. Based on the dark matter field fluid model, and the above result, the closest distance of the Moon to the Earth was about 259000 km (center to center) or 250900 km (surface to surface) at 4.5 billion years ago, this is far beyond the Roche's limit. In the modern astronomy textbook by Chaisson and McMillan (Chaisson and McMillan, 1993, p.173), the estimated distance at 4.5 billion years ago was 250000 km, this is probably the most reasonable number that most astronomers believe and it agrees excellently with the result of this model. The closest distance of the Moon to the Earth by Hansen’s models was about 38 Earth radii or 242000 km (Hansen, 1982). According to this model, the length of day of the Earth was about 8 hours at 4.5 billion years ago. Fig. 1 shows the evolution of the distance of Moon to the Earth and the length of day of the Earth with the age of the Earth-Moon system described by this model along with data from Kvale et al. (1999), Sonett et al. (1996) and Scrutton (1978). One can see that those data fit this model very well in their time range. Fig. 2 shows the geological data of solar days year-1 from Wells (1963) and from Sonett et al. (1996) and the description (solid line) by this dark matter field fluid model for past 900 million years. One can see that the model agrees with the geological and fossil data beautifully. The important difference of this model with early models in describing the early evolution of the Earth-Moon system is that this model is based only on current data of the Moon-Earth system and there are no assumptions about the conditions of earlier Earth rotation and continental drifting. Based on this model, the Earth-Moon system has been smoothly evolving to the current position since it was established and the recessional rate of the Moon has been gradually increasing, however, this description does not take it into account that there might be special events happened in the past to cause the suddenly significant changes in the motions of the Earth and the Moon, such as strong impacts by giant asteroids and comets, etc, because those impacts are very common in the universe. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological evidence. Based on Eq. 9, the recessional rate exponentially increases with time. One may then imagine that the recessional rate will quickly become very large. The increase is in fact extremely slow. The Moon's recessional rate will be 3.04 × 10-9 m s-1 after 10 billion years and 7.64 × 10-9 m s-1 after 20 billion years. However, whether the Moon's recession will continue or at some time later another mechanism will take over is not known. It should be understood that the tidal friction does affect the evolution of the Earth itself such as the surface crust structure, continental drifting and evolution of bio-system, etc; it may also play a role in slowing the Earth’s rotation, however, such role is not a dominant mechanism. Unfortunately, there is no data available for the changes of the Earth's orbital motion and all other members of solar system. According to this model and above results, the recessional rate of the Earth should be 6.86 × 10-7 m s-1 = 21.6 m year-1 = 2.16 km century-1, the length of a year increases about 6.8 ms and the change of the temperature is -1.8 × 10-8 K year-1 with constant radiation level of the Sun and the stable environment on the Earth. The length of a year at 1 billion years ago would be 80% of the current length of the year. However, much evidence (growth-bands of corals and shellfish as well as some other evidences) suggest that there has been no apparent change in the length of the year over the billion years and the Earth's orbital motion is more stable than its rotation. This suggests that dark matter field fluid is circulating around Sun with the same direction and similar speed of Earth (at least in the Earth's orbital range). Therefore, the Earth's orbital motion experiences very little or no dragging force from the dark matter field fluid. However, this is a conjecture, extensive research has to be conducted to verify if this is the case. 6. Skeptical description of the evolution of the Mars The Moon does not have liquid fluid on its surface, even there is no air, therefore, there is no ocean-like tidal friction force to slow its rotation; however, the rotation of the Moon is still slowing at significant rate of (-1.260 ± 0.024) × 10-23 rad s-2, which agrees with the model very well. Based on this, one may reasonably think that the Mars’s rotation should be slowing also. The Mars is our nearest neighbor which has attracted human’s great attention since ancient time. The exploration of the Mars has been heating up in recent decades. NASA, Russian and Europe Space Agency sent many space crafts to the Mars to collect data and study this mysterious planet. So far there is still not enough data about the history of this planet to describe its evolution. Same as the Earth, the Mars rotates about its central axis and revolves around the Sun, however, the Mars does not have a massive moon circulating it (Mars has two small satellites: Phobos and Deimos) and there is no liquid fluid on its surface, therefore, there is no apparent ocean-like tidal friction force to slow its rotation by tidal friction theories. Based on the above result and current the Mars's data, this model predicts that the angular acceleration of the Mars should be about -4.38 × 10-22 rad s-2. Figure 3 describes the possible evolution of the length of day and the solar days/Mars year, the vertical dash line marks the current age of the Mars with assumption that the Mars was formed in a similar time period of the Earth formation. Such description was not given before according to the author’s knowledge and is completely skeptical due to lack of reliable data. However, with further expansion of the research and exploration on the Mars, we shall feel confident that the reliable data about the angular rotation acceleration of the Mars will be available in the near future which will provide a vital test for the prediction of this model. There are also other factors which may affect the Mars’s rotation rate such as mass redistribution due to season change, winds, possible volcano eruptions and Mars quakes. Therefore, the data has to be carefully analyzed. 7. Discussion about the model From the above results, one can see that the current Earth-Moon data and the geological and fossil data agree with the model very well and the past evolution of the Earth-Moon system can be described by the model without introducing any additional parameters; this model reveals the interesting relationship between the rotation and receding (Eq. 17 and Eq. 18) of the same celestial body or different celestial bodies in the same gravitational system, such relationship is not known before. Such success can not be explained by “coincidence” or “luck” because of so many data involved (current Earth’s and Moon’s data and geological and fossil data) if one thinks that this is just a “ad hoc” or a wrong model, although the chance for the natural happening of such “coincidence” or “luck” could be greater than wining a jackpot lottery; the future Mars’s data will clarify this; otherwise, a new theory from different approach can be developed to give the same or better description as this model does. It is certain that this model is not perfect and may have defects, further development may be conducted. James Clark Maxwell said in the 1873 “ The vast interplanetary and interstellar regions will no longer be regarded as waste places in the universe, which the Creator has not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find them to be already full of this wonderful medium; so full, that no human power can remove it from the smallest portion of space, or produce the slightest flaw in its infinite continuity. It extends unbroken from star to star ….” The medium that Maxwell talked about is the aether which was proposed as the carrier of light wave propagation. The Michelson-Morley experiment only proved that the light wave propagation does not depend on such medium and did not reject the existence of the medium in the interstellar space. In fact, the concept of the interstellar medium has been developed dramatically recently such as the dark matter, dark energy, cosmic fluid, etc. The dark matter field fluid is just a part of such wonderful medium and “precisely” described by Maxwell. 7. Conclusion The evolution of the Earth-Moon system can be described by the dark matter field fluid model with non-Newtonian approach and the current data of the Earth and the Moon fits this model very well. At 4.5 billion years ago, the closest distance of the Moon to the Earth could be about 259000 km, which is far beyond the Roche’s limit and the length of day was about 8 hours. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The Mars’s rotation is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2. References S. G. Brush, 1983. L. R. Godfrey (editor), Ghost from the Nineteenth century: Creationist Arguments for a young Earth. Scientists confront creationism. W. W. Norton & Company, New York, London, pp 49. E. Chaisson and S. McMillan. 1993. Astronomy Today, Prentice Hall, Englewood Cliffs, NJ 07632. J. O. Dickey, et al., 1994. Science, 265, 482. D. G. Finch, 1981. Earth, Moon, and Planets, 26(1), 109. K. S. Hansen, 1982. Rev. Geophys. and Space Phys. 20(3), 457. W. K. Hartmann, D. R. Davis, 1975. Icarus, 24, 504. B. A. Kagan, N. B. Maslova, 1994. Earth, Moon and Planets 66, 173. B. A. Kagan, 1997. Prog. Oceanog. 40, 109. E. P. Kvale, H. W. Johnson, C. O. Sonett, A. W. Archer, and A. Zawistoski, 1999, J. Sediment. Res. 69(6), 1154. K. Lang, 1992. Astrophysical Data: Planets and Stars, Springer-Verlag, New York. H. Pan, 2005. Internat. J. Modern Phys. A, 20(14), 3135. R. D. Ray, B. G. Bills, B. F. Chao, 1999. J. Geophys. Res. 104(B8), 17653. C. T. Scrutton, 1978. P. Brosche, J. Sundermann, (Editors.), Tidal Friction and the Earth’s Rotation. Springer-Verlag, Berlin, pp. 154. L. B. Slichter, 1963. J. Geophys. Res. 68, 14. C. P. Sonett, E. P. Kvale, M. A. Chan, T. M. Demko, 1996. Science, 273, 100. F. D. Stacey, 1977. Physics of the Earth, second edition. John Willey & Sons. J. W. Wells, 1963. Nature, 197, 948. Caption Figure 1, the evolution of Moon’s distance and the length of day of the earth with the age of the Earth-Moon system. Solid lines are calculated according to the dark matter field fluid model. Data sources: the Moon distances are from Kvale and et al. and for the length of day: (a and b) are from Scrutton ( page 186, fig. 8), c is from Sonett and et al. The dash line marks the current age of the Earth-Moon system. Figure 2, the evolution of Solar days of year with the age of the Earth-Moon system. The solid line is calculated according to dark matter field fluid model. The data are from Wells (3.9 ~ 4.435 billion years range), Sonett (3.6 billion years) and current age (4.5 billion years). Figure 3, the skeptical description of the evolution of Mars’s length of day and the solar days/Mars year with the age of the Mars (assuming that the Mars’s age is about 4.5 billion years). The vertical dash line marks the current age of Mars. Figure 1, Moon's distance and the length of day of Earth change with the age of Earth-Moon system The age of Earth-Moon system (109 years) 0 1 2 3 4 5 Distance Length of day Roche's limit Hansen's result Figure 2, the solar days / year vs. the age of the Earth The age of the Earth (109 years) 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6
La evolución del sistema Tierra-Luna basado en el modelo de fluido oscuro La evolución del sistema Tierra-Luna basado en el modelo de fluido de campo de materia oscura Hongjun Pan Departamento de Química Universidad del Norte de Texas, Denton, Texas 76203, U.S. A. Resumen La evolución del sistema Tierra-Luna es descrita por el fluido del campo de materia oscura modelo con un enfoque no newtoniano propuesto en la Reunión de la División de Partículas y Field 2004, American Physical Society. El comportamiento actual de la Luna-Tierra sistema está de acuerdo con este modelo muy bien y el patrón general de la evolución de la El sistema Luna-Tierra descrito por este modelo concuerda con la evidencia geológica y fósil. La distancia más cercana de la Luna a la Tierra era de unos 259000 km en 4.500 millones de años atrás, que está mucho más allá del límite del Roche. El resultado sugiere que la fricción de marea puede no ser la causa principal de la evolución del sistema Tierra-Luna. La oscuridad media La constante de fluido de campo de materia derivada de los datos del sistema Tierra-Luna es 4,39 × 10-22 s-1m-1. Este modelo predice que la rotación de Marte también se está desacelerando con la aceleración angular tasa alrededor de -4.38 × 10-22 rad s-2. Palabras clave. materia oscura, fluido, evolución, Tierra, Luna, Marte 1. Introducción La teoría aceptada popularmente para la formación del sistema Tierra-Luna es que la Luna se formó a partir de escombros de un fuerte impacto por un gigante planetesimal con el La Tierra al final del período de formación del planeta (Hartmann y Davis 1975). Desde el formación del sistema Tierra-Luna, que ha estado evolucionando en toda escala de tiempo. Está bien. sabe que la Luna se está alejando de nosotros y de la rotación de la Tierra y de la Luna La rotación se está desacelerando. La teoría popular es que la fricción de mareas causa todos esos cambios. basado en la conservación del impulso angular del sistema Tierra-Luna. Los la situación se complica al describir la evolución pasada de la Luna-Tierra sistema. Debido a que la Luna se está alejando de nosotros y la rotación de la Tierra se está desacelerando, esto significa que la Luna estaba más cerca y la rotación de la Tierra era más rápida en el pasado. Creacionistas argumentan que sobre la base de la teoría de la fricción de mareas, la fricción de mareas debe ser más fuerte y la la tasa de recesión de la Luna debe ser mayor en el pasado, la distancia de la Luna caería rápidamente dentro del límite de Roche (para la tierra, 15500 km) en el que la Luna sería desgarrado por la gravedad en 1 a 2 mil millones de años atrás. Sin embargo, las pruebas geológicas indica que la recesión de la Luna en el pasado fue más lenta que la tasa actual, es decir, la recesión se ha acelerado con el tiempo. Por lo tanto, debe concluirse que las mareas la fricción fue mucho menos en el pasado remoto de lo que deduciríamos sobre la base de Observaciones actuales (Stacey 1977). Esto se llamó “escala de tiempo geológica dificultad” o “crisis lunar” y es uno de los principales argumentos de los creacionistas contra el teoría de la fricción de mareas (Brush 1983). Pero tenemos que considerar el caso cuidadosamente en varios aspectos. Una posible escenario es que la Tierra ha estado experimentando una evolución dinámica en toda escala de tiempo desde su creación, las condiciones geológicas y físicas (como las posiciones del continente y a la deriva, la corteza, fluctuación de la temperatura superficial como el efecto glacial/snowball, etc.) pasado remoto podría ser sustancialmente diferente de la actual, en la que la fricción de mareas podría ser mucho menos; por lo tanto, la tasa de descenso de la Luna podría ser más lenta. Varios En el pasado se propusieron modelos de fricción de mareas para describir la evolución de la Tierra- Sistema lunar para evitar tal dificultad o crisis y poner a la Luna en un lugar bastante cómodo distancia de la Tierra hace 4.500 millones de años (Hansen 1982, Kagan y Maslova 1994, Ray et al. 1999, Finch 1981, Slichter 1963). Las teorías de la fricción de marea explican que el presente la tasa de disipación de las mareas es anomalosamente alta porque la fuerza de las mareas está cerca de una resonancia en la función de respuesta del océano (Brush 1983). Kagan dio una revisión detallada sobre los modelos de fricción de mareas (Kagan 1997). Estos modelos se basan en muchos supuestos sobre condiciones geológicas (posición continental y deriva) y físicas en el pasado, y muchos parámetros (como el ángulo de retardo de fase, la aproximación multimodo con el tiempo frecuencias dependientes de los modos de resonancia, etc.) tienen que ser introducidos y cuidadosamente ajustados para hacer sus predicciones cerca de la evidencia geológica. Sin embargo, los los supuestos y parámetros siguen siendo cuestionados, en cierta medida, como brebaje. El segundo escenario posible es que otro mecanismo podría dominar el la evolución del sistema Tierra-Luna y el papel de la fricción de mareas no es significativo. In la Reunión de la División de Partículas y Campo 2004, American Physical Society, Universidad de California en Riverside, el autor propuso un modelo de fluido de campo de materia oscura (Pan 2005) con un enfoque no newtoniano, los datos actuales de la Luna y la Tierra están de acuerdo con este modelo muy bien. Este documento demostrará que la evolución pasada de la Luna-Tierra sistema puede ser descrito por el modelo de fluido de campo de materia oscura sin ninguna suposición sobre las condiciones geológicas y físicas del pasado. Aunque el tema de la evolución de el sistema Tierra-Luna ha sido ampliamente estudiado analítica o numéricamente, a la conocimiento del autor, no hay teorías similares o equivalentes a este modelo. 2. Materia invisible En la cosmología moderna, se propuso que la materia visible en el universo es aproximadamente el 2 ~ 10 % de la materia total y alrededor del 90 ~ 98% de la materia total es actualmente invisible que se llama materia oscura y energía oscura, tal materia invisible tiene un anti- propiedad de gravedad para hacer que el universo se expanda más y más rápido. Si la proporción de los componentes de materia del universo está cerca de esta hipótesis, entonces, la evolución del universo debe ser dominada por el mecanismo físico de tal materia invisible, tal mecanismo físico podría estar mucho más allá de la corriente La física newtoniana y la física Einsteiniana, y la física Newtoniana y la Einsteiniana la física podría reflejar sólo un rincón del iceberg de la física mayor. Si la proporción de los componentes de materia del universo está cerca de esta hipótesis, entonces, debería ser más razonable pensar que tal materia invisible dominante se propaga en en todas partes del universo (la densidad de la materia invisible puede variar de un lugar a otro lugar); en otras palabras, todos los objetos de materia visible deben estar rodeados por tales invisibles materia y el movimiento de la materia visible objetos deben ser afectados por el invisible materia si hay interacciones entre la materia visible y la materia invisible. Si la proporción de los componentes de materia del universo está cerca de esta hipótesis, entonces, el tamaño de las partículas de la materia invisible debe ser muy pequeño y por debajo de la límite de detección de la tecnología actual; de lo contrario, se detectaría hace mucho tiempo con tal cantidad dominante. Con esta materia invisible en mente, nos movemos a la siguiente sección para desarrollar la Modelo de fluido de campo de materia oscura con enfoque no newtoniano. Para la simplicidad, todos invisibles materia (materia oscura, energía oscura y otros términos posibles) se llama materia oscura aquí. 3. El modelo de fluido de campo de materia oscura En este modelo propuesto, se supone que: 1. Un cuerpo celeste gira y se mueve en el espacio, que, para la simplicidad, es uniforme lleno de la materia oscura que está en estado de quiescencia relativa al movimiento del cuerpo celeste. La materia oscura posee una propiedad de campo y una propiedad fluida; puede interactúe con el cuerpo celeste con sus propiedades de fluido y campo; por lo tanto, puede tener intercambio de energía con el cuerpo celeste, y afectan el movimiento del cuerpo celeste. 2. La propiedad del fluido sigue el principio general de la mecánica del fluido. La materia oscura partículas de líquido de campo pueden ser tan pequeñas que fácilmente pueden impregnarse en ordinario materia “barionica”; es decir, los objetos de materia ordinaria podrían estar saturados con tal materia oscura fluido de campo. Por lo tanto, todo el cuerpo celestial interactúa con el fluido del campo de materia oscura, en el forma de una esponja que se mueve a través del agua. La naturaleza de la propiedad de campo de la materia oscura se desconoce el líquido del campo. Se asume aquí que la interacción del campo asociado con el fluido del campo de materia oscura con el cuerpo celestial es proporcional a la masa del cuerpo celeste. El fluido del campo de materia oscura se supone que tiene una fuerza repulsiva contra el fuerza gravitatoria hacia la materia bariónica. La naturaleza y el mecanismo de tal repulsivo La fuerza es desconocida. Con las suposiciones anteriores, uno puede estudiar cómo el fluido del campo de materia oscura puede influir en el movimiento de un cuerpo celeste y comparar los resultados con las observaciones. Los la forma común de los cuerpos celestes es esférica. Según la ley de Stokes, un rígido no- esfera permeable que se mueve a través de un líquido quiescente con un Reynolds suficientemente bajo número experimenta una fuerza de resistencia F rvF 6−= (1) donde v es la velocidad de movimiento, r es el radio de la esfera, y μ es la viscosidad del fluido constante. La dirección de la fuerza de resistencia F en Eq. 1 es opuesto a la dirección de la velocidad v. Para una esfera rígida que se mueve a través del fluido del campo de materia oscura, debido al doble propiedades del fluido del campo de materia oscura y su permeación en la esfera, la fuerza F puede no ser proporcional al radio de la esfera. Además, F puede ser proporcional a la masa de la esfera debido a la interacción de campo. Por lo tanto, con los efectos combinados de fluido y campo, la fuerza ejercida en la esfera por el fluido del campo de materia oscura es se supone que es de la forma escalonada (2) mvrF n= 16 donde n es un parámetro derivado de la saturación por el fluido de campo de materia oscura, el r1-n puede ser visto como el radio efectivo con la misma unidad que r, m es la masa de la esfera, y η es la constante del fluido del campo de materia oscura, que es equivalente a μ. La dirección de la Fuerza de resistencia F en Eq. 2 es opuesto a la dirección de la velocidad v. La fuerza descrita por Eq. 2 es dependiente de la velocidad y causa una aceleración negativa. De acuerdo con Segunda ley del movimiento de Newton, la ecuación del movimiento para la esfera es mvr m n= 16 (3) Entonces (4) )6exp( 10 vtrv No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. donde v0 es la velocidad inicial (t = 0) de la esfera. Si la esfera gira alrededor de un centro gravitacional masivo, hay tres fuerzas en la línea entre la esfera y el centro gravitacional: (1) la fuerza gravitatoria, (2) la fuerza de aceleración centrípeta; y (3) la fuerza repulsiva del fluido del campo de materia oscura. La fuerza de arrastre en Eq. 3 reduce la velocidad orbital y hace que la esfera se mueva hacia el centro gravitacional. Sin embargo, si la suma de la fuerza de aceleración centrípeta y la fuerza repulsiva es más fuerte que la fuerza gravitacional, entonces, la esfera se moverá hacia afuera y se retirará de el centro gravitacional. Este es el caso del interés aquí. Si la velocidad cambia en Eq. 3 es suficientemente lento y la fuerza repulsiva es pequeña en comparación con la fuerza gravitacional y fuerza de aceleración centrípeta, entonces la tasa de retroceso será en consecuencia relativamente Lentamente. Por lo tanto, la fuerza gravitacional y la fuerza de aceleración centrípeta puede ser aproximadamente tratados en equilibrio en cualquier momento. La ecuación pseudo equilibrio es GMm 2 2 = (5) donde G es la constante gravitacional, M es la masa del centro gravitacional, y R es el radio de la órbita. Insertar v de Eq. 4 en Eq. 5 rendimientos )12exp( 1 R n−= (6) (7) )12exp( 10 trRR n−= donde R = (8) R0 es la distancia inicial al centro gravitacional. Tenga en cuenta que R aumenta exponencialmente con el tiempo. El aumento de la energía orbital con el retroceso proviene del repulsivo fuerza del fluido de campo de materia oscura. La tasa de recesión de la esfera es dR n−= 112 (9) La aceleración de la recesión es ( Rr Rd n 21 12 − = ). (10) La aceleración recesiva es positiva y proporcional a su distancia a la centro gravitacional, así que la recesión es cada vez más rápida. Según la mecánica de los fluidos, para una esfera rígida no permeable giratoria alrededor de su eje central en el fluido quiescente, el par T ejercido por el fluido en la esfera 38 rT − = (11) donde • es la velocidad angular de la esfera. La dirección del par en Eq. 11 es opuesta a la dirección de la rotación. En el caso de una esfera que gira en el quiescente Líquido de campo de materia oscura con velocidad angular, similar a Eq. 2, la T propuesta ejerció en la esfera es ( ) mrT n 318 = (12) La dirección del par en Eq. 12 es opuesto a la dirección de la rotación. Los el par causa la aceleración angular negativa = (13) donde estoy el momento de inercia de la esfera en el fluido del campo de materia oscura ( )21 2 nrmI = (14) Por lo tanto, la ecuación de rotación para la esfera en el fluido del campo de materia oscura es * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * d = 120 (15) Resolver esta ecuación produce (16) )20exp( 10 tr No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. donde ­0 es la velocidad angular inicial. Uno puede ver que la velocidad angular de la esfera disminuye exponencialmente con el tiempo y la desaceleración angular es proporcional a su velocidad angular. Para la misma esfera celestial, combinando Eq. 9 y Eq. 15 rendimientos (17) El significado de Eq. 17 es que sólo contiene datos observados sin suposiciones y parámetros indeterminados; por lo tanto, es una prueba crítica para este modelo. Para dos esferas celestes diferentes en el mismo sistema, combinando Eq. 9 y Eq. 15 rendimientos 67,1 1 −=−= (18) Esta es otra prueba crítica para este modelo. 4. El comportamiento actual del sistema Tierra-Luna concuerda con el modelo El sistema Luna-Tierra es el sistema gravitacional más simple. El sistema solar es complejo, la Tierra y la Luna experimentan no sólo la interacción del Sol, sino también interacciones de otros planetas. Consideremos el sistema gravitacional Tierra-Luna local como un sistema gravitacional local aislado, es decir, la influencia del Sol y otros planetas sobre la rotación y el movimiento orbital de la Luna y sobre la rotación de la Tierra asumido insignificante en comparación con las fuerzas ejercidas por la luna y la tierra en el otro. Además, la excentricidad de la órbita de la Luna es lo suficientemente pequeña como para ser ignorada. Los datos sobre la Luna y la Tierra a partir de referencias (Dickey et.al., 1994, y Lang, 1992) listado a continuación para la conveniencia de los lectores para verificar el cálculo porque los datos pueden varían ligeramente con diferentes fuentes de datos. Luna: Radio medio: r = 1738,0 km Masa: m = 7,3483 × 1025 gramos Período de rotación = 27,321661 días Velocidad angular de la Luna = 2,6617 × 10-6 rad s-1 Distancia media a la Tierra Rm= 384400 km Velocidad orbital media v = 1.023 km s-1 Excentricidad de la órbita e = 0,0549 Velocidad de aceleración de rotación angular = -25,88 ± 0,5 arcoseg siglo-2 = (-1,255 ± 0,024) × siglo rad 10-4-2 = (-1.260 ± 0.024) × 10-23 rad s-2 Tasa de retroceso de la Tierra = 3,82 ± 0,07 cm año-1 = (1,21 ± 0,02) × 10-9 m s-1 Tierra: Radio medio: r = 6371,0 km Masa: m = 5,9742 × 1027 gramos Período de rotación = 23 h 56m 04.098904s = 86164.098904s Velocidad angular de rotación = 7,292115 × 10-5 rad s-1 Distancia media al Sol Rm= 149.597.870,61 km Velocidad orbital media v = 29,78 km s-1 Aceleración angular de la Tierra = (-5,5 ± 0,5) × 10-22 rad s-2 Velocidad angular de rotación de la Luna y aumento de la distancia media a la Tierra (tasa de descenso) se obtuvieron de la gama de láser lunar del Programa Apollo (Dickey et.al., 1994). Insertando los datos de la rotación y recesión de la Luna en Eq. 17, el resultado es 039.054,1 10662,2121,1 1092509.31026.1 (19) La distancia R en Eq. 19 es desde el centro de la Luna hasta el centro de la Tierra y el número 384400 km se supone que es la distancia de la superficie de la Luna a la superficie de la Tierra. Eq. 19 está en buen acuerdo con el valor teórico de -1.67. El resultado está de acuerdo con el modelo utilizado aquí. La diferencia (alrededor del 7,8%) entre los valores de -1,54 y - 1.67 pueden provenir de varias fuentes: 1. El orbital de la Luna no es un círculo perfecto 2. La Luna no es una esfera rígida perfecta. 3. El efecto del Sol y otros planetas. 4. Errores en los datos. 5. Posibles otras razones desconocidas. Los dos parámetros n y η en Eq. 9 y Eq. 15 se puede determinar con dos datos Sets. El tercer conjunto de datos se puede utilizar para seguir probando el modelo. Si este modelo es correcto describe la situación actual, debe dar resultados coherentes para diferentes movimientos. Los los valores de n y η calculados a partir de tres conjuntos de datos diferentes se enumeran a continuación (Nota: la distancia media de la Luna a la Tierra y los radios medios de la Luna y la Tierra son utilizado en el cálculo). El valor de n: n = 0,64 De la rotación de la Luna: η = 4,27 × 10-22 s-1 m-1 De la rotación de la Tierra: η = 4,26 × 10-22 s-1 m-1 De la recesión de la Luna: η = 4,64 × 10-22 s-1 m-1 Se puede ver que los tres valores de η son consistentes dentro del rango de error en los datos. El valor medio de η: η = (4,39 ± 0,22) × 10-22 s-1 m-1 Al insertar los datos de la rotación de la Tierra, la recesión de la Luna y el valor de n en Eq. 18, el resultado es 14.053.1 6371000 1738000 1021.11029.7 1092509,3105.5 )64.01( (20) Esto también está de acuerdo con el modelo utilizado aquí. La fuerza de arrastre ejercida sobre el movimiento orbital de la Luna por el campo de materia oscura fluido es -1.11 × 108 N, esto es insignificantemente pequeño en comparación con la fuerza gravitacional entre la Luna y la Tierra ~ 1,90 × 1020 N; y el torque ejercido por el campo de materia oscura fluido en las rotaciones de la Tierra y la Luna es T = -5,49 × 1016 Nm y -1,15 × 1012 Nm, respectivamente. 5. La evolución del sistema Tierra-Luna Sonett et al. encontró que la longitud del día terrestre hace 900 millones de años fue alrededor de 19,2 horas sobre la base de los sedimentos de marea laminadas en la Tierra (Sonett y otros, 1996). De acuerdo con el modelo presentado aquí, en ese tiempo, la duración del día fue alrededor de 19,2 horas, esto concuerda muy bien con Sonett et al.El resultado. Otro aspecto crítico de modelar la evolución del sistema Tierra-Luna es: dar una estimación razonable de la distancia más cercana de la Luna a la Tierra cuando la El sistema se estableció hace 4.500 millones de años. Basado en el fluido del campo de materia oscura modelo, y el resultado anterior, la distancia más cercana de la Luna a la Tierra fue 259000 km (centro a centro) o 250900 km (superficie a superficie) en 4.500 millones de años atrás, Esto está mucho más allá del límite del Roche. En el moderno libro de texto de astronomía de Chaisson y McMillan (Chaisson y McMillan, 1993, p.173), la distancia estimada en 4.500 millones hace 250000 km, este es probablemente el número más razonable que Los astrónomos creen y concuerdan excelentemente con el resultado de este modelo. El más cercano distancia de la Luna a la Tierra por los modelos de Hansen era de unos 38 radios de la Tierra o 242000 km (Hansen, 1982). De acuerdo con este modelo, la longitud del día de la Tierra fue de aproximadamente 8 horas a 4.5 Hace miles de millones de años. Fig. 1 muestra la evolución de la distancia de la Luna a la Tierra y el longitud del día de la Tierra con la edad del sistema Tierra-Luna descrito por este modelo junto con datos de Kvale et al. (1999), Sonett y otros (1996) y Scrutton (1978). Uno puede ver que esos datos encajan muy bien en este modelo en su rango de tiempo. Fig. 2 muestra los datos geológicos de los días solares año-1 de Wells (1963) y de Sonett et al. (1996) y la descripción (línea sólida) de este modelo de fluido de campo de materia oscura desde hace 900 millones de años. Se puede ver que el modelo está de acuerdo con el datos fósiles maravillosamente. La diferencia importante de este modelo con los modelos tempranos en la descripción de la la evolución del sistema Tierra-Luna es que este modelo se basa sólo en los datos actuales de la Sistema Luna-Tierra y no hay suposiciones sobre las condiciones de la Tierra anterior rotación y deriva continental. Basado en este modelo, el sistema Tierra-Luna ha sido evolución a la situación actual desde que se estableció y la tasa de recesión de la Luna ha ido aumentando gradualmente, sin embargo, esta descripción no lo toma en cuenta que podría haber acontecimientos especiales sucedidos en el pasado para causar el repentino cambios significativos en los movimientos de la Tierra y la Luna, tales como fuertes impactos por asteroides y cometas gigantes, etc., porque esos impactos son muy comunes en el universo. El patrón general de la evolución del sistema Luna-Tierra descrito por este modelo está de acuerdo con las pruebas geológicas. Basado en Eq. 9, la tasa de recesión exponencialmente aumenta con el tiempo. Se puede imaginar entonces que la tasa de recesión se convertirá rápidamente Muy grande. De hecho, el aumento es extremadamente lento. La tasa de recesión de la Luna será 3,04 × 10-9 m s-1 después de 10 mil millones de años y 7,64 × 10-9 m s-1 después de 20 mil millones de años. Sin embargo, si la recesión de la Luna continuará o en algún momento más tarde otro No se sabe si el mecanismo asumirá el control. Se debe entender que la fricción de mareas afecta a la evolución de la propia Tierra, como la estructura de la corteza superficial, continental la deriva y la evolución del biosistema, etc; también puede jugar un papel en la desaceleración de la Tierra la rotación, sin embargo, ese papel no es un mecanismo dominante. Desafortunadamente, no hay datos disponibles sobre los cambios en la órbita de la Tierra. movimiento y todos los demás miembros del sistema solar. De acuerdo con este modelo y los resultados anteriores, la tasa de recesión de la Tierra debe ser de 6,86 × 10-7 m s-1 = 21,6 m año-1 = 2,16 km siglo-1, la longitud de un año aumenta alrededor de 6,8 ms y el cambio de la temperatura es -1.8 × 10-8 K año-1 con constante nivel de radiación del Sol y el entorno estable en la Tierra. La duración de un año, hace mil millones de años, sería el 80% de la duración actual. del año. Sin embargo, muchas pruebas (bandas de crecimiento de corales y mariscos, así como de otras pruebas) sugieren que no ha habido ningún cambio aparente en la duración de la año sobre los mil millones de años y el movimiento orbital de la Tierra es más estable que su rotación. Esto sugiere que el líquido del campo de materia oscura está circulando alrededor del Sol con el mismo dirección y velocidad similar de la Tierra (al menos en el rango orbital de la Tierra). Por lo tanto, el El movimiento orbital de la Tierra experimenta muy poca o ninguna fuerza de arrastre de la materia oscura fluido de campo. Sin embargo, se trata de una conjetura, hay que llevar a cabo una amplia investigación para verificar Si este es el caso. 6. Descripción escéptica de la evolución del Marte La Luna no tiene líquido líquido en su superficie, incluso no hay aire, por lo tanto, no hay una fuerza de fricción mareomotriz similar al océano para ralentizar su rotación; sin embargo, la rotación de la La Luna todavía se está desacelerando a un ritmo significativo de (-1.260 ± 0.024) × 10-23 rad s-2, lo que está de acuerdo con el modelo muy bien. En base a esto, uno puede pensar razonablemente que los la rotación también debería ser más lenta. El Marte es nuestro vecino más cercano que ha atraído la gran atención de los humanos Desde la antigüedad. La exploración de Marte se ha estado calentando en las últimas décadas. NASA, Agencia Espacial Rusa y Europa enviaron muchas naves espaciales a Marte para recolectar datos y estudiar este misterioso planeta. Hasta ahora todavía no hay suficientes datos sobre el historia de este planeta para describir su evolución. Igual que la Tierra, el Marte gira alrededor su eje central y gira alrededor del Sol, sin embargo, el Marte no tiene una masa (Marte tiene dos pequeños satélites: Fobos y Deimos) y no hay líquido líquido en su superficie, por lo tanto, no hay aparente fuerza de fricción mareo-como el océano a ralentizar su rotación por teorías de fricción de mareas. Sobre la base del resultado anterior y actual Los datos de Marte, este modelo predice que la aceleración angular del Marte debería ser alrededor de -4.38 × 10-22 rad s-2. La figura 3 describe la posible evolución de la duración del día y la días solares / año de Marte, la línea vertical marca la edad actual del Marte con asumir que el Marte se formó en un período de tiempo similar de la formación de la Tierra. Tal descripción no fue dada antes de acuerdo con el conocimiento del autor y es completamente escéptico debido a la falta de datos confiables. Sin embargo, con una mayor expansión de la investigación y exploración en Marte, nos sentiremos seguros de que los datos confiables sobre la aceleración angular de rotación del Marte estará disponible en el futuro próximo que proporcionará una prueba vital para la predicción de este modelo. También hay otros factores que puede afectar a la tasa de rotación de Marte, como la redistribución de masa debido a la temporada cambio, vientos, posibles erupciones volcánicas y terremotos de Marte. Por lo tanto, los datos deben ser cuidadosamente analizados. 7. Discusión sobre el modelo De los resultados anteriores, se puede ver que los datos actuales Tierra-Luna y el datos geológicos y fósiles están de acuerdo con el modelo muy bien y la evolución pasada de la Sistema Tierra-Luna puede ser descrito por el modelo sin introducir ningún adicional parámetros; este modelo revela la interesante relación entre la rotación y Retirada (Eq. 17 y Eq. 18) del mismo cuerpo celestial o diferentes cuerpos celestes en el mismo sistema gravitacional, tal relación no se conoce antes. Tal éxito puede no debe explicarse por “coincidencia” o “suerte” debido a la gran cantidad de datos Los datos de la Tierra y la Luna y los datos geológicos y fósiles) si uno piensa que esto es sólo un “ad hoc” o un modelo equivocado, aunque la posibilidad de que “coincidencia” o “suerte” podría ser mayor que ganar un premio mayor de la lotería; el futuro de Marte los datos aclararán esto; de lo contrario, se puede desarrollar una nueva teoría a partir de un enfoque diferente dar la misma o mejor descripción como lo hace este modelo. Es cierto que este modelo es no perfecto y puede tener defectos, se puede llevar a cabo un mayor desarrollo. James Clark Maxwell dijo en el 1873 “El vasto interplanetario e interestelar regiones ya no serán considerados como lugares de desecho en el universo, que el Creador tiene no se considera apto para llenar con los símbolos de la orden múltiple de Su reino. Encontraremos estar ya llenos de este maravilloso medio; tan lleno, que ningún poder humano puede quitarlo de la porción más pequeña del espacio, o producir el más mínimo defecto en su infinito continuidad. Se extiende ininterrumpidamente de estrella a estrella...”. El medio que habló Maxwell alrededor es el éter que fue propuesto como portador de la propagación de la onda de luz. Los El experimento Michelson-Morley sólo demostró que la propagación de la onda de luz no depende de tal medio y no rechaza la existencia del medio en el interestelar espacio. De hecho, el concepto de medio interestelar se ha desarrollado dramáticamente recientemente como la materia oscura, la energía oscura, el fluido cósmico, etc. El campo de la materia oscura fluido es sólo una parte de tan maravilloso medio y “precisamente” descrito por Maxwell. 7. Conclusión La evolución del sistema Tierra-Luna puede ser descrita por el campo de materia oscura modelo fluido con enfoque no newtoniano y los datos actuales de la Tierra y la Luna Se adapta muy bien a este modelo. Hace 4.500 millones de años, la distancia más cercana de la Luna La Tierra podría estar a unos 259000 km, que está muy por encima del límite de Roche y de la longitud de El día era alrededor de 8 horas. El patrón general de la evolución del sistema Luna-Tierra descrita por este modelo concuerda con la evidencia geológica y fósil. La fricción de mareas puede no sea la causa principal de la evolución del sistema Tierra-Luna. La rotación de Marte también se está desacelerando con la velocidad de aceleración angular alrededor de -4,38 × 10-22 rad s-2. Bibliografía S. G. Brush, 1983. L. R. Godfrey (editor), Fantasma del siglo XIX: Argumentos creacionistas para una Tierra joven. Los científicos se enfrentan al creacionismo. W. W. Norton & Company, Nueva York, Londres, pp. E. Chaisson y S. McMillan. 1993. Astronomía Hoy, Sala Prentice, Englewood Cliffs, NJ 07632. J. O. Dickey, et al., 1994. Ciencia, 265, 482. D. G. Finch, 1981. Tierra, Luna y Planetas, 26(1), 109. K. S. Hansen, 1982. Rev. Geophys. y Space Phys. 20(3), 457. W. K. Hartmann, D. R. Davis, 1975. Ícaro, 24, 504. B. A. Kagan, N. B. Maslova, 1994. Tierra, Luna y Planetas 66, 173. B. A. Kagan, 1997. Prog. Oceanog. 40, 109. E. P. Kvale, H. W. Johnson, C. O. Sonett, A. W. Archer, y A. Zawistoski, 1999, J. Sedimento. Res. 69(6), 1154. K. Lang, 1992. Datos Astrofísicos: Planetas y Estrellas, Springer-Verlag, Nueva York. H. Pan, 2005. Internat. J. Phys modernos. A, 20(14), 3135. R. D. Ray, B. G. Bills, B. F. Chao, 1999. J. Geophys. Res. 104 (B8), 17653. C. T. Scrutton, 1978. P. Brosche, J. Sundermann, (Editors.), la fricción de mareas y el La rotación de la Tierra. Springer-Verlag, Berlín, pp. 154. L. B. Slichter, 1963. J. Geophys. Res. 68, 14. C. P. Sonett, E. P. Kvale, M. A. Chan, T. M. Demko, 1996. Ciencia, 273, 100. F. D. Stacey, 1977. Física de la Tierra, segunda edición. John Willey & Sons. J. W. Wells, 1963. Naturaleza, 197, 948. Título Figura 1, la evolución de la distancia de la Luna y la longitud del día de la tierra con la era del sistema Tierra-Luna. Las líneas sólidas se calculan según la materia oscura modelo de fluido de campo. Fuentes de datos: las distancias de la Luna son de Kvale y et al. y para el longitud del día: (a y b) son de Scrutton (página 186, fig. 8), c es de Sonett y et al. La línea marca la edad actual del sistema Tierra-Luna. Figura 2, la evolución de los días solares del año con la edad de la Luna-Tierra sistema. La línea sólida se calcula según el modelo de fluido de campo de materia oscura. Los datos son de Wells (3.9 ~ 4.435 millones de años de rango), Sonett (3.600 millones de años) y actual edad (4.500 millones de años). Figura 3, la descripción escéptica de la evolución de la longitud del día de Marte y el días solares/año de Marte con la edad de Marte (suponiendo que la edad de Marte es de aproximadamente 4.5 miles de millones de años). La línea vertical marca la edad actual de Marte. Figura 1, distancia de la Luna y la longitud del día de la Tierra cambio con la era del sistema Tierra-Luna La edad del sistema Tierra-Luna (109 años) 0 1 2 3 4 5 Distancia Duración del día Límite de Roche Resultado de Hansen Figura 2, los días solares / año vs. la edad de la Tierra La edad de la Tierra (109 años) 3,5 3,6 3,7 3,8 3,9 4,0 4,1 4,2 4,3 4,4 4,5 4,6
704
A determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata
We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the determinant.
A Determinant of Stirling Cycle Numbers Counts Unlabeled Acyclic Single-Source Automata DAVID CALLAN Department of Statistics University of Wisconsin-Madison 1300 University Ave Madison, WI 53706-1532 callan@stat.wisc.edu March 30, 2007 Abstract We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the deter- minant. 1 Introduction The chief purpose of this paper is to show bijectively that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. Specifically, let Ak(n) denote the kn × kn matrix with (i, j) entry [ ⌊ i−1 ⌊ i−1 ⌋+1+i−j , where is the Stirling cycle number, the number of permutations on [i] with j cycles. For example, A2(5) = 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 3 2 0 0 0 0 0 0 0 0 1 3 2 0 0 0 0 0 0 0 0 1 6 11 6 0 0 0 0 0 0 0 1 6 11 6 0 0 0 0 0 0 0 1 10 35 50 24 0 0 0 0 0 0 1 10 35 50 0 0 0 0 0 0 0 1 15 85 0 0 0 0 0 0 0 0 1 15 http://arxiv.org/abs/0704.0004v1 As evident in the example, Ak(n) is formed from k copies of each of rows 2 through n+1 of the Stirling cycle triangle, arranged so that the first nonzero entry in each row is a 1 and, after the first row, this 1 occurs just before the main diagonal; in other words, Ak(n) is a Hessenberg matrix with 1s on the infra-diagonal. We will show Main Theorem. The determinant of Ak(n) is the number of unlabeled acyclic single- source automata with n transient states on a (k + 1)-letter input alphabet. Section 2 reviews basic terminology for automata and recurrence relations to count finite acyclic automata. Section 3 introduces column-marked subdiagonal paths, which play an intermediate role, and a way to code them. Section 4 presents a bijection from these column-marked subdiagonal paths to unlabeled acyclic single-source automata. Fi- nally, Section 5 evaluates detAk(n) using a sign-reversing involution and shows that the determinant counts the codes for column-marked subdiagonal paths. 2 Automata A (complete, deterministic) automaton consists of a set of states and an input alphabet whose letters transform the states among themselves: a letter and a state produce another state (possibly the same one). A finite automaton (finite set of states, finite input alphabet of, say, k letters) can be represented as a k-regular directed multigraph with ordered edges: the vertices represent the states and the first, second, . . . edge from a vertex give the effect of the first, second, . . . alphabet letter on that state. A finite automaton cannot be acyclic in the usual sense of no cycles: pick a vertex and follow any path from it. This path must ultimately hit a previously encountered vertex, thereby creating a cycle. So the term acyclic is used in the looser sense that only one vertex, called the sink, is involved in cycles. This means that all edges from the sink loop back to itself (and may safely be omitted) and all other paths feed into the sink. A non-sink state is called transient. The size of an acyclic automaton is the number of transient states. An acyclic automaton of size n thus has transient states which we label 1, 2, . . . , n and a sink, labeled n + 1. Liskovets [1] uses the inclusion-exclusion principle (more about this below) to obtain the following recurrence relation for the number ak(n) of acyclic automata of size n on a k-letter input alphabet (k ≥ 1): ak(0) = 1; ak(n) = (−1)n−j−1 (j + 1)k(n−j)ak(j), n ≥ 1. A source is a vertex with no incoming edges. A finite acyclic automaton has at least one source because a path traversed backward v1 ← v2 ← v3 ← . . . must have distinct vertices and so cannot continue indefinitely. An automaton is single-source (or initially connected) if it has only one source. Let Bk(n) denote the set of single-source acyclic finite (SAF) automata on a k-letter input alphabet with vertices 1, 2, . . . , n + 1 where 1 is the source and n + 1 is the sink, and set bk(n) = | Bk(n) |. The two-line representation of an automaton in Bk(n) is the 2× kn matrix whose columns list the edges in order. For example, 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 2 4 6 6 6 6 6 6 6 3 5 3 2 2 6 is in B3(5) and the source-to-sink paths in B include 1 → 6, 1 → 6, 1 → 6, where the alphabet is {a, b, c}. Proposition 1. The number bk(n) of SAF automata of size n on a k-letter input alphabet (n, k ≥ 1) is given by bk(n) = (−1)n−i (i+ 1)k(n−i)ak(i) Remark This formula is a bit more succinct than the the recurrence in [1, Theorem 3.2]. Proof Consider the setA of acyclic automata with transient vertices [n] = {1, 2, . . . , n} in which 1 is a source. Call 2, 3, . . . , n the interior vertices. For X ⊆ [2, n], let f(X) = # automata in A whose set of interior vertices includes X, g(X) = # automata in A whose set of interior vertices is precisely X. Then f(X) = Y :X⊆Y⊆[2,n] g(Y ) and by Möbius inversion [2] on the lattice of subsets of [2, n], g(X) = Y :X⊆Y⊆[2,n] µ(X, Y )f(Y ) where µ(X, Y ) is the Möbius function for this lattice. Since µ(X, Y ) = (−1)|Y |−|X| if X ⊆ Y , we have in particular that g(∅) = Y⊆[2,n] (−1)| Y |f(Y ). (1) Let | Y | = n − i so that 1 ≤ i ≤ n. When Y consists entirely of sources, the vertices in [n+ 1]\Y and their incident edges form a subautomaton with i transient states; there are ak(i) such. Also, all edges from the n − i vertices comprising Y go directly into [n + 1]\Y : (i + 1)k(n−i) choices. Thus f(Y ) = (i + 1)k(n−i)ak(i). By definition, g(∅) is the number of automata in A for which 1 is the only source, that is, g(∅) = bk(n) and the Proposition now follows from (1). An unlabeled SAF automaton is an equivalence class of SAF automata under relabeling of the interior vertices. Liskovets notes [1] (and we prove below) that Bk(n) has no nontrivial automorphisms, that is, each of the (n− 1)! relabelings of the interior vertices of B ∈ Bk(n) produces a different automaton. So unlabeled SAF automata of size n on a k-letter alphabet are counted by 1 (n−1)! bk(n). The next result establishes a canonical representative in each relabeling class. Proposition 2. Each equivalence class in Bk(n) under relabeling of interior vertices has size (n− 1)! and contains exactly one SAF automaton with the “last occurrences increas- ing” property: the last occurrences of the interior vertices—2, 3, . . . , n—in the bottom row of its two-line representation occur in that order. Proof The first assertion follows from the fact that the interior vertices of an au- tomatonB ∈ bk(n) can be distinguished intrinsically, that is, independent of their labeling. To see this, first mark the source, namely 1, with a mark (new label) v1 and observe that there exists at least one interior vertex whose only incoming edge(s) are from the source (the only currently marked vertex) for otherwise a cycle would be present. For each such interior vertex v, choose the last edge from the marked vertex to v using the built-in ordering of these edges. This determines an order on these vertices; mark them in order v2, v3, . . . , vj (j ≥ 2). If there still remain unmarked interior vertices, at least one of them has incoming edges only from a marked vertex or again a cycle would be present. For each such vertex, use the last incoming edge from a marked vertex, where now edges are arranged in order of initial vertex vi with the built-in order breaking ties, to order and mark these vertices vj+1, vj+2, . . .. Proceed similarly until all interior vertices are marked. For example, for 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 2 4 6 6 6 6 6 6 6 3 5 3 2 2 6 v1 = 1 and there is just one interior vertex, namely 4, whose only incoming edge is from the source, and so v2 = 4 and 4 becomes a marked vertex. Now all incoming edges to both 3 and 5 are from marked vertices and the last such edges (built-in order comes into play) are 4 → 5 and 4 → 3 putting vertices 3, 5 in the order 5, 3. So v3 = 5 and v4 = 3. Finally, v5 = 2. This proves the first assertion. By construction of the vs, relabeling each interior vertex i with the subscript of its corresponding v produces an automaton in Bk(n) with the “last occurrences increasing” property and is the only relabeling that does so. The example yields 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 5 2 6 4 3 4 5 5 6 6 6 6 6 6 6 Now let Ck(n) denote the set of canonical SAF automata in Bk(n) representing un- labeled automata; thus | Ck(n) | = (n−1)! bk(n). Henceforth, we identify an unlabeled au- tomaton with its canonical representative. 3 Column-Marked Subdiagonal Paths A subdiagonal (k, n, p)-path is a lattice path of steps E = (1, 0) and N = (0, 1), E for east and N for north, from (0, 0) to (kn, p) that never rise above the line y = 1 x. Let Ck(n, p) denote the set of such paths.For k ≥ 1, it is clear that Ck(n, p) is nonempty only for 0 ≤ p ≤ n and it is known (generalized ballot theorem) that |Ck(n, p) | = kn− kp+ 1 kn+ p+ 1 kn+ p + 1 A path P in Ck(n, n) can be coded by the heights of its E steps above the line y = −1; this gives a a sequence (bi) i=1 subject to the restrictions 1 ≤ b1 ≤ b2 ≤ . . . ≤ bkn and bi ≤ ⌈i/k⌉ for all i. A column-marked subdiagonal (k, n, p)-path is one in which, for each i ∈ [1, kn], one of the lattice squares below the ith E step and above the horizontal line y = −1 is marked, say with a ‘ ∗ ’. Let C k(n, p) denote the set of such marked paths. b b b b b b b b b b b ∗ ∗ ∗ (0,0) (8,4) y = −1 y = 1 A path in C 2(4, 3) A marked path P ∗ in C k(n, n) can be coded by a sequence of pairs (ai, bi) where i=1 is the code for the underlying path P and ai ∈ [1, bi] gives the position of the ∗ in the ith column. The example is coded by (1, 1), (1, 1), (1, 2), (2, 2), (1, 2), (3, 3), (1, 3), (2, 3). An explicit sum for |C k(n, n) | is k(n, n) | = 1≤b1≤b2≤...≤bkn, bi ≤ ⌈i/k⌉ for all i b1b2 . . . bkn, because the summand b1b2 . . . bkn is the number of ways to insert the ‘ ∗ ’s in the underlying path coded by (bi) It is also possible to obtain a recurrence for |C k(n, p) |, and then, using Prop. 1, to show analytically that |C k(n, n) | = | Ck+1(n) |. However, it is much more pleasant to give a bijection and in the next section we will do so. In particular, the number of SAF automata on a 2-letter alphabet is | C2(n) | = |C 1(n, n) | = 1≤b1≤b2≤...≤bn bi ≤ i for all i b1b2 . . . bn = (1, 3, 16, 127, 1363, . . .)n≥1, sequence A082161 in [3]. 4 Bijection from Paths to Automata In this section we exhibit a bijection from C k(n, n) to Ck+1(n). Using the illustrated path as a working example with k = 2 and n = 4, b b b b b b b b b b b ∗ ∗ ∗ (0,0) (8,4) y = −1 y = 1 first construct the top row of a two-line representation consisting of k + 1 each 1s, 2s, . . . ,n s and number them left to right: The last step in the path is necessarily anN step. For the second last, third last,. . .N steps in the path, count the number of steps following it. This gives a sequence i1, i2, . . . , in−1 satisfying 1 ≤ i1 < i2 < . . . < in−1 and ij ≤ (k + 1)j for all j. Circle the positions i1, i2, . . . , in−1 in the two-line representation and then insert (in boldface) 2, 3, . . . , n in the second row in the circled positions: 2 3 4 These will be the last occurrences of 2, 3, . . . , n in the second row. Working from the last column in the path back to the first, fill in the blanks in the second row left to right as follows. Count the number of squares from the ∗ up to the path (including the ∗ square) http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A082161 and add this number to the nearest boldface number to the left of the current blank entry (if there are no boldface numbers to the left, add this number to 1) and insert the result in the current blank square. In the example the numbers of squares are 2,3,1,2,1,2,1,1 yielding 2 4 5 3 3 5 4 5 4 5 5 This will fill all blank entries except the last. Note that ∗ s in the bottom row correspond to sink (that is, n+1) labels in the second row. Finally, insert n+1 into the last remaining blank space to give the image automaton: 1 1 1 2 2 2 3 3 3 4 4 4 2 4 5 3 3 5 4 5 4 5 5 5 This process is fully reversible and the map is a bijection. 5 Evaluation of detAk(n) For simplicity, we treat the case k = 1, leaving the generalization to arbitrary k as a not-too-difficult exercise for the interested reader. Write A(n) for A1(n). Thus A(n) = 1≤i,j≤n . From the definition of detA(n) as a sum of signed products, we show that detA(n) is the total weight of certain lists of permutations, each list carrying weight ±1. Then a weight-reversing involution cancels all −1 weights and reduces the problem to counting the surviving lists. These surviving lists are essentially the codes for paths in C 1(n, p), and the Main Theorem follows from §4. To describe the permutations giving a nonzero contribution to detA(n) = σ sgn σ× i=1 ai,σ(i), define the code of a permutation σ on [n] to be the list c = (ci) i=1 with ci = σ(i)−(i−1). Since the (i, j) entry of A(n), , is 0 unless j ≥ i−1, we must have σ(i) ≥ i−1 for all i. It is well known that there are 2n−1 such permutations, corresponding to compositions of n, with codes characterized by the following four conditions: (i) ci ≥ 0 for all i, (ii) c1 ≥ 1, (iii) each ci ≥ 1 is immediately followed by ci − 1 zeros in the list, i=1 ci = n. Let us call such a list a padded composition of n: deleting the zeros is a bijection to ordinary compositions of n. For example, (3, 0, 0, 1, 2, 0) is a padded composition of 6. For a permutation σ with padded composition code c, the nonzero entries in c give the cycle lengths of σ. Hence sgnσ, which is the parity of “n−#cycles in σ”, is given by (−1)#0s in c. We have detA(n) = σ sgn σ i=1 ai,σ(i) = σ sgn σ 2i−σ(i) , and so detA(n) = (−1)#0s in c i+ 1− ci where the sum is restricted to padded compositions c of n with ci ≤ i for all i (A002083) because i+1−ci = 0 unless ci ≤ i. Henceforth, let us write all permutations in standard cycle form whereby the smallest entry occurs first in each cycle and these smallest entries increase left to right. Thus, with dashes separating cycles, 154-2-36 is the standard cycle form of the permutation ( 1 2 3 4 5 65 2 6 1 4 3 ). We define a nonfirst entry to be one that does not start a cycle. Thus the preceding permutation has 3 nonfirst entries: 5,4,6. Note that the number of nonfirst entries is 0 only for the identity permutation. We denote an identity permutation (of any size) by ǫ. By definition of Stirling cycle number, the product in (2) counts lists (πi) i=1 of permu- tations where πi is a permutation on [i+1] with i+1− ci cycles, equivalently, with ci ≤ i nonfirst entries. So define Ln to be the set all lists of permutations π = (πi) i=1 where πi is a permutation on [i + 1], #nonfirst entries in πi is ≤ i, π1 is the transposition (1,2), each nonidentity permutation πi is immediately followed by ci − 1 ǫ’s where ci ≥ 1 is the number of nonfirst entries in πi (so the total number of nonfirst entries is n). Assign a weight to π ∈ Ln by wt(π) = (−1) # ǫ’s in π. Then detA(n) = wt(π). We now define a weight-reversing involution on (most of) Ln. Given π ∈ Ln, scan the list of its component permutations π1 = (1, 2), π2, π3, . . . left to right. Stop at the first one that either (i) has more than one nonfirst entry, or (ii) has only one nonfirst entry, b say, and b > maximum nonfirst entry m of the next permutation in the list. Say πk is the permutation where we stop. http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002083 In case (i) decrement (i.e. decrease by 1) the number of ǫ’s in the list by splitting πk into two nonidentity permutations as follows. Let m be the largest nonfirst entry of πk and let ℓ be its predecessor. Replace πk and its successor in the list (necessarily an ǫ) by the following two permutations: first the transposition (ℓ,m) and second the permutation obtained from πk by erasing m from its cycle and turning it into a singleton. Here are two examples of this case (recall permutations are in standard cycle form and, for clarity, singleton cycles are not shown). i 1 2 3 4 5 6 πi 12 13 23 14-253 ǫ ǫ i 1 2 3 4 5 6 πi 12 13 23 25 14-23 ǫ i 1 2 3 4 5 6 πi 12 23 14 13-24 ǫ 23 i 1 2 3 4 5 6 πi 12 23 14 24 13 23 The reader may readily check that this sends case (i) to case (ii). In case (ii), πk is a transposition (a, b) with b > maximum nonfirst entry m of πk+1. In this case, increment the number of ǫ’s in the list by combining πk and πk+1 into a single permutation followed by an ǫ: in πk+1, b is a singleton; delete this singleton and insert b immediately after a in πk+1 (in the same cycle). The reader may check that this reverses the result in the two examples above and, in general, sends case (ii) to case (i). Since the map alters the number of ǫ’s in the list by 1, it is clearly weight-reversing. The map fails only for lists that both consist entirely of transpositions and have the form (a1, b1), (a2, b2), . . . , (an, bn) with b1 ≤ b2 ≤ . . . ≤ bn. Such lists have weight 1. Hence detA(n) is the number of lists (ai, bi) satisfying 1 ≤ ai < bi ≤ i+ 1 for 1 ≤ i ≤ n, and b1 ≤ b2 ≤ . . . ≤ bn. After subtracting 1 from each bi, these lists code the paths in C 1(n, n) and, using §4, detA(n) = |C 1(n, n) | = | C2(n) |. References [1] Valery A. Liskovets, Exact enumeration of acyclic deterministic au- tomata, Disc. Appl. Math., in press, 2006. Earlier version available at http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html [2] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd ed., Cambridge University Press, NY, 2001. [3] Neil J. Sloane (founder and maintainer), The On-Line Encyclopedia of Integer Se- quences http://www.research.att.com:80/ njas/sequences/index.html?blank=1 http://www.research.att.com:80/~njas/sequences/index.html?blank=1
Introduction The chief purpose of this paper is to show bijectively that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. Specifically, let Ak(n) denote the kn × kn matrix with (i, j) entry [ ⌊ i−1 ⌊ i−1 ⌋+1+i−j , where is the Stirling cycle number, the number of permutations on [i] with j cycles. For example, A2(5) = 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 3 2 0 0 0 0 0 0 0 0 1 3 2 0 0 0 0 0 0 0 0 1 6 11 6 0 0 0 0 0 0 0 1 6 11 6 0 0 0 0 0 0 0 1 10 35 50 24 0 0 0 0 0 0 1 10 35 50 0 0 0 0 0 0 0 1 15 85 0 0 0 0 0 0 0 0 1 15 http://arxiv.org/abs/0704.0004v1 As evident in the example, Ak(n) is formed from k copies of each of rows 2 through n+1 of the Stirling cycle triangle, arranged so that the first nonzero entry in each row is a 1 and, after the first row, this 1 occurs just before the main diagonal; in other words, Ak(n) is a Hessenberg matrix with 1s on the infra-diagonal. We will show Main Theorem. The determinant of Ak(n) is the number of unlabeled acyclic single- source automata with n transient states on a (k + 1)-letter input alphabet. Section 2 reviews basic terminology for automata and recurrence relations to count finite acyclic automata. Section 3 introduces column-marked subdiagonal paths, which play an intermediate role, and a way to code them. Section 4 presents a bijection from these column-marked subdiagonal paths to unlabeled acyclic single-source automata. Fi- nally, Section 5 evaluates detAk(n) using a sign-reversing involution and shows that the determinant counts the codes for column-marked subdiagonal paths. 2 Automata A (complete, deterministic) automaton consists of a set of states and an input alphabet whose letters transform the states among themselves: a letter and a state produce another state (possibly the same one). A finite automaton (finite set of states, finite input alphabet of, say, k letters) can be represented as a k-regular directed multigraph with ordered edges: the vertices represent the states and the first, second, . . . edge from a vertex give the effect of the first, second, . . . alphabet letter on that state. A finite automaton cannot be acyclic in the usual sense of no cycles: pick a vertex and follow any path from it. This path must ultimately hit a previously encountered vertex, thereby creating a cycle. So the term acyclic is used in the looser sense that only one vertex, called the sink, is involved in cycles. This means that all edges from the sink loop back to itself (and may safely be omitted) and all other paths feed into the sink. A non-sink state is called transient. The size of an acyclic automaton is the number of transient states. An acyclic automaton of size n thus has transient states which we label 1, 2, . . . , n and a sink, labeled n + 1. Liskovets [1] uses the inclusion-exclusion principle (more about this below) to obtain the following recurrence relation for the number ak(n) of acyclic automata of size n on a k-letter input alphabet (k ≥ 1): ak(0) = 1; ak(n) = (−1)n−j−1 (j + 1)k(n−j)ak(j), n ≥ 1. A source is a vertex with no incoming edges. A finite acyclic automaton has at least one source because a path traversed backward v1 ← v2 ← v3 ← . . . must have distinct vertices and so cannot continue indefinitely. An automaton is single-source (or initially connected) if it has only one source. Let Bk(n) denote the set of single-source acyclic finite (SAF) automata on a k-letter input alphabet with vertices 1, 2, . . . , n + 1 where 1 is the source and n + 1 is the sink, and set bk(n) = | Bk(n) |. The two-line representation of an automaton in Bk(n) is the 2× kn matrix whose columns list the edges in order. For example, 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 2 4 6 6 6 6 6 6 6 3 5 3 2 2 6 is in B3(5) and the source-to-sink paths in B include 1 → 6, 1 → 6, 1 → 6, where the alphabet is {a, b, c}. Proposition 1. The number bk(n) of SAF automata of size n on a k-letter input alphabet (n, k ≥ 1) is given by bk(n) = (−1)n−i (i+ 1)k(n−i)ak(i) Remark This formula is a bit more succinct than the the recurrence in [1, Theorem 3.2]. Proof Consider the setA of acyclic automata with transient vertices [n] = {1, 2, . . . , n} in which 1 is a source. Call 2, 3, . . . , n the interior vertices. For X ⊆ [2, n], let f(X) = # automata in A whose set of interior vertices includes X, g(X) = # automata in A whose set of interior vertices is precisely X. Then f(X) = Y :X⊆Y⊆[2,n] g(Y ) and by Möbius inversion [2] on the lattice of subsets of [2, n], g(X) = Y :X⊆Y⊆[2,n] µ(X, Y )f(Y ) where µ(X, Y ) is the Möbius function for this lattice. Since µ(X, Y ) = (−1)|Y |−|X| if X ⊆ Y , we have in particular that g(∅) = Y⊆[2,n] (−1)| Y |f(Y ). (1) Let | Y | = n − i so that 1 ≤ i ≤ n. When Y consists entirely of sources, the vertices in [n+ 1]\Y and their incident edges form a subautomaton with i transient states; there are ak(i) such. Also, all edges from the n − i vertices comprising Y go directly into [n + 1]\Y : (i + 1)k(n−i) choices. Thus f(Y ) = (i + 1)k(n−i)ak(i). By definition, g(∅) is the number of automata in A for which 1 is the only source, that is, g(∅) = bk(n) and the Proposition now follows from (1). An unlabeled SAF automaton is an equivalence class of SAF automata under relabeling of the interior vertices. Liskovets notes [1] (and we prove below) that Bk(n) has no nontrivial automorphisms, that is, each of the (n− 1)! relabelings of the interior vertices of B ∈ Bk(n) produces a different automaton. So unlabeled SAF automata of size n on a k-letter alphabet are counted by 1 (n−1)! bk(n). The next result establishes a canonical representative in each relabeling class. Proposition 2. Each equivalence class in Bk(n) under relabeling of interior vertices has size (n− 1)! and contains exactly one SAF automaton with the “last occurrences increas- ing” property: the last occurrences of the interior vertices—2, 3, . . . , n—in the bottom row of its two-line representation occur in that order. Proof The first assertion follows from the fact that the interior vertices of an au- tomatonB ∈ bk(n) can be distinguished intrinsically, that is, independent of their labeling. To see this, first mark the source, namely 1, with a mark (new label) v1 and observe that there exists at least one interior vertex whose only incoming edge(s) are from the source (the only currently marked vertex) for otherwise a cycle would be present. For each such interior vertex v, choose the last edge from the marked vertex to v using the built-in ordering of these edges. This determines an order on these vertices; mark them in order v2, v3, . . . , vj (j ≥ 2). If there still remain unmarked interior vertices, at least one of them has incoming edges only from a marked vertex or again a cycle would be present. For each such vertex, use the last incoming edge from a marked vertex, where now edges are arranged in order of initial vertex vi with the built-in order breaking ties, to order and mark these vertices vj+1, vj+2, . . .. Proceed similarly until all interior vertices are marked. For example, for 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 2 4 6 6 6 6 6 6 6 3 5 3 2 2 6 v1 = 1 and there is just one interior vertex, namely 4, whose only incoming edge is from the source, and so v2 = 4 and 4 becomes a marked vertex. Now all incoming edges to both 3 and 5 are from marked vertices and the last such edges (built-in order comes into play) are 4 → 5 and 4 → 3 putting vertices 3, 5 in the order 5, 3. So v3 = 5 and v4 = 3. Finally, v5 = 2. This proves the first assertion. By construction of the vs, relabeling each interior vertex i with the subscript of its corresponding v produces an automaton in Bk(n) with the “last occurrences increasing” property and is the only relabeling that does so. The example yields 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 5 2 6 4 3 4 5 5 6 6 6 6 6 6 6 Now let Ck(n) denote the set of canonical SAF automata in Bk(n) representing un- labeled automata; thus | Ck(n) | = (n−1)! bk(n). Henceforth, we identify an unlabeled au- tomaton with its canonical representative. 3 Column-Marked Subdiagonal Paths A subdiagonal (k, n, p)-path is a lattice path of steps E = (1, 0) and N = (0, 1), E for east and N for north, from (0, 0) to (kn, p) that never rise above the line y = 1 x. Let Ck(n, p) denote the set of such paths.For k ≥ 1, it is clear that Ck(n, p) is nonempty only for 0 ≤ p ≤ n and it is known (generalized ballot theorem) that |Ck(n, p) | = kn− kp+ 1 kn+ p+ 1 kn+ p + 1 A path P in Ck(n, n) can be coded by the heights of its E steps above the line y = −1; this gives a a sequence (bi) i=1 subject to the restrictions 1 ≤ b1 ≤ b2 ≤ . . . ≤ bkn and bi ≤ ⌈i/k⌉ for all i. A column-marked subdiagonal (k, n, p)-path is one in which, for each i ∈ [1, kn], one of the lattice squares below the ith E step and above the horizontal line y = −1 is marked, say with a ‘ ∗ ’. Let C k(n, p) denote the set of such marked paths. b b b b b b b b b b b ∗ ∗ ∗ (0,0) (8,4) y = −1 y = 1 A path in C 2(4, 3) A marked path P ∗ in C k(n, n) can be coded by a sequence of pairs (ai, bi) where i=1 is the code for the underlying path P and ai ∈ [1, bi] gives the position of the ∗ in the ith column. The example is coded by (1, 1), (1, 1), (1, 2), (2, 2), (1, 2), (3, 3), (1, 3), (2, 3). An explicit sum for |C k(n, n) | is k(n, n) | = 1≤b1≤b2≤...≤bkn, bi ≤ ⌈i/k⌉ for all i b1b2 . . . bkn, because the summand b1b2 . . . bkn is the number of ways to insert the ‘ ∗ ’s in the underlying path coded by (bi) It is also possible to obtain a recurrence for |C k(n, p) |, and then, using Prop. 1, to show analytically that |C k(n, n) | = | Ck+1(n) |. However, it is much more pleasant to give a bijection and in the next section we will do so. In particular, the number of SAF automata on a 2-letter alphabet is | C2(n) | = |C 1(n, n) | = 1≤b1≤b2≤...≤bn bi ≤ i for all i b1b2 . . . bn = (1, 3, 16, 127, 1363, . . .)n≥1, sequence A082161 in [3]. 4 Bijection from Paths to Automata In this section we exhibit a bijection from C k(n, n) to Ck+1(n). Using the illustrated path as a working example with k = 2 and n = 4, b b b b b b b b b b b ∗ ∗ ∗ (0,0) (8,4) y = −1 y = 1 first construct the top row of a two-line representation consisting of k + 1 each 1s, 2s, . . . ,n s and number them left to right: The last step in the path is necessarily anN step. For the second last, third last,. . .N steps in the path, count the number of steps following it. This gives a sequence i1, i2, . . . , in−1 satisfying 1 ≤ i1 < i2 < . . . < in−1 and ij ≤ (k + 1)j for all j. Circle the positions i1, i2, . . . , in−1 in the two-line representation and then insert (in boldface) 2, 3, . . . , n in the second row in the circled positions: 2 3 4 These will be the last occurrences of 2, 3, . . . , n in the second row. Working from the last column in the path back to the first, fill in the blanks in the second row left to right as follows. Count the number of squares from the ∗ up to the path (including the ∗ square) http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A082161 and add this number to the nearest boldface number to the left of the current blank entry (if there are no boldface numbers to the left, add this number to 1) and insert the result in the current blank square. In the example the numbers of squares are 2,3,1,2,1,2,1,1 yielding 2 4 5 3 3 5 4 5 4 5 5 This will fill all blank entries except the last. Note that ∗ s in the bottom row correspond to sink (that is, n+1) labels in the second row. Finally, insert n+1 into the last remaining blank space to give the image automaton: 1 1 1 2 2 2 3 3 3 4 4 4 2 4 5 3 3 5 4 5 4 5 5 5 This process is fully reversible and the map is a bijection. 5 Evaluation of detAk(n) For simplicity, we treat the case k = 1, leaving the generalization to arbitrary k as a not-too-difficult exercise for the interested reader. Write A(n) for A1(n). Thus A(n) = 1≤i,j≤n . From the definition of detA(n) as a sum of signed products, we show that detA(n) is the total weight of certain lists of permutations, each list carrying weight ±1. Then a weight-reversing involution cancels all −1 weights and reduces the problem to counting the surviving lists. These surviving lists are essentially the codes for paths in C 1(n, p), and the Main Theorem follows from §4. To describe the permutations giving a nonzero contribution to detA(n) = σ sgn σ× i=1 ai,σ(i), define the code of a permutation σ on [n] to be the list c = (ci) i=1 with ci = σ(i)−(i−1). Since the (i, j) entry of A(n), , is 0 unless j ≥ i−1, we must have σ(i) ≥ i−1 for all i. It is well known that there are 2n−1 such permutations, corresponding to compositions of n, with codes characterized by the following four conditions: (i) ci ≥ 0 for all i, (ii) c1 ≥ 1, (iii) each ci ≥ 1 is immediately followed by ci − 1 zeros in the list, i=1 ci = n. Let us call such a list a padded composition of n: deleting the zeros is a bijection to ordinary compositions of n. For example, (3, 0, 0, 1, 2, 0) is a padded composition of 6. For a permutation σ with padded composition code c, the nonzero entries in c give the cycle lengths of σ. Hence sgnσ, which is the parity of “n−#cycles in σ”, is given by (−1)#0s in c. We have detA(n) = σ sgn σ i=1 ai,σ(i) = σ sgn σ 2i−σ(i) , and so detA(n) = (−1)#0s in c i+ 1− ci where the sum is restricted to padded compositions c of n with ci ≤ i for all i (A002083) because i+1−ci = 0 unless ci ≤ i. Henceforth, let us write all permutations in standard cycle form whereby the smallest entry occurs first in each cycle and these smallest entries increase left to right. Thus, with dashes separating cycles, 154-2-36 is the standard cycle form of the permutation ( 1 2 3 4 5 65 2 6 1 4 3 ). We define a nonfirst entry to be one that does not start a cycle. Thus the preceding permutation has 3 nonfirst entries: 5,4,6. Note that the number of nonfirst entries is 0 only for the identity permutation. We denote an identity permutation (of any size) by ǫ. By definition of Stirling cycle number, the product in (2) counts lists (πi) i=1 of permu- tations where πi is a permutation on [i+1] with i+1− ci cycles, equivalently, with ci ≤ i nonfirst entries. So define Ln to be the set all lists of permutations π = (πi) i=1 where πi is a permutation on [i + 1], #nonfirst entries in πi is ≤ i, π1 is the transposition (1,2), each nonidentity permutation πi is immediately followed by ci − 1 ǫ’s where ci ≥ 1 is the number of nonfirst entries in πi (so the total number of nonfirst entries is n). Assign a weight to π ∈ Ln by wt(π) = (−1) # ǫ’s in π. Then detA(n) = wt(π). We now define a weight-reversing involution on (most of) Ln. Given π ∈ Ln, scan the list of its component permutations π1 = (1, 2), π2, π3, . . . left to right. Stop at the first one that either (i) has more than one nonfirst entry, or (ii) has only one nonfirst entry, b say, and b > maximum nonfirst entry m of the next permutation in the list. Say πk is the permutation where we stop. http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002083 In case (i) decrement (i.e. decrease by 1) the number of ǫ’s in the list by splitting πk into two nonidentity permutations as follows. Let m be the largest nonfirst entry of πk and let ℓ be its predecessor. Replace πk and its successor in the list (necessarily an ǫ) by the following two permutations: first the transposition (ℓ,m) and second the permutation obtained from πk by erasing m from its cycle and turning it into a singleton. Here are two examples of this case (recall permutations are in standard cycle form and, for clarity, singleton cycles are not shown). i 1 2 3 4 5 6 πi 12 13 23 14-253 ǫ ǫ i 1 2 3 4 5 6 πi 12 13 23 25 14-23 ǫ i 1 2 3 4 5 6 πi 12 23 14 13-24 ǫ 23 i 1 2 3 4 5 6 πi 12 23 14 24 13 23 The reader may readily check that this sends case (i) to case (ii). In case (ii), πk is a transposition (a, b) with b > maximum nonfirst entry m of πk+1. In this case, increment the number of ǫ’s in the list by combining πk and πk+1 into a single permutation followed by an ǫ: in πk+1, b is a singleton; delete this singleton and insert b immediately after a in πk+1 (in the same cycle). The reader may check that this reverses the result in the two examples above and, in general, sends case (ii) to case (i). Since the map alters the number of ǫ’s in the list by 1, it is clearly weight-reversing. The map fails only for lists that both consist entirely of transpositions and have the form (a1, b1), (a2, b2), . . . , (an, bn) with b1 ≤ b2 ≤ . . . ≤ bn. Such lists have weight 1. Hence detA(n) is the number of lists (ai, bi) satisfying 1 ≤ ai < bi ≤ i+ 1 for 1 ≤ i ≤ n, and b1 ≤ b2 ≤ . . . ≤ bn. After subtracting 1 from each bi, these lists code the paths in C 1(n, n) and, using §4, detA(n) = |C 1(n, n) | = | C2(n) |. References [1] Valery A. Liskovets, Exact enumeration of acyclic deterministic au- tomata, Disc. Appl. Math., in press, 2006. Earlier version available at http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html [2] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd ed., Cambridge University Press, NY, 2001. [3] Neil J. Sloane (founder and maintainer), The On-Line Encyclopedia of Integer Se- quences http://www.research.att.com:80/ njas/sequences/index.html?blank=1 http://www.research.att.com:80/~njas/sequences/index.html?blank=1
Un determinante de los números de ciclo de Stirling cuenta sin etiqueta Automata de una sola fuente acíclica DAVID CALLAN Departamento de Estadística Universidad de Wisconsin-Madison 1300 University Ave Madison, WI 53706-1532 callen@stat.wisc.edu 30 de marzo de 2007 Resumen Demostramos que un determinante de los números de ciclo Stirling cuenta sin etiqueta acíclica autómatas de una sola fuente. La prueba implica una bijección de estos autómatas a algunos caminos de celosía marcados y una involución de inversión de signos para evaluar la disuasión Minant. 1 Introducción El propósito principal de este artículo es mostrar bijectamente que un determinante de los números de ciclo Stirling cuenta autómatas acíclicos de una sola fuente sin etiqueta. Específicamente, deje que Ak(n) denote la matriz kn × kn con (i, j) entrada [ i−1 i−1 1+i−j , donde es el número del ciclo de Stirling, el número de permutaciones en [i] con ciclos j. Por ejemplo, A2(5) = 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 3 2 0 0 0 0 0 0 0 0 1 3 2 0 0 0 0 0 0 0 0 1 6 11 6 0 0 0 0 0 0 0 1 6 11 6 0 0 0 0 0 0 0 1 10 35 50 24 0 0 0 0 0 1 10 35 50 0 0 0 0 0 0 1 15 85 0 0 0 0 0 0 0 1 15 http://arxiv.org/abs/0704.004v1 Como es evidente en el ejemplo, Ak(n) se forma a partir de k copias de cada una de las filas 2 a n+1 del triángulo del ciclo de Stirling, dispuesto de modo que la primera entrada no cero en cada fila es un 1 y, después de la primera fila, este 1 ocurre justo antes de la diagonal principal; en otras palabras, Ak(n) es una matriz de Hessenberg con 1s en la infradiagonal. Vamos a mostrar Teorema Principal. El determinante de Ak(n) es el número de mono- autómatas de origen con n estados transitorios en un alfabeto de entrada (k + 1) letras. En la sección 2 se examina la terminología básica para contar las relaciones automatizadas y recurrentes autómatas acíclicos finitos. En la sección 3 se introducen las vías subdiagonales marcadas con columnas, que jugar un papel intermedio, y una manera de codificarlos. En la sección 4 se presenta una estos caminos subdiagonales marcados con columnas a autómatas acíclicos sin etiquetar de una sola fuente. Fi- nally, la sección 5 evalúa detAk(n) usando una involución de inversión de signos y muestra que la determinante cuenta los códigos para las rutas subdiagonales con marca de columna. 2 Automatas Un autómata (completa, determinista) consiste en un conjunto de estados y un alfabeto de entrada cuyas letras transforman los estados entre sí: una carta y un estado producen otro Estado (posiblemente el mismo). Un autómata finito (conjunto finito de estados, alfabeto de entrada finito de, digamos, k letras) se puede representar como un multógrafo dirigido k-regular con bordes ordenados: los vértices representan los estados y el primero, segundo,. .. borde de un vértice dan el efecto del primero, segundo,. .. letra del alfabeto en ese estado. Un autómata finito no puede ser acíclico en el sentido habitual de no ciclos: elegir un vértice y seguir cualquier camino de él. Este camino debe finalmente golpeó un vértice previamente encontrado, creando así un ciclo. Así que el término acíclico se utiliza en el sentido más suelto que sólo un vértice, llamado el fregadero, está involucrado en ciclos. Esto significa que todos los bordes del lazo del fregadero de nuevo a sí mismo (y puede ser omitida) y todos los otros caminos se alimentan en el fregadero. Un estado no-sumidero se llama transitorio. El tamaño de un autómata acíclico es el número de estados transitorios. Un autómata acíclico de tamaño n por lo tanto tiene estados transitorios que etiquetamos 1, 2,........................................................................................................................................................................................... Liskovets [1] utiliza el principio de inclusión-exclusión (más sobre esto a continuación) para obtener la siguiente relación de recurrencia para el número ak(n) de autómatas acíclicos del tamaño n en un alfabeto de entrada de letra k (k ≥ 1): ak(0) = 1; ak(n) = (−1)n−j−1 (j + 1)k(n−j)ak(j), n ≥ 1. Una fuente es un vértice sin bordes entrantes. Un autómata acíclico finito tiene al menos una fuente porque un camino atravesó hacia atrás v1 ← v2 ← v3 ←. .. debe tener distinto vértices y así no pueden continuar indefinidamente. Un autómata es de una sola fuente (o inicialmente conectado) si sólo tiene una fuente. Deje que Bk(n) denote el conjunto de una fuente acíclica autómatas finitos (SAF) en un alfabeto de entrada de letra k con vértices 1, 2,...., n + 1 donde 1 es la fuente y n + 1 es el fregadero, y set bk(n) = Bk(n). La representación en dos líneas de un autómata en Bk(n) es la matriz de 2×kn cuyas columnas listan los bordes en orden. Por ejemplo, 1 1 1 2 2 2 3 3 4 4 4 5 5 2 4 6 6 6 6 6 6 3 5 3 2 2 6 está en B3(5) y las rutas de origen a fregadero en B incluyen 1 → 6, 1 → 6, 1 → 6, donde el alfabeto es {a, b, c}. Proposición 1. El número bk(n) de SAF autómata del tamaño n en un alfabeto de entrada de letra k (n, k ≥ 1) bk(n) = (−1)n−i (i+1)k(n-i)ak(i) Nota Esta fórmula es un poco más sucinta que la recurrencia en [1, Teorema 3.2]. Prueba Considere el conjuntoA de autómatas acíclicos con vértices transitorios [n] = {1, 2,..., n} en la que 1 es una fuente. Llamar 2, 3,..., n los vértices interiores. Para X [2, n], vamos f(X) = # autómatas en A cuyo conjunto de vértices interiores incluye X, g(X) = # autómata en A cuyo conjunto de vértices interiores es precisamente X. Entonces f(X) = Y:XY[2,n] g(Y) y por Möbius inversión [2] en la celosía de subconjuntos de [2, n], g(X) = Y:XY[2,n] μ(X, Y)f(Y) donde μ(X, Y) es la función Möbius para esto Enrejado. Desde μ(X, Y) = (−1)Y X si X Y, tenemos en particular que g(­) = Y[2,n] (−1) Y f(Y ). 1).......................................................................................................................................................... Dejar Y = n − i de modo que 1 ≤ i ≤ n. Cuando Y consiste enteramente de fuentes, los vértices en [n+1]\Y y sus bordes de incidente forman un subautomatón con i estados transitorios; allí son ak(i) tales. También, todos los bordes de los n − i vértices que componen Y ir directamente en [n + 1]\Y : (i + 1)k(n-i) opciones. Así f(Y) = (i + 1)k(n-i)ak(i). Por definición, g(­) es el número de autómatas en A para los cuales 1 es la única fuente, es decir, g(­) = bk(n) y la La propuesta se deriva ahora de (1). Un autómata SAF sin etiquetar es una clase de equivalencia de autómatas SAF bajo reetiquetado de los vértices interiores. Liskovets nota [1] (y demostramos a continuación) que Bk(n) no tiene automorfismos no triviales, es decir, cada uno de los (n− 1)! reetiquetados de los vértices interiores de B-Bk(n) produce un autómata diferente. Así que autómatas SAF sin etiqueta de tamaño n en un alfabeto de letra-k se cuenta por 1 (n−1)! bk(n). El siguiente resultado establece un canon representante en cada clase de reetiquetado. Proposición 2. Cada clase de equivalencia en Bk(n) bajo reetiquetado de vértices interiores tiene ¡Tamaño (n− 1)! y contiene exactamente un autómata SAF con las “últimas ocurrencias ing” propiedad: las últimas ocurrencias de los vértices interiores—2, 3,..., n—en la fila inferior de su representación de dos líneas se producen en ese orden. Prueba La primera afirmación se deriva del hecho de que los vértices interiores de un au- bk(n) se puede distinguir intrínsecamente, es decir, independientemente de su etiquetado. Para ver esto, primero marque la fuente, a saber, 1, con una marca (nueva etiqueta) v1 y observe que existe al menos un vértice interior cuyo único borde(s) entrante(s) son de la fuente (el único vértice actualmente marcado) para de lo contrario un ciclo estaría presente. Para cada uno de ellos vértice interior v, elija el último borde del vértice marcado a v utilizando el incorporado orden de estos bordes. Esto determina un orden en estos vértices; marquelos en orden v2, v3,. .., vj (j ≥ 2). Si aún quedan vértices interiores sin marcar, al menos uno de ellos tiene bordes entrantes sólo de un vértice marcado o de nuevo un ciclo estaría presente. Por cada tal vértice, utilizar el último borde entrante de un vértice marcado, donde ahora los bordes son arreglados en orden de vértice inicial vi con los lazos de ruptura incorporados en orden, a orden y marca estos vértices vj+1, vj+2,.... Proceda de manera similar hasta que todos los vértices interiores estén marcados. Por ejemplo, para 1 1 1 2 2 2 3 3 4 4 4 5 5 2 4 6 6 6 6 6 6 3 5 3 2 2 6 v1 = 1 y sólo hay un vértice interior, a saber, 4, cuyo único borde entrante es de la fuente, y así v2 = 4 y 4 se convierte en un vértice marcado. Ahora todos los bordes entrantes a 3 y 5 son de vértices marcados y los últimos tales bordes (construido en orden entra en jugar) son 4 → 5 y 4 → 3 poner vértices 3, 5 en el orden 5, 3. Así v3 = 5 y v4 = 3. Finalmente, v5 = 2. Esto demuestra la primera afirmación. Por la construcción de la vs, reetiquetando cada uno vértice interior i con el subíndice de su correspondiente v produce un autómata en Bk(n) con la propiedad “últimas ocurrencias aumentando” y es el único reetiquetado que lo hace. El ejemplo da resultados 1 1 1 2 2 2 3 3 4 4 4 5 5 5 2 6 4 3 4 5 5 6 6 6 6 6 6 Ahora deje que Ck(n) denote el conjunto de autómatas canónicos SAF en Bk(n) que representan un- etiquetada autómata; así Ck(n) = (n−1)! bk(n). De ahora en adelante, identificamos un au- tomate con su representante canónico. 3 Rutas subdiagonales marcadas por columnas Un camino subdiagonal (k, n, p) es una trayectoria de celosía de los pasos E = (1, 0) y N = (0, 1), E para Este y N para el norte, de (0, 0) a (kn, p) que nunca se elevan por encima de la línea y = 1 x. Vamos. Ck(n, p) indica el conjunto de tales rutas.Para k ≥ 1, está claro que Ck(n, p) es no vacío solamente para 0 ≤ p ≤ n y se conoce (teorema de votación generalizada) que Ck(n, p) = kn− kp+ 1 kn+ p+ 1 kn+ p + 1 Una ruta P en Ck(n, n) puede ser codificada por las alturas de sus pasos de E por encima de la línea y = −1; esto da una secuencia (bi) i=1 sujeto a las restricciones 1 ≤ b1 ≤ b2 ≤. ≤ bkn y b ≤ i/k para todos los i. Un camino subdiagonal marcado por la columna (k, n, p) es uno en el que, para cada i+ [1, kn], uno de se marcan los cuadrados de celosía por debajo del paso ith E y por encima de la línea horizontal y = −1, decir con un ‘ * ’. Let C k(n, p) denota el conjunto de estas rutas marcadas. b b b b b b b b b b b * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * (0,0) (8,4) y = −1 y = 1 Un camino en C 2, 4, 3) Una trayectoria marcada P* en C k(n, n) se puede codificar por una secuencia de pares (ai, bi) donde i=1 es el código para la ruta subyacente P y ai â € [1, bi] da la posición de la â € en la ith columna. El ejemplo está codificado por (1, 1), (1, 1), (1, 2), (2, 2), (1, 2), (3, 3), (1, 3), (2, 3). Una suma explícita para C k(n, n) es k(n, n) = 1≤b1≤b2≤...≤bkn, b ≤ i/k para todos los i b1b2. .. bkn, porque la suma b1b2. .. bkn es el número de maneras de insertar los ‘ * ’ en el subyacente ruta codificada por (bi) También es posible obtener una recurrencia para C k(n, p), y luego, usando Prop. 1, a mostrar analíticamente que C k(n, n) = Ck+1(n). Sin embargo, es mucho más agradable a dar una bijección y en la siguiente sección lo haremos. En particular, el número de FAS autómatas en un alfabeto de 2 letras es C2(n) = C 1 n, n) = 1≤b1≤b2≤...≤bn b ≤ i para todos los i b1b2. .. bn = (1, 3, 16, 127, 1363,.......................................................................................................................................... secuencia A082161 en [3]. 4 Biyección de Rutas a Automata En esta sección exhibimos una bijección de C k(n, n) a Ck+1(n). Usando la ilustración ruta como ejemplo de trabajo con k = 2 y n = 4, b b b b b b b b b b b * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * (0,0) (8,4) y = −1 y = 1 primero construir la fila superior de una representación de dos líneas que consta de k + 1 cada 1s, 2s, . ............................................................................................................................................................................................................................................................... El último paso en el camino es necesariamente un paso N. Para el segundo último, tercer último,...N pasos en el camino, cuente el número de pasos que lo siguen. Esto da una secuencia i1, i2,. ............................................................................ que cumplan 1 ≤ i1 < i2 <. .. < in−1 e ij ≤ (k + 1)j para todos j. Círculo de las posiciones i1, i2,. ............................................ la segunda fila en las posiciones en círculo: 2 3 4 Estas serán las últimas ocurrencias de 2, 3,...., n en la segunda fila. Trabajando desde el último columna en la ruta de vuelta a la primera, rellenar los espacios en blanco en la segunda fila de izquierda a derecha como sigue. Contar el número de cuadrados desde el * hasta el camino (incluyendo el * cuadrado) http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A082161 y añadir este número al número negrita más cercano a la izquierda de la entrada en blanco actual (si no hay números en negrita a la izquierda, añada este número a 1) e inserte el resultado en el cuadrado en blanco actual. En el ejemplo los números de cuadrados son 2,3,1,2,1,2,1,1 rendimiento 2 4 5 3 3 5 4 5 4 5 5 Esto llenará todas las entradas en blanco excepto la última. Tenga en cuenta que * s en la fila inferior corresponden para hundir (es decir, n+1) etiquetas en la segunda fila. Por último, insertar n+1 en el último resto espacio en blanco para dar el autómata de la imagen: 1 1 1 2 2 2 3 3 3 4 4 2 4 5 3 3 5 4 5 4 5 5 5 Este proceso es completamente reversible y el mapa es una bijección. 5 Evaluación de detAk(n) Para simplificar, tratamos el caso k = 1, dejando la generalización a arbitrario k como un ejercicio no demasiado difícil para el lector interesado. Escriba A(n) para A1(n). Por lo tanto A(n) = 1≤i,j≤n . A partir de la definición de detA(n) como una suma de productos firmados, nosotros mostrar que detA(n) es el peso total de ciertas listas de permutaciones, cada lista que lleva peso ±1. Entonces una involución que invierte el peso cancela todos los −1 pesos y reduce el problema para contar las listas de sobrevivientes. Estas listas supervivientes son esencialmente los códigos para rutas en C 1 (n, p), y el Teorema Principal sigue del § 4. Para describir las permutaciones dando una contribución no cero a detA(n) =  sgn i=1 ai,(i), definir el código de una permutación  en [n] para ser la lista c = (ci) i=1 con ci = (i)−(i−1). Desde la entrada (i, j) de A(n), , es 0 a menos que j ≥ i−1, debemos tener (i) ≥ i−1 para todos los i. Es bien sabido que hay 2n−1 tales permutaciones, correspondientes a las composiciones de n, con códigos caracterizados por las cuatro condiciones siguientes: i) ci ≥ 0 para todos los i, ii) c1 ≥ 1, iii) cada ci ≥ 1 es inmediatamente seguido de ci − 1 ceros en la lista, i=1 ci = n. Llamemos a tal lista una composición acolchada de n: borrar los ceros es una bijección a composiciones ordinarias de n. Por ejemplo, (3, 0, 0, 1, 2, 0) es un acolchado composición de 6. Para una permutación  con código de composición acolchado c, el no cero las entradas en c dan las longitudes del ciclo de . Por lo tanto sgnđ, que es la paridad de “nciclos (−1)#0s in c. Tenemos detA(n) =  sgn  i=1 ai,(i) =  sgn  2i(i) , y así detA(n) = (−1)#0s in c i+ 1− ci cuando la suma se limite a composiciones acolchadas c de n con ci ≤ i para todos i (A002083) porque i+1−ci = 0 a menos que ci ≤ i. A partir de ahora, escribamos todas las permutaciones en forma de ciclo estándar por el cual el más pequeño la entrada se produce primero en cada ciclo y estas entradas más pequeñas aumentan de izquierda a derecha. Por lo tanto, con los ciclos de separación de guiones, 154-2-36 es la forma estándar del ciclo de la permutación ( 1 2 3 4 5 65 2 6 1 4 3 ). Definimos una entrada no primera para ser una que no comienza un ciclo. Por lo tanto, la la permutación anterior tiene 3 entradas no primeras: 5,4,6. Tenga en cuenta que el número de nonfirst entradas es 0 sólo para la permutación de identidad. Denotamos una permutación de identidad (de cualquier (tamaño) por. Por definición del número de ciclo de Stirling, el producto en (2) listas de recuentos ( i=1 de permu- En los casos en que se trate de una permutación en [i+1] con ciclos i+1− ci, equivalentemente, con ci ≤ i nonfirst entrys. Definir Ln para ser el conjunto de todas las listas de permutaciones i=1 donde πi es una permutación en [i + 1], #nonfirst en cada permutación de no identidad πi es seguida inmediatamente por ci − 1 • s donde ci ≥ 1 es la número de entradas que no son las primeras (por lo que el número total de entradas que no son las primeras es n). Asignar a peso en peso a η ° Ln por wt(η) = (−1) # Está en π. Entonces detA(n) = wt(l). Ahora definimos una involución de inversión de peso en (la mayoría de) Ln. Teniendo en cuenta el número de Ln, escanee el lista de sus permutaciones de componentes η1 = (1, 2), η2, η3,. .. de izquierda a derecha. Detente al principio. una que: i) tenga más de una entrada que no sea la primera, o ii) tenga sólo una entrada que no sea la primera, b decir, y b > máximo nonfirst entrada m de la siguiente permutación en la lista. Digamos que lk es el Permutación donde nos detenemos. http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002083 En el caso i) decremento (es decir, Disminuir en 1) el número de personas que figuran en la lista dividiendo la cantidad de personas que figuran en la lista en dos permutaciones de no identidad como sigue. Deje m ser la entrada más grande nonfirst de ηk Y que yo sea su predecesor. Sustitúyase ηk y su sucesor en la lista (necesariamente un ) por las dos permutaciones siguientes: primero la transposición (l,m) y segundo la permutación obtenido de ηk borrando m de su ciclo y convirtiéndola en un singleton. Aquí están. dos ejemplos de este caso (recordar las permutaciones están en forma de ciclo estándar y, para mayor claridad, ciclos singleton no se muestran). i 1 2 3 4 5 6 12 13 23 14-253 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # i 1 2 3 4 5 6 12 13 23 25 14-23 i 1 2 3 4 5 6 12 23 14 13-24 23 i 1 2 3 4 5 6 12 23 14 24 13 23 El lector puede comprobar fácilmente que esto envía el caso (i) al caso (ii). En el caso ii), ηk es una transposición (a, b) con b > máximo nonfirst entry m de ηk+1. In en este caso, aumentar el número de los de la lista mediante la combinación de ηk y ηk+1 en un solo permutación seguida de un : en ηk+1, b es un singleton; borrar este singleton e insertar b inmediatamente después de una in ηk+1 (en el mismo ciclo). El lector puede comprobar que esto invierte el resultado en los dos ejemplos anteriores y, en general, envía el caso ii) al caso i). Desde el mapa altera el número de los de la lista por 1, que es claramente el peso-reversing. El mapa falla sólo para las listas que consisten en su totalidad de transposiciones y tienen la forma (a1, b1), (a2, b2),. ............................................................................................................................................................................................................................................................... ≤ bn. Tales listas tienen peso 1. Por lo tanto detA(n) es el número de listas (ai, bi) Satisfacción 1 ≤ ai < bi ≤ i+ 1 para 1 ≤ i ≤ n, y b1 ≤ b2 ≤. ≤ bn. Después de restar 1 de cada uno bi, estas listas codifican las rutas en C 1 (n, n) y utilizando §4, detA(n) = C 1 n, n) = C2(n). Bibliografía [1] Valery A. Liskovets, enumeración exacta Tomata, Disc. Appl. Math., en la prensa, 2006. Versión anterior disponible en http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html [2] J. H. van Lint y R. M. Wilson, A Course in Combinatorics, 2nd ed., Cambridge University Press, NY, 2001. [3] Neil J. Sloane (fundador y encargado), La Enciclopedia en Línea de Integer Se- quences http://www.research.att.com:80/njas/sequences/index.html?blank=1 http://www.research.att.com:80/~njas/sequences/index.html?blank=1
704.001
From dyadic $\Lambda_{\alpha}$ to $\Lambda_{\alpha}$
In this paper we show how to compute the $\Lambda_{\alpha}$ norm, $\alpha\ge 0$, using the dyadic grid. This result is a consequence of the description of the Hardy spaces $H^p(R^N)$ in terms of dyadic and special atoms.
FROM DYADIC Λα TO Λα WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY Abstract. In this paper we show how to compute the Λα norm , α ≥ 0, using the dyadic grid. This result is a consequence of the description of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms. Recently, several novel methods for computing the BMO norm of a function f in two dimensions were discussed in [9]. Given its importance, it is also of interest to explore the possibility of computing the norm of a BMO function, or more generally a function in the Lipschitz class Λα, using the dyadic grid in RN . It turns out that the BMO question is closely related to that of approximating functions in the Hardy space H1(RN ) by the Haar system. The approximation in H1(RN ) by affine systems was proved in [2], but this result does not apply to the Haar system. Now, if HA(R) denotes the closure of the Haar system in H1(R), it is not hard to see that the distance d(f,HA) of f ∈ H1(R) to HA is ∼ f(x) dx ∣, see [1]. Thus, neither dyadic atoms suffice to describe the Hardy spaces, nor the evaluation of the norm in BMO can be reduced to a straightforward computation using the dyadic intervals. In this paper we address both of these issues. First, we give a characterization of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms, and then, by a duality argument, we show how to compute the norm in Λα(R N ), α ≥ 0, using the dyadic grid. We begin by introducing some notations. Let J denote a family of cubes Q in RN , and Pd the collection of polynomials in R N of degree less than or equal to d. Given α ≥ 0, Q ∈ J , and a locally integrable function g, let pQ(g) denote the unique polynomial in P[α] such that [g − pQ(g)]χQ has vanishing moments up to order [α]. For a locally square-integrable function g, we consider the maximal function α,J g(x) given by α,J g(x) = sup x∈Q,Q∈J |Q|α/N |g(y)− pQ(g)(y)| 1991 Mathematics Subject Classification. 42B30,42B35. http://arxiv.org/abs/0704.0005v1 2 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY The Lipschitz space Λα,J consists of those functions g such that M α,J g is in L∞, ‖g‖Λα,J = ‖M α,J g‖∞; when the family in question contains all cubes in RN , we simply omit the subscript J . Of course, Λ0 = BMO. Two other families, of dyadic nature, are of interest to us. Intervals in R of the form In,k = [ (k−1)2 n, k2n], where k and n are arbitrary integers, positive, negative or 0, are said to be dyadic. In RN , cubes which are the product of dyadic intervals of the same length, i.e., of the form Qn,k = In,k1 ×· · ·×In,kN , are called dyadic, and the collection of all such cubes is denoted D. There is also the family D0. Let I n,k = [(k− 1)2 n, (k+ 1)2n], where k and n are arbitrary integers. Clearly I ′n,k is dyadic if k is odd, but not if k is even. Now, the collection {I ′n,k : n, k integers} contains all dyadic intervals as well as the shifts [(k − 1)2n + 2n−1, k 2n + 2n−1] of the dyadic intervals by their half length. In RN , put D0 = {Q n,k : Q n,k = I × · · · × I ′n,kN }; Q n,k is called a special cube. Note that D0 contains D properly. Finally, given I ′n,k, let I n,k = [(k − 1)2 n, k2n], and I n,k = [k2 n, (k + 1)2n]. The 2N subcubes of Q′n,k = I × · · · × I ′n,kN of the form I × · · · × I Sj = L or R, 1 ≤ j ≤ N , are called the dyadic subcubes of Q Let Q0 denote the special cube [−1, 1] N . Given α ≥ 0, we construct a family Sα of piecewise polynomial splines in L 2(Q0) that will be useful in characterizing Λα. Let A be the subspace of L 2(Q0) consisting of all functions with vanishing moments up to order [α] which coincide with a polynomial in P[α] on each of the 2 N dyadic subcubes of Q0. A is a finite dimensional subspace of L2(Q0), and, therefore, by the Graham-Schmidt orthogonalization process, say, A has an orthonormal basis in L2(Q0) consisting of functions p1, . . . , pM with vanishing moments up to order [α], which coincide with a polynomial in P[α] on each dyadic subinterval of Q0. Together with each p we also consider all dyadic dilations and integer translations given by pLn,k,α(x) = 2 n(N+α)pL(2nx1 + k1, . . . , 2 nxN + kN ) , 1 ≤ L ≤ M , and let Sα = {p n,k,α : n, k integers, 1 ≤ L ≤ M} . Our first result shows how the dyadic grid can be used to compute the norm in Λα. Theorem A. Let g be a locally square-integrable function and α ≥ 0. Then, g ∈ Λα if, and only if, g ∈ Λα,D and Aα(g) = supp∈Sα ∣〈g, p〉 ∣ < ∞. Moreover, ‖g‖Λα ∼ ‖g‖Λα,D +Aα(g) . Furthermore, it is also true, and the proof is given in Proposition 2.1 be- low, that ‖g‖Λα ∼ ‖g‖Λα,D0 . However, in this simpler formulation, the tree structure of the cubes in D has been lost. FROM DYADIC Λα TO Λα 3 The proof of Theorem A relies on a close investigation of the predual of Λα, namely, the Hardy space H p(RN ) with 0 < p = (α + N)/N ≤ 1. In the process we characterize Hp in terms of simpler subspaces: H , or dyadic Hp, and H , the space generated by the special atoms in Sα. Specifically, we Theorem B. Let 0 < p ≤ 1, and α = N(1/p− 1). We then have Hp = H where the sum is understood in the sense of quasinormed Banach spaces. The paper is organized as follows. In Section 1 we show that individual Hp atoms can be written as a superposition of dyadic and special atoms; this fact may be thought of as an extension of the one-dimensional result of Fridli concerning L∞ 1- atoms, see [5] and [1]. Then, we prove Theorem B. In Section 2 we discuss how to pass from Λα,D, and Λα,D0 , to the Lipschitz space Λα. 1. Characterization of the Hardy spaces Hp We adopt the atomic definition of the Hardy spaces Hp, 0 < p ≤ 1, see [6] and [10]. Recall that a compactly supported function a with [N(1/p− 1)] vanishing moments is an L2 p -atom with defining cube Q if supp(a) ⊆ Q, and |Q|1/p | a(x) |2dx ≤ 1 . The Hardy space Hp(RN ) = Hp consists of those distributions f that can be written as f = λjaj , where the aj ’s are H p atoms, |λj | p < ∞, and the convergence is in the sense of distributions as well as in Hp. Furthermore, ‖f‖Hp ∼ inf |λj | where the infimum is taken over all possible atomic decompositions of f . This last expression has traditionally been called the atomic Hp norm of f . Collections of atoms with special properties can be used to gain a better understanding of the Hardy spaces. Formally, let A be a non-empty subset of L2 p -atoms in the unit ball of Hp. The atomic space H spanned by A consists of those ϕ in Hp of the form λjaj , aj ∈ A , |λj | p < ∞ . It is readily seen that, endowed with the atomic norm ‖ϕ‖Hp = inf |λj | : ϕ = λj aj , aj ∈ A becomes a complete quasinormed space. Clearly, H ⊆ Hp, and, for f ∈ H , ‖f‖Hp ≤ ‖f‖Hp 4 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY Two families are of particular interest to us. When A is the collection of all L2 p -atoms whose defining cube is dyadic, the resulting space is H or dyadic Hp. Now, although ‖f‖Hp ≤ ‖f‖Hp , the two quasinorms are not equivalent on H . Indeed, for p = 1 and N = 1, the functions fn(x) = 2 n[χ[1−2−n,1](x) − χ[1,1+2−n](x)] , satisfy ‖fn‖H1 = 1, but ‖fn‖H1 ∼ |n| tends to infinity with n. Next, when Sα is the family of piecewise polynomial splines constructed above with α = N(1/p − 1), in analogy with the one-dimensional results in [4] and [1], H is referred to as the space generated by special atoms. We are now ready to describe Hp atoms as a superposition of dyadic and special atoms. Lemma 1.1. Let a be an L2 p -atom with defining cube Q, 0 < p ≤ 1, and α = N(1/p − 1). Then a can be written as a linear combination of 2N dyadic atoms ai, each supported in one of the dyadic subcubes of the smallest special cube Qn,k containing Q, and a special atom b in Sα. More precisely, a(x) = i=1 di ai(x) + L=1 cL p −n,−k,α(x), with |di| , |cL| ≤ c. Proof. Suppose first that the defining cube of a is Q0, and let Q1, . . . , Q2N denote the dyadic subcubes of Q0. Furthermore, let {e i , . . . , e i } denote an orthonormal basis of the subspace Ai of L 2(Qi) consisting of polynomials in P[α], 1 ≤ i ≤ 2 N . Put αi(x) = a(x)χQi (x)− 〈aχQi , e j(x) , 1 ≤ i ≤ 2 and observe that 〈αi, e j〉 = 0 for 1 ≤ j ≤ M . Therefore, αi has [α] vanishing moments, is supported in Qi, and ‖αi‖2 ≤ ‖aχQi‖2 + ‖aχQi‖2 ≤ (M + 1) ‖aχQi‖2 . ai(x) = 2N(1/2−1/p) M + 1 αi(x) , 1 ≤ i ≤ N , is an L2 p - dyadic atom. Finally, put b(x) = a(x) − M + 1 2N(1/2−1/p) ai(x) . FROM DYADIC Λα TO Λα 5 Clearly b has [α] vanishing moments, is supported in Q0, coincides with a polynomial in P[α] on each dyadic subcube of Q0, and ‖b‖22 ≤ |〈aχQi , e 2 ≤ M ‖a‖22 . So, b ∈ A, and, consequently, b(x) = L=1 cL p L(x), where |cL| = |〈b, p L〉| ≤ c , 1 ≤ L ≤ M . In the general case, let Q be the defining cube of a, side-length Q = ℓ, and let n and k = (k1, . . . , kN ) be chosen so that 2 n−1 ≤ ℓ < 2n, and Q ⊂ [(k1 − 1)2 n, (k1 + 1)2 n]× · · · × [(kN − 1)2 n, (kN + 1)2 Then, (1/2)N ≤ |Q|/2nN < 1. Now, given x ∈ Q0, let a ′ be the translation and dilation of a given by a′(x) = 2nN/pa(2nx1 − k1, . . . , 2 nxN − kN ) . Clearly, [α] moments of a′ vanish, and ‖a′‖2 = 2 nN/p 2−nN/2‖a‖2 ≤ c |Q| 1/p|Q|−1/2‖a‖2 ≤ c . Thus, a′ is a multiple of an atom with defining cube Q0. By the first part of the proof, a′(x) = i(x) + L(x) , x ∈ Q0 . The support of each a′i is contained in one of the dyadic subcubes of Q0, and, consequently, there is a k such that ai(x) = 2 −nN/pa′i(2 −nx1 − k1, . . . , 2 −nxN − kN ) ai is an L 2p -atom supported in one of the dyadic subcubes of Q. Similarly for the pL’s. Thus, a(x) = di ai(x) + −n,−k,N(1/p−1)(x) , and we have finished. � Theorem B follows readily from Lemma 1.1. Clearly, H →֒ Hp. Conversely, let f = j λj aj be in H p. By Lemma 1.1 each aj can be written as a sum of dyadic and special atoms, and, by distributing the sum, we can write f = fd + fs, with fd in H , fs in H , and ‖fd‖Hp , ‖fs‖Hp |λj | Taking the infimum over the decompositions of f we get ‖f‖Hp c ‖f‖Hp , and H p →֒ H . This completes the proof. 6 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY The meaning of this decomposition is the following. Cubes in D are con- tained in one of the 2N non-overlapping quadrants of RN . To allow for the information carried by a dyadic cube to be transmitted to an adjacent dyadic cube, they must be connected. The pLn,k,α’s channel information across ad- jacent dyadic cubes which would otherwise remain disconnected. The reader will have no difficulty in proving the quantitative version of this observation: Let T be a linear mapping defined on Hp, 0 < p ≤ 1, that assumes values in a quasinormed Banach space X . Then, T is continuous if, and only if, the restrictions of T to H and H are continuous. 2. Characterizations of Λα Theorem A describes how to pass from Λα,D to Λα, and we prove it next. Since (Hp)∗ = Λα and (H )∗ = Λα,D, from Theorem B it follows readily that Λα = Λα,D ∩ (H )∗, so it only remains to show that (H )∗ is characterized by the condition Aα(g) < ∞. First note that if g is a locally square-integrable function with Aα(g) < ∞ and f = j,L cj,L p nj ,kj ,α , since 0 < p ≤ 1, |〈g, f〉| ≤ |cj,L| |〈g, p nj ,kj ,α ≤ Aα(g) |cj,L| and, consequently, taking the infimum over all atomic decompositions of f in , we get g ∈ (H )∗ and ‖g‖(Hp )∗ ≤ Aα(g). To prove the converse we proceed as in [3]. Let Qn = [−2 n, 2n]N . We begin by observing that functions f in L2(Qn) that have vanishing moments up to order [α] and coincide with polynomials of degree [α] on the dyadic subcubes of Qn belong to H ‖f‖Hp ≤ |Qn| 1/p−1/2‖f‖2 . Given ℓ ∈ (H )∗, for a fixed n let us consider the restriction of ℓ to the space of L2 functions f with [α] vanishing moments that are supported in Qn. Since |ℓ(f)| ≤ ‖ℓ‖ ‖f‖Hp ≤ ‖ℓ‖ |Qn| 1/p−1/2‖f‖2 , this restriction is continuous with respect to the norm in L2 and, consequently, it can be extended to a continuous linear functional in L2 and represented as ℓ(f) = f(x) gn(x) dx , FROM DYADIC Λα TO Λα 7 where gn ∈ L 2(Qn) and satisfies ‖gn‖2 ≤ ‖ℓ‖ |Qn| 1/p−1/2. Clearly, gn is uniquely determined in Qn up to a polynomial pn in P[α]. Therefore, gn(x) − pn(x) = gm(x)− pm(x) , a.e. x ∈ Qmin(n,m) . Consequently, if g(x) = gn(x)− pn(x) , x ∈ Qn , g(x) is well defined a.e. and, if f ∈ L2 has [α] vanishing moments and is supported in Qn, we have ℓ(f) = f(x) gn(x) dx f(x) [gn(x)− pn(x)] dx f(x) g(x) dx . Moreover, since each 2nN/ppL(2n ·+k) is an L2 p-atom, 1 ≤ L ≤ M , it readily follows that Aα(g) = sup 1≤L≤M n,k∈Z |〈g, 2−n/ppL(2n ·+k)〉| ≤ ‖ℓ‖ sup ‖pL‖Hp ≤ ‖ℓ‖ , and, consequently, Aα(g) ≤ ‖ℓ‖ , and (H )∗ is the desired space. � The reader will have no difficulty in showing that this result implies the following: Let T be a bounded linear operator from a quasinormed space X into Λα,D. Then, T is bounded from X into Λα if, and only if, Aα(Tx) ≤ c ‖x‖X for every x ∈ X . The process of averaging the translates of dyadic BMO functions leads to BMO, and is an important tool in obtaining results in BMO once they are known to be true in its dyadic counterpart, BMOd, see [7]. It is also known that BMO can be obtained as the intersection of BMOd and one of its shifted counterparts, see [8]. These results motivate our next proposition, which essentially says that g ∈ Λα if, and only if, g ∈ Λα,D and g is in the Lipschitz class obtained from the shifted dyadic grid. Note that the shifts involved in this class are in all directions parallel to the coordinate axis and depend on the side-length of the cube. Proposition 2.1. Λα = Λα,D0 , and ‖g‖Λα ∼ ‖g‖Λα,D0 . Proof. It is obvious that ‖g‖Λα,D0 ≤ ‖g‖Λα . To show the other inequality we invoke Theorem A. Since D ⊂ D0, it suffices to estimate Aα(g), or, equiva- lently, |〈g, p〉| for p ∈ Sα, α = N(1/p − 1). So, pick p = p n,k,α in Sα. The defining cube Q of pLn,k,α is in D0, and, since p n,k,α has [α] vanishing moments, 8 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY 〈pLn,k,α, pQ(g)〉 = 0. Therefore, |〈g, pLn,k,α〉| = |〈g − pQ(g), p n,k,α〉| ≤ ‖pLn,k,α‖2 ‖g − pQ(g)‖L2(Q) ≤ |Q|α/N |Q|1/2‖pLn,k,α‖2 ‖g‖Λα,D0 . Now, a simple change of variables gives |Q|α/N |Q|1/2‖pLn,k,α‖2 ≤ 1, and, con- sequently, also Aα(g) ≤ ‖g‖Λα,D0 . � References [1] W. Abu-Shammala, J.-L. Shiu, and A. Torchinsky, Characterizations of the Hardy space H1 and BMO, preprint. [2] H.-Q. Bui and R. S. Laugesen, Approximation and spanning in the Hardy space, by affine systems, Constr. Approx., to appear. [3] A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distibution, II, Advances in Math., 24 (1977), 101–171. [4] G. S. de Souza, Spaces formed by special atoms, I, Rocky Mountain J. Math. 14 (1984), no. 2, 423–431. [5] S. Fridli, Transition from the dyadic to the real nonperiodic Hardy space, Acta Math. Acad. Paedagog. Niházi (N.S.) 16 (2000), 1–8, (electronic). [6] J. Garćıa-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, Notas de Matemática 116, North Holland, Amsterdam, 1985. [7] J. Garnett and P. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. [8] T. Mei, BMO is the intersection of two translates of dyadic BMO, C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1003–1006. [9] T. M. Le and L. A. Vese, Image decomposition using total variation and div( BMO)∗, Multiscale Model. Simul. 4, (2005), no. 2, 390–423. [10] A. Torchinsky, Real-variable methods in harmonic analysis, Dover Publications, Inc., Mineola, NY, 2004. Department of Mathematics, Indiana University, Bloomington IN 47405 E-mail address: wabusham@indiana.edu Department of Mathematics, Indiana University, Bloomington IN 47405 E-mail address: torchins@indiana.edu 1. Characterization of the Hardy spaces Hp 2. Characterizations of References
FROM DYADIC Λα TO Λα WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY Abstract. In this paper we show how to compute the Λα norm , α ≥ 0, using the dyadic grid. This result is a consequence of the description of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms. Recently, several novel methods for computing the BMO norm of a function f in two dimensions were discussed in [9]. Given its importance, it is also of interest to explore the possibility of computing the norm of a BMO function, or more generally a function in the Lipschitz class Λα, using the dyadic grid in RN . It turns out that the BMO question is closely related to that of approximating functions in the Hardy space H1(RN ) by the Haar system. The approximation in H1(RN ) by affine systems was proved in [2], but this result does not apply to the Haar system. Now, if HA(R) denotes the closure of the Haar system in H1(R), it is not hard to see that the distance d(f,HA) of f ∈ H1(R) to HA is ∼ f(x) dx ∣, see [1]. Thus, neither dyadic atoms suffice to describe the Hardy spaces, nor the evaluation of the norm in BMO can be reduced to a straightforward computation using the dyadic intervals. In this paper we address both of these issues. First, we give a characterization of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms, and then, by a duality argument, we show how to compute the norm in Λα(R N ), α ≥ 0, using the dyadic grid. We begin by introducing some notations. Let J denote a family of cubes Q in RN , and Pd the collection of polynomials in R N of degree less than or equal to d. Given α ≥ 0, Q ∈ J , and a locally integrable function g, let pQ(g) denote the unique polynomial in P[α] such that [g − pQ(g)]χQ has vanishing moments up to order [α]. For a locally square-integrable function g, we consider the maximal function α,J g(x) given by α,J g(x) = sup x∈Q,Q∈J |Q|α/N |g(y)− pQ(g)(y)| 1991 Mathematics Subject Classification. 42B30,42B35. http://arxiv.org/abs/0704.0005v1 2 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY The Lipschitz space Λα,J consists of those functions g such that M α,J g is in L∞, ‖g‖Λα,J = ‖M α,J g‖∞; when the family in question contains all cubes in RN , we simply omit the subscript J . Of course, Λ0 = BMO. Two other families, of dyadic nature, are of interest to us. Intervals in R of the form In,k = [ (k−1)2 n, k2n], where k and n are arbitrary integers, positive, negative or 0, are said to be dyadic. In RN , cubes which are the product of dyadic intervals of the same length, i.e., of the form Qn,k = In,k1 ×· · ·×In,kN , are called dyadic, and the collection of all such cubes is denoted D. There is also the family D0. Let I n,k = [(k− 1)2 n, (k+ 1)2n], where k and n are arbitrary integers. Clearly I ′n,k is dyadic if k is odd, but not if k is even. Now, the collection {I ′n,k : n, k integers} contains all dyadic intervals as well as the shifts [(k − 1)2n + 2n−1, k 2n + 2n−1] of the dyadic intervals by their half length. In RN , put D0 = {Q n,k : Q n,k = I × · · · × I ′n,kN }; Q n,k is called a special cube. Note that D0 contains D properly. Finally, given I ′n,k, let I n,k = [(k − 1)2 n, k2n], and I n,k = [k2 n, (k + 1)2n]. The 2N subcubes of Q′n,k = I × · · · × I ′n,kN of the form I × · · · × I Sj = L or R, 1 ≤ j ≤ N , are called the dyadic subcubes of Q Let Q0 denote the special cube [−1, 1] N . Given α ≥ 0, we construct a family Sα of piecewise polynomial splines in L 2(Q0) that will be useful in characterizing Λα. Let A be the subspace of L 2(Q0) consisting of all functions with vanishing moments up to order [α] which coincide with a polynomial in P[α] on each of the 2 N dyadic subcubes of Q0. A is a finite dimensional subspace of L2(Q0), and, therefore, by the Graham-Schmidt orthogonalization process, say, A has an orthonormal basis in L2(Q0) consisting of functions p1, . . . , pM with vanishing moments up to order [α], which coincide with a polynomial in P[α] on each dyadic subinterval of Q0. Together with each p we also consider all dyadic dilations and integer translations given by pLn,k,α(x) = 2 n(N+α)pL(2nx1 + k1, . . . , 2 nxN + kN ) , 1 ≤ L ≤ M , and let Sα = {p n,k,α : n, k integers, 1 ≤ L ≤ M} . Our first result shows how the dyadic grid can be used to compute the norm in Λα. Theorem A. Let g be a locally square-integrable function and α ≥ 0. Then, g ∈ Λα if, and only if, g ∈ Λα,D and Aα(g) = supp∈Sα ∣〈g, p〉 ∣ < ∞. Moreover, ‖g‖Λα ∼ ‖g‖Λα,D +Aα(g) . Furthermore, it is also true, and the proof is given in Proposition 2.1 be- low, that ‖g‖Λα ∼ ‖g‖Λα,D0 . However, in this simpler formulation, the tree structure of the cubes in D has been lost. FROM DYADIC Λα TO Λα 3 The proof of Theorem A relies on a close investigation of the predual of Λα, namely, the Hardy space H p(RN ) with 0 < p = (α + N)/N ≤ 1. In the process we characterize Hp in terms of simpler subspaces: H , or dyadic Hp, and H , the space generated by the special atoms in Sα. Specifically, we Theorem B. Let 0 < p ≤ 1, and α = N(1/p− 1). We then have Hp = H where the sum is understood in the sense of quasinormed Banach spaces. The paper is organized as follows. In Section 1 we show that individual Hp atoms can be written as a superposition of dyadic and special atoms; this fact may be thought of as an extension of the one-dimensional result of Fridli concerning L∞ 1- atoms, see [5] and [1]. Then, we prove Theorem B. In Section 2 we discuss how to pass from Λα,D, and Λα,D0 , to the Lipschitz space Λα. 1. Characterization of the Hardy spaces Hp We adopt the atomic definition of the Hardy spaces Hp, 0 < p ≤ 1, see [6] and [10]. Recall that a compactly supported function a with [N(1/p− 1)] vanishing moments is an L2 p -atom with defining cube Q if supp(a) ⊆ Q, and |Q|1/p | a(x) |2dx ≤ 1 . The Hardy space Hp(RN ) = Hp consists of those distributions f that can be written as f = λjaj , where the aj ’s are H p atoms, |λj | p < ∞, and the convergence is in the sense of distributions as well as in Hp. Furthermore, ‖f‖Hp ∼ inf |λj | where the infimum is taken over all possible atomic decompositions of f . This last expression has traditionally been called the atomic Hp norm of f . Collections of atoms with special properties can be used to gain a better understanding of the Hardy spaces. Formally, let A be a non-empty subset of L2 p -atoms in the unit ball of Hp. The atomic space H spanned by A consists of those ϕ in Hp of the form λjaj , aj ∈ A , |λj | p < ∞ . It is readily seen that, endowed with the atomic norm ‖ϕ‖Hp = inf |λj | : ϕ = λj aj , aj ∈ A becomes a complete quasinormed space. Clearly, H ⊆ Hp, and, for f ∈ H , ‖f‖Hp ≤ ‖f‖Hp 4 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY Two families are of particular interest to us. When A is the collection of all L2 p -atoms whose defining cube is dyadic, the resulting space is H or dyadic Hp. Now, although ‖f‖Hp ≤ ‖f‖Hp , the two quasinorms are not equivalent on H . Indeed, for p = 1 and N = 1, the functions fn(x) = 2 n[χ[1−2−n,1](x) − χ[1,1+2−n](x)] , satisfy ‖fn‖H1 = 1, but ‖fn‖H1 ∼ |n| tends to infinity with n. Next, when Sα is the family of piecewise polynomial splines constructed above with α = N(1/p − 1), in analogy with the one-dimensional results in [4] and [1], H is referred to as the space generated by special atoms. We are now ready to describe Hp atoms as a superposition of dyadic and special atoms. Lemma 1.1. Let a be an L2 p -atom with defining cube Q, 0 < p ≤ 1, and α = N(1/p − 1). Then a can be written as a linear combination of 2N dyadic atoms ai, each supported in one of the dyadic subcubes of the smallest special cube Qn,k containing Q, and a special atom b in Sα. More precisely, a(x) = i=1 di ai(x) + L=1 cL p −n,−k,α(x), with |di| , |cL| ≤ c. Proof. Suppose first that the defining cube of a is Q0, and let Q1, . . . , Q2N denote the dyadic subcubes of Q0. Furthermore, let {e i , . . . , e i } denote an orthonormal basis of the subspace Ai of L 2(Qi) consisting of polynomials in P[α], 1 ≤ i ≤ 2 N . Put αi(x) = a(x)χQi (x)− 〈aχQi , e j(x) , 1 ≤ i ≤ 2 and observe that 〈αi, e j〉 = 0 for 1 ≤ j ≤ M . Therefore, αi has [α] vanishing moments, is supported in Qi, and ‖αi‖2 ≤ ‖aχQi‖2 + ‖aχQi‖2 ≤ (M + 1) ‖aχQi‖2 . ai(x) = 2N(1/2−1/p) M + 1 αi(x) , 1 ≤ i ≤ N , is an L2 p - dyadic atom. Finally, put b(x) = a(x) − M + 1 2N(1/2−1/p) ai(x) . FROM DYADIC Λα TO Λα 5 Clearly b has [α] vanishing moments, is supported in Q0, coincides with a polynomial in P[α] on each dyadic subcube of Q0, and ‖b‖22 ≤ |〈aχQi , e 2 ≤ M ‖a‖22 . So, b ∈ A, and, consequently, b(x) = L=1 cL p L(x), where |cL| = |〈b, p L〉| ≤ c , 1 ≤ L ≤ M . In the general case, let Q be the defining cube of a, side-length Q = ℓ, and let n and k = (k1, . . . , kN ) be chosen so that 2 n−1 ≤ ℓ < 2n, and Q ⊂ [(k1 − 1)2 n, (k1 + 1)2 n]× · · · × [(kN − 1)2 n, (kN + 1)2 Then, (1/2)N ≤ |Q|/2nN < 1. Now, given x ∈ Q0, let a ′ be the translation and dilation of a given by a′(x) = 2nN/pa(2nx1 − k1, . . . , 2 nxN − kN ) . Clearly, [α] moments of a′ vanish, and ‖a′‖2 = 2 nN/p 2−nN/2‖a‖2 ≤ c |Q| 1/p|Q|−1/2‖a‖2 ≤ c . Thus, a′ is a multiple of an atom with defining cube Q0. By the first part of the proof, a′(x) = i(x) + L(x) , x ∈ Q0 . The support of each a′i is contained in one of the dyadic subcubes of Q0, and, consequently, there is a k such that ai(x) = 2 −nN/pa′i(2 −nx1 − k1, . . . , 2 −nxN − kN ) ai is an L 2p -atom supported in one of the dyadic subcubes of Q. Similarly for the pL’s. Thus, a(x) = di ai(x) + −n,−k,N(1/p−1)(x) , and we have finished. � Theorem B follows readily from Lemma 1.1. Clearly, H →֒ Hp. Conversely, let f = j λj aj be in H p. By Lemma 1.1 each aj can be written as a sum of dyadic and special atoms, and, by distributing the sum, we can write f = fd + fs, with fd in H , fs in H , and ‖fd‖Hp , ‖fs‖Hp |λj | Taking the infimum over the decompositions of f we get ‖f‖Hp c ‖f‖Hp , and H p →֒ H . This completes the proof. 6 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY The meaning of this decomposition is the following. Cubes in D are con- tained in one of the 2N non-overlapping quadrants of RN . To allow for the information carried by a dyadic cube to be transmitted to an adjacent dyadic cube, they must be connected. The pLn,k,α’s channel information across ad- jacent dyadic cubes which would otherwise remain disconnected. The reader will have no difficulty in proving the quantitative version of this observation: Let T be a linear mapping defined on Hp, 0 < p ≤ 1, that assumes values in a quasinormed Banach space X . Then, T is continuous if, and only if, the restrictions of T to H and H are continuous. 2. Characterizations of Λα Theorem A describes how to pass from Λα,D to Λα, and we prove it next. Since (Hp)∗ = Λα and (H )∗ = Λα,D, from Theorem B it follows readily that Λα = Λα,D ∩ (H )∗, so it only remains to show that (H )∗ is characterized by the condition Aα(g) < ∞. First note that if g is a locally square-integrable function with Aα(g) < ∞ and f = j,L cj,L p nj ,kj ,α , since 0 < p ≤ 1, |〈g, f〉| ≤ |cj,L| |〈g, p nj ,kj ,α ≤ Aα(g) |cj,L| and, consequently, taking the infimum over all atomic decompositions of f in , we get g ∈ (H )∗ and ‖g‖(Hp )∗ ≤ Aα(g). To prove the converse we proceed as in [3]. Let Qn = [−2 n, 2n]N . We begin by observing that functions f in L2(Qn) that have vanishing moments up to order [α] and coincide with polynomials of degree [α] on the dyadic subcubes of Qn belong to H ‖f‖Hp ≤ |Qn| 1/p−1/2‖f‖2 . Given ℓ ∈ (H )∗, for a fixed n let us consider the restriction of ℓ to the space of L2 functions f with [α] vanishing moments that are supported in Qn. Since |ℓ(f)| ≤ ‖ℓ‖ ‖f‖Hp ≤ ‖ℓ‖ |Qn| 1/p−1/2‖f‖2 , this restriction is continuous with respect to the norm in L2 and, consequently, it can be extended to a continuous linear functional in L2 and represented as ℓ(f) = f(x) gn(x) dx , FROM DYADIC Λα TO Λα 7 where gn ∈ L 2(Qn) and satisfies ‖gn‖2 ≤ ‖ℓ‖ |Qn| 1/p−1/2. Clearly, gn is uniquely determined in Qn up to a polynomial pn in P[α]. Therefore, gn(x) − pn(x) = gm(x)− pm(x) , a.e. x ∈ Qmin(n,m) . Consequently, if g(x) = gn(x)− pn(x) , x ∈ Qn , g(x) is well defined a.e. and, if f ∈ L2 has [α] vanishing moments and is supported in Qn, we have ℓ(f) = f(x) gn(x) dx f(x) [gn(x)− pn(x)] dx f(x) g(x) dx . Moreover, since each 2nN/ppL(2n ·+k) is an L2 p-atom, 1 ≤ L ≤ M , it readily follows that Aα(g) = sup 1≤L≤M n,k∈Z |〈g, 2−n/ppL(2n ·+k)〉| ≤ ‖ℓ‖ sup ‖pL‖Hp ≤ ‖ℓ‖ , and, consequently, Aα(g) ≤ ‖ℓ‖ , and (H )∗ is the desired space. � The reader will have no difficulty in showing that this result implies the following: Let T be a bounded linear operator from a quasinormed space X into Λα,D. Then, T is bounded from X into Λα if, and only if, Aα(Tx) ≤ c ‖x‖X for every x ∈ X . The process of averaging the translates of dyadic BMO functions leads to BMO, and is an important tool in obtaining results in BMO once they are known to be true in its dyadic counterpart, BMOd, see [7]. It is also known that BMO can be obtained as the intersection of BMOd and one of its shifted counterparts, see [8]. These results motivate our next proposition, which essentially says that g ∈ Λα if, and only if, g ∈ Λα,D and g is in the Lipschitz class obtained from the shifted dyadic grid. Note that the shifts involved in this class are in all directions parallel to the coordinate axis and depend on the side-length of the cube. Proposition 2.1. Λα = Λα,D0 , and ‖g‖Λα ∼ ‖g‖Λα,D0 . Proof. It is obvious that ‖g‖Λα,D0 ≤ ‖g‖Λα . To show the other inequality we invoke Theorem A. Since D ⊂ D0, it suffices to estimate Aα(g), or, equiva- lently, |〈g, p〉| for p ∈ Sα, α = N(1/p − 1). So, pick p = p n,k,α in Sα. The defining cube Q of pLn,k,α is in D0, and, since p n,k,α has [α] vanishing moments, 8 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY 〈pLn,k,α, pQ(g)〉 = 0. Therefore, |〈g, pLn,k,α〉| = |〈g − pQ(g), p n,k,α〉| ≤ ‖pLn,k,α‖2 ‖g − pQ(g)‖L2(Q) ≤ |Q|α/N |Q|1/2‖pLn,k,α‖2 ‖g‖Λα,D0 . Now, a simple change of variables gives |Q|α/N |Q|1/2‖pLn,k,α‖2 ≤ 1, and, con- sequently, also Aα(g) ≤ ‖g‖Λα,D0 . � References [1] W. Abu-Shammala, J.-L. Shiu, and A. Torchinsky, Characterizations of the Hardy space H1 and BMO, preprint. [2] H.-Q. Bui and R. S. Laugesen, Approximation and spanning in the Hardy space, by affine systems, Constr. Approx., to appear. [3] A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distibution, II, Advances in Math., 24 (1977), 101–171. [4] G. S. de Souza, Spaces formed by special atoms, I, Rocky Mountain J. Math. 14 (1984), no. 2, 423–431. [5] S. Fridli, Transition from the dyadic to the real nonperiodic Hardy space, Acta Math. Acad. Paedagog. Niházi (N.S.) 16 (2000), 1–8, (electronic). [6] J. Garćıa-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, Notas de Matemática 116, North Holland, Amsterdam, 1985. [7] J. Garnett and P. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. [8] T. Mei, BMO is the intersection of two translates of dyadic BMO, C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1003–1006. [9] T. M. Le and L. A. Vese, Image decomposition using total variation and div( BMO)∗, Multiscale Model. Simul. 4, (2005), no. 2, 390–423. [10] A. Torchinsky, Real-variable methods in harmonic analysis, Dover Publications, Inc., Mineola, NY, 2004. Department of Mathematics, Indiana University, Bloomington IN 47405 E-mail address: wabusham@indiana.edu Department of Mathematics, Indiana University, Bloomington IN 47405 E-mail address: torchins@indiana.edu 1. Characterization of the Hardy spaces Hp 2. Characterizations of References
DE DÍA A DÍA WAEL ABU-SHAMMALA Y ALBERTO TORCHINSKY Resumen. En este artículo mostramos cómo calcular la norma â € ¢, α ≥ 0, usando la cuadrícula dyádica. Este resultado es una consecuencia de la descripción de el Hardy espacios Hp(RN) en términos de átomos dyadic y especiales. Recientemente, varios métodos novedosos para calcular la norma BMO de una función f en dos dimensiones fueron discutidos en [9]. Dada su importancia, es también de interés por explorar la posibilidad de calcular la norma de una función de OMG, o más generalmente una función en la clase Lipschitz, usando la cuadrícula dyádica en RN. Resulta que la cuestión de los OMG está estrechamente relacionada con la de los OMG. funciones de aproximación en el espacio Hardy H1(RN) por el sistema Haar. La aproximación en H1(RN ) por los sistemas afín se demostró en [2], pero este el resultado no se aplica al sistema Haar. Ahora, si HA(R) denota el cierre del sistema Haar en H1(R), no es difícil ver que la distancia d(f,HA) de f-H1(R) a HA f(x) dx •, véase [1]. Por lo tanto, ni los átomos dyádicos suficiente para describir los espacios Hardy, ni la evaluación de la norma en BMO puede reducirse a un cálculo sencillo utilizando los intervalos dyádicos. En este documento abordamos ambas cuestiones. Primero, damos una caracterización de los espacios Hardy Hp(RN ) en términos de átomos dyadic y especiales, y luego, por un argumento de dualidad, mostramos cómo calcular la norma en â € (R N ), α ≥ 0, usando la cuadrícula dyádica. Comenzamos por introducir algunas anotaciones. Deja que J denote una familia de cubos Q en RN, y Pd la colección de polinomios en R N de grado inferior o igual a igual a d. Dado α ≥ 0, Q â € J, y una función localmente integrable g, dejar pQ(g) denotar el polinomio único en P[α] de tal manera que [g − pQ(g)]χQ ha desaparecido momentos hasta el orden [α]. Para una función localmente integrable cuadrado g, consideramos la función máxima α,J g(x) dado por α,J g(x) = sup X-Q-Q-J. Q/N g(y)− pQ(g(y) 1991 Clasificación del sujeto de las matemáticas. 42B30,42B35. http://arxiv.org/abs/0704.0005v1 2 WAEL ABU-SHAMMALA Y ALBERTO TORCHINSKY El espacio Lipschitz,J consiste en esas funciones g tal que M α,J g es en L­, g,J = M α,J g; cuando la familia en cuestión contiene todos los cubos en RN, simplemente omitimos el subíndice J. Por supuesto, 0 = BMO. Otras dos familias, de naturaleza dyádica, son de interés para nosotros. Intervalos en R de la forma In,k = [ (k−1)2 n, k2n], donde k y n son enteros arbitrarios, positivos, negativo o 0, se dice que es dyádico. En RN, cubos que son el producto de intervalos dyádicos de la misma longitud, es decir, de la forma Qn,k = In,k1 · In,kN, se llaman dyádicos, y la colección de todos estos cubos se denota D. También está la familia D0. Deja que yo... n,k = [(k− 1)2 n, (k+ 1)2n], donde k y n son enteros arbitrarios. Claramente ′n,k es dyadic si k es impar, pero no si k es par. Ahora, la colección {I ′n,k : n, k enteros} contiene todos los intervalos dyádicos también como los cambios [(k − 1)2n + 2n−1, k 2n + 2n−1] de los intervalos dyádicos por su La mitad de largo. En RN, poner D0 = {Q n,k : Q n,k = I × · · × I ′n,kN }; Q n,k es llamado cubo especial. Tenga en cuenta que D0 contiene D correctamente. Por último, dado I ′n,k, dejar que yo n,k = [(k − 1)2 n, k2n], y I n,k = [k2 n, (k + 1)2n]. Los subcubos 2N de Q′n,k = I × · · · × I ′n,kN del formulario I × · · · × I Sj = L o R, 1 ≤ j ≤ N, se llaman subcubes dyádicos de Q Que Q0 denote el cubo especial [−1, 1] N. Dado α ≥ 0, construimos un familia Sα de splines polinomios a trozos en L 2 (Q0) que será útil en caracterizando a â € â € TM. Dejar A ser el subespacio de L 2-Q0) que consiste en todas las funciones con momentos de desaparición hasta el orden [α] que coinciden con un polinomio en P[α] sobre cada uno de los 2 N subcubes dyádicos de Q0. A es una dimensión finita subespacio de L2(Q0), y, por lo tanto, por la ortogonalización Graham-Schmidt proceso, digamos, A tiene una base ortonormal en L2(Q0) que consiste en funciones p1,. ..., pM con momentos de desaparición hasta el orden [α], que coinciden con un polinomio en P[α] en cada subintervalo diádico de Q0. Junto con cada p también consideramos todas las dilaciones dyádicas y traducciones enteras dadas por pLn,k,α(x) = 2 n(N)pL(2nx1 + k1,. 2........................................................... nxN + kN ), 1 ≤ L ≤ M, y dejar que Sα = {p n,k,α : n, k enteros, 1 ≤ L ≤ M}. Nuestro primer resultado muestra cómo se puede utilizar la cuadrícula dyadic para calcular la norma en la letra a). Teorema A. Dejar g ser una función localmente integrable cuadrado y α ≥ 0. Entonces, g + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + g, p # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # Además, D + Aα(g). Además, también es cierto, y la prueba se da en la Proposición 2.1 ser- bajo, que â â € â € â € â € â € â € â € â â € â € â â € â â â â â â â â â â â â â â â â â â â â â, D0. Sin embargo, en esta formulación más simple, el árbol la estructura de los cubos en D se ha perdido. DE DÍA A DÍA 3 La prueba de Teorema A se basa en una investigación a fondo de la predual de â € ¢, a saber, el espacio Hardy H p(RN) con 0 < p = (α + N)/N ≤ 1. En el proceso que caracterizamos Hp en términos de subespacios más simples: H , o Hp dyádico, y H , el espacio generado por los átomos especiales en Sα. Específicamente, nosotros Teorema B. Let 0 < p ≤ 1, y α = N(1/p− 1). Entonces tenemos Hp = H donde la suma se entiende en el sentido de espacios Banach cuasinormed. El documento se organiza de la siguiente manera. En la Sección 1 mostramos que el individuo Los átomos de Hp pueden ser escritos como una superposición de átomos dyádicos y especiales; este hecho puede ser considerado como una extensión del resultado unidimensional de Fridli relativo a los átomos L- 1, véase [5] y [1]. Entonces, probamos Teorema B. En la sección 2 se discute cómo pasar de â € € TM, D, y â €, D0, a la Lipschitz espacio. 1. Caracterización de los espacios Hardy Hp Adoptamos la definición atómica de los espacios Hardy Hp, 0 < p ≤ 1, ver [6] y [10]. Recuerde que una función de soporte compacto a con [N(1/p− 1)] momentos de desaparición es un L2 p -átomo con el cubo definitorio Q si supp(a) Q, y Q1/p a(x) 2dx ≤ 1. El espacio Hardy Hp(RN) = Hp consiste en las distribuciones f que pueden ser escrito como f = ♥jaj, donde los aj’s son H p átomos, j p < فارسى, y la convergencia es en el sentido de distribuciones, así como en Hp. Además, # FHp # # Inf # j donde el infimum es tomado sobre todas las posibles descomposiciones atómicas de f. última expresión se ha llamado tradicionalmente la norma atómica Hp de f. Las colecciones de átomos con propiedades especiales se pueden utilizar para obtener un mejor comprensión de los espacios Hardy. Formalmente, dejar A ser un subconjunto no vacío de L2 p -átomos en la bola de unidad de Hp. El espacio atómico H Ampliado por A consiste en los ♥ en Hp de la forma *Jaj, aj* *A* j p < فارسى. Se ve fácilmente que, dotado de la norma atómica Hp = inf j • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, se convierte en un espacio cuasinombrado completo. Claramente, H Hp, y, para f • H , + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 4 WAEL ABU-SHAMMALA Y ALBERTO TORCHINSKY Dos familias son de especial interés para nosotros. Cuando A es la colección de todos los L2 p -átomos cuyo cubo definidor es dyádico, el espacio resultante es H o Hp dyádico. Ahora, aunque "f"Hp ≤ "f"Hp , las dos cuasinormas no son equivalente en H . De hecho, para p = 1 y N = 1, las funciones fn(x) = 2 n[χ[1−2−n,1](x) − χ[1,1+2−n](x)], satisfacer «fn»H1 = 1, pero «fn»H1 n tiende a la infinidad con n. A continuación, cuando Sα es la familia de splines polinomios a trozos construidos arriba con α = N(1/p − 1), en analogía con los resultados unidimensionales en [4] y [1], H se conoce como el espacio generado por átomos especiales. Ahora estamos listos para describir los átomos de Hp como una superposición de dyádico y átomos especiales. Lemma 1.1. Dejar ser un L2 p -átomo con el cubo definitorio Q, 0 < p ≤ 1, y α = N(1/p − 1). A continuación, una se puede escribir como una combinación lineal de 2N átomos dyádicos ai, cada uno apoyado en uno de los subcubes dyádicos de los más pequeños cubo especial Qn,k que contiene Q, y un átomo especial b en Sα. Más precisamente, a(x) = i=1 di ai(x) + L=1 cL p −n,−k,α(x), con di, cL ≤ c. Prueba. Supongamos primero que el cubo definitorio de a es Q0, y dejar Q1,. .., Q2N denotan los subcubos dyádicos de Q0. Además, {e) i,. .., e i } denotar un base ortonormal del Ai subespacial de L 2-Qi) compuesto de polinomios en P[α], 1 ≤ i ≤ 2 N. Pon αi(x) = a(x)χQi (x)− * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * j(x), 1 ≤ i ≤ 2 y observar que i, e i = 0 para 1 ≤ j ≤ M. Por lo tanto, αi ha desaparecido [α] momentos, se apoya en Qi, y 2 ≤ 1 × 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + ≤ (M + 1) ≤ (M + 1) ≤ (Qi+2) ≤ (M + 1) ≤ (M + 1) ≤ (M + 1) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ≤ (M) ai(x) = 2N(1/2−1/p) M + 1 αi(x), 1 ≤ i ≤ N, es un átomo L2 p - dyádico. Por último, poner b(x) = a(x) − M + 1 2N(1/2−1/p) ai(x). DE DÍA A 5 Claramente b tiene [α] momentos de desaparición, se apoya en Q0, coincide con un polinomio en P[α] en cada subcubo diádico de Q0, y â € TM bâ € 22 ≤ aχQi, e 2 ≤ M â € a € 22. Por lo tanto, b A, y, en consecuencia, b (x) = L=1 cL p L(x), donde cL = b, p L ≤ c, 1 ≤ L ≤ M. En el caso general, que Q sea el cubo definitorio de a, la longitud lateral Q = l, y dejar n y k = (k1,. .., kN ) ser elegido de modo que 2 n−1 ≤ l < 2n, y Q â € [(k1 − 1)2 n, (k1 + 1)2 n]× ·· · × [(kN − 1)2 n, (kN + 1)2 Entonces, (1/2)N ≤ Q/2nN < 1. Ahora, dado x â € ¢ Q0, dejar un ′ ser la traducción y la dilatación de un dado por a′(x) = 2nN/pa(2nx1 − k1,. 2........................................................... nxN − kN ). Claramente, [α] los momentos de un ′ desaparecen, y 2 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 nN/p 2−nN/2+a+2 ≤ c Q 1/pQ1/2â > 2 ≤ c. Por lo tanto, a′ es un múltiplo de un átomo con el cubo que define Q0. Por la primera parte de la prueba, a′(x) = i(x) + L(x), x(+) Q0. El soporte de cada a′i está contenido en uno de los subcubos dyadic de Q0, y, En consecuencia, hay una k tal que ai(x) = 2 −nN/pa′i(2 − nx1 − k1,. 2........................................................... − nxN − kN ) ai es una L 2p -átomo apoyado en uno de los subcubos dyadic de Q. Del mismo modo para los pL. Por lo tanto, a(x) = di ai(x) + − n,− k,N(1/p−1)(x), y hemos terminado. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. Teorema B sigue fácilmente de Lemma 1.1. Claramente, H Hp. Por el contrario, dejar f = j j aj ser en H p. Por Lemma 1.1 cada aj se puede escribir como una suma de átomos dyádicos y especiales, y, al distribuir la suma, podemos escribir f = fd + fs, con fd en H , fs en H , y â € € TM TM fdâ € TM Hp , â € ¢fsâ € ¢Hp j Tomando el infimum sobre las descomposiciones de f obtenemos â € â € â € TM TM Hp c + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + p H . Esto completa la prueba. 6 WAEL ABU-SHAMMALA Y ALBERTO TORCHINSKY El significado de esta descomposición es el siguiente. Los cubos en D son con- contenido en uno de los cuadrantes 2N no superpuestos de RN. Para permitir la información transportada por un cubo dyádico para ser transmitida a un dyádico adyacente cubo, deben estar conectados. El pLn,k,α canal de información a través de anuncios cubos dyádicos jacent que de otro modo permanecerían desconectados. El lector no tendrá dificultad alguna para demostrar la versión cuantitativa de esta observación: Que T sea una asignación lineal definida en Hp, 0 < p ≤ 1, que asume valores en un espacio de Banach cuasinombrado X. Entonces, T es continua si, y sólo si, la restricciones de T a H y H son continuas. 2. Caracterizaciones de Teorema A describe cómo pasar de â € ¢, D a â € TM, y lo probamos a continuación. Desde (Hp)* = y (H) )* =,D, del Teorema B se sigue fácilmente que # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # )*, por lo que sólo queda por demostrar que (H )* se caracteriza por por la condición Aα(g) < فارسى. Primera nota que si g es una función localmente integrable cuadrado con Aα(g) < y f = j,L cj,L p nj,kj,α , desde 0 < p ≤ 1, g, f ≤ cj,L g, p nj,kj,α ≤ Aα(g) cj,L y, en consecuencia, tomar el ínfimo sobre todas las descomposiciones atómicas de f en , obtenemos g # (H )* (Hp)* (Hp)****************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************** )* ≤ Aα(g). Para probar lo contrario procedemos como en [3]. Que Qn = [−2 N, 2n]N. Comenzamos observando que las funciones f en L2(Qn) que han desaparecido momentos hasta orden [α] y coinciden con polinomios de grado [α] en los subcubos dyadic de Qn pertenecen a H â € â € TM € TM TM TM TM Hp ≤ Qn 1/p-1/2°f+2. Teniendo en cuenta la letra h) del apartado 1 del artículo 4 del Reglamento (CEE) n° 1408/71 del Consejo, de 17 de diciembre de 1971, por el que se establece la organización común de mercados en el sector de la leche y de los productos lácteos y por el que se deroga el Reglamento (CEE) n° 1408/71 del Consejo, de 17 de diciembre de 1971, por el que se establecen disposiciones de aplicación del Reglamento (CEE) n° 1408/71 del Consejo, se deroga el Reglamento (CEE) n° 1408/71 del Consejo, por el que se establecen disposiciones de aplicación del Reglamento (CEE) n° 1408/71 del Consejo, por el que se establecen disposiciones de aplicación del Reglamento (CEE) n° 1408/71 del Consejo, por el que se establece la organización común de mercados en el sector de la leche y de los productos lácteos )*, para un n fijo consideremos la restricción de l al espacio funciones de L2 f con [α] momentos de desaparición que se soportan en Qn. Desde l(f) ≤ â â € â € â € € TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM TM ≤ Qn 1/p-1/2°f+2, esta restricción es continua con respecto a la norma en L2 y, en consecuencia, se puede extender a una función lineal continua en L2 y se representa como l(f) = f(x) gn(x) dx, DE DÍA A DÍA 7 donde gn â € L 2(Qn) y satisface las condiciones siguientes: 1/p−1/2. Claramente, gn es determinado exclusivamente en Qn hasta un pn polinomio en P[α]. Por lo tanto, gn(x) − pn(x) = gm(x)− pm(x), a.e. x Qmin(n,m). En consecuencia, si g(x) = gn(x)− pn(x), x • Qn, g(x) está bien definido a.e. y, si f L2 tiene [α] momentos de desaparición y es apoyado en Qn, tenemos l(f) = f(x) gn(x) dx f(x) [gn(x)− pn(x)] dx f(x) g(x) dx. Además, dado que cada 2nN/ppL(2n k) es un L2 p-átomo, 1 ≤ L ≤ M, fácilmente De ello se desprende que: Aα(g) = sup 1≤L≤M n,kÃ3z g, 2−n/ppL(2n k) ≤ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + «pL», «Hp» ≤ «l», y, en consecuencia, Aα(g) ≤ , y (H) )* es el espacio deseado. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. El lector no tendrá ninguna dificultad en demostrar que este resultado implica la siguiente: Dejar que T sea un operador lineal limitado de un espacio cuasinombrado X en â € TM a â € TM a â € TM a â TM a â TM a â TM a â TM a â TM a, D. Entonces, T se limita de X a si, y sólo si, Aα(Tx) ≤ c x x x por cada x x x. El proceso de promedio de las traducciones de funciones de BMO dyadic conduce a BMO, y es una herramienta importante para obtener resultados en BMO una vez que son Se sabe que es cierto en su homólogo dyádico, BMOd, véase [7]. También se conoce que BMO se puede obtener como la intersección de BMOd y uno de sus desplazados homólogas, véase [8]. Estos resultados motivan nuestra próxima propuesta, que esencialmente dice que g â € ¬ si, y sólo si, g â € €, D y g está en el Lipschitz clase obtenida de la cuadrícula dyádica desplazada. Tenga en cuenta que los cambios involucrados en esta clase están en todas las direcciones paralelas al eje de coordenadas y dependen de la longitud lateral del cubo. Proposición 2.1. # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # Prueba. Es obvio que "g", D0 ≤ "g". Para mostrar la otra desigualdad nosotros Invoque el Teorema A. Dado que D • D0, basta con estimar Aα(g), o equiva- lenty, g, p para p Sα, α = N(1/p − 1). Por lo tanto, pick p = p n,k,α en Sα. Los definir cubo Q de pLn,k,α está en D0, y, desde p n,k,α tiene [α] momentos de desaparición, 8 WAEL ABU-SHAMMALA Y ALBERTO TORCHINSKY PLn,k,α, pQ(g) = 0. Por lo tanto, g, pLn,k, = g − pQ(g), p n,k, ≤ pLn,k,2 °g − pQ(g)°L2(Q) ≤ Q/N Q1/2pLn,k,2 g,D0. Ahora, un simple cambio de variables da Q/N Q1/2pLn,k,2 ≤ 1, y, con- Secuencialmente, también Aα(g) ≤ g,D0. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. Bibliografía [1] W. Abu-Shammala, J.-L. Shiu, y A. Torchinsky, Caracterizaciones del Hardy espacio H1 y BMO, preimpresión. [2] H.-Q. Bui y R. S. Laugesen, Aproximación y extensión en el espacio Hardy, por sistemas de afina, Constr. Aprox., para aparecer. [3] A. P. Calderón y A. Torchinsky, Funciones máximas parabólicas asociadas a un distibución, II, Avances en matemáticas., 24 (1977), 101–171. [4] G. S. de Souza, Espacios formados por átomos especiales, I, Rocky Mountain J. Matemáticas. 14 (1984), No. 2, 423-431. [5] S. Fridli, Transición del diádico al verdadero espacio Hardy no periódico, Acta Math. Acad. Pedagogo. Niházi (N.S.) 16 (2000), 1–8, (electrónica). [6] J. Gara-Cuerva y J. L. Rubio de Francia, Desigualdades de normas ponderadas y relacionadas temas, Notas de Matemáticas 116, Holanda del Norte, Amsterdam, 1985. [7] J. Garnett y P. Jones, BMO de dyadic BMO, Pacific J. Matemáticas. 99 (1982), No. 2, 351–371. [8] T. Mei, BMO es la intersección de dos traducciones de BMO dyádico, C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1003–1006. [9] T. M. Le y L. A. Vese, descomposición de la imagen utilizando variación total y div( BMO)*, Modelo multiescala. Simul. 4, (2005), no. 2, 390-423. [10] A. Torchinsky, Métodos reales variables en el análisis armónico, Dover Publications, Inc., Mineola, NY, 2004. Departamento de Matemáticas, Universidad de Indiana, Bloomington IN 47405 Dirección de correo electrónico: wabusham@indiana.edu Departamento de Matemáticas, Universidad de Indiana, Bloomington IN 47405 Dirección de correo electrónico: torchins@indiana.edu 1. Caracterización de los espacios Hardy Hp 2. Caracterizaciones de Bibliografía
704.001
Polymer Quantum Mechanics and its Continuum Limit
A rather non-standard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schroedinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schroedinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schroedinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle and a simple cosmological model.
Polymer Quantum Mechanics and its Continuum Limit Alejandro Corichi,1, 2, 3, ∗ Tatjana Vukašinac,4, † and José A. Zapata1, ‡ Instituto de Matemáticas, Unidad Morelia, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, A. Postal 61-3, Morelia, Michoacán 58090, Mexico Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D.F. 04510, Mexico Institute for Gravitational Physics and Geometry, Physics Department, Pennsylvania State University, University Park PA 16802, USA Facultad de Ingenieŕıa Civil, Universidad Michoacana de San Nicolas de Hidalgo, Morelia, Michoacán 58000, Mexico A rather non-standard quantum representation of the canonical commutation relations of quan- tum mechanics systems, known as the polymer representation has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schrödinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schrödinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schrödinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle and a simple cosmological model. PACS numbers: 04.60.Pp, 04.60.Ds, 04.60.Nc 11.10.Gh. I. INTRODUCTION The so-called polymer quantum mechanics, a non- regular and somewhat ‘exotic’ representation of the canonical commutation relations (CCR) [1], has been used to explore both mathematical and physical issues in background independent theories such as quantum grav- ity [2, 3]. A notable example of this type of quantization, when applied to minisuperspace models has given way to what is known as loop quantum cosmology [4, 5]. As in any toy model situation, one hopes to learn about the subtle technical and conceptual issues that are present in full quantum gravity by means of simple, finite di- mensional examples. This formalism is not an exception in this regard. Apart from this motivation coming from physics at the Planck scale, one can independently ask for the relation between the standard continuous repre- sentations and their polymer cousins at the level of math- ematical physics. A deeper understanding of this relation becomes important on its own. The polymer quantization is made of several steps. The first one is to build a representation of the Heisenberg-Weyl algebra on a Kinematical Hilbert space that is “background independent”, and that is sometimes referred to as the polymeric Hilbert space Hpoly. The second and most important part, the implementation of dynamics, deals with the definition of a Hamiltonian (or Hamiltonian constraint) on this space. In the examples ∗Electronic address: corichi@matmor.unam.mx †Electronic address: tatjana@shi.matmor.unam.mx ‡Electronic address: zapata@matmor.unam.mx studied so far, the first part is fairly well understood, yielding the kinematical Hilbert space Hpoly that is, how- ever, non-separable. For the second step, a natural im- plementation of the dynamics has proved to be a bit more difficult, given that a direct definition of the Hamiltonian Ĥ of, say, a particle on a potential on the space Hpoly is not possible since one of the main features of this repre- sentation is that the operators q̂ and p̂ cannot be both simultaneously defined (nor their analogues in theories involving more elaborate variables). Thus, any operator that involves (powers of) the not defined variable has to be regulated by a well defined operator which normally involves introducing some extra structure on the configu- ration (or momentum) space, namely a lattice. However, this new structure that plays the role of a regulator can not be removed when working in Hpoly and one is left with the ambiguity that is present in any regularization. The freedom in choosing it can be sometimes associated with a length scale (the lattice spacing). For ordinary quantum systems such as a simple harmonic oscillator, that has been studied in detail from the polymer view- point, it has been argued that if this length scale is taken to be ‘sufficiently small’, one can arbitrarily approximate standard Schrödinger quantum mechanics [2, 3]. In the case of loop quantum cosmology, the minimum area gap A0 of the full quantum gravity theory imposes such a scale, that is then taken to be fundamental [4]. A natural question is to ask what happens when we change this scale and go to even smaller ‘distances’, that is, when we refine the lattice on which the dynamics of the theory is defined. Can we define consistency con- ditions between these scales? Or even better, can we take the limit and find thus a continuum limit? As it http://arxiv.org/abs/0704.0007v2 mailto:corichi@matmor.unam.mx mailto:tatjana@shi.matmor.unam.mx mailto:zapata@matmor.unam.mx has been shown recently in detail, the answer to both questions is in the affirmative [6]. There, an appropriate notion of scale was defined in such a way that one could define refinements of the theory and pose in a precise fashion the question of the continuum limit of the theory. These results could also be seen as handing a procedure to remove the regulator when working on the appropri- ate space. The purpose of this paper is to further explore different aspects of the relation between the continuum and the polymer representation. In particular in the first part we put forward a novel way of deriving the polymer representation from the ordinary Schrödinger represen- tation as an appropriate limit. In Sec. II we derive two versions of the polymer representation as different lim- its of the Schrödinger theory. In Sec. III we show that these two versions can be seen as different polarizations of the ‘abstract’ polymer representation. These results, to the best of our knowledge, are new and have not been reported elsewhere. In Sec. IV we pose the problem of implementing the dynamics on the polymer representa- tion. In Sec. V we motivate further the question of the continuum limit (i.e. the proper removal of the regulator) and recall the basic constructions of [6]. Several exam- ples are considered in Sec. VI. In particular a simple harmonic oscillator, the polymer free particle and a sim- ple quantum cosmology model are considered. The free particle and the cosmological model represent a general- ization of the results obtained in [6] where only systems with a discrete and non-degenerate spectrum where con- sidered. We end the paper with a discussion in Sec. VII. In order to make the paper self-contained, we will keep the level of rigor in the presentation to that found in the standard theoretical physics literature. II. QUANTIZATION AND POLYMER REPRESENTATION In this section we derive the so called polymer repre- sentation of quantum mechanics starting from a specific reformulation of the ordinary Schrödinger representation. Our starting point will be the simplest of all possible phase spaces, namely Γ = R2 corresponding to a particle living on the real line R. Let us choose coordinates (q, p) thereon. As a first step we shall consider the quantization of this system that leads to the standard quantum theory in the Schrödinger description. A convenient route is to introduce the necessary structure to define the Fock rep- resentation of such system. From this perspective, the passage to the polymeric case becomes clearest. Roughly speaking by a quantization one means a passage from the classical algebraic bracket, the Poisson bracket, {q, p} = 1 (1) to a quantum bracket given by the commutator of the corresponding operators, [ q̂, p̂] = i~ 1̂ (2) These relations, known as the canonical commutation re- lation (CCR) become the most common corner stone of the (kinematics of the) quantum theory; they should be satisfied by the quantum system, when represented on a Hilbert space H. There are alternative points of departure for quantum kinematics. Here we consider the algebra generated by the exponentiated versions of q̂ and p̂ that are denoted U(α) = ei(α q̂)/~ ; V (β) = ei(β p̂)/~ where α and β have dimensions of momentum and length, respectively. The CCR now become U(α) · V (β) = e(−iα β)/~V (β) · U(α) (3) and the rest of the product is U(α1)·U(α2) = U(α1+α2) ; V (β1)·V (β2) = V (β1+β2) The Weyl algebra W is generated by taking finite linear combinations of the generators U(αi) and V (βi) where the product (3) is extended by linearity, (Ai U(αi) +Bi V (βi)) From this perspective, quantization means finding an unitary representation of the Weyl algebra W on a Hilbert space H′ (that could be different from the ordi- nary Schrödinger representation). At first it might look weird to attempt this approach given that we know how to quantize such a simple system; what do we need such a complicated object as W for? It is infinite dimensional, whereas the set S = {1̂, q̂, p̂}, the starting point of the ordinary Dirac quantization, is rather simple. It is in the quantization of field systems that the advantages of the Weyl approach can be fully appreciated, but it is also useful for introducing the polymer quantization and comparing it to the standard quantization. This is the strategy that we follow. A question that one can ask is whether there is any freedom in quantizing the system to obtain the ordinary Schrödinger representation. On a first sight it might seem that there is none given the Stone-Von Neumann unique- ness theorem. Let us review what would be the argument for the standard construction. Let us ask that the repre- sentation we want to build up is of the Schrödinger type, namely, where states are wave functions of configuration space ψ(q). There are two ingredients to the construction of the representation, namely the specification of how the basic operators (q̂, p̂) will act, and the nature of the space of functions that ψ belongs to, that is normally fixed by the choice of inner product on H, or measure µ on R. The standard choice is to select the Hilbert space to be, H = L2(R, dq) the space of square-integrable functions with respect to the Lebesgue measure dq (invariant under constant trans- lations) on R. The operators are then represented as, q̂ · ψ(q) = (q ψ)(q) and p̂ · ψ(q) = −i ~ ∂ ψ(q) (4) Is it possible to find other representations? In order to appreciate this freedom we go to the Weyl algebra and build the quantum theory thereon. The representation of the Weyl algebra that can be called of the ‘Fock type’ involves the definition of an extra structure on the phase space Γ: a complex structure J . That is, a linear map- ping from Γ to itself such that J2 = −1. In 2 dimen- sions, all the freedom in the choice of J is contained in the choice of a parameter d with dimensions of length. It is also convenient to define: k = p/~ that has dimensions of 1/L. We have then, Jd : (q, k) 7→ (−d2 k, q/d2) This object together with the symplectic structure: Ω((q, p); (q′, p′)) = q p′ − p q′ define an inner product on Γ by the formula gd(· ; ·) = Ω(· ; Jd ·) such that: gd((q, p); (q ′, p′)) = q q′ + which is dimension-less and positive definite. Note that with this quantities one can define complex coordinates (ζ, ζ̄) as usual: q + i p ; ζ̄ = q − i d from which one can build the standard Fock representa- tion. Thus, one can alternatively view the introduction of the length parameter d as the quantity needed to de- fine (dimensionless) complex coordinates on the phase space. But what is the relevance of this object (J or d)? The definition of complex coordinates is useful for the construction of the Fock space since from them one can define, in a natural way, creation and annihilation operators. But for the Schrödinger representation we are interested here, it is a bit more subtle. The subtlety is that within this approach one uses the algebraic prop- erties of W to construct the Hilbert space via what is known as the Gel’fand-Naimark-Segal (GNS) construc- tion. This implies that the measure in the Schrödinger representation becomes non trivial and thus the momen- tum operator acquires an extra term in order to render the operator self-adjoint. The representation of the Weyl algebra is then, when acting on functions φ(q) [7]: Û(α) · φ(q) := (eiα q/~ φ)(q) V̂ (β) · φ(q) := e (q−β/2) φ(q − β) The Hilbert space structure is introduced by the defini- tion of an algebraic state (a positive linear functional) ωd : W → C, that must coincide with the expectation value in the Hilbert space taken on a special state ref- ered to as the vacuum: ωd(a) = 〈â〉vac, for all a ∈ W . In our case this specification of J induces such a unique state ωd that yields, 〈Û(α)〉vac = e− d2 α2 ~2 (5) 〈V̂ (β)〉vac = e− d2 (6) Note that the exponents in the vacuum expectation values correspond to the metric constructed out of J : d2 α2 = gd((0, α); (0, α)) and = gd((β, 0); (β, 0)). Wave functions belong to the space L2(R, dµd), where the measure that dictates the inner product in this rep- resentation is given by, dµd = d2 dq In this representation, the vacuum is given by the iden- tity function φ0(q) = 1 that is, just as any plane wave, normalized. Note that for each value of d > 0, the rep- resentation is well defined and continuous in α and β. Note also that there is an equivalence between the q- representation defined by d and the k-representation de- fined by 1/d. How can we recover then the standard representation in which the measure is given by the Lebesgue measure and the operators are represented as in (4)? It is easy to see that there is an isometric isomorphism K that maps the d-representation in Hd to the standard Schrödinger representation in Hschr by: ψ(q) = K · φ(q) = e d1/2π1/4 φ(q) ∈ Hschr = L2(R, dq) Thus we see that all d-representations are unitarily equiv- alent. This was to be expected in view of the Stone-Von Neumann uniqueness result. Note also that the vacuum now becomes ψ0(q) = d1/2π1/4 2 d2 , so even when there is no information about the param- eter d in the representation itself, it is contained in the vacuum state. This procedure for constructing the GNS- Schrödinger representation for quantum mechanics has also been generalized to scalar fields on arbitrary curved space in [8]. Note, however that so far the treatment has all been kinematical, without any knowledge of a Hamil- tonian. For the Simple Harmonic Oscillator of mass m and frequency ω, there is a natural choice compatible with the dynamics given by d = , in which some calculations simplify (for instance for coherent states), but in principle one can use any value of d. Our study will be simplified by focusing on the funda- mental entities in the Hilbert Space Hd , namely those states generated by acting with Û(α) on the vacuum φ0(q) = 1. Let us denote those states by, φα(q) = Û(α) · φ0(q) = ei The inner product between two such states is given by 〈φα, φλ〉d = dµd e ~ = e− (λ−α)2 d2 4 ~2 (7) Note incidentally that, contrary to some common belief, the ‘plane waves’ in this GNS Hilbert space are indeed normalizable. Let us now consider the polymer representation. For that, it is important to note that there are two possible limiting cases for the parameter d: i) The limit 1/d 7→ 0 and ii) The case d 7→ 0. In both cases, we have ex- pressions that become ill defined in the representation or measure, so one needs to be careful. A. The 1/d 7→ 0 case. The first observation is that from the expressions (5) and (6) for the algebraic state ωd, we see that the limiting cases are indeed well defined. In our case we get, ωA := lim1/d→0 ωd such that, ωA(Û(α)) = δα,0 and ωA(V̂ (β)) = 1 (8) From this, we can indeed construct the representation by means of the GNS construction. In order to do that and to show how this is obtained we shall consider several expressions. One has to be careful though, since the limit has to be taken with care. Let us consider the measure on the representation that behaves as: dµd = d2 dq 7→ 1 so the measures tends to an homogeneous measure but whose ‘normalization constant’ goes to zero, so the limit becomes somewhat subtle. We shall return to this point later. Let us now see what happens to the inner product between the fundamental entities in the Hilbert Space Hd given by (7). It is immediate to see that in the 1/d 7→ 0 limit the inner product becomes, 〈φα, φλ〉d 7→ δα,λ (9) with δα,λ being Kronecker’s delta. We see then that the plane waves φα(q) become an orthonormal basis for the new Hilbert space. Therefore, there is a delicate interplay between the two terms that contribute to the measure in order to maintain the normalizability of these functions; we need the measure to become damped (by 1/d) in order to avoid that the plane waves acquire an infinite norm (as happens with the standard Lebesgue measure), but on the other hand the measure, that for any finite value of d is a Gaussian, becomes more and more spread. It is important to note that, in this limit, the operators Û(α) become discontinuous with respect to α, given that for any given α1 and α2 (different), its action on a given basis vector ψλ(q) yields orthogonal vectors. Since the continuity of these operators is one of the hypotesis of the Stone-Von Neumann theorem, the uniqueness result does not apply here. The representation is inequivalent to the standard one. Let us now analyze the other operator, namely the action of the operator V̂ (β) on the basis φα(q): V̂ (β) · φα(q) = e− ~ e(β/d 2+iα/~)q which in the limit 1/d 7→ 0 goes to, V̂ (β) · φα(q) 7→ ei ~ φα(q) that is continuous on β. Thus, in the limit, the operator p̂ = −i~∂q is well defined. Also, note that in this limit the operator p̂ has φα(q) as its eigenstate with eigenvalue given by α: p̂ · φα(q) 7→ αφα(q) To summarize, the resulting theory obtained by taking the limit 1/d 7→ 0 of the ordinary Schrödinger descrip- tion, that we shall call the ‘polymer representation of type A’, has the following features: the operators U(α) are well defined but not continuous in α, so there is no generator (no operator associated to q). The basis vec- tors φα are orthonormal (for α taking values on a contin- uous set) and are eigenvectors of the operator p̂ that is well defined. The resulting Hilbert space HA will be the (A-version of the) polymer representation. Let us now consider the other case, namely, the limit when d 7→ 0. B. The d 7→ 0 case Let us now explore the other limiting case of the Schrödinger/Fock representations labelled by the param- eter d. Just as in the previous case, the limiting algebraic state becomes, ωB := limd→0 ωd such that, ωB(Û(α)) = 1 and ωB(V̂ (β)) = δβ,0 (10) From this positive linear function, one can indeed con- struct the representation using the GNS construction. First let us note that the measure, even when the limit has to be taken with due care, behaves as: dµd = d2 dq 7→ δ(q) dq That is, as Dirac’s delta distribution. It is immediate to see that, in the d 7→ 0 limit, the inner product between the fundamental states φα(q) becomes, 〈φα, φλ〉d 7→ 1 (11) This in fact means that the vector ξ = φα − φλ belongs to the Kernel of the limiting inner product, so one has to mod out by these (and all) zero norm states in order to get the Hilbert space. Let us now analyze the other operator, namely the action of the operator V̂ (β) on the vacuum φ0(q) = 1, which for arbitrary d has the form, φ̃β := V̂ (β) · φ0(q) = e (q−β/2) The inner product between two such states is given by 〈φ̃α, φ̃β〉d = e− (α−β)2 In the limit d → 0, 〈φ̃α, φ̃β〉d → δα,β. We can see then that it is these functions that become the orthonormal, ‘discrete basis’ in the theory. However, the function φ̃β(q) in this limit becomes ill defined. For example, for β > 0, it grows unboundedly for q > β/2, is equal to one if q = β/2 and zero otherwise. In order to overcome these difficulties and make more transparent the resulting the- ory, we shall consider the other form of the representation in which the measure is incorporated into the states (and the resulting Hilbert space is L2(R, dq)). Thus the new state ψβ(q) := K · (V̂ (β) · φ0(q)) = (q−β)2 We can now take the limit and what we get is d 7→0 ψβ(q) := δ 1/2(q, β) where by δ1/2(q, β) we mean something like ‘the square root of the Dirac distribution’. What we really mean is an object that satisfies the following property: δ1/2(q, β) · δ1/2(q, α) = δ(q, β) δβ,α That is, if α = β then it is just the ordinary delta, other- wise it is zero. In a sense these object can be regarded as half-densities that can not be integrated by themselves, but whose product can. We conclude then that the inner product is, 〈ψβ , ψα〉 = dq ψβ(q)ψα(q) = dq δ(q, α) δβ,α = δβ,α which is just what we expected. Note that in this repre- sentation, the vacuum state becomes ψ0(q) := δ 1/2(q, 0), namely, the half-delta with support in the origin. It is important to note that we are arriving in a natural way to states as half-densities, whose squares can be integrated without the need of a nontrivial measure on the configu- ration space. Diffeomorphism invariance arises then in a natural but subtle manner. Note that as the end result we recover the Kronecker delta inner product for the new fundamental states: χβ(q) := δ 1/2(q, β). Thus, in this new B-polymer representation, the Hilbert space HB is the completion with respect to the inner product (13) of the states generated by taking (finite) linear combinations of basis elements of the form χβ : Ψ(q) = bi χβi(q) (14) Let us now introduce an equivalent description of this Hilbert space. Instead of having the basis elements be half-deltas as elements of the Hilbert space where the inner product is given by the ordinary Lebesgue measure dq, we redefine both the basis and the measure. We could consider, instead of a half-delta with support β, a Kronecker delta or characteristic function with support on β: χ′β(q) := δq,β These functions have a similar behavior with respect to the product as the half-deltas, namely: χ′β(q) · χ′α(q) = δβ,α. The main difference is that neither χ ′ nor their squares are integrable with respect to the Lebesgue mea- sure (having zero norm). In order to fix that problem we have to change the measure so that we recover the basic inner product (13) with our new basis. The needed mea- sure turns out to be the discrete counting measure on R. Thus any state in the ‘half density basis’ can be written (using the same expression) in terms of the ‘Kronecker basis’. For more details and further motivation see the next section. Note that in this B-polymer representation, both Û and V̂ have their roles interchanged with that of the A-polymer representation: while U(α) is discontinuous and thus q̂ is not defined in the A-representation, we have that it is V (β) in the B-representation that has this property. In this case, it is the operator p̂ that can not be defined. We see then that given a physical system for which the configuration space has a well defined physi- cal meaning, within the possible representation in which wave-functions are functions of the configuration variable q, the A and B polymer representations are radically dif- ferent and inequivalent. Having said this, it is also true that the A and B representations are equivalent in a different sense, by means of the duality between q and p representations and the d↔ 1/d duality: The A-polymer representation in the “q-representation” is equivalent to the B-polymer representation in the “p-representation”, and conversely. When studying a problem, it is important to decide from the beginning which polymer representation (if any) one should be using (for instance in the q-polarization). This has as a consequence an implication on which variable is naturally “quantized” (even if continuous): p for A and q for B. There could be for instance a physical criteria for this choice. For example a fundamental symmetry could suggest that one representation is more natural than an- other one. This indeed has been recently noted by Chiou in [10], where the Galileo group is investigated and where it is shown that the B representation is better behaved. In the other polarization, namely for wavefunctions of p, the picture gets reversed: q is discrete for the A- representation, while p is for the B-case. Let us end this section by noting that the procedure of obtaining the polymer quantization by means of an appropriate limit of Fock-Schrödinger representations might prove useful in more general settings in field theory or quantum gravity. III. POLYMER QUANTUM MECHANICS: KINEMATICS In previous sections we have derived what we have called the A and B polymer representations (in the q- polarization) as limiting cases of ordinary Fock repre- sentations. In this section, we shall describe, without any reference to the Schrödinger representation, the ‘ab- stract’ polymer representation and then make contact with its two possible realizations, closely related to the A and B cases studied before. What we will see is that one of them (the A case) will correspond to the p-polarization while the other one corresponds to the q−representation, when a choice is made about the physical significance of the variables. We can start by defining abstract kets |µ〉 labelled by a real number µ. These shall belong to the Hilbert space Hpoly. From these states, we define a generic ‘cylinder states’ that correspond to a choice of a finite collection of numbers µi ∈ R with i = 1, 2, . . . , N . Associated to this choice, there are N vectors |µi〉, so we can take a linear combination of them |ψ〉 = ai |µi〉 (15) The polymer inner product between the fundamental kets is given by, 〈ν|µ〉 = δν,µ (16) That is, the kets are orthogonal to each other (when ν 6= µ) and they are normalized (〈µ|µ〉 = 1). Immediately, this implies that, given any two vectors |φ〉 = j=1 bj |νj〉 and |ψ〉 = i=1 ai |µi〉, the inner product between them is given by, 〈φ|ψ〉 = b̄j ai 〈νj |µi〉 = b̄k ak where the sum is over k that labels the intersection points between the set of labels {νj} and {µi}. The Hilbert space Hpoly is the Cauchy completion of finite linear com- bination of the form (15) with respect to the inner prod- uct (16). Hpoly is non-separable. There are two basic operators on this Hilbert space: the ‘label operator’ ε̂: ε̂ |µ〉 := µ |µ〉 and the displacement operator ŝ (λ), ŝ (λ) |µ〉 := |µ+ λ〉 The operator ε̂ is symmetric and the operator(s) ŝ(λ) defines a one-parameter family of unitary operators on Hpoly, where its adjoint is given by ŝ† (λ) = ŝ (−λ). This action is however, discontinuous with respect to λ given that |µ〉 and |µ + λ〉 are always orthogonal, no matter how small is λ. Thus, there is no (Hermitian) operator that could generate ŝ (λ) by exponentiation. So far we have given the abstract characterization of the Hilbert space, but one would like to make contact with concrete realizations as wave functions, or by iden- tifying the abstract operators ε̂ and ŝ with physical op- erators. Suppose we have a system with a configuration space with coordinate given by q, and p denotes its canonical conjugate momenta. Suppose also that for physical rea- sons we decide that the configuration coordinate q will have some “discrete character” (for instance, if it is to be identified with position, one could say that there is an underlying discreteness in position at a small scale). How can we implement such requirements by means of the polymer representation? There are two possibilities, depending on the choice of ‘polarizations’ for the wave- functions, namely whether they will be functions of con- figuration q or momenta p. Let us the divide the discus- sion into two parts. A. Momentum polarization In this polarization, states will be denoted by, ψ(p) = 〈p|ψ〉 where ψµ(p) = 〈p|µ〉 = ei How are then the operators ε̂ and ŝ represented? Note that if we associate the multiplicative operator V̂ (λ) · ψµ(p) = ei ~ = ei (µ+λ) p = ψ(µ+λ)(p) we see then that the operator V̂ (λ) corresponds precisely to the shift operator ŝ (λ). Thus we can also conclude that the operator p̂ does not exist. It is now easy to identify the operator q̂ with: q̂ · ψµ(p) = −i~ ψµ(p) = µ e ~ = µψµ(p) namely, with the abstract operator ε̂. The reason we say that q̂ is discrete is because this operator has as its eigenvalue the label µ of the elementary state ψµ(p), and this label, even when it can take value in a continuum of possible values, is to be understood as a discrete set, given that the states are orthonormal for all values of µ. Given that states are now functions of p, the inner product (16) should be defined by a measure µ on the space on which the wave-functions are defined. In order to know what these two objects are, namely, the quan- tum “configuration” space C and the measure thereon1, we have to make use of the tools available to us from the theory of C∗-algebras. If we consider the operators V̂ (λ), together with their natural product and ∗-relation given by V̂ ∗(λ) = V̂ (−λ), they have the structure of an Abelian C∗-algebra (with unit) A. We know from the representation theory of such objects that A is iso- morphic to the space of continuous functions C0(∆) on a compact space ∆, the spectrum of A. Any representation of A on a Hilbert space as multiplication operator will be on spaces of the form L2(∆, dµ). That is, our quantum configuration space is the spectrum of the algebra, which in our case corresponds to the Bohr compactification Rb of the real line [11]. This space is a compact group and there is a natural probability measure defined on it, the Haar measure µH. Thus, our Hilbert space Hpoly will be isomorphic to the space, Hpoly,p = L2(Rb, dµH) (17) In terms of ‘quasi periodic functions’ generated by ψµ(p), the inner product takes the form 〈ψµ|ψλ〉 := dµH ψµ(p)ψλ(p) := = lim L 7→∞ dpψµ(p)ψλ(p) = δµ,λ (18) note that in the p-polarization, this characterization cor- responds to the ‘A-version’ of the polymer representation of Sec. II (where p and q are interchanged). B. q-polarization Let us now consider the other polarization in which wave functions will depend on the configuration coordinate q: ψ(q) = 〈q|ψ〉 The basic functions, that now will be called ψ̃µ(q), should be, in a sense, the dual of the functions ψµ(p) of the previous subsection. We can try to define them via a ‘Fourier transform’: ψ̃µ(q) := 〈q|µ〉 = 〈q| dµH|p〉〈p|µ〉 which is given by ψ̃µ(q) := dµH〈q|p〉ψµ(p) = dµH e −i p q ~ = δq,µ (19) 1 here we use the standard terminology of ‘configuration space’ to denote the domain of the wave function even when, in this case, it corresponds to the physical momenta p. That is, the basic objects in this representation are Kro- necker deltas. This is precisely what we had found in Sec. II for the B-type representation. How are now the basic operators represented and what is the form of the inner product? Regarding the operators, we expect that they are represented in the opposite manner as in the previous p-polarization case, but that they preserve the same features: p̂ does not exist (the derivative of the Kro- necker delta is ill defined), but its exponentiated version V̂ (λ) does: V̂ (λ) · ψ(q) = ψ(q + λ) and the operator q̂ that now acts as multiplication has as its eigenstates, the functions ψ̃ν(q) = δν,q: q̂ · ψ̃µ(q) := µ ψ̃µ(q) What is now the nature of the quantum configurations space Q? And what is the measure thereon dµq? that defines the inner product we should have: 〈ψ̃µ(q), ψ̃λ(q)〉 = δµ,λ The answer comes from one of the characterizations of the Bohr compactification: we know that it is, in a precise sense, dual to the real line but when equipped with the discrete topology Rd. Furthermore, the measure on Rd will be the ‘counting measure’. In this way we recover the same properties we had for the previous characterization of the polymer Hilbert space. We can thus write: Hpoly,x := L2(Rd, dµc) (20) This completes a precise construction of the B-type poly- mer representation sketched in the previous section. Note that if we had chosen the opposite physical situation, namely that q, the configuration observable, be the quan- tity that does not have a corresponding operator, then we would have had the opposite realization: In the q- polarization we would have had the type-A polymer rep- resentation and the type-B for the p-polarization. As we shall see both scenarios have been considered in the literature. Up to now we have only focused our discussion on the kinematical aspects of the quantization process. Let us now consider in the following section the issue of dynam- ics and recall the approach that had been adopted in the literature, before the issue of the removal of the regulator was reexamined in [6]. IV. POLYMER QUANTUM MECHANICS: DYNAMICS As we have seen the construction of the polymer representation is rather natural and leads to a quan- tum theory with different properties than the usual Schrödinger counterpart such as its non-separability, the non-existence of certain operators and the existence of normalized eigen-vectors that yield a precise value for one of the phase space coordinates. This has been done without any regard for a Hamiltonian that endows the system with a dynamics, energy and so on. First let us consider the simplest case of a particle of mass m in a potential V (q), in which the Hamiltonian H takes the form, p2 + V (q) Suppose furthermore that the potential is given by a non- periodic function, such as a polynomial or a rational func- tion. We can immediately see that a direct implementa- tion of the Hamiltonian is out of our reach, for the simple reason that, as we have seen, in the polymer representa- tion we can either represent q or p, but not both! What has been done so far in the literature? The simplest thing possible: approximate the non-existing term by a well defined function that can be quantized and hope for the best. As we shall see in next sections, there is indeed more that one can do. At this point there is also an important decision to be made: which variable q or p should be regarded as “dis- crete”? Once this choice is made, then it implies that the other variable will not exist: if q is regarded as dis- crete, then p will not exist and we need to approximate the kinetic term p2/2m by something else; if p is to be the discrete quantity, then q will not be defined and then we need to approximate the potential V (q). What hap- pens with a periodic potential? In this case one would be modelling, for instance, a particle on a regular lattice such as a phonon living on a crystal, and then the natural choice is to have q not well defined. Furthermore, the po- tential will be well defined and there is no approximation needed. In the literature both scenarios have been considered. For instance, when considering a quantum mechanical system in [2], the position was chosen to be discrete, so p does not exist, and one is then in the A type for the momentum polarization (or the type B for the q- polarization). With this choice, it is the kinetic term the one that has to be approximated, so once one has done this, then it is immediate to consider any potential that will thus be well defined. On the other hand, when con- sidering loop quantum cosmology (LQC), the standard choice is that the configuration variable is not defined [4]. This choice is made given that LQC is regarded as the symmetric sector of full loop quantum gravity where the connection (that is regarded as the configuration vari- able) can not be promoted to an operator and one can only define its exponentiated version, namely, the holon- omy. In that case, the canonically conjugate variable, closely related to the volume, becomes ‘discrete’, just as in the full theory. This case is however, different from the particle in a potential example. First we could mention that the functional form of the Hamiltonian constraint that implements dynamics has a different structure, but the more important difference lies in that the system is constrained. Let us return to the case of the particle in a po- tential and for definiteness, let us start with the aux- iliary kinematical framework in which: q is discrete, p can not be promoted and thus we have to approximate the kinetic term p̂2/2m. How is this done? The stan- dard prescription is to define, on the configuration space C, a regular ‘graph’ γµ0 . This consists of a numerable set of points, equidistant, and characterized by a pa- rameter µ0 that is the (constant) separation between points. The simplest example would be to consider the set γµ0 = {q ∈ R | q = nµ0 , ∀ n ∈ Z}. This means that the basic kets that will be considered |µn〉 will correspond precisely to labels µn belonging to the graph γµ0 , that is, µn = nµ0. Thus, we shall only consider states of the form, |ψ〉 = bn |µn〉 . (21) This ‘small’ Hilbert space Hγµ0 , the graph Hilbert space, is a subspace of the ‘large’ polymer Hilbert space Hpoly but it is separable. The condition for a state of the form (21) to belong to the Hilbert space Hγµ0 is that the co- efficients bn satisfy: n |bn|2 <∞. Let us now consider the kinetic term p̂2/2m. We have to approximate it by means of trigonometric functions, that can be built out of the functions of the form eiλ p/~. As we have seen in previous sections, these functions can indeed be promoted to operators and act as translation operators on the kets |µ〉. If we want to remain in the graph γ, and not create ‘new points’, then one is con- strained to considering operators that displace the kets by just the right amount. That is, we want the basic shift operator V̂ (λ) to be such that it maps the ket with label |µn〉 to the next ket, namely |µn+1〉. This can in- deed achieved by fixing, once and for all, the value of the allowed parameter λ to be λ = µ0. We have then, V̂ (µ0) · |µn〉 = |µn + µ0〉 = |µn+1〉 which is what we wanted. This basic ‘shift operator’ will be the building block for approximating any (polynomial) function of p. In order to do that we notice that the function p can be approximated by, p ≈ ~ (µ0 p ~ − e−i where the approximation is good for p << ~/µ0. Thus, one can define a regulated operator p̂µ0 that depends on the ‘scale’ µ0 as: p̂µ0 · |µn〉 := [V (µ0) − V (−µ0)] · |µn〉 = (|µn+1〉 − |µn−1〉) (22) In order to regulate the operator p̂2, there are (at least) two possibilities, namely to compose the operator p̂µ0 with itself or to define a new approximation. The oper- ator p̂µ0 · p̂µ0 has the feature that shifts the states two steps in the graph to both sides. There is however an- other operator that only involves shifting once: p̂2µ0 · |νn〉 := [2 − V̂ (µ0) − V̂ (−µ0)] · |νn〉 = (2|νn〉 − |νn+1〉 − |νn−1〉) (23) which corresponds to the approximation p2 ≈ 2~ cos(µ0 p/~)), valid also in the regime p << ~/µ0. With these considerations, one can define the operator Ĥµ0 , the Hamiltonian at scale µ0, that in practice ‘lives’ on the space Hγµ0 as, Ĥµ0 := p̂2µ0 + V̂ (q) , (24) that is a well defined, symmetric operator on Hγµ0 . No- tice that the operator is also defined on Hpoly, but there its physical interpretation is problematic. For example, it turns out that the expectation value of the kinetic term calculated on most states (states which are not tailored to the exact value of the parameter µ0) is zero. Even if one takes a state that gives “reasonable“ expectation values of the µ0-kinetic term and uses it to calculate the expectation value of the kinetic term corresponding to a slight perturbation of the parameter µ0 one would get zero. This problem, and others that arise when working on Hpoly, forces one to assign a physical interpretation to the Hamiltonian Ĥµ0 only when its action is restricted to the subspace Hγµ0 . Let us now explore the form that the Hamiltonian takes in the two possible polarizations. In the q-polarization, the basis, labelled by n is given by the functions χn(q) = δq,µn . That is, the wave functions will only have sup- port on the set γµ0 . Alternatively, one can think of a state as completely characterized by the ‘Fourier coeffi- cients’ an: ψ(q) ↔ an, which is the value that the wave function ψ(q) takes at the point q = µn = nµ0. Thus, the Hamiltonian takes the form of a difference equation when acting on a general state ψ(q). Solving the time independent Schrödinger equation Ĥ · ψ = E ψ amounts to solving the difference equation for the coefficients an. The momentum polarization has a different structure. In this case, the operator p̂2µ0 acts as a multiplication operator, p̂2µ0 · ψ(p) = 1 − cos (µ0 p ψ(p) (25) The operator corresponding to q will be represented as a derivative operator q̂ · ψ(p) := i~ ∂p ψ(p). For a generic potential V (q), it has to be defined by means of spectral theory defined now on a circle. Why on a circle? For the simple reason that by restricting ourselves to a regular graph γµ0 , the functions of p that preserve it (when acting as shift operators) are of the form e(i m µ0 p/~) for m integer. That is, what we have are Fourier modes, labelled by m, of period 2π ~/µ0 in p. Can we pretend then that the phase space variable p is now compactified? The answer is in the affirmative. The inner product on periodic functions ψµ0(p) of p coming from the full Hilbert space Hpoly and given by 〈φ(p)|ψ(p)〉poly = lim L 7→∞ dp φ(p)ψ(p) is precisely equivalent to the inner product on the circle given by the uniform measure 〈φ(p)|ψ(p)〉µ0 = ∫ π~/µ0 −π~/µ0 dp φ(p)ψ(p) with p ∈ (−π~/µ0, π~/µ0). As long as one restricts at- tention to the graph γµ0 , one can work in this separable Hilbert space Hγµ0 of square integrable functions on S Immediately, one can see the limitations of this descrip- tion. If the mechanical system to be quantized is such that its orbits have values of the momenta p that are not small compared with π~/µ0 then the approximation taken will be very poor, and we don’t expect neither the effective classical description nor its quantization to be close to the standard one. If, on the other hand, one is al- ways within the region in which the approximation can be regarded as reliable, then both classical and quantum de- scriptions should approximate the standard description. What does ‘close to the standard description’ exactly mean needs, of course, some further clarification. In particular one is assuming the existence of the usual Schrödinger representation in which the system has a be- havior that is also consistent with observations. If this is the case, the natural question is: How can we approxi- mate such description from the polymer picture? Is there a fine enough graph γµ0 that will approximate the system in such a way that all observations are indistinguishable? Or even better, can we define a procedure, that involves a refinement of the graph γµ0 such that one recovers the standard picture? It could also happen that a continuum limit can be de- fined but does not coincide with the ‘expected one’. But there might be also physical systems for which there is no standard description, or it just does not make sense. Can in those cases the polymer representation, if it ex- ists, provide the correct physical description of the sys- tem under consideration? For instance, if there exists a physical limitation to the minimum scale set by µ0, as could be the case for a quantum theory of gravity, then the polymer description would provide a true physical bound on the value of certain quantities, such as p in our example. This could be the case for loop quantum cosmology, where there is a minimum value for physical volume (coming from the full theory), and phase space points near the ‘singularity’ lie at the region where the approximation induced by the scale µ0 departs from the standard classical description. If in that case the poly- mer quantum system is regarded as more fundamental than the classical system (or its standard Wheeler-De Witt quantization), then one would interpret this dis- crepancies in the behavior as a signal of the breakdown of classical description (or its ‘naive’ quantization). In the next section we present a method to remove the regulator µ0 which was introduced as an intermedi- ate step to construct the dynamics. More precisely, we shall consider the construction of a continuum limit of the polymer description by means of a renormalization procedure. V. THE CONTINUUM LIMIT This section has two parts. In the first one we motivate the need for a precise notion of the continuum limit of the polymeric representation, explaining why the most direct, and naive approach does not work. In the sec- ond part, we shall present the main ideas and results of the paper [6], where the Hamiltonian and the physical Hilbert space in polymer quantum mechanics are con- structed as a continuum limit of effective theories, follow- ing Wilson’s renormalization group ideas. The resulting physical Hilbert space turns out to be unitarily isomor- phic to the ordinary Hs = L2(R, dq) of the Schrödinger theory. Before describing the results of [6] we should discuss the precise meaning of reaching a theory in the contin- uum. Let us for concreteness consider the B-type repre- sentation in the q-polarization. That is, states are func- tions of q and the orthonormal basis χµ(q) is given by characteristic functions with support on q = µ. Let us now suppose we have a Schrödinger state Ψ(q) ∈ Hs = L2(R, dq). What is the relation between Ψ(q) and a state in Hpoly,x? We are also interested in the opposite ques- tion, that is, we would like to know if there is a preferred state in Hs that is approximated by an arbitrary state ψ(q) in Hpoly,x. The first obvious observation is that a Schödinger state Ψ(q) does not belong to Hpoly,x since it would have an infinite norm. To see that note that even when the would-be state can be formally expanded in the χµ basis as, Ψ(q) = Ψ(µ) χµ(q) where the sum is over the parameter µ ∈ R. Its associ- ated norm in Hpoly,x would be: |Ψ(q)|2poly = |Ψ(µ)|2 → ∞ which blows up. Note that in order to define a mapping P : Hs → Hpoly,x, there is a huge ambiguity since the values of the function Ψ(q) are needed in order to expand the polymer wave function. Thus we can only define a mapping in a dense subset D of Hs where the values of the functions are well defined (recall that in Hs the value of functions at a given point has no meaning since states are equivalence classes of functions). We could for instance ask that the mapping be defined for representatives of the equivalence classes in Hs that are piecewise continuous. From now on, when we refer to an element of the space Hs we shall be refereeing to one of those representatives. Notice then that an element of Hs does define an element of Cyl∗γ , the dual to the space Cylγ , that is, the space of cylinder functions with support on the (finite) lattice γ = {µ1, µ2, . . . , µN}, in the following way: Ψ(q) : Cylγ −→ C such that Ψ(q)[ψ(q)] = (Ψ|ψ〉 := Ψ(µ) 〈χµ| ψi χµi〉polyγ Ψ(µi)ψi < ∞ (26) Note that this mapping could be seen as consisting of two parts: First, a projection Pγ : Cyl ∗ → Cylγ such that Pγ(Ψ) = Ψγ(q) := i Ψ(µi)χµi(q) ∈ Cylγ . The state Ψγ is sometimes refereed to as the ‘shadow of Ψ(q) on the lattice γ’. The second step is then to take the inner product between the shadow Ψγ(q) and the state ψ(q) with respect to the polymer inner product 〈Ψγ |ψ〉polyγ . Now this inner product is well defined. Notice that for any given lattice γ the corresponding projector Pγ can be intuitively interpreted as some kind of ‘coarse graining map’ from the continuum to the lattice γ. In terms of functions of q the projection is replacing a continuous function defined on R with a function over the lattice γ ⊂ R which is a discrete set simply by restricting Ψ to γ. The finer the lattice the more points that we have on the curve. As we shall see in the second part of this section, there is indeed a precise notion of coarse graining that implements this intuitive idea in a concrete fashion. In particular, we shall need to replace the lattice γ with a decomposition of the real line in intervals (having the lattice points as end points). Let us now consider a system in the polymer represen- tation in which a particular lattice γ0 was chosen, say with points of the form {qk ∈ R |qk = ka0 , ∀ k ∈ Z}, namely a uniform lattice with spacing equal to a0. In this case, any Schrödinger wave function (of the type that we consider) will have a unique shadow on the lattice γ0. If we refine the lattice γ 7→ γn by dividing each interval in 2n new intervals of length an = a0/2 n we have new shad- ows that have more and more points on the curve. Intu- itively, by refining infinitely the graph we would recover the original function Ψ(q). Even when at each finite step the corresponding shadow has a finite norm in the poly- mer Hilbert space, the norm grows unboundedly and the limit can not be taken, precisely because we can not em- bed Hs into Hpoly. Suppose now that we are interested in the reverse process, namely starting from a polymer theory on a lattice and asking for the ‘continuum wave function’ that is best approximated by a wave function over a graph. Suppose furthermore that we want to con- sider the limit of the graph becoming finer. In order to give precise answers to these (and other) questions we need to introduce some new technology that will allow us to overcome these apparent difficulties. In the remaining of this section we shall recall these constructions for the benefit of the reader. Details can be found in [6] (which is an application of the general formalism discussed in [9]). The starting point in this construction is the concept of a scale C, which allows us to define the effective the- ories and the concept of continuum limit. In our case a scale is a decomposition of the real line in the union of closed-open intervals, that cover the whole line and do not intersect. Intuitively, we are shifting the emphasis from the lattice points to the intervals defined by the same points with the objective of approximating con- tinuous functions defined on R with functions that are constant on the intervals defined by the lattice. To be precise, we define an embedding, for each scale Cn from Hpoly to Hs by means of a step function: Ψ(man) χman(q) → Ψ(man) χαm(q) ∈ Hs with χαn(q) a characteristic function on the interval αm = [man, (m + 1)an). Thus, the shadows (living on the lattice) were just an intermediate step in the con- struction of the approximating function; this function is piece-wise constant and can be written as a linear com- bination of step functions with the coefficients provided by the shadows. The challenge now is to define in an appropriate sense how one can approximate all the aspects of the theory by means of this constant by pieces functions. Then the strategy is that, for any given scale, one can define an effective theory by approximating the kinetic operator by a combination of the translation operators that shift between the vertices of the given decomposition, in other words by a periodic function in p. As a result one has a set of effective theories at given scales which are mutually related by coarse graining maps. This framework was developed in [6]. For the convenience of the reader we briefly recall part of that framework. Let us denote the kinematic polymer Hilbert space at the scale Cn as HCn , and its basis elements as eαi,Cn , where αi = [ian, (i + 1)an) ∈ Cn. By construction this basis is orthonormal. The basis elements in the dual Hilbert space H∗Cn are denoted by ωαi,Cn ; they are also orthonormal. The states ωαi,Cn have a simple action on Cyl, ωαi,Cn(δx0,q) = χαi,Cn(x0). That is, if x0 is in the interval αi of Cn the result is one and it is zero if it is not there. Given any m ≤ n, we define d∗m,n : H∗Cn → H as the ‘coarse graining’ map between the dual Hilbert spaces, that sends the part of the elements of the dual basis to zero while keeping the information of the rest: d∗m,n(ωαi,Cn) = ωβj ,Cm if i = j2 n−m, in the opposite case d∗m,n(ωαi,Cn) = 0. At every scale the corresponding effective theory is given by the hamiltonian Hn. These Hamiltonians will be treated as quadratic forms, hn : HCn → R, given by hn(ψ) = λ (ψ,Hnψ) , (27) where λ2Cn is a normalizaton factor. We will see later that this rescaling of the inner product is necessary in order to guarantee the convergence of the renormalized theory. The completely renormalized theory at this scale is obtained as hrenm := lim d⋆m,nhn. (28) and the renormalized Hamiltonians are compatible with each other, in the sense that d⋆m,nh n = h In order to analyze the conditions for the convergence in (28) let us express the Hamiltonian in terms of its eigen-covectors end eigenvalues. We will work with effec- tive Hamiltonians that have a purely discrete spectrum (labelled by ν) Hn · Ψν,Cn = Eν,Cn Ψν,Cn . We shall also introduce, as an intermediate step, a cut-off in the energy levels. The origin of this cut-off is in the approximation of the Hamiltonian of our system at a given scale with a Hamiltonian of a periodic system in a regime of small energies, as we explained earlier. Thus, we can write hνcut−offm = νcut−off Eν,CmΨν,Cm ⊗ Ψν,Cm , (29) where the eigen covectors Ψν,Cm are normalized accord- ing to the inner product rescaled by 1 , and the cut- off can vary up to a scale dependent bound, νcut−off ≤ νmax(Cm). The Hilbert space of covectors together with such inner product will be called H⋆renCm . In the presence of a cut-off, the convergence of the microscopically corrected Hamiltonians, equation (28) is equivalent to the existence of the following two limits. The first one is the convergence of the energy levels, Eν,Cn = E ν . (30) Second is the existence of the completely renormalized eigen covectors, d⋆m,n Ψν,Cn = Ψ ∈ H⋆renCm ⊂ Cyl ⋆ . (31) We clarify that the existence of the above limit means that Ψrenν,Cm(δx0,q) is well defined for any δx0,q ∈ Cyl. No- tice that this point-wise convergence, if it can take place at all, will require the tuning of the normalization factors λ2Cn . Now we turn to the question of the continuum limit of the renormalized covectors. First we can ask for the existence of the limit Ψrenν,Cn(δx0,q) (32) for any δx0,q ∈ Cyl. When this limits exists there is a natural action of the eigen covectors in the continuum limit. Below we consider another notion of the continuum limit of the renormalized eigen covectors. When the completely renormalized eigen covectors exist, they form a collection that is d⋆-compatible, d⋆m,nΨ = Ψrenν,Cm . A sequence of d ⋆-compatible nor- malizable covectors define an element of , which is the projective limit of the renormalized spaces of covec- H⋆renCn . (33) The inner product in this space is defined by ({ΨCn}, {ΦCn})renR := lim (ΨCn ,ΦCn) The natural inclusion of C∞0 in is by an antilinear map which assigns to any Ψ ∈ C∞0 the d⋆-compatible collection ΨshadCn := ωαiΨ̄(L(αi)) ∈ H⋆renCn ⊂ Cyl ΨshadCn will be called the shadow of Ψ at scale Cn and acts in Cyl as a piecewise constant function. Clearly other types of test functions like Schwartz functions are also naturally included in . In this context a shadow is a state of the effective theory that approximates a state in the continuum theory. Since the inner product in is degenerate, the physical Hilbert space is defined as H⋆phys := / ker(·, ·)ren Hphys := H⋆⋆phys The nature of the physical Hilbert space, whether it is isomorphic to the Schrödinger Hilber space, Hs, or not, is determined by the normalization factors λ2Cn which can be obtained from the conditions asking for compatibil- ity of the dynamics of the effective theories at different scales. The dynamics of the system under consideration selects the continuum limit. Let us now return to the definition of the Hamilto- nian in the continuum limit. First consider the contin- uum limit of the Hamiltonian (with cut-off) in the sense of its point-wise convergence as a quadratic form. It turns out that if the limit of equation (32) exists for all the eigencovectors allowed by the cut-off, we have νcut−off ren : Hpoly,x → R defined by νcut−off ren (δx0,q) := lim hνcut−off renn ([δx0,q]Cn). (34) This Hamiltonian quadratic form in the continuum can be coarse grained to any scale and, as can be ex- pected, it yields the completely renormalized Hamilto- nian quadratic forms at that scale. However, this is not a completely satisfactory continuum limit because we can not remove the auxiliary cut-off νcut−off . If we tried, as we include more and more eigencovectors in the Hamilto- nian the calculations done at a given scale would diverge and doing them in the continuum is just as divergent. Below we explore a more successful path. We can use the renormalized inner product to induce an action of the cut–off Hamiltonians on νcut−off ren ({ΨCn}) := lim hνcut−off renn ((ΨCn , ·)renCn ), where we have used the fact that (ΨCn , ·)renCn ∈ HCn . The existence of this limit is trivial because the renormalized Hamiltonians are finite sums and the limit exists term by term. These cut-off Hamiltonians descend to the physical Hilbert space νcut−off ren ([{ΨCn}]) := h νcut−off ren ({ΨCn}) for any representative {ΨCn} ∈ [{ΨCn}] ∈ H⋆phys. Finally we can address the issue of removal of the cut- off. The Hamiltonian hren → R is defined by the limit := lim νcut−off→∞ νcut−off ren when the limit exists. Its corresponding Hermitian form in Hphys is defined whenever the above limit exists. This concludes our presentation of the main results of [6]. Let us now consider several examples of systems for which the continuum limit can be investigated. VI. EXAMPLES In this section we shall develop several examples of systems that have been treated with the polymer quanti- zation. These examples are simple quantum mechanical systems, such as the simple harmonic oscillator and the free particle, as well as a quantum cosmological model known as loop quantum cosmology. A. The Simple Harmonic Oscillator In this part, let us consider the example of a Simple Har- monic Oscillator (SHO) with parameters m and ω, clas- sically described by the following Hamiltonian mω2 x2. Recall that from these parameters one can define a length scale D = ~/mω. In the standard treatment one uses this scale to define a complex structure JD (and an in- ner product from it), as we have described in detail that uniquely selects the standard Schrödinger representation. At scale Cn we have an effective Hamiltonian for the Simple Harmonic Oscillator (SHO) given by HCn = 1 − cos anp mω2x2 . (35) If we interchange position and momentum, this Hamilto- nian is exactly that of a pendulum of mass m, length l and subject to a constant gravitational field g: ĤCn = − +mgl(1 − cos θ) where those quantities are related to our system by, mω an , g = , θ = That is, we are approximating, for each scale Cn the SHO by a pendulum. There is, however, an important difference. From our knowledge of the pendulum system, we know that the quantum system will have a spectrum for the energy that has two different asymptotic behav- iors, the SHO for low energies and the planar rotor in the higher end, corresponding to oscillating and rotating solutions respectively2. As we refine our scale and both the length of the pendulum and the height of the periodic potential increase, we expect to have an increasing num- ber of oscillating states (for a given pendulum system, there is only a finite number of such states). Thus, it is justified to consider the cut-off in the energy eigenval- ues, as discussed in the last section, given that we only expect a finite number of states of the pendulum to ap- proximate SHO eigenstates. With these consideration in mind, the relevant question is whether the conditions for the continuum limit to exist are satisfied. This question has been answered in the affirmative in [6]. What was shown there was that the eigen-values and eigen func- tions of the discrete systems, which represent a discrete and non-degenerate set, approximate those of the contin- uum, namely, of the standard harmonic oscillator when the inner product is renormalized by a factor λ2Cn = 1/2 This convergence implies that the continuum limit exists as we understand it. Let us now consider the simplest possible system, a free particle, that has nevertheless the particular feature that the spectrum of the energy is con- tinuous. 2 Note that both types of solutions are, in the phase space, closed. This is the reason behind the purely discrete spectrum. The distinction we are making is between those solutions inside the separatrix, that we call oscillating, and those that are above it that we call rotating. B. Free Polymer Particle In the limit ω → 0, the Hamiltonian of the Simple Harmonic oscillator (35) goes to the Hamiltonian of a free particle and the corresponding time independent Schrödinger equation, in the p−polarization, is given by (1 − cos anp ) − ECn ψ̃(p) = 0 where we now have that p ∈ S1, with p ∈ (−π~ Thus, we have ECn = 1 − cos ≤ ECn,max ≡ 2 . (36) At each scale the energy of the particle we can describe is bounded from above and the bound depends on the scale. Note that in this case the spectrum is continu- ous, which implies that the ordinary eigenfunctions of the Hilbert are not normalizable. This imposes an upper bound in the value that the energy of the particle can have, in addition to the bound in the momentum due to its “compactification”. Let us first look for eigen-solutions to the time inde- pendent Schrödinger equation, that is, for energy eigen- states. In the case of the ordinary free particle, these correspond to constant momentum plane waves of the form e±( ) and such that the ordinary dispersion re- lation p2/2m = E is satisfied. These plane waves are not square integrable and do not belong to the ordinary Hilbert space of the Schrödinger theory but they are still useful for extracting information about the system. For the polymer free particle we have, ψ̃Cn(p) = c1δ(p− PCn) + c2δ(p+ PCn) where PCn is a solution of the previous equation consid- ering a fixed value of ECn . That is, PCn = P (ECn) = arccos 1 − ma The inverse Fourier transform yields, in the ‘x represen- tation’, ψCn(xj) = ∫ π~/an −π~/an ψ̃(p) e p j dp = ixjPCn /~ + c2e −ixjPCn /~ .(37) with xj = an j for j ∈ Z. Note that the eigenfunctions are still delta functions (in the p representation) and thus not (square) normalizable with respect to the polymer inner product, that in the p polarization is just given by the ordinary Haar measure on S1, and there is no quantization of the momentum (its spectrum is still truly continuous). Let us now consider the time dependent Schrödinger equation, i~ ∂t Ψ̃(p, t) = Ĥ · Ψ̃(p, t). Which now takes the form, Ψ̃(p, t) = (1 − cos (an p/~)) Ψ̃(p, t) that has as its solution, Ψ̃(p, t) = e− (1−cos (an p/~)) t ψ̃(p) = e(−iECn /~) t ψ̃(p) for any initial function ψ̃(p), where ECn satisfy the dis- persion relation (36). The wave function Ψ(xj , t), the xj-representation of the wave function, can be obtained for any given time t by Fourier transforming with (37) the wave function Ψ̃(p, t). In order to check out the convergence of the micro- scopically corrected Hamiltonians we should analyze the convergence of the energy levels and of the proper cov- ectors. In the limit n → ∞, ECn → E = p2/2m so we can be certain that the eigen-values for the energy converge (when fixing the value of p). Let us write the proper covector as ΨCn = (ψCn , ·)renCn ∈ H . Then we can bring microscopic corrections to scale Cm and look for convergence of such corrections ΨrenCm = lim d⋆m,nΨCn . It is easy to see that given any basis vector eαi ∈ HCm the following limit ΨrenCm(eαi,Cm) = limCn→∞ ΨCn(dn,m(eαi,Cm)) exists and is equal to ΨshadCm (eαi,Cm) = [d ⋆ΨSchr](eαi,Cm) = Ψ Schr(iam) where ΨshadCm is calculated using the free particle Hamilto- nian in the Schrödinger representation. This expression defines the completely renormalized proper covector at the scale Cm. C. Polymer Quantum Cosmology In this section we shall present a version of quantum cosmology that we call polymer quantum cosmology. The idea behind this name is that the main input in the quan- tization of the corresponding mini-superspace model is the use of a polymer representation as here understood. Another important input is the choice of fundamental variables to be used and the definition of the Hamiltonian constraint. Different research groups have made differ- ent choices. We shall take here a simple model that has received much attention recently, namely an isotropic, homogeneous FRW cosmology with k = 0 and coupled to a massless scalar field ϕ. As we shall see, a proper treatment of the continuum limit of this system requires new tools under development that are beyond the scope of this work. We will thus restrict ourselves to the intro- duction of the system and the problems that need to be solved. The system to be quantized corresponds to the phase space of cosmological spacetimes that are homogeneous and isotropic and for which the homogeneous spatial slices have a flat intrinsic geometry (k = 0 condition). The only matter content is a mass-less scalar field ϕ. In this case the spacetime geometry is given by metrics of the form: ds2 = −dt2 + a2(t) (dx2 + dy2 + dz2) where the function a(t) carries all the information and degrees of freedom of the gravity part. In terms of the coordinates (a, pa, ϕ, pϕ) for the phase space Γ of the the- ory, all the dynamics is captured in the Hamiltonian con- straint C := −3 + 8πG 2|a|3 The first step is to define the constraint on the kine- matical Hilbert space to find physical states and then a physical inner product to construct the physical Hilbert space. First note that one can rewrite the equation as: p2a a 2 = 8πG If, as is normally done, one chooses ϕ to act as an in- ternal time, the right hand side would be promoted, in the quantum theory, to a second derivative. The left hand side is, furthermore, symmetric in a and pa. At this point we have the freedom in choosing the variable that will be quantized and the variable that will not be well defined in the polymer representation. The standard choice is that pa is not well defined and thus, a and any geometrical quantity derived from it, is quantized. Fur- thermore, we have the choice of polarization on the wave function. In this respect the standard choice is to select the a-polarization, in which a acts as multiplication and the approximation of pa, namely sin(λ pa)/λ acts as a difference operator on wave functions of a. For details of this particular choice see [5]. Here we shall adopt the op- posite polarization, that is, we shall have wave functions Ψ(pa, ϕ). Just as we did in the previous cases, in order to gain intuition about the behavior of the polymer quantized theory, it is convenient to look at the equivalent prob- lem in the classical theory, namely the classical system we would get be approximating the non-well defined ob- servable (pa in our present case) by a well defined object (made of trigonometric functions). Let us for simplicity choose to replace pa 7→ sin(λ pa)/λ. With this choice we get an effective classical Hamiltonian constraint that depends on λ: Cλ := − sin(λ pa) λ2|a| + 8πG 2|a|3 We can now compute effective equations of motion by means of the equations: Ḟ := {F, Cλ}, for any observable F ∈ C∞(Γ), and where we are using the effective (first order) action: dτ(pa ȧ+ pϕ ϕ̇−N Cλ) with the choice N = 1. The first thing to notice is that the quantity pϕ is a constant of the motion, given that the variable ϕ is cyclic. The second observation is that ϕ̇ = 8πG has the same sign as pϕ and never vanishes. Thus ϕ can be used as a (n internal) time variable. The next observation is that the equation for , namely the effective Friedman equation, will have a zero for a non-zero value of a given by λ2p2ϕ. This is the value at which there will be bounce if the trajectory started with a large value of a and was con- tracting. Note that the ‘size’ of the universe when the bounce occurs depends on both the constant pϕ (that dictates the matter density) and the value of the lattice size λ. Here it is important to stress that for any value of pϕ (that uniquely fixes the trajectory in the (a, pa) plane), there will be a bounce. In the original description in terms of Einstein’s equations (without the approxima- tion that depends on λ), there in no such bounce. If ȧ < 0 initially, it will remain negative and the universe collapses, reaching the singularity in a finite proper time. What happens within the effective description if we re- fine the lattice and go from λ to λn := λ/2 n? The only thing that changes, for the same classical orbit labelled by pϕ, is that the bounce occurs at a ‘later time’ and for a smaller value of a∗ but the qualitative picture remains the same. This is the main difference with the systems considered before. In those cases, one could have classical trajecto- ries that remained, for a given choice of parameter λ, within the region where sin(λp)/λ is a good approxima- tion to p. Of course there were also classical trajectories that were outside this region but we could then refine the lattice and find a new value λ′ for which the new clas- sical trajectory is well approximated. In the case of the polymer cosmology, this is never the case: Every classical trajectory will pass from a region where the approxima- tion is good to a region where it is not; this is precisely where the ‘quantum corrections’ kick in and the universes bounces. Given that in the classical description, the ‘original’ and the ‘corrected’ descriptions are so different we expect that, upon quantization, the corresponding quantum the- ories, namely the polymeric and the Wheeler-DeWitt will be related in a non-trivial way (if at all). In this case, with the choice of polarization and for a particular factor ordering we have, sin(λpa) · Ψ(pa, ϕ) = 0 as the Polymer Wheeler-DeWitt equation. In order to approach the problem of the continuum limit of this quantum theory, we have to realize that the task is now somewhat different than before. This is so given that the system is now a constrained system with a constraint operator rather than a regular non-singular system with an ordinary Hamiltonian evolution. Fortu- nately for the system under consideration, the fact that the variable ϕ can be regarded as an internal time allows us to interpret the quantum constraint as a generalized Klein-Gordon equation of the form Ψ = Θλ · Ψ where the operator Θλ is ‘time independent’. This al- lows us to split the space of solutions into ‘positive and negative frequency’, introduce a physical inner product on the positive frequency solutions of this equation and a set of physical observables in terms of which to de- scribe the system. That is, one reduces in practice the system to one very similar to the Schrödinger case by taking the positive square root of the previous equation: Θλ · Ψ. The question we are interested is whether the continuum limit of these theories (labelled by λ) exists and whether it corresponds to the Wheeler- DeWitt theory. A complete treatment of this problem lies, unfortunately, outside the scope of this work and will be reported elsewhere [12]. VII. DISCUSSION Let us summarize our results. In the first part of the article we showed that the polymer representation of the canonical commutation relations can be obtained as the limiting case of the ordinary Fock-Schrödinger represen- tation in terms of the algebraic state that defines the representation. These limiting cases can also be inter- preted in terms of the naturally defined coherent states associated to each representation labelled by the param- eter d, when they become infinitely ‘squeezed’. The two possible limits of squeezing lead to two different polymer descriptions that can nevertheless be identified, as we have also shown, with the two possible polarizations for an abstract polymer representation. This resulting the- ory has, however, very different behavior as the standard one: The Hilbert space is non-separable, the representa- tion is unitarily inequivalent to the Schrödinger one, and natural operators such as p̂ are no longer well defined. This particular limiting construction of the polymer the- ory can shed some light for more complicated systems such as field theories and gravity. In the regular treatments of dynamics within the poly- mer representation, one needs to introduce some extra structure, such as a lattice on configuration space, to con- struct a Hamiltonian and implement the dynamics for the system via a regularization procedure. How does this re- sulting theory compare to the original continuum theory one had from the beginning? Can one hope to remove the regulator in the polymer description? As they stand there is no direct relation or mapping from the polymer to a continuum theory (in case there is one defined). As we have shown, one can indeed construct in a systematic fashion such relation by means of some appropriate no- tions related to the definition of a scale, closely related to the lattice one had to introduce in the regularization. With this important shift in perspective, and an appro- priate renormalization of the polymer inner product at each scale one can, subject to some consistency condi- tions, define a procedure to remove the regulator, and arrive to a Hamiltonian and a Hilbert space. As we have seen, for some simple examples such as a free particle and the harmonic oscillator one indeed recovers the Schrödinger description back. For other sys- tems, such as quantum cosmological models, the answer is not as clear, since the structure of the space of classi- cal solutions is such that the ‘effective description’ intro- duced by the polymer regularization at different scales is qualitatively different from the original dynamics. A proper treatment of these class of systems is underway and will be reported elsewhere [12]. Perhaps the most important lesson that we have learned here is that there indeed exists a rich inter- play between the polymer description and the ordinary Schrödinger representation. 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Polymer Quantum Mechanics and its Continuum Limit Alejandro Corichi,1, 2, 3, ∗ Tatjana Vukašinac,4, † and José A. Zapata1, ‡ Instituto de Matemáticas, Unidad Morelia, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, A. Postal 61-3, Morelia, Michoacán 58090, Mexico Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D.F. 04510, Mexico Institute for Gravitational Physics and Geometry, Physics Department, Pennsylvania State University, University Park PA 16802, USA Facultad de Ingenieŕıa Civil, Universidad Michoacana de San Nicolas de Hidalgo, Morelia, Michoacán 58000, Mexico A rather non-standard quantum representation of the canonical commutation relations of quan- tum mechanics systems, known as the polymer representation has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schrödinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schrödinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schrödinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle and a simple cosmological model. PACS numbers: 04.60.Pp, 04.60.Ds, 04.60.Nc 11.10.Gh. I. INTRODUCTION The so-called polymer quantum mechanics, a non- regular and somewhat ‘exotic’ representation of the canonical commutation relations (CCR) [1], has been used to explore both mathematical and physical issues in background independent theories such as quantum grav- ity [2, 3]. A notable example of this type of quantization, when applied to minisuperspace models has given way to what is known as loop quantum cosmology [4, 5]. As in any toy model situation, one hopes to learn about the subtle technical and conceptual issues that are present in full quantum gravity by means of simple, finite di- mensional examples. This formalism is not an exception in this regard. Apart from this motivation coming from physics at the Planck scale, one can independently ask for the relation between the standard continuous repre- sentations and their polymer cousins at the level of math- ematical physics. A deeper understanding of this relation becomes important on its own. The polymer quantization is made of several steps. The first one is to build a representation of the Heisenberg-Weyl algebra on a Kinematical Hilbert space that is “background independent”, and that is sometimes referred to as the polymeric Hilbert space Hpoly. The second and most important part, the implementation of dynamics, deals with the definition of a Hamiltonian (or Hamiltonian constraint) on this space. In the examples ∗Electronic address: corichi@matmor.unam.mx †Electronic address: tatjana@shi.matmor.unam.mx ‡Electronic address: zapata@matmor.unam.mx studied so far, the first part is fairly well understood, yielding the kinematical Hilbert space Hpoly that is, how- ever, non-separable. For the second step, a natural im- plementation of the dynamics has proved to be a bit more difficult, given that a direct definition of the Hamiltonian Ĥ of, say, a particle on a potential on the space Hpoly is not possible since one of the main features of this repre- sentation is that the operators q̂ and p̂ cannot be both simultaneously defined (nor their analogues in theories involving more elaborate variables). Thus, any operator that involves (powers of) the not defined variable has to be regulated by a well defined operator which normally involves introducing some extra structure on the configu- ration (or momentum) space, namely a lattice. However, this new structure that plays the role of a regulator can not be removed when working in Hpoly and one is left with the ambiguity that is present in any regularization. The freedom in choosing it can be sometimes associated with a length scale (the lattice spacing). For ordinary quantum systems such as a simple harmonic oscillator, that has been studied in detail from the polymer view- point, it has been argued that if this length scale is taken to be ‘sufficiently small’, one can arbitrarily approximate standard Schrödinger quantum mechanics [2, 3]. In the case of loop quantum cosmology, the minimum area gap A0 of the full quantum gravity theory imposes such a scale, that is then taken to be fundamental [4]. A natural question is to ask what happens when we change this scale and go to even smaller ‘distances’, that is, when we refine the lattice on which the dynamics of the theory is defined. Can we define consistency con- ditions between these scales? Or even better, can we take the limit and find thus a continuum limit? As it http://arxiv.org/abs/0704.0007v2 mailto:corichi@matmor.unam.mx mailto:tatjana@shi.matmor.unam.mx mailto:zapata@matmor.unam.mx has been shown recently in detail, the answer to both questions is in the affirmative [6]. There, an appropriate notion of scale was defined in such a way that one could define refinements of the theory and pose in a precise fashion the question of the continuum limit of the theory. These results could also be seen as handing a procedure to remove the regulator when working on the appropri- ate space. The purpose of this paper is to further explore different aspects of the relation between the continuum and the polymer representation. In particular in the first part we put forward a novel way of deriving the polymer representation from the ordinary Schrödinger represen- tation as an appropriate limit. In Sec. II we derive two versions of the polymer representation as different lim- its of the Schrödinger theory. In Sec. III we show that these two versions can be seen as different polarizations of the ‘abstract’ polymer representation. These results, to the best of our knowledge, are new and have not been reported elsewhere. In Sec. IV we pose the problem of implementing the dynamics on the polymer representa- tion. In Sec. V we motivate further the question of the continuum limit (i.e. the proper removal of the regulator) and recall the basic constructions of [6]. Several exam- ples are considered in Sec. VI. In particular a simple harmonic oscillator, the polymer free particle and a sim- ple quantum cosmology model are considered. The free particle and the cosmological model represent a general- ization of the results obtained in [6] where only systems with a discrete and non-degenerate spectrum where con- sidered. We end the paper with a discussion in Sec. VII. In order to make the paper self-contained, we will keep the level of rigor in the presentation to that found in the standard theoretical physics literature. II. QUANTIZATION AND POLYMER REPRESENTATION In this section we derive the so called polymer repre- sentation of quantum mechanics starting from a specific reformulation of the ordinary Schrödinger representation. Our starting point will be the simplest of all possible phase spaces, namely Γ = R2 corresponding to a particle living on the real line R. Let us choose coordinates (q, p) thereon. As a first step we shall consider the quantization of this system that leads to the standard quantum theory in the Schrödinger description. A convenient route is to introduce the necessary structure to define the Fock rep- resentation of such system. From this perspective, the passage to the polymeric case becomes clearest. Roughly speaking by a quantization one means a passage from the classical algebraic bracket, the Poisson bracket, {q, p} = 1 (1) to a quantum bracket given by the commutator of the corresponding operators, [ q̂, p̂] = i~ 1̂ (2) These relations, known as the canonical commutation re- lation (CCR) become the most common corner stone of the (kinematics of the) quantum theory; they should be satisfied by the quantum system, when represented on a Hilbert space H. There are alternative points of departure for quantum kinematics. Here we consider the algebra generated by the exponentiated versions of q̂ and p̂ that are denoted U(α) = ei(α q̂)/~ ; V (β) = ei(β p̂)/~ where α and β have dimensions of momentum and length, respectively. The CCR now become U(α) · V (β) = e(−iα β)/~V (β) · U(α) (3) and the rest of the product is U(α1)·U(α2) = U(α1+α2) ; V (β1)·V (β2) = V (β1+β2) The Weyl algebra W is generated by taking finite linear combinations of the generators U(αi) and V (βi) where the product (3) is extended by linearity, (Ai U(αi) +Bi V (βi)) From this perspective, quantization means finding an unitary representation of the Weyl algebra W on a Hilbert space H′ (that could be different from the ordi- nary Schrödinger representation). At first it might look weird to attempt this approach given that we know how to quantize such a simple system; what do we need such a complicated object as W for? It is infinite dimensional, whereas the set S = {1̂, q̂, p̂}, the starting point of the ordinary Dirac quantization, is rather simple. It is in the quantization of field systems that the advantages of the Weyl approach can be fully appreciated, but it is also useful for introducing the polymer quantization and comparing it to the standard quantization. This is the strategy that we follow. A question that one can ask is whether there is any freedom in quantizing the system to obtain the ordinary Schrödinger representation. On a first sight it might seem that there is none given the Stone-Von Neumann unique- ness theorem. Let us review what would be the argument for the standard construction. Let us ask that the repre- sentation we want to build up is of the Schrödinger type, namely, where states are wave functions of configuration space ψ(q). There are two ingredients to the construction of the representation, namely the specification of how the basic operators (q̂, p̂) will act, and the nature of the space of functions that ψ belongs to, that is normally fixed by the choice of inner product on H, or measure µ on R. The standard choice is to select the Hilbert space to be, H = L2(R, dq) the space of square-integrable functions with respect to the Lebesgue measure dq (invariant under constant trans- lations) on R. The operators are then represented as, q̂ · ψ(q) = (q ψ)(q) and p̂ · ψ(q) = −i ~ ∂ ψ(q) (4) Is it possible to find other representations? In order to appreciate this freedom we go to the Weyl algebra and build the quantum theory thereon. The representation of the Weyl algebra that can be called of the ‘Fock type’ involves the definition of an extra structure on the phase space Γ: a complex structure J . That is, a linear map- ping from Γ to itself such that J2 = −1. In 2 dimen- sions, all the freedom in the choice of J is contained in the choice of a parameter d with dimensions of length. It is also convenient to define: k = p/~ that has dimensions of 1/L. We have then, Jd : (q, k) 7→ (−d2 k, q/d2) This object together with the symplectic structure: Ω((q, p); (q′, p′)) = q p′ − p q′ define an inner product on Γ by the formula gd(· ; ·) = Ω(· ; Jd ·) such that: gd((q, p); (q ′, p′)) = q q′ + which is dimension-less and positive definite. Note that with this quantities one can define complex coordinates (ζ, ζ̄) as usual: q + i p ; ζ̄ = q − i d from which one can build the standard Fock representa- tion. Thus, one can alternatively view the introduction of the length parameter d as the quantity needed to de- fine (dimensionless) complex coordinates on the phase space. But what is the relevance of this object (J or d)? The definition of complex coordinates is useful for the construction of the Fock space since from them one can define, in a natural way, creation and annihilation operators. But for the Schrödinger representation we are interested here, it is a bit more subtle. The subtlety is that within this approach one uses the algebraic prop- erties of W to construct the Hilbert space via what is known as the Gel’fand-Naimark-Segal (GNS) construc- tion. This implies that the measure in the Schrödinger representation becomes non trivial and thus the momen- tum operator acquires an extra term in order to render the operator self-adjoint. The representation of the Weyl algebra is then, when acting on functions φ(q) [7]: Û(α) · φ(q) := (eiα q/~ φ)(q) V̂ (β) · φ(q) := e (q−β/2) φ(q − β) The Hilbert space structure is introduced by the defini- tion of an algebraic state (a positive linear functional) ωd : W → C, that must coincide with the expectation value in the Hilbert space taken on a special state ref- ered to as the vacuum: ωd(a) = 〈â〉vac, for all a ∈ W . In our case this specification of J induces such a unique state ωd that yields, 〈Û(α)〉vac = e− d2 α2 ~2 (5) 〈V̂ (β)〉vac = e− d2 (6) Note that the exponents in the vacuum expectation values correspond to the metric constructed out of J : d2 α2 = gd((0, α); (0, α)) and = gd((β, 0); (β, 0)). Wave functions belong to the space L2(R, dµd), where the measure that dictates the inner product in this rep- resentation is given by, dµd = d2 dq In this representation, the vacuum is given by the iden- tity function φ0(q) = 1 that is, just as any plane wave, normalized. Note that for each value of d > 0, the rep- resentation is well defined and continuous in α and β. Note also that there is an equivalence between the q- representation defined by d and the k-representation de- fined by 1/d. How can we recover then the standard representation in which the measure is given by the Lebesgue measure and the operators are represented as in (4)? It is easy to see that there is an isometric isomorphism K that maps the d-representation in Hd to the standard Schrödinger representation in Hschr by: ψ(q) = K · φ(q) = e d1/2π1/4 φ(q) ∈ Hschr = L2(R, dq) Thus we see that all d-representations are unitarily equiv- alent. This was to be expected in view of the Stone-Von Neumann uniqueness result. Note also that the vacuum now becomes ψ0(q) = d1/2π1/4 2 d2 , so even when there is no information about the param- eter d in the representation itself, it is contained in the vacuum state. This procedure for constructing the GNS- Schrödinger representation for quantum mechanics has also been generalized to scalar fields on arbitrary curved space in [8]. Note, however that so far the treatment has all been kinematical, without any knowledge of a Hamil- tonian. For the Simple Harmonic Oscillator of mass m and frequency ω, there is a natural choice compatible with the dynamics given by d = , in which some calculations simplify (for instance for coherent states), but in principle one can use any value of d. Our study will be simplified by focusing on the funda- mental entities in the Hilbert Space Hd , namely those states generated by acting with Û(α) on the vacuum φ0(q) = 1. Let us denote those states by, φα(q) = Û(α) · φ0(q) = ei The inner product between two such states is given by 〈φα, φλ〉d = dµd e ~ = e− (λ−α)2 d2 4 ~2 (7) Note incidentally that, contrary to some common belief, the ‘plane waves’ in this GNS Hilbert space are indeed normalizable. Let us now consider the polymer representation. For that, it is important to note that there are two possible limiting cases for the parameter d: i) The limit 1/d 7→ 0 and ii) The case d 7→ 0. In both cases, we have ex- pressions that become ill defined in the representation or measure, so one needs to be careful. A. The 1/d 7→ 0 case. The first observation is that from the expressions (5) and (6) for the algebraic state ωd, we see that the limiting cases are indeed well defined. In our case we get, ωA := lim1/d→0 ωd such that, ωA(Û(α)) = δα,0 and ωA(V̂ (β)) = 1 (8) From this, we can indeed construct the representation by means of the GNS construction. In order to do that and to show how this is obtained we shall consider several expressions. One has to be careful though, since the limit has to be taken with care. Let us consider the measure on the representation that behaves as: dµd = d2 dq 7→ 1 so the measures tends to an homogeneous measure but whose ‘normalization constant’ goes to zero, so the limit becomes somewhat subtle. We shall return to this point later. Let us now see what happens to the inner product between the fundamental entities in the Hilbert Space Hd given by (7). It is immediate to see that in the 1/d 7→ 0 limit the inner product becomes, 〈φα, φλ〉d 7→ δα,λ (9) with δα,λ being Kronecker’s delta. We see then that the plane waves φα(q) become an orthonormal basis for the new Hilbert space. Therefore, there is a delicate interplay between the two terms that contribute to the measure in order to maintain the normalizability of these functions; we need the measure to become damped (by 1/d) in order to avoid that the plane waves acquire an infinite norm (as happens with the standard Lebesgue measure), but on the other hand the measure, that for any finite value of d is a Gaussian, becomes more and more spread. It is important to note that, in this limit, the operators Û(α) become discontinuous with respect to α, given that for any given α1 and α2 (different), its action on a given basis vector ψλ(q) yields orthogonal vectors. Since the continuity of these operators is one of the hypotesis of the Stone-Von Neumann theorem, the uniqueness result does not apply here. The representation is inequivalent to the standard one. Let us now analyze the other operator, namely the action of the operator V̂ (β) on the basis φα(q): V̂ (β) · φα(q) = e− ~ e(β/d 2+iα/~)q which in the limit 1/d 7→ 0 goes to, V̂ (β) · φα(q) 7→ ei ~ φα(q) that is continuous on β. Thus, in the limit, the operator p̂ = −i~∂q is well defined. Also, note that in this limit the operator p̂ has φα(q) as its eigenstate with eigenvalue given by α: p̂ · φα(q) 7→ αφα(q) To summarize, the resulting theory obtained by taking the limit 1/d 7→ 0 of the ordinary Schrödinger descrip- tion, that we shall call the ‘polymer representation of type A’, has the following features: the operators U(α) are well defined but not continuous in α, so there is no generator (no operator associated to q). The basis vec- tors φα are orthonormal (for α taking values on a contin- uous set) and are eigenvectors of the operator p̂ that is well defined. The resulting Hilbert space HA will be the (A-version of the) polymer representation. Let us now consider the other case, namely, the limit when d 7→ 0. B. The d 7→ 0 case Let us now explore the other limiting case of the Schrödinger/Fock representations labelled by the param- eter d. Just as in the previous case, the limiting algebraic state becomes, ωB := limd→0 ωd such that, ωB(Û(α)) = 1 and ωB(V̂ (β)) = δβ,0 (10) From this positive linear function, one can indeed con- struct the representation using the GNS construction. First let us note that the measure, even when the limit has to be taken with due care, behaves as: dµd = d2 dq 7→ δ(q) dq That is, as Dirac’s delta distribution. It is immediate to see that, in the d 7→ 0 limit, the inner product between the fundamental states φα(q) becomes, 〈φα, φλ〉d 7→ 1 (11) This in fact means that the vector ξ = φα − φλ belongs to the Kernel of the limiting inner product, so one has to mod out by these (and all) zero norm states in order to get the Hilbert space. Let us now analyze the other operator, namely the action of the operator V̂ (β) on the vacuum φ0(q) = 1, which for arbitrary d has the form, φ̃β := V̂ (β) · φ0(q) = e (q−β/2) The inner product between two such states is given by 〈φ̃α, φ̃β〉d = e− (α−β)2 In the limit d → 0, 〈φ̃α, φ̃β〉d → δα,β. We can see then that it is these functions that become the orthonormal, ‘discrete basis’ in the theory. However, the function φ̃β(q) in this limit becomes ill defined. For example, for β > 0, it grows unboundedly for q > β/2, is equal to one if q = β/2 and zero otherwise. In order to overcome these difficulties and make more transparent the resulting the- ory, we shall consider the other form of the representation in which the measure is incorporated into the states (and the resulting Hilbert space is L2(R, dq)). Thus the new state ψβ(q) := K · (V̂ (β) · φ0(q)) = (q−β)2 We can now take the limit and what we get is d 7→0 ψβ(q) := δ 1/2(q, β) where by δ1/2(q, β) we mean something like ‘the square root of the Dirac distribution’. What we really mean is an object that satisfies the following property: δ1/2(q, β) · δ1/2(q, α) = δ(q, β) δβ,α That is, if α = β then it is just the ordinary delta, other- wise it is zero. In a sense these object can be regarded as half-densities that can not be integrated by themselves, but whose product can. We conclude then that the inner product is, 〈ψβ , ψα〉 = dq ψβ(q)ψα(q) = dq δ(q, α) δβ,α = δβ,α which is just what we expected. Note that in this repre- sentation, the vacuum state becomes ψ0(q) := δ 1/2(q, 0), namely, the half-delta with support in the origin. It is important to note that we are arriving in a natural way to states as half-densities, whose squares can be integrated without the need of a nontrivial measure on the configu- ration space. Diffeomorphism invariance arises then in a natural but subtle manner. Note that as the end result we recover the Kronecker delta inner product for the new fundamental states: χβ(q) := δ 1/2(q, β). Thus, in this new B-polymer representation, the Hilbert space HB is the completion with respect to the inner product (13) of the states generated by taking (finite) linear combinations of basis elements of the form χβ : Ψ(q) = bi χβi(q) (14) Let us now introduce an equivalent description of this Hilbert space. Instead of having the basis elements be half-deltas as elements of the Hilbert space where the inner product is given by the ordinary Lebesgue measure dq, we redefine both the basis and the measure. We could consider, instead of a half-delta with support β, a Kronecker delta or characteristic function with support on β: χ′β(q) := δq,β These functions have a similar behavior with respect to the product as the half-deltas, namely: χ′β(q) · χ′α(q) = δβ,α. The main difference is that neither χ ′ nor their squares are integrable with respect to the Lebesgue mea- sure (having zero norm). In order to fix that problem we have to change the measure so that we recover the basic inner product (13) with our new basis. The needed mea- sure turns out to be the discrete counting measure on R. Thus any state in the ‘half density basis’ can be written (using the same expression) in terms of the ‘Kronecker basis’. For more details and further motivation see the next section. Note that in this B-polymer representation, both Û and V̂ have their roles interchanged with that of the A-polymer representation: while U(α) is discontinuous and thus q̂ is not defined in the A-representation, we have that it is V (β) in the B-representation that has this property. In this case, it is the operator p̂ that can not be defined. We see then that given a physical system for which the configuration space has a well defined physi- cal meaning, within the possible representation in which wave-functions are functions of the configuration variable q, the A and B polymer representations are radically dif- ferent and inequivalent. Having said this, it is also true that the A and B representations are equivalent in a different sense, by means of the duality between q and p representations and the d↔ 1/d duality: The A-polymer representation in the “q-representation” is equivalent to the B-polymer representation in the “p-representation”, and conversely. When studying a problem, it is important to decide from the beginning which polymer representation (if any) one should be using (for instance in the q-polarization). This has as a consequence an implication on which variable is naturally “quantized” (even if continuous): p for A and q for B. There could be for instance a physical criteria for this choice. For example a fundamental symmetry could suggest that one representation is more natural than an- other one. This indeed has been recently noted by Chiou in [10], where the Galileo group is investigated and where it is shown that the B representation is better behaved. In the other polarization, namely for wavefunctions of p, the picture gets reversed: q is discrete for the A- representation, while p is for the B-case. Let us end this section by noting that the procedure of obtaining the polymer quantization by means of an appropriate limit of Fock-Schrödinger representations might prove useful in more general settings in field theory or quantum gravity. III. POLYMER QUANTUM MECHANICS: KINEMATICS In previous sections we have derived what we have called the A and B polymer representations (in the q- polarization) as limiting cases of ordinary Fock repre- sentations. In this section, we shall describe, without any reference to the Schrödinger representation, the ‘ab- stract’ polymer representation and then make contact with its two possible realizations, closely related to the A and B cases studied before. What we will see is that one of them (the A case) will correspond to the p-polarization while the other one corresponds to the q−representation, when a choice is made about the physical significance of the variables. We can start by defining abstract kets |µ〉 labelled by a real number µ. These shall belong to the Hilbert space Hpoly. From these states, we define a generic ‘cylinder states’ that correspond to a choice of a finite collection of numbers µi ∈ R with i = 1, 2, . . . , N . Associated to this choice, there are N vectors |µi〉, so we can take a linear combination of them |ψ〉 = ai |µi〉 (15) The polymer inner product between the fundamental kets is given by, 〈ν|µ〉 = δν,µ (16) That is, the kets are orthogonal to each other (when ν 6= µ) and they are normalized (〈µ|µ〉 = 1). Immediately, this implies that, given any two vectors |φ〉 = j=1 bj |νj〉 and |ψ〉 = i=1 ai |µi〉, the inner product between them is given by, 〈φ|ψ〉 = b̄j ai 〈νj |µi〉 = b̄k ak where the sum is over k that labels the intersection points between the set of labels {νj} and {µi}. The Hilbert space Hpoly is the Cauchy completion of finite linear com- bination of the form (15) with respect to the inner prod- uct (16). Hpoly is non-separable. There are two basic operators on this Hilbert space: the ‘label operator’ ε̂: ε̂ |µ〉 := µ |µ〉 and the displacement operator ŝ (λ), ŝ (λ) |µ〉 := |µ+ λ〉 The operator ε̂ is symmetric and the operator(s) ŝ(λ) defines a one-parameter family of unitary operators on Hpoly, where its adjoint is given by ŝ† (λ) = ŝ (−λ). This action is however, discontinuous with respect to λ given that |µ〉 and |µ + λ〉 are always orthogonal, no matter how small is λ. Thus, there is no (Hermitian) operator that could generate ŝ (λ) by exponentiation. So far we have given the abstract characterization of the Hilbert space, but one would like to make contact with concrete realizations as wave functions, or by iden- tifying the abstract operators ε̂ and ŝ with physical op- erators. Suppose we have a system with a configuration space with coordinate given by q, and p denotes its canonical conjugate momenta. Suppose also that for physical rea- sons we decide that the configuration coordinate q will have some “discrete character” (for instance, if it is to be identified with position, one could say that there is an underlying discreteness in position at a small scale). How can we implement such requirements by means of the polymer representation? There are two possibilities, depending on the choice of ‘polarizations’ for the wave- functions, namely whether they will be functions of con- figuration q or momenta p. Let us the divide the discus- sion into two parts. A. Momentum polarization In this polarization, states will be denoted by, ψ(p) = 〈p|ψ〉 where ψµ(p) = 〈p|µ〉 = ei How are then the operators ε̂ and ŝ represented? Note that if we associate the multiplicative operator V̂ (λ) · ψµ(p) = ei ~ = ei (µ+λ) p = ψ(µ+λ)(p) we see then that the operator V̂ (λ) corresponds precisely to the shift operator ŝ (λ). Thus we can also conclude that the operator p̂ does not exist. It is now easy to identify the operator q̂ with: q̂ · ψµ(p) = −i~ ψµ(p) = µ e ~ = µψµ(p) namely, with the abstract operator ε̂. The reason we say that q̂ is discrete is because this operator has as its eigenvalue the label µ of the elementary state ψµ(p), and this label, even when it can take value in a continuum of possible values, is to be understood as a discrete set, given that the states are orthonormal for all values of µ. Given that states are now functions of p, the inner product (16) should be defined by a measure µ on the space on which the wave-functions are defined. In order to know what these two objects are, namely, the quan- tum “configuration” space C and the measure thereon1, we have to make use of the tools available to us from the theory of C∗-algebras. If we consider the operators V̂ (λ), together with their natural product and ∗-relation given by V̂ ∗(λ) = V̂ (−λ), they have the structure of an Abelian C∗-algebra (with unit) A. We know from the representation theory of such objects that A is iso- morphic to the space of continuous functions C0(∆) on a compact space ∆, the spectrum of A. Any representation of A on a Hilbert space as multiplication operator will be on spaces of the form L2(∆, dµ). That is, our quantum configuration space is the spectrum of the algebra, which in our case corresponds to the Bohr compactification Rb of the real line [11]. This space is a compact group and there is a natural probability measure defined on it, the Haar measure µH. Thus, our Hilbert space Hpoly will be isomorphic to the space, Hpoly,p = L2(Rb, dµH) (17) In terms of ‘quasi periodic functions’ generated by ψµ(p), the inner product takes the form 〈ψµ|ψλ〉 := dµH ψµ(p)ψλ(p) := = lim L 7→∞ dpψµ(p)ψλ(p) = δµ,λ (18) note that in the p-polarization, this characterization cor- responds to the ‘A-version’ of the polymer representation of Sec. II (where p and q are interchanged). B. q-polarization Let us now consider the other polarization in which wave functions will depend on the configuration coordinate q: ψ(q) = 〈q|ψ〉 The basic functions, that now will be called ψ̃µ(q), should be, in a sense, the dual of the functions ψµ(p) of the previous subsection. We can try to define them via a ‘Fourier transform’: ψ̃µ(q) := 〈q|µ〉 = 〈q| dµH|p〉〈p|µ〉 which is given by ψ̃µ(q) := dµH〈q|p〉ψµ(p) = dµH e −i p q ~ = δq,µ (19) 1 here we use the standard terminology of ‘configuration space’ to denote the domain of the wave function even when, in this case, it corresponds to the physical momenta p. That is, the basic objects in this representation are Kro- necker deltas. This is precisely what we had found in Sec. II for the B-type representation. How are now the basic operators represented and what is the form of the inner product? Regarding the operators, we expect that they are represented in the opposite manner as in the previous p-polarization case, but that they preserve the same features: p̂ does not exist (the derivative of the Kro- necker delta is ill defined), but its exponentiated version V̂ (λ) does: V̂ (λ) · ψ(q) = ψ(q + λ) and the operator q̂ that now acts as multiplication has as its eigenstates, the functions ψ̃ν(q) = δν,q: q̂ · ψ̃µ(q) := µ ψ̃µ(q) What is now the nature of the quantum configurations space Q? And what is the measure thereon dµq? that defines the inner product we should have: 〈ψ̃µ(q), ψ̃λ(q)〉 = δµ,λ The answer comes from one of the characterizations of the Bohr compactification: we know that it is, in a precise sense, dual to the real line but when equipped with the discrete topology Rd. Furthermore, the measure on Rd will be the ‘counting measure’. In this way we recover the same properties we had for the previous characterization of the polymer Hilbert space. We can thus write: Hpoly,x := L2(Rd, dµc) (20) This completes a precise construction of the B-type poly- mer representation sketched in the previous section. Note that if we had chosen the opposite physical situation, namely that q, the configuration observable, be the quan- tity that does not have a corresponding operator, then we would have had the opposite realization: In the q- polarization we would have had the type-A polymer rep- resentation and the type-B for the p-polarization. As we shall see both scenarios have been considered in the literature. Up to now we have only focused our discussion on the kinematical aspects of the quantization process. Let us now consider in the following section the issue of dynam- ics and recall the approach that had been adopted in the literature, before the issue of the removal of the regulator was reexamined in [6]. IV. POLYMER QUANTUM MECHANICS: DYNAMICS As we have seen the construction of the polymer representation is rather natural and leads to a quan- tum theory with different properties than the usual Schrödinger counterpart such as its non-separability, the non-existence of certain operators and the existence of normalized eigen-vectors that yield a precise value for one of the phase space coordinates. This has been done without any regard for a Hamiltonian that endows the system with a dynamics, energy and so on. First let us consider the simplest case of a particle of mass m in a potential V (q), in which the Hamiltonian H takes the form, p2 + V (q) Suppose furthermore that the potential is given by a non- periodic function, such as a polynomial or a rational func- tion. We can immediately see that a direct implementa- tion of the Hamiltonian is out of our reach, for the simple reason that, as we have seen, in the polymer representa- tion we can either represent q or p, but not both! What has been done so far in the literature? The simplest thing possible: approximate the non-existing term by a well defined function that can be quantized and hope for the best. As we shall see in next sections, there is indeed more that one can do. At this point there is also an important decision to be made: which variable q or p should be regarded as “dis- crete”? Once this choice is made, then it implies that the other variable will not exist: if q is regarded as dis- crete, then p will not exist and we need to approximate the kinetic term p2/2m by something else; if p is to be the discrete quantity, then q will not be defined and then we need to approximate the potential V (q). What hap- pens with a periodic potential? In this case one would be modelling, for instance, a particle on a regular lattice such as a phonon living on a crystal, and then the natural choice is to have q not well defined. Furthermore, the po- tential will be well defined and there is no approximation needed. In the literature both scenarios have been considered. For instance, when considering a quantum mechanical system in [2], the position was chosen to be discrete, so p does not exist, and one is then in the A type for the momentum polarization (or the type B for the q- polarization). With this choice, it is the kinetic term the one that has to be approximated, so once one has done this, then it is immediate to consider any potential that will thus be well defined. On the other hand, when con- sidering loop quantum cosmology (LQC), the standard choice is that the configuration variable is not defined [4]. This choice is made given that LQC is regarded as the symmetric sector of full loop quantum gravity where the connection (that is regarded as the configuration vari- able) can not be promoted to an operator and one can only define its exponentiated version, namely, the holon- omy. In that case, the canonically conjugate variable, closely related to the volume, becomes ‘discrete’, just as in the full theory. This case is however, different from the particle in a potential example. First we could mention that the functional form of the Hamiltonian constraint that implements dynamics has a different structure, but the more important difference lies in that the system is constrained. Let us return to the case of the particle in a po- tential and for definiteness, let us start with the aux- iliary kinematical framework in which: q is discrete, p can not be promoted and thus we have to approximate the kinetic term p̂2/2m. How is this done? The stan- dard prescription is to define, on the configuration space C, a regular ‘graph’ γµ0 . This consists of a numerable set of points, equidistant, and characterized by a pa- rameter µ0 that is the (constant) separation between points. The simplest example would be to consider the set γµ0 = {q ∈ R | q = nµ0 , ∀ n ∈ Z}. This means that the basic kets that will be considered |µn〉 will correspond precisely to labels µn belonging to the graph γµ0 , that is, µn = nµ0. Thus, we shall only consider states of the form, |ψ〉 = bn |µn〉 . (21) This ‘small’ Hilbert space Hγµ0 , the graph Hilbert space, is a subspace of the ‘large’ polymer Hilbert space Hpoly but it is separable. The condition for a state of the form (21) to belong to the Hilbert space Hγµ0 is that the co- efficients bn satisfy: n |bn|2 <∞. Let us now consider the kinetic term p̂2/2m. We have to approximate it by means of trigonometric functions, that can be built out of the functions of the form eiλ p/~. As we have seen in previous sections, these functions can indeed be promoted to operators and act as translation operators on the kets |µ〉. If we want to remain in the graph γ, and not create ‘new points’, then one is con- strained to considering operators that displace the kets by just the right amount. That is, we want the basic shift operator V̂ (λ) to be such that it maps the ket with label |µn〉 to the next ket, namely |µn+1〉. This can in- deed achieved by fixing, once and for all, the value of the allowed parameter λ to be λ = µ0. We have then, V̂ (µ0) · |µn〉 = |µn + µ0〉 = |µn+1〉 which is what we wanted. This basic ‘shift operator’ will be the building block for approximating any (polynomial) function of p. In order to do that we notice that the function p can be approximated by, p ≈ ~ (µ0 p ~ − e−i where the approximation is good for p << ~/µ0. Thus, one can define a regulated operator p̂µ0 that depends on the ‘scale’ µ0 as: p̂µ0 · |µn〉 := [V (µ0) − V (−µ0)] · |µn〉 = (|µn+1〉 − |µn−1〉) (22) In order to regulate the operator p̂2, there are (at least) two possibilities, namely to compose the operator p̂µ0 with itself or to define a new approximation. The oper- ator p̂µ0 · p̂µ0 has the feature that shifts the states two steps in the graph to both sides. There is however an- other operator that only involves shifting once: p̂2µ0 · |νn〉 := [2 − V̂ (µ0) − V̂ (−µ0)] · |νn〉 = (2|νn〉 − |νn+1〉 − |νn−1〉) (23) which corresponds to the approximation p2 ≈ 2~ cos(µ0 p/~)), valid also in the regime p << ~/µ0. With these considerations, one can define the operator Ĥµ0 , the Hamiltonian at scale µ0, that in practice ‘lives’ on the space Hγµ0 as, Ĥµ0 := p̂2µ0 + V̂ (q) , (24) that is a well defined, symmetric operator on Hγµ0 . No- tice that the operator is also defined on Hpoly, but there its physical interpretation is problematic. For example, it turns out that the expectation value of the kinetic term calculated on most states (states which are not tailored to the exact value of the parameter µ0) is zero. Even if one takes a state that gives “reasonable“ expectation values of the µ0-kinetic term and uses it to calculate the expectation value of the kinetic term corresponding to a slight perturbation of the parameter µ0 one would get zero. This problem, and others that arise when working on Hpoly, forces one to assign a physical interpretation to the Hamiltonian Ĥµ0 only when its action is restricted to the subspace Hγµ0 . Let us now explore the form that the Hamiltonian takes in the two possible polarizations. In the q-polarization, the basis, labelled by n is given by the functions χn(q) = δq,µn . That is, the wave functions will only have sup- port on the set γµ0 . Alternatively, one can think of a state as completely characterized by the ‘Fourier coeffi- cients’ an: ψ(q) ↔ an, which is the value that the wave function ψ(q) takes at the point q = µn = nµ0. Thus, the Hamiltonian takes the form of a difference equation when acting on a general state ψ(q). Solving the time independent Schrödinger equation Ĥ · ψ = E ψ amounts to solving the difference equation for the coefficients an. The momentum polarization has a different structure. In this case, the operator p̂2µ0 acts as a multiplication operator, p̂2µ0 · ψ(p) = 1 − cos (µ0 p ψ(p) (25) The operator corresponding to q will be represented as a derivative operator q̂ · ψ(p) := i~ ∂p ψ(p). For a generic potential V (q), it has to be defined by means of spectral theory defined now on a circle. Why on a circle? For the simple reason that by restricting ourselves to a regular graph γµ0 , the functions of p that preserve it (when acting as shift operators) are of the form e(i m µ0 p/~) for m integer. That is, what we have are Fourier modes, labelled by m, of period 2π ~/µ0 in p. Can we pretend then that the phase space variable p is now compactified? The answer is in the affirmative. The inner product on periodic functions ψµ0(p) of p coming from the full Hilbert space Hpoly and given by 〈φ(p)|ψ(p)〉poly = lim L 7→∞ dp φ(p)ψ(p) is precisely equivalent to the inner product on the circle given by the uniform measure 〈φ(p)|ψ(p)〉µ0 = ∫ π~/µ0 −π~/µ0 dp φ(p)ψ(p) with p ∈ (−π~/µ0, π~/µ0). As long as one restricts at- tention to the graph γµ0 , one can work in this separable Hilbert space Hγµ0 of square integrable functions on S Immediately, one can see the limitations of this descrip- tion. If the mechanical system to be quantized is such that its orbits have values of the momenta p that are not small compared with π~/µ0 then the approximation taken will be very poor, and we don’t expect neither the effective classical description nor its quantization to be close to the standard one. If, on the other hand, one is al- ways within the region in which the approximation can be regarded as reliable, then both classical and quantum de- scriptions should approximate the standard description. What does ‘close to the standard description’ exactly mean needs, of course, some further clarification. In particular one is assuming the existence of the usual Schrödinger representation in which the system has a be- havior that is also consistent with observations. If this is the case, the natural question is: How can we approxi- mate such description from the polymer picture? Is there a fine enough graph γµ0 that will approximate the system in such a way that all observations are indistinguishable? Or even better, can we define a procedure, that involves a refinement of the graph γµ0 such that one recovers the standard picture? It could also happen that a continuum limit can be de- fined but does not coincide with the ‘expected one’. But there might be also physical systems for which there is no standard description, or it just does not make sense. Can in those cases the polymer representation, if it ex- ists, provide the correct physical description of the sys- tem under consideration? For instance, if there exists a physical limitation to the minimum scale set by µ0, as could be the case for a quantum theory of gravity, then the polymer description would provide a true physical bound on the value of certain quantities, such as p in our example. This could be the case for loop quantum cosmology, where there is a minimum value for physical volume (coming from the full theory), and phase space points near the ‘singularity’ lie at the region where the approximation induced by the scale µ0 departs from the standard classical description. If in that case the poly- mer quantum system is regarded as more fundamental than the classical system (or its standard Wheeler-De Witt quantization), then one would interpret this dis- crepancies in the behavior as a signal of the breakdown of classical description (or its ‘naive’ quantization). In the next section we present a method to remove the regulator µ0 which was introduced as an intermedi- ate step to construct the dynamics. More precisely, we shall consider the construction of a continuum limit of the polymer description by means of a renormalization procedure. V. THE CONTINUUM LIMIT This section has two parts. In the first one we motivate the need for a precise notion of the continuum limit of the polymeric representation, explaining why the most direct, and naive approach does not work. In the sec- ond part, we shall present the main ideas and results of the paper [6], where the Hamiltonian and the physical Hilbert space in polymer quantum mechanics are con- structed as a continuum limit of effective theories, follow- ing Wilson’s renormalization group ideas. The resulting physical Hilbert space turns out to be unitarily isomor- phic to the ordinary Hs = L2(R, dq) of the Schrödinger theory. Before describing the results of [6] we should discuss the precise meaning of reaching a theory in the contin- uum. Let us for concreteness consider the B-type repre- sentation in the q-polarization. That is, states are func- tions of q and the orthonormal basis χµ(q) is given by characteristic functions with support on q = µ. Let us now suppose we have a Schrödinger state Ψ(q) ∈ Hs = L2(R, dq). What is the relation between Ψ(q) and a state in Hpoly,x? We are also interested in the opposite ques- tion, that is, we would like to know if there is a preferred state in Hs that is approximated by an arbitrary state ψ(q) in Hpoly,x. The first obvious observation is that a Schödinger state Ψ(q) does not belong to Hpoly,x since it would have an infinite norm. To see that note that even when the would-be state can be formally expanded in the χµ basis as, Ψ(q) = Ψ(µ) χµ(q) where the sum is over the parameter µ ∈ R. Its associ- ated norm in Hpoly,x would be: |Ψ(q)|2poly = |Ψ(µ)|2 → ∞ which blows up. Note that in order to define a mapping P : Hs → Hpoly,x, there is a huge ambiguity since the values of the function Ψ(q) are needed in order to expand the polymer wave function. Thus we can only define a mapping in a dense subset D of Hs where the values of the functions are well defined (recall that in Hs the value of functions at a given point has no meaning since states are equivalence classes of functions). We could for instance ask that the mapping be defined for representatives of the equivalence classes in Hs that are piecewise continuous. From now on, when we refer to an element of the space Hs we shall be refereeing to one of those representatives. Notice then that an element of Hs does define an element of Cyl∗γ , the dual to the space Cylγ , that is, the space of cylinder functions with support on the (finite) lattice γ = {µ1, µ2, . . . , µN}, in the following way: Ψ(q) : Cylγ −→ C such that Ψ(q)[ψ(q)] = (Ψ|ψ〉 := Ψ(µ) 〈χµ| ψi χµi〉polyγ Ψ(µi)ψi < ∞ (26) Note that this mapping could be seen as consisting of two parts: First, a projection Pγ : Cyl ∗ → Cylγ such that Pγ(Ψ) = Ψγ(q) := i Ψ(µi)χµi(q) ∈ Cylγ . The state Ψγ is sometimes refereed to as the ‘shadow of Ψ(q) on the lattice γ’. The second step is then to take the inner product between the shadow Ψγ(q) and the state ψ(q) with respect to the polymer inner product 〈Ψγ |ψ〉polyγ . Now this inner product is well defined. Notice that for any given lattice γ the corresponding projector Pγ can be intuitively interpreted as some kind of ‘coarse graining map’ from the continuum to the lattice γ. In terms of functions of q the projection is replacing a continuous function defined on R with a function over the lattice γ ⊂ R which is a discrete set simply by restricting Ψ to γ. The finer the lattice the more points that we have on the curve. As we shall see in the second part of this section, there is indeed a precise notion of coarse graining that implements this intuitive idea in a concrete fashion. In particular, we shall need to replace the lattice γ with a decomposition of the real line in intervals (having the lattice points as end points). Let us now consider a system in the polymer represen- tation in which a particular lattice γ0 was chosen, say with points of the form {qk ∈ R |qk = ka0 , ∀ k ∈ Z}, namely a uniform lattice with spacing equal to a0. In this case, any Schrödinger wave function (of the type that we consider) will have a unique shadow on the lattice γ0. If we refine the lattice γ 7→ γn by dividing each interval in 2n new intervals of length an = a0/2 n we have new shad- ows that have more and more points on the curve. Intu- itively, by refining infinitely the graph we would recover the original function Ψ(q). Even when at each finite step the corresponding shadow has a finite norm in the poly- mer Hilbert space, the norm grows unboundedly and the limit can not be taken, precisely because we can not em- bed Hs into Hpoly. Suppose now that we are interested in the reverse process, namely starting from a polymer theory on a lattice and asking for the ‘continuum wave function’ that is best approximated by a wave function over a graph. Suppose furthermore that we want to con- sider the limit of the graph becoming finer. In order to give precise answers to these (and other) questions we need to introduce some new technology that will allow us to overcome these apparent difficulties. In the remaining of this section we shall recall these constructions for the benefit of the reader. Details can be found in [6] (which is an application of the general formalism discussed in [9]). The starting point in this construction is the concept of a scale C, which allows us to define the effective the- ories and the concept of continuum limit. In our case a scale is a decomposition of the real line in the union of closed-open intervals, that cover the whole line and do not intersect. Intuitively, we are shifting the emphasis from the lattice points to the intervals defined by the same points with the objective of approximating con- tinuous functions defined on R with functions that are constant on the intervals defined by the lattice. To be precise, we define an embedding, for each scale Cn from Hpoly to Hs by means of a step function: Ψ(man) χman(q) → Ψ(man) χαm(q) ∈ Hs with χαn(q) a characteristic function on the interval αm = [man, (m + 1)an). Thus, the shadows (living on the lattice) were just an intermediate step in the con- struction of the approximating function; this function is piece-wise constant and can be written as a linear com- bination of step functions with the coefficients provided by the shadows. The challenge now is to define in an appropriate sense how one can approximate all the aspects of the theory by means of this constant by pieces functions. Then the strategy is that, for any given scale, one can define an effective theory by approximating the kinetic operator by a combination of the translation operators that shift between the vertices of the given decomposition, in other words by a periodic function in p. As a result one has a set of effective theories at given scales which are mutually related by coarse graining maps. This framework was developed in [6]. For the convenience of the reader we briefly recall part of that framework. Let us denote the kinematic polymer Hilbert space at the scale Cn as HCn , and its basis elements as eαi,Cn , where αi = [ian, (i + 1)an) ∈ Cn. By construction this basis is orthonormal. The basis elements in the dual Hilbert space H∗Cn are denoted by ωαi,Cn ; they are also orthonormal. The states ωαi,Cn have a simple action on Cyl, ωαi,Cn(δx0,q) = χαi,Cn(x0). That is, if x0 is in the interval αi of Cn the result is one and it is zero if it is not there. Given any m ≤ n, we define d∗m,n : H∗Cn → H as the ‘coarse graining’ map between the dual Hilbert spaces, that sends the part of the elements of the dual basis to zero while keeping the information of the rest: d∗m,n(ωαi,Cn) = ωβj ,Cm if i = j2 n−m, in the opposite case d∗m,n(ωαi,Cn) = 0. At every scale the corresponding effective theory is given by the hamiltonian Hn. These Hamiltonians will be treated as quadratic forms, hn : HCn → R, given by hn(ψ) = λ (ψ,Hnψ) , (27) where λ2Cn is a normalizaton factor. We will see later that this rescaling of the inner product is necessary in order to guarantee the convergence of the renormalized theory. The completely renormalized theory at this scale is obtained as hrenm := lim d⋆m,nhn. (28) and the renormalized Hamiltonians are compatible with each other, in the sense that d⋆m,nh n = h In order to analyze the conditions for the convergence in (28) let us express the Hamiltonian in terms of its eigen-covectors end eigenvalues. We will work with effec- tive Hamiltonians that have a purely discrete spectrum (labelled by ν) Hn · Ψν,Cn = Eν,Cn Ψν,Cn . We shall also introduce, as an intermediate step, a cut-off in the energy levels. The origin of this cut-off is in the approximation of the Hamiltonian of our system at a given scale with a Hamiltonian of a periodic system in a regime of small energies, as we explained earlier. Thus, we can write hνcut−offm = νcut−off Eν,CmΨν,Cm ⊗ Ψν,Cm , (29) where the eigen covectors Ψν,Cm are normalized accord- ing to the inner product rescaled by 1 , and the cut- off can vary up to a scale dependent bound, νcut−off ≤ νmax(Cm). The Hilbert space of covectors together with such inner product will be called H⋆renCm . In the presence of a cut-off, the convergence of the microscopically corrected Hamiltonians, equation (28) is equivalent to the existence of the following two limits. The first one is the convergence of the energy levels, Eν,Cn = E ν . (30) Second is the existence of the completely renormalized eigen covectors, d⋆m,n Ψν,Cn = Ψ ∈ H⋆renCm ⊂ Cyl ⋆ . (31) We clarify that the existence of the above limit means that Ψrenν,Cm(δx0,q) is well defined for any δx0,q ∈ Cyl. No- tice that this point-wise convergence, if it can take place at all, will require the tuning of the normalization factors λ2Cn . Now we turn to the question of the continuum limit of the renormalized covectors. First we can ask for the existence of the limit Ψrenν,Cn(δx0,q) (32) for any δx0,q ∈ Cyl. When this limits exists there is a natural action of the eigen covectors in the continuum limit. Below we consider another notion of the continuum limit of the renormalized eigen covectors. When the completely renormalized eigen covectors exist, they form a collection that is d⋆-compatible, d⋆m,nΨ = Ψrenν,Cm . A sequence of d ⋆-compatible nor- malizable covectors define an element of , which is the projective limit of the renormalized spaces of covec- H⋆renCn . (33) The inner product in this space is defined by ({ΨCn}, {ΦCn})renR := lim (ΨCn ,ΦCn) The natural inclusion of C∞0 in is by an antilinear map which assigns to any Ψ ∈ C∞0 the d⋆-compatible collection ΨshadCn := ωαiΨ̄(L(αi)) ∈ H⋆renCn ⊂ Cyl ΨshadCn will be called the shadow of Ψ at scale Cn and acts in Cyl as a piecewise constant function. Clearly other types of test functions like Schwartz functions are also naturally included in . In this context a shadow is a state of the effective theory that approximates a state in the continuum theory. Since the inner product in is degenerate, the physical Hilbert space is defined as H⋆phys := / ker(·, ·)ren Hphys := H⋆⋆phys The nature of the physical Hilbert space, whether it is isomorphic to the Schrödinger Hilber space, Hs, or not, is determined by the normalization factors λ2Cn which can be obtained from the conditions asking for compatibil- ity of the dynamics of the effective theories at different scales. The dynamics of the system under consideration selects the continuum limit. Let us now return to the definition of the Hamilto- nian in the continuum limit. First consider the contin- uum limit of the Hamiltonian (with cut-off) in the sense of its point-wise convergence as a quadratic form. It turns out that if the limit of equation (32) exists for all the eigencovectors allowed by the cut-off, we have νcut−off ren : Hpoly,x → R defined by νcut−off ren (δx0,q) := lim hνcut−off renn ([δx0,q]Cn). (34) This Hamiltonian quadratic form in the continuum can be coarse grained to any scale and, as can be ex- pected, it yields the completely renormalized Hamilto- nian quadratic forms at that scale. However, this is not a completely satisfactory continuum limit because we can not remove the auxiliary cut-off νcut−off . If we tried, as we include more and more eigencovectors in the Hamilto- nian the calculations done at a given scale would diverge and doing them in the continuum is just as divergent. Below we explore a more successful path. We can use the renormalized inner product to induce an action of the cut–off Hamiltonians on νcut−off ren ({ΨCn}) := lim hνcut−off renn ((ΨCn , ·)renCn ), where we have used the fact that (ΨCn , ·)renCn ∈ HCn . The existence of this limit is trivial because the renormalized Hamiltonians are finite sums and the limit exists term by term. These cut-off Hamiltonians descend to the physical Hilbert space νcut−off ren ([{ΨCn}]) := h νcut−off ren ({ΨCn}) for any representative {ΨCn} ∈ [{ΨCn}] ∈ H⋆phys. Finally we can address the issue of removal of the cut- off. The Hamiltonian hren → R is defined by the limit := lim νcut−off→∞ νcut−off ren when the limit exists. Its corresponding Hermitian form in Hphys is defined whenever the above limit exists. This concludes our presentation of the main results of [6]. Let us now consider several examples of systems for which the continuum limit can be investigated. VI. EXAMPLES In this section we shall develop several examples of systems that have been treated with the polymer quanti- zation. These examples are simple quantum mechanical systems, such as the simple harmonic oscillator and the free particle, as well as a quantum cosmological model known as loop quantum cosmology. A. The Simple Harmonic Oscillator In this part, let us consider the example of a Simple Har- monic Oscillator (SHO) with parameters m and ω, clas- sically described by the following Hamiltonian mω2 x2. Recall that from these parameters one can define a length scale D = ~/mω. In the standard treatment one uses this scale to define a complex structure JD (and an in- ner product from it), as we have described in detail that uniquely selects the standard Schrödinger representation. At scale Cn we have an effective Hamiltonian for the Simple Harmonic Oscillator (SHO) given by HCn = 1 − cos anp mω2x2 . (35) If we interchange position and momentum, this Hamilto- nian is exactly that of a pendulum of mass m, length l and subject to a constant gravitational field g: ĤCn = − +mgl(1 − cos θ) where those quantities are related to our system by, mω an , g = , θ = That is, we are approximating, for each scale Cn the SHO by a pendulum. There is, however, an important difference. From our knowledge of the pendulum system, we know that the quantum system will have a spectrum for the energy that has two different asymptotic behav- iors, the SHO for low energies and the planar rotor in the higher end, corresponding to oscillating and rotating solutions respectively2. As we refine our scale and both the length of the pendulum and the height of the periodic potential increase, we expect to have an increasing num- ber of oscillating states (for a given pendulum system, there is only a finite number of such states). Thus, it is justified to consider the cut-off in the energy eigenval- ues, as discussed in the last section, given that we only expect a finite number of states of the pendulum to ap- proximate SHO eigenstates. With these consideration in mind, the relevant question is whether the conditions for the continuum limit to exist are satisfied. This question has been answered in the affirmative in [6]. What was shown there was that the eigen-values and eigen func- tions of the discrete systems, which represent a discrete and non-degenerate set, approximate those of the contin- uum, namely, of the standard harmonic oscillator when the inner product is renormalized by a factor λ2Cn = 1/2 This convergence implies that the continuum limit exists as we understand it. Let us now consider the simplest possible system, a free particle, that has nevertheless the particular feature that the spectrum of the energy is con- tinuous. 2 Note that both types of solutions are, in the phase space, closed. This is the reason behind the purely discrete spectrum. The distinction we are making is between those solutions inside the separatrix, that we call oscillating, and those that are above it that we call rotating. B. Free Polymer Particle In the limit ω → 0, the Hamiltonian of the Simple Harmonic oscillator (35) goes to the Hamiltonian of a free particle and the corresponding time independent Schrödinger equation, in the p−polarization, is given by (1 − cos anp ) − ECn ψ̃(p) = 0 where we now have that p ∈ S1, with p ∈ (−π~ Thus, we have ECn = 1 − cos ≤ ECn,max ≡ 2 . (36) At each scale the energy of the particle we can describe is bounded from above and the bound depends on the scale. Note that in this case the spectrum is continu- ous, which implies that the ordinary eigenfunctions of the Hilbert are not normalizable. This imposes an upper bound in the value that the energy of the particle can have, in addition to the bound in the momentum due to its “compactification”. Let us first look for eigen-solutions to the time inde- pendent Schrödinger equation, that is, for energy eigen- states. In the case of the ordinary free particle, these correspond to constant momentum plane waves of the form e±( ) and such that the ordinary dispersion re- lation p2/2m = E is satisfied. These plane waves are not square integrable and do not belong to the ordinary Hilbert space of the Schrödinger theory but they are still useful for extracting information about the system. For the polymer free particle we have, ψ̃Cn(p) = c1δ(p− PCn) + c2δ(p+ PCn) where PCn is a solution of the previous equation consid- ering a fixed value of ECn . That is, PCn = P (ECn) = arccos 1 − ma The inverse Fourier transform yields, in the ‘x represen- tation’, ψCn(xj) = ∫ π~/an −π~/an ψ̃(p) e p j dp = ixjPCn /~ + c2e −ixjPCn /~ .(37) with xj = an j for j ∈ Z. Note that the eigenfunctions are still delta functions (in the p representation) and thus not (square) normalizable with respect to the polymer inner product, that in the p polarization is just given by the ordinary Haar measure on S1, and there is no quantization of the momentum (its spectrum is still truly continuous). Let us now consider the time dependent Schrödinger equation, i~ ∂t Ψ̃(p, t) = Ĥ · Ψ̃(p, t). Which now takes the form, Ψ̃(p, t) = (1 − cos (an p/~)) Ψ̃(p, t) that has as its solution, Ψ̃(p, t) = e− (1−cos (an p/~)) t ψ̃(p) = e(−iECn /~) t ψ̃(p) for any initial function ψ̃(p), where ECn satisfy the dis- persion relation (36). The wave function Ψ(xj , t), the xj-representation of the wave function, can be obtained for any given time t by Fourier transforming with (37) the wave function Ψ̃(p, t). In order to check out the convergence of the micro- scopically corrected Hamiltonians we should analyze the convergence of the energy levels and of the proper cov- ectors. In the limit n → ∞, ECn → E = p2/2m so we can be certain that the eigen-values for the energy converge (when fixing the value of p). Let us write the proper covector as ΨCn = (ψCn , ·)renCn ∈ H . Then we can bring microscopic corrections to scale Cm and look for convergence of such corrections ΨrenCm = lim d⋆m,nΨCn . It is easy to see that given any basis vector eαi ∈ HCm the following limit ΨrenCm(eαi,Cm) = limCn→∞ ΨCn(dn,m(eαi,Cm)) exists and is equal to ΨshadCm (eαi,Cm) = [d ⋆ΨSchr](eαi,Cm) = Ψ Schr(iam) where ΨshadCm is calculated using the free particle Hamilto- nian in the Schrödinger representation. This expression defines the completely renormalized proper covector at the scale Cm. C. Polymer Quantum Cosmology In this section we shall present a version of quantum cosmology that we call polymer quantum cosmology. The idea behind this name is that the main input in the quan- tization of the corresponding mini-superspace model is the use of a polymer representation as here understood. Another important input is the choice of fundamental variables to be used and the definition of the Hamiltonian constraint. Different research groups have made differ- ent choices. We shall take here a simple model that has received much attention recently, namely an isotropic, homogeneous FRW cosmology with k = 0 and coupled to a massless scalar field ϕ. As we shall see, a proper treatment of the continuum limit of this system requires new tools under development that are beyond the scope of this work. We will thus restrict ourselves to the intro- duction of the system and the problems that need to be solved. The system to be quantized corresponds to the phase space of cosmological spacetimes that are homogeneous and isotropic and for which the homogeneous spatial slices have a flat intrinsic geometry (k = 0 condition). The only matter content is a mass-less scalar field ϕ. In this case the spacetime geometry is given by metrics of the form: ds2 = −dt2 + a2(t) (dx2 + dy2 + dz2) where the function a(t) carries all the information and degrees of freedom of the gravity part. In terms of the coordinates (a, pa, ϕ, pϕ) for the phase space Γ of the the- ory, all the dynamics is captured in the Hamiltonian con- straint C := −3 + 8πG 2|a|3 The first step is to define the constraint on the kine- matical Hilbert space to find physical states and then a physical inner product to construct the physical Hilbert space. First note that one can rewrite the equation as: p2a a 2 = 8πG If, as is normally done, one chooses ϕ to act as an in- ternal time, the right hand side would be promoted, in the quantum theory, to a second derivative. The left hand side is, furthermore, symmetric in a and pa. At this point we have the freedom in choosing the variable that will be quantized and the variable that will not be well defined in the polymer representation. The standard choice is that pa is not well defined and thus, a and any geometrical quantity derived from it, is quantized. Fur- thermore, we have the choice of polarization on the wave function. In this respect the standard choice is to select the a-polarization, in which a acts as multiplication and the approximation of pa, namely sin(λ pa)/λ acts as a difference operator on wave functions of a. For details of this particular choice see [5]. Here we shall adopt the op- posite polarization, that is, we shall have wave functions Ψ(pa, ϕ). Just as we did in the previous cases, in order to gain intuition about the behavior of the polymer quantized theory, it is convenient to look at the equivalent prob- lem in the classical theory, namely the classical system we would get be approximating the non-well defined ob- servable (pa in our present case) by a well defined object (made of trigonometric functions). Let us for simplicity choose to replace pa 7→ sin(λ pa)/λ. With this choice we get an effective classical Hamiltonian constraint that depends on λ: Cλ := − sin(λ pa) λ2|a| + 8πG 2|a|3 We can now compute effective equations of motion by means of the equations: Ḟ := {F, Cλ}, for any observable F ∈ C∞(Γ), and where we are using the effective (first order) action: dτ(pa ȧ+ pϕ ϕ̇−N Cλ) with the choice N = 1. The first thing to notice is that the quantity pϕ is a constant of the motion, given that the variable ϕ is cyclic. The second observation is that ϕ̇ = 8πG has the same sign as pϕ and never vanishes. Thus ϕ can be used as a (n internal) time variable. The next observation is that the equation for , namely the effective Friedman equation, will have a zero for a non-zero value of a given by λ2p2ϕ. This is the value at which there will be bounce if the trajectory started with a large value of a and was con- tracting. Note that the ‘size’ of the universe when the bounce occurs depends on both the constant pϕ (that dictates the matter density) and the value of the lattice size λ. Here it is important to stress that for any value of pϕ (that uniquely fixes the trajectory in the (a, pa) plane), there will be a bounce. In the original description in terms of Einstein’s equations (without the approxima- tion that depends on λ), there in no such bounce. If ȧ < 0 initially, it will remain negative and the universe collapses, reaching the singularity in a finite proper time. What happens within the effective description if we re- fine the lattice and go from λ to λn := λ/2 n? The only thing that changes, for the same classical orbit labelled by pϕ, is that the bounce occurs at a ‘later time’ and for a smaller value of a∗ but the qualitative picture remains the same. This is the main difference with the systems considered before. In those cases, one could have classical trajecto- ries that remained, for a given choice of parameter λ, within the region where sin(λp)/λ is a good approxima- tion to p. Of course there were also classical trajectories that were outside this region but we could then refine the lattice and find a new value λ′ for which the new clas- sical trajectory is well approximated. In the case of the polymer cosmology, this is never the case: Every classical trajectory will pass from a region where the approxima- tion is good to a region where it is not; this is precisely where the ‘quantum corrections’ kick in and the universes bounces. Given that in the classical description, the ‘original’ and the ‘corrected’ descriptions are so different we expect that, upon quantization, the corresponding quantum the- ories, namely the polymeric and the Wheeler-DeWitt will be related in a non-trivial way (if at all). In this case, with the choice of polarization and for a particular factor ordering we have, sin(λpa) · Ψ(pa, ϕ) = 0 as the Polymer Wheeler-DeWitt equation. In order to approach the problem of the continuum limit of this quantum theory, we have to realize that the task is now somewhat different than before. This is so given that the system is now a constrained system with a constraint operator rather than a regular non-singular system with an ordinary Hamiltonian evolution. Fortu- nately for the system under consideration, the fact that the variable ϕ can be regarded as an internal time allows us to interpret the quantum constraint as a generalized Klein-Gordon equation of the form Ψ = Θλ · Ψ where the operator Θλ is ‘time independent’. This al- lows us to split the space of solutions into ‘positive and negative frequency’, introduce a physical inner product on the positive frequency solutions of this equation and a set of physical observables in terms of which to de- scribe the system. That is, one reduces in practice the system to one very similar to the Schrödinger case by taking the positive square root of the previous equation: Θλ · Ψ. The question we are interested is whether the continuum limit of these theories (labelled by λ) exists and whether it corresponds to the Wheeler- DeWitt theory. A complete treatment of this problem lies, unfortunately, outside the scope of this work and will be reported elsewhere [12]. VII. DISCUSSION Let us summarize our results. In the first part of the article we showed that the polymer representation of the canonical commutation relations can be obtained as the limiting case of the ordinary Fock-Schrödinger represen- tation in terms of the algebraic state that defines the representation. These limiting cases can also be inter- preted in terms of the naturally defined coherent states associated to each representation labelled by the param- eter d, when they become infinitely ‘squeezed’. The two possible limits of squeezing lead to two different polymer descriptions that can nevertheless be identified, as we have also shown, with the two possible polarizations for an abstract polymer representation. This resulting the- ory has, however, very different behavior as the standard one: The Hilbert space is non-separable, the representa- tion is unitarily inequivalent to the Schrödinger one, and natural operators such as p̂ are no longer well defined. This particular limiting construction of the polymer the- ory can shed some light for more complicated systems such as field theories and gravity. In the regular treatments of dynamics within the poly- mer representation, one needs to introduce some extra structure, such as a lattice on configuration space, to con- struct a Hamiltonian and implement the dynamics for the system via a regularization procedure. How does this re- sulting theory compare to the original continuum theory one had from the beginning? Can one hope to remove the regulator in the polymer description? As they stand there is no direct relation or mapping from the polymer to a continuum theory (in case there is one defined). As we have shown, one can indeed construct in a systematic fashion such relation by means of some appropriate no- tions related to the definition of a scale, closely related to the lattice one had to introduce in the regularization. With this important shift in perspective, and an appro- priate renormalization of the polymer inner product at each scale one can, subject to some consistency condi- tions, define a procedure to remove the regulator, and arrive to a Hamiltonian and a Hilbert space. As we have seen, for some simple examples such as a free particle and the harmonic oscillator one indeed recovers the Schrödinger description back. For other sys- tems, such as quantum cosmological models, the answer is not as clear, since the structure of the space of classi- cal solutions is such that the ‘effective description’ intro- duced by the polymer regularization at different scales is qualitatively different from the original dynamics. A proper treatment of these class of systems is underway and will be reported elsewhere [12]. Perhaps the most important lesson that we have learned here is that there indeed exists a rich inter- play between the polymer description and the ordinary Schrödinger representation. The full structure of such re- lation still needs to be unravelled. We can only hope that a full understanding of these issues will shed some light in the ultimate goal of treating the quantum dynamics of background independent field systems such as general relativity. Acknowledgments We thank A. Ashtekar, G. Hossain, T. Pawlowski and P. Singh for discussions. This work was in part supported by CONACyT U47857-F and 40035-F grants, by NSF PHY04-56913, by the Eberly Research Funds of Penn State, by the AMC-FUMEC exchange program and by funds of the CIC-Universidad Michoacana de San Nicolás de Hidalgo. [1] R. Beaume, J. Manuceau, A. Pellet and M. Sirugue, “Translation Invariant States In Quantum Mechanics,” Commun. Math. Phys. 38, 29 (1974); W. E. Thirring and H. Narnhofer, “Covariant QED without indefinite met- ric,” Rev. Math. Phys. 4, 197 (1992); F. Acerbi, G. Mor- chio and F. 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La mecánica cuántica de polímeros y su límite de continuidad Alejandro Corichi,1, 2, 3, ∗ Tatjana Vukašinac,4, † y José A. Zapata1, ‡ Instituto de Matemáticas, Unidad Morelia, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, A. Postal 61-3, Morelia, Michoacán 58090, México Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D.F. 04510, México Instituto de Física Gravitacional y Geometría, Departamento de Física, Pennsylvania State University, University Park PA 16802, EE.UU. Facultad de Ingeniería Civil, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58000, México Una representación cuántica bastante no estándar de las relaciones canónicas de conmutación de sistemas mecánicos de tom, conocidos como la representación del polímero ha ganado cierta atención en los últimos años, debido a su posible relación con la física a escala de Planck. En particular, este enfoque ha sido el siguiente: seguido en un sector simétrico de la gravedad cuántica del bucle conocido como cosmología cuántica del bucle. Aquí vamos. explorar diferentes aspectos de la relación entre la teoría ordinaria de Schrödinger y el polímero descripción. El periódico tiene dos partes. En el primero, derivamos la mecánica cuántica del polímero a partir de la teoría ordinaria de Schrödinger y mostrar que la descripción del polímero surge como un límite adecuado. En la segunda parte consideramos el límite continuo de esta teoría, a saber, el proceso inverso en el que se parte de la teoría discreta e intenta recuperar de nuevo lo ordinario Schrödinger mecánica cuántica. Consideramos varios ejemplos de interés, incluyendo el armónico oscilador, la partícula libre y un modelo cosmológico simple. Números PACS: 04.60.Pp, 04.60.Ds, 04.60.Nc 11.10.Gh. I. INTRODUCCIÓN La llamada mecánica cuántica polimérica, una no- representación regular y algo «exótica» de la las relaciones canónicas de conmutación (CCR) [1], utilizado para explorar tanto las cuestiones matemáticas y físicas en teorías independientes de fondo tales como la grav cuántica- ity [2, 3]. Un ejemplo notable de este tipo de cuantificación, cuando se aplica a modelos minisuperespacio ha dado paso a lo que se conoce como cosmología cuántica de bucle [4, 5]. Al igual que en cualquier situación modelo de juguete, uno espera aprender sobre el sutiles cuestiones técnicas y conceptuales que están presentes en la gravedad cuántica completa por medio de di- simple, finito Ejemplos mensionales. Este formalismo no es una excepción a este respecto. Aparte de esta motivación que viene de física en la escala de Planck, uno puede preguntar independientemente para la relación entre la representación continua estándar las sentaciones y sus primos poliméricos a nivel de matemáticas... Física emática. Una comprensión más profunda de esta relación se vuelve importante por sí solo. La cuantificación del polímero está hecha de varios pasos. El primero es construir una representación de la álgebra Heisenberg-Weyl en un espacio Kinematical Hilbert que es “independiente en el fondo”, y que a veces es conocido como el poliespacial polimérico Hilbert Hpoly. Los la segunda y más importante parte, la aplicación de dinámica, se ocupa de la definición de un Hamiltonian (o Constreñimiento hamiltoniano) en este espacio. En los ejemplos * Dirección electrónica: corichi@matmor.unam.mx †Dirección electrónica: tatjana@shi.matmor.unam.mx ‡Dirección electrónica: zapata@matmor.unam.mx estudiado hasta ahora, la primera parte es bastante bien entendido, dando el espacio cinemático Hilbert Hpoly es decir, cómo- Nunca, no-separable. Para el segundo paso, un im natural la aplicación de la dinámica ha demostrado ser un poco más difícil, dado que una definición directa de la Hamiltonian de, digamos, una partícula en un potencial en el espacio Hpoly es no es posible ya que una de las principales características de esta representación sentation es que los operadores qâ € y pâ € no pueden ser a la vez definido simultáneamente (ni sus análogos en las teorías con variables más elaboradas). Por lo tanto, cualquier operador que implica (poderes de) la variable no definida tiene que estar regulados por un operador bien definido que normalmente implica la introducción de una estructura adicional en la configuración ración (o impulso) espacio, es decir, una celosía. Sin embargo, esta nueva estructura que juega el papel de un regulador puede no se retira cuando se trabaja en Hpoly y se deja uno con la ambigüedad que está presente en cualquier regularización. La libertad a la hora de elegirla puede ser asociada a veces con una escala de longitud (el espaciado de celosía). En el caso de las personas de edad ordinaria sistemas cuánticos tales como un oscilador armónico simple, que se ha estudiado en detalle desde el punto de vista del polímero punto, se ha argumentado que si se toma esta escala de longitud para ser «suficientemente pequeño», se puede aproximar arbitrariamente Mecánica cuántica estándar de Schrödinger [2, 3]. En el caso de cosmología cuántica de bucle, la brecha de área mínima A0 de la teoría de la gravedad cuántica completa impone tal escala, que entonces se considera fundamental [4]. Una pregunta natural es preguntar qué sucede cuando nosotros cambiar esta escala e ir a ‘distancias’ aún más pequeñas, que es, cuando refinamos la celosía en la que la dinámica de la teoría está definida. ¿Podemos definir la consistencia con- ¿diciones entre estas escalas? O incluso mejor, ¿podemos tomar el límite y encontrar así un límite continuo? Como ella. http://arxiv.org/abs/0704.0007v2 mailto:corichi@matmor.unam.mx mailto:tatjana@shi.matmor.unam.mx mailto:zapata@matmor.unam.mx se ha mostrado recientemente en detalle, la respuesta a ambos las preguntas son afirmativas [6]. En este caso, una la noción de escala se definía de tal manera que se podía definir los refinamientos de la teoría y posar en un preciso forma la cuestión del límite continuo de la teoría. Estos resultados también podrían ser vistos como la entrega de un procedimiento para eliminar el regulador cuando se trabaja en el apro- se comió el espacio. El propósito de este documento es explorar más a fondo diferentes aspectos de la relación entre el continuum y la representación del polímero. En particular, en la primera parte planteamos una nueva manera de derivar el polímero representación del ordinario Schrödinger represen- sión como límite adecuado. In Sec. II derivamos dos versiones de la representación del polímero como diferente lim- es de la teoría de Schrödinger. In Sec. III mostramos que estas dos versiones pueden ser vistas como diferentes polarizaciones de la representación «abstracta» del polímero. Estos resultados, a lo mejor de nuestro conocimiento, son nuevos y no han sido notificada en otro lugar. In Sec. IV planteamos el problema de la aplicación de la dinámica en el polímero representa- tion. In Sec. V motivamos aún más la cuestión de la límite continuo (es decir, la eliminación adecuada del regulador) y recordar las construcciones básicas de [6]. Varios exámenes... ples se consideran en Sec. VI. En particular, un simple oscilador armónico, la partícula libre de polímero y un sim- Se considera el modelo cuántico de cosmología. El libre la partícula y el modelo cosmológico representan un lización de los resultados obtenidos en [6], en los que sólo los sistemas con un espectro discreto y no degenerado, Sidered. Terminamos el trabajo con una discusión en Sec. VII. Con el fin de hacer el papel autónomo, vamos a mantener el nivel de rigor en la presentación a la que se encuentra en el literatura física teórica estándar. II. CUANTIZACIÓN Y POLÍMER REPRESENTACIÓN En esta sección derivamos el llamado repre- envío de la mecánica cuántica a partir de un reformulación de la representación ordinaria de Schrödinger. Nuestro punto de partida será el más simple de todos los posibles espacios de fase, a saber, • = R2 correspondientes a una partícula viviendo en la línea real R. Elijamos las coordenadas (q, p) sobre el mismo. Como primer paso consideraremos la cuantificación de este sistema que conduce a la teoría cuántica estándar en la descripción de Schrödinger. Una ruta conveniente es a introducir la estructura necesaria para definir el Fock rep- el resentimiento de tal sistema. Desde esta perspectiva, el el paso al caso polimérico se vuelve más claro. Aproximadamente hablando por una cuantificación uno significa un pasaje del soporte algebraico clásico, el soporte Poisson, {q, p} = 1 (1) a un soporte cuántico dado por el conmutador de la los operadores correspondientes, [ qâ, pâ €] = i~ 1â € (2) Estas relaciones, conocidas como la conmutación canónica re- ración (CCR) se convierten en la piedra más común de la esquina de la (kinemática de la) teoría cuántica; deben ser satisfecho por el sistema cuántico, cuando se representa en un Hilbert Space H. Hay puntos de partida alternativos para el cuántico cinemática. Aquí consideramos el álgebra generada por las versiones exponenteadas de qâ € y pâ € que se denotan U(α) = ei(α q)/~ ; V (β) = ei(β p)/~ donde α y β tienen dimensiones de impulso y longitud, respectivamente. El CCR ahora se convierte en U(α) · V (β) = e(−iα β)/~V (β) · U(α) (3) y el resto del producto es U(α1)·U(α2) = U(α12) ; V (β1)·V (β2) = V (β1+2) El álgebra W de Weyl se genera tomando lineal finito combinaciones de los generadores U(αi) y V (βi) donde el producto (3) se amplía por linealidad, (Ai U(αi) +Bi V (βi)) Desde esta perspectiva, la cuantificación significa encontrar un representación unitaria del álgebra W de Weyl en una Hilbert espacio H′ (que podría ser diferente de los ordi- nary Schrödinger representación). Al principio podría parecer raro para intentar este enfoque dado que sabemos cómo para cuantificar un sistema tan sencillo; ¿qué necesitamos? ¿Un objeto complicado como W? Es infinitamente dimensional, mientras que el conjunto S = {1», q», p, el punto de partida de la la cuantificación ordinaria de Dirac, es bastante simple. Está en la cuantificación de sistemas de campo que las ventajas de el enfoque de Weyl se puede apreciar plenamente, pero es también útil para la introducción de la cuantificación del polímero y comparándolo con la cuantificación estándar. Esta es la estrategia que seguimos. Una pregunta que uno puede hacer es si hay alguna libertad en la cuantificación del sistema para obtener lo ordinario Representación de Schrödinger. A primera vista podría parecer que no hay ninguno dado el único Stone-Von Neumann- Teorema de ness. Repasemos cuál sería el argumento. para la construcción estándar. Pidamos que el representante... El envío que queremos construir es del tipo Schrödinger, a saber, donde los estados son funciones de onda de configuración espacio (q). Hay dos ingredientes en la construcción de la representación, a saber, la especificación de cómo la los operadores básicos (qá, pá) actuarán, y la naturaleza del espacio de las funciones a las que • pertenece, que normalmente se fija por la elección del producto interior en H, o la medida μ en R. La opción estándar es seleccionar el espacio Hilbert a ser, H = L2(R, dq) el espacio de funciones integrables cuadradas con respecto a la medida de Lebesgue dq (invariante bajo constante trans- lations) en R. Los operadores se representan entonces como, qâ · â € (q) = (q â €)(q) y pâ · â € (q) = −i ~ •(q) (4) ¿Es posible encontrar otras representaciones? Con el fin de apreciar esta libertad vamos al álgebra de Weyl y construir la teoría cuántica al respecto. La representación del álgebra de Weyl que se puede llamar del ‘tipo Fock’ implica la definición de una estructura adicional en la fase espacio: una estructura compleja J. Es decir, un mapa lineal. Ping de a sí mismo de tal manera que J2 = −1. En dos dimensiones. sions, toda la libertad en la elección de J está contenida en la elección de un parámetro d con dimensiones de longitud. Lo siento. También es conveniente definir: k = p/~ que tiene dimensiones de 1/L. Tenemos entonces, Jd : (q, k) 7→ (−d2 k, q/d2) Este objeto junto con la estructura simpléctica: (q′, p′)) = q p′ − p q′ define un producto interior en * por la fórmula gd(· ; ·) = (· ; Jd ·) de tal manera que: gd(q, p); (q ′, p′)) = q q′ + que es sin dimensión y positiva definida. Tenga en cuenta que con estas cantidades se puede definir coordenadas complejas (, ) como de costumbre: q + i p ; = q − i d a partir de la cual se puede construir el estándar Fock representa- tion. Por lo tanto, se puede ver alternativamente la introducción del parámetro de longitud d como la cantidad necesaria para de- Coordenadas complejas finas (sin dimensión) en la fase espacio. Pero ¿cuál es la relevancia de este objeto (J o d)? La definición de coordenadas complejas es útil para la construcción del espacio Fock ya que de ellos uno puede definir, de una manera natural, la creación y la aniquilación operadores. Pero para la representación de Schrödinger somos Interesado aquí, es un poco más sutil. La sutileza es que dentro de este enfoque se utiliza la prop algebraica erties de W para construir el espacio Hilbert a través de lo que es conocido como el Gel’fand-Naimark-Segal (GNS) tion. Esto implica que la medida en el asunto Schrödinger representación se convierte en no trivial y por lo tanto la momen- el operador adquiere un término adicional con el fin de renderizar el operador autoadjunto. La representación del Weyl álgebra es entonces, cuando se actúa sobre las funciones فارسى(q) [7]: *(α) ·*(q) := (eiα q/~ ♥)(q) (β) · (q) := e (q/2) (q − β) La estructura espacial de Hilbert es introducida por el defini- ión de un estado algebraico (un funcional lineal positivo) D : W → C, que debe coincidir con la expectativa valor en el espacio Hilbert tomado en un estado especial ref- ered a como el vacío: d(a) = vac, para todos un W. En nuestro caso, esta especificación de J induce a un único de que los rendimientos, (α)vac = e− d2 α2 ~2 (5) Vócalo (β)Vócalo = e− d2 (6) Tenga en cuenta que los exponentes en la expectativa de vacío los valores corresponden a la métrica construida a partir de J : d2 α2 = gd(0, α); (0, α)) y = gd(β, 0); (β, 0). Las funciones de onda pertenecen al espacio L2(R, dμd), donde la medida que dicta el producto interior en este rep- la resensión es dada por, dμd = d2 dq En esta representación, el vacío es dado por el iden- función de la tity Ł0(q) = 1 es decir, al igual que cualquier onda de plano, normalizado. Tenga en cuenta que para cada valor de d > 0, el rep- la resención es bien definida y continua en α y β. Tenga en cuenta también que hay una equivalencia entre la q- representación definida por d y la k-representación de- multado por 1/d. ¿Cómo podemos recuperar entonces la representación estándar en la que la medida es dada por la medida Lebesgue y los operadores están representados como en (4)? Es fácil de ver que hay un isomorfismo isométrico K que mapea la d-representación en Hd a la norma Schrödinger representación en Hschr por: (q) = K · (q) = e d1/2η1/4 Hschr = L2(R, dq) Así vemos que todas las representaciones d son unitariamente equiv- Alent. Esto era de esperar en vista de la Stone-Von Resultado de la singularidad de Neumann. Tenga en cuenta también que el vacío ahora se convierte en 0(q) = d1/2η1/4 2 d2, Así que incluso cuando no hay información sobre el param- eter d en la representación en sí, está contenida en el estado de vacío. Este procedimiento para la construcción del GNS- Schrödinger representación para la mecánica cuántica ha también se generalizó a los campos escalares sobre curvas arbitrarias espacio en [8]. Nótese, sin embargo, que hasta ahora el tratamiento ha todos fueron cinemáticos, sin ningún conocimiento de un Hamil- Tonian. Para el Oscilador Armónico Simple de masa m y la frecuencia, hay una opción natural compatible con la dinámica dada por d = , en el que algunos los cálculos se simplifican (por ejemplo, para los estados coherentes), pero en principio se puede utilizar cualquier valor de d. Nuestro estudio se simplificará concentrándose en la las entidades mentales en el Hilbert Space Hd, a saber, los los estados generados por la acción con فارسى(α) en el vacío 0(q) = 1. Vamos a denotar esos estados por, (q) = ­(α) · ­0(q) = ei El producto interno entre dos de estos estados es dado por , d = dμd e ~ = e− ()2 d2 4 ~2 (7) Note, por cierto, que, contrariamente a alguna creencia común, las ‘ondas del avión’ en este espacio GNS Hilbert son de hecho normalizable. Consideremos ahora la representación del polímero. Por que, es importante tener en cuenta que hay dos posibles casos límite para el parámetro d: i) El límite 1/d 7→ 0 y ii) El caso d 7→ 0. En ambos casos, tenemos ex- presiones que se definan mal en la representación o medida, por lo que uno necesita tener cuidado. A. El caso 1/d 7→ 0. La primera observación es que de las expresiones (5) y (6) para el estado algebraico........................................................................................................................................................................................................................................................... En efecto, los casos están bien definidos. En nuestro caso obtenemos, A := lim1/d→0 A (α) =,0 y A (β) = 1 (8) A partir de esto, de hecho podemos construir la representación mediante la construcción del GNS. Con el fin de hacer eso y para mostrar cómo se obtiene esto vamos a considerar varios expresiones. Sin embargo, hay que tener cuidado, ya que el límite tiene que ser tomado con cuidado. Consideremos la medida sobre la representación que se comporta como: dμd = d2 dq 7→ 1 por lo que las medidas tienden a una medida homogénea, pero cuya ‘normalización constante’ va a cero, por lo que el límite se vuelve algo sutil. Volveremos a este punto. Más tarde. Veamos ahora qué pasa con el producto interior. entre las entidades fundamentales en el Hilbert Space Hd dado por (7). Es inmediato ver que en el 1/d 7→ 0 limitar el producto interior se convierte, , d 7→, con Kronecker como el delta de Kronecker. Vemos entonces que el ondas planas (q) se convierten en una base ortonormal para el nuevo espacio Hilbert. Por lo tanto, hay una interacción delicada entre los dos términos que contribuyen a la medida en mantener la normalidad de estas funciones; Necesitamos que la medida se humedezca (por 1/d) en orden evitar que las ondas planas adquieran una norma infinita (como sucede con la medida estándar de Lebesgue), pero por otro lado la medida, que para cualquier valor finito de d es un gaussiano, se vuelve cada vez más extendido. Es importante señalar que, en este límite, los operadores • (α) llegar a ser discontinuo con respecto a α, dado que para cualquier α1 y α2 (diferente), su acción en un determinado vector base (q) produce vectores ortogonales. Desde el continuidad de estos operadores es uno de los hipotesis de el teorema de Stone-Von Neumann, el resultado de la singularidad no se aplica aquí. La representación es inequivalente al estándar. Analicemos ahora el otro operador, a saber, el acción del operador Vó (β) sobre la base de (q): (β) · (q) = e− ~ e(β/d 2+iα/~)q que en el límite 1/d 7→ 0 va a, (β) · (q) 7→ ei ~ (q) que es continuo en β. Por lo tanto, en el límite, el operador = −iq está bien definido. Además, tenga en cuenta que en este límite el operador tiene (q) como su propio estado con valor propio dado por α: · (q) 7→ (q) Para resumir, la teoría resultante obtenida por el límite 1/d 7→ 0 de la descripción ordinaria de Schrödinger sión, que llamaremos la «representación de polímeros de tipo A», tiene las siguientes características: los operadores U(α) están bien definidos pero no continuos en α, por lo que no hay generador (sin operador asociado a q). La base vec- tors son ortonormales (para α tomando valores en un contin- y son autovectores del operador que es bien definido. El espacio resultante Hilbert HA será el (A-versión de la) representación del polímero. Vamos ahora. considerar el otro caso, a saber, el límite cuando d 7→ 0. B. El caso d 7→ 0 Exploremos ahora el otro caso limitante de la representaciones de Schrödinger/Fock etiquetadas por el eter d. Al igual que en el caso anterior, la limitación algebraica el estado se convierte, B := limd→0 •d de tal manera que, B(α) = 1 y B(V® (β)) = 0 (10) A partir de esta función lineal positiva, uno puede de hecho con- structe la representación usando la construcción GNS. Primero tomemos nota de que la medida, incluso cuando el límite debe ser tomado con el debido cuidado, se comporta como: dμd = d2 dq 7→ (q) dq Es decir, como distribución delta de Dirac. Es inmediato a ver que, en el límite d 7→ 0, el producto interior entre los estados fundamentales (q) se convierte, , d 7→ 1 (11) Esto de hecho significa que el vector = − pertenece al Kernel del producto interior limitante, por lo que uno tiene que mod hacia fuera por estos (y todos) estados de la norma cero con el fin de Conseguir el espacio de Hilbert. Analicemos ahora el otro operador, a saber, el acción del operador Vâr (β) sobre el vacío Ø0(q) = 1, que para arbitrario d tiene la forma, := Vó (β) · 0(q) = e (q/2) El producto interno entre dos de estos estados es dado por , d = e− ()2 En el límite d → 0,, d →,β. Podemos ver entonces. que son estas funciones las que se vuelven ortonormales, ‘bases discretas’ en la teoría. Sin embargo, la función (q) en este límite se vuelve mal definido. Por ejemplo, para β > 0, crece sin límite para q > β/2, es igual a uno si q = β/2 y cero de lo contrario. Con el fin de superar estos las dificultades y hacer más transparente el resultado de ory, vamos a considerar la otra forma de la representación en la que la medida se incorpora a los Estados (y el espacio resultante de Hilbert es L2(R, dq). Por lo tanto, la nueva estado (q) := K · (Vâ ° (β) · Ø0(q)) = (q)2 Ahora podemos tomar el límite y lo que obtenemos es d 7→0 (q) := ♥ 1/2(q, β) donde por Ł1/2(q, β) nos referimos a algo como ‘el cuadrado raíz de la distribución de Dirac». Lo que realmente queremos decir es un objeto que satisface la siguiente propiedad: 1/2(q, β) · 1/2(q, α) = 1/2(q, β) Es decir, si α = β entonces es sólo el delta ordinario, otro- sabiamente es cero. En cierto sentido, este objeto puede ser considerado como medias densidades que no pueden integrarse por sí mismas, pero cuyo producto puede. Concluimos entonces que el interior el producto es, , = dq (q)(q) = dq (q, α), α = α,α que es justo lo que esperábamos. Nótese que en esta repre- sentation, el estado de vacío se convierte en 0(q) := 1/2(q, 0), a saber, la mitad delta con apoyo en el origen. Lo es. importante tener en cuenta que estamos llegando de una manera natural a estados como medias densidades, cuyos cuadrados se pueden integrar sin necesidad de una medida no trivial en la configuración espacio de racionamiento. La invarianza del difeomorfismo surge entonces en un natural pero sutil. Tenga en cuenta que como resultado final recuperamos el Kronecker producto interior delta para los nuevos estados fundamentales: (q) := ♥ 1/2(q, β). Así, en esta nueva representación de B-polímero, el Hilbert espacio HB es la terminación con respecto al interior producto (13) de los estados generados por la toma (finito) combinaciones lineales de elementos de base de la forma : (q) = bi i(q) (14) Ahora vamos a introducir una descripción equivalente de esto Espacio Hilbert. En lugar de tener los elementos de base ser medio-deltas como elementos del espacio Hilbert donde el producto interior es dado por la medida ordinaria de Lebesgue dq, redefinimos tanto la base como la medida. Nosotros podría considerar, en lugar de una media-delta con soporte β, una Kronecker delta o función característica con soporte sobre β: (q) := ♥q,β Estas funciones tienen un comportamiento similar con respecto a el producto como media delta, a saber: (q) · (q) = # # # # #, # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # La principal diferencia es que ninguna de las dos χ ′ ni sus Los cuadrados son integrables con respecto al Lebesgue mea- seguro (tiene cero norma). Con el fin de solucionar ese problema nosotros tiene que cambiar la medida para que recuperemos la base producto interior (13) con nuestra nueva base. La mea necesaria... seguro resulta ser la medida de conteo discreto en R. Por lo tanto, cualquier estado en la «base de la media densidad» se puede escribir (utilizando la misma expresión) en términos de «Kronecker base». Para más detalles y más motivación vea el siguiente sección. Nótese que en esta representación de polímero B, ambos sus roles intercambiados con el de la Representación de A-polímero: mientras que U(α) es discontinuo y, por lo tanto, qâ € no se define en la representación-A, nosotros tener que es V (β) en la representación B que tiene este propiedad. En este caso, es el operador el que no puede se definan. Vemos entonces que dado un sistema físico para que el espacio de configuración tiene un fisico bien definido en la posible representación en la que funciones de onda son funciones de la variable de configuración q, las representaciones de polímeros A y B son radicalmente dif- Ferent e inequivalente. Dicho esto, también es cierto que la A y B representaciones son equivalentes en un sentido diferente, por medios de la dualidad entre q y p representaciones y la dualidad d↔ 1/d: La representación de A-polímero en la “representación q” es equivalente al polímero B representación en la "p-representación", e inversamente. Cuando se estudia un problema, es importante decidir desde el comienzo de la representación del polímero (en su caso) debe ser utilizado (por ejemplo en la q-polarización). Esto tiene como consecuencia una implicación sobre qué variable es naturalmente “cuantificada” (incluso si continua): p para A y q para B. Podría haber, por ejemplo, un criterio físico para esta elección. Por ejemplo, una simetría fundamental podría Sugiere que una representación es más natural que una... otro. Esto ha sido observado recientemente por Chiou. en [10], donde se investiga el grupo Galileo y se demuestra que la representación B se comporta mejor. En la otra polarización, es decir, para las funciones de onda de p, la imagen se invierte: q es discreto para el A- representación, mientras que p es para el caso B. Terminemos con esto. , señalando que el procedimiento de obtención de la cuantificación del polímero mediante un límite adecuado de las representaciones Fock-Schrödinger podrían resultar útiles en ajustes más generales en teoría de campo o gravedad cuántica. III. MECANISMOS DE CUANTO POLÍMICO: KINEMÁTICAS En secciones anteriores hemos derivado lo que tenemos las llamadas representaciones poliméricas A y B (en la polarización) como casos limitantes de la representación ordinaria de Fock Enviaciones. En esta sección, describiremos, sin cualquier referencia a la representación de Schrödinger, la «ab- representación de polímero de estrazo y luego hacer contacto con sus dos posibles realizaciones, estrechamente relacionadas con la A y B casos estudiados anteriormente. Lo que vamos a ver es que uno de ellos (el caso A) corresponderá a la p-polarización mientras que el otro corresponde a la representación q, cuando se toma una decisión sobre el significado físico de las variables. Podemos empezar por definir kets abstractos etiquetados por un número real μ. Estos pertenecerán al espacio de Hilbert. Hpoly. A partir de estos estados, definimos un 'cilindro genérico estados’ que corresponden a una elección de una colección finita de Números μi-R con i = 1, 2,...., N. Asociados a esto elección, hay N vectores i, por lo que podemos tomar un lineal combinación de ellos = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 ai i (15) El producto interior del polímero entre los kets fundamentales es administrado por, =,μ (16) Es decir, los kets son ortogonales entre sí (cuando ν 6= μ) y se normalizan ( = 1). Inmediatamente, esto implica que, dado que cualquier dos vectores = j=1 bj j y = i=1 ai i, el producto interior entre ellos es administrado por, = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * b̄k ak donde la suma es sobre k que etiqueta los puntos de intersección entre el conjunto de etiquetas j} y i}. El Hilbert espacio Hpoly es la terminación Cauchy de finito lineal com- de la forma (15) con respecto al pro blema interno uct (16). Hpoly no es separable. Hay dos básicos operadores en este espacio Hilbert: el «operador de etiquetas» : := μ y el operador de desplazamiento.. (.......................................................................................................................................................................................................................................................... # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # El operador es simétrico y el(los) operador(es) define una familia de un parámetro de operadores unitarios en Hpoly, donde su contiguo es dado por () = (). Esto la acción es, sin embargo, discontinuo con respecto a que y + son siempre ortogonales, no importa que tan pequeño es. Por lo tanto, no hay operador (hermitano) que podría generar... (.......................................................................................................................................................................................................................................................... Hasta ahora hemos dado la caracterización abstracta de el espacio Hilbert, pero uno quisiera hacer contacto con realizaciones concretas como funciones de onda, o por iden- • la adaptación de los operadores abstractos a las condiciones físicas de trabajo; Erators. Supongamos que tenemos un sistema con un espacio de configuración con coordenada dada por q, y p denota su canónico conjugate momenta. Supongamos también que para la física rea- hijos decidimos que la configuración coordin q will tienen algún “carácter discreto” (por ejemplo, si se trata de se identifican con la posición, se podría decir que hay una posición discreta subyacente en pequeña escala). ¿Cómo podemos aplicar estos requisitos por medio de ¿La representación del polímero? Hay dos posibilidades, dependiendo de la elección de las ‘polarizaciones’ para la ola- funciones, a saber, si serán funciones de Figuración q o momenta p. Vamos a dividir el disco- sión en dos partes. A. Polarización momentánea En esta polarización, los estados serán denotados por, (p) = (p) donde (p) = p = ei ¿Cómo están representados entonces los operadores y? Nota que si asociamos el operador multiplicativo Vá r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r s r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r s r r r r r r r r r r r r r r r r r r r r r r r r ~ = ei () p = ()(p) Vemos entonces que el operador VÃ3r (­) corresponde exactamente para el operador de turnos. Por lo tanto, también podemos concluir que el operador no existe. Ahora es fácil identificar al operador qÃ3r con: q · (p) = −i~ (p) = μ e ~ = (p) a saber, con el operador abstracto. La razón por la que decir que qâ € es discreto es porque este operador tiene como su eigenvalue la etiqueta μ del estado elemental (p), y esta etiqueta, incluso cuando puede tomar valor en un continuum de los posibles valores, debe entenderse como un conjunto discreto, dado que los estados son ortonormales para todos los valores de μ. Dado que los estados son ahora funciones de p, el interior el producto (16) debe definirse mediante una medida μ en la espacio en el que se definen las funciones de onda. En orden saber cuáles son estos dos objetos, a saber, el quan- el espacio "configuración" C y la medida correspondiente1, tenemos que hacer uso de las herramientas disponibles para nosotros desde la teoría de C*-álgebras. Si consideramos a los operadores Vócalo, junto con su producto natural y su relación con dado por Váš ∗() = Váš (), que tienen la estructura de a Abelian C*-álgebra (con unidad) A. Sabemos por la teoría de la representación de tales objetos que A es iso- mórfico al espacio de las funciones continuas C0(­) en una espacio compacto, el espectro de A. Cualquier representación de A en un espacio Hilbert como operador de multiplicación será sobre los espacios de la forma L2(­, dμ). Es decir, nuestro cuántico espacio de configuración es el espectro del álgebra, que en nuestro caso corresponde a la compactación de Bohr Rb de la línea real [11]. Este espacio es un grupo compacto y hay una medida de probabilidad natural definida en ella, el Medida Haar μH. Por lo tanto, nuestro Hilbert espacio Hpoly será isomórfico al espacio, Hpoly,p = L2(Rb, dμH) (17) En términos de «funciones periódicas cuasi» generadas por (p), el producto interior toma la forma := dμH (p)(p) := = lim L 7° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° dp(p)(p) =, nota que en la p-polarización, esta caracterización cor- responde a la «versión A» de la representación del polímero de Sec. II (donde se intercambian p y q). B. Q-polarización Consideremos ahora la otra polarización en la que la ola las funciones dependerán de la coordenada de configuración q: (q) = (q) + (q) + (q) + (q) = (q) + (q) + (q) = (q) + (q) = (q) + (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) = (q) Las funciones básicas, que ahora se llamará (q), debe ser, en cierto sentido, el dual de las funciones (p) de la subsección anterior. Podemos tratar de definirlos a través de un «Fourier transform»: (q) := q = q dμHpp que es dada por (q) := dμHqp(p) = dμH e −i p q ~ = q,μ (19) 1 aquí utilizamos la terminología estándar de ‘espacio de configuración’ para denotar el dominio de la función de onda incluso cuando, en este caso, corresponde al momento físicoa p. Es decir, los objetos básicos en esta representación son Kro- cuello deltas. Esto es precisamente lo que habíamos encontrado en Sec. II para la representación del tipo B. ¿Cómo está ahora el los operadores básicos representados y cuál es la forma de la ¿Producto interior? En cuanto a los operadores, esperamos que están representados de la manera opuesta como en el p-polarización anterior, pero que preservan el las mismas características: p® no existe (la derivada de la Kro- cuello delta está mal definido), pero su versión exponencial En el caso de Vázquez, se entiende por: VÃ3r (­) · (­) = (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) (­) y el operador qâ € que ahora actúa como multiplicación ha como sus propios estados, las funciones (q) =,q: q · (q) := μ (q) ¿Cuál es ahora la naturaleza de las configuraciones cuánticas espacio Q? ¿Y cuál es la medida sobre dμq? que define el producto interior que deberíamos tener: (q), (q) =, La respuesta viene de una de las caracterizaciones de la compactación de Bohr: sabemos que es, en un preciso sentido, dual a la línea real, pero cuando está equipado con el topología discreta Rd. Además, la medida relativa a Rd será la «medida de contabilidad». De esta manera recuperamos el las mismas propiedades que teníamos para la caracterización anterior del espacio del polímero Hilbert. Así podemos escribir: Hpoly,x := L2(Rd, dμc) (20) Esto completa una construcción precisa del poli-tipo B representación mer bosquejada en la sección anterior. Nota que si hubiéramos elegido la situación física opuesta, a saber q, la configuración observable, ser el quan- dad que no tiene un operador correspondiente, entonces habríamos tenido la realización opuesta: en el q- polarización habríamos tenido el polímero tipo A rep- resentimiento y el tipo-B para la p-polarización. As veremos que ambos escenarios han sido considerados en el literatura. Hasta ahora sólo hemos centrado nuestra discusión en el Aspectos cinemáticos del proceso de cuantificación. Déjanos Ahora considere en la siguiente sección la cuestión de la dinam- y recordar el enfoque que se había adoptado en el informe de la Comisión de Asuntos Económicos y Monetarios y de Política Industrial. literatura, antes de la cuestión de la eliminación del regulador fue reexaminado en [6]. IV. MECANISMOS DE CUANTO POLÍMICO: DINÁMICA Como hemos visto la construcción del polímero la representación es bastante natural y conduce a un teoría de tum con diferentes propiedades que la habitual Schrödinger homólogo como su no separabilidad, la no existencia de determinados operadores y la existencia de eigen-vectores normalizados que dan un valor preciso para una de las coordenadas espaciales de fase. Esto se ha hecho. sin tener en cuenta a un Hamiltoniano que dota a la sistema con una dinámica, energía y así sucesivamente. Primero consideremos el caso más simple de una partícula de masa m en un potencial V (q), en el que el Hamiltonian H adopta la forma, p2 + V (q) Supongamos además que el potencial es dado por un no- función periódica, como un polinomio o una función racional tion. Podemos ver inmediatamente que una implementación directa- de los Hamiltonianos está fuera de nuestro alcance, para el simple la razón de que, como hemos visto, en el polímero representa- ¡Podemos representar q o p, pero no los dos! ¿Qué? ¿Se ha hecho hasta ahora en la literatura? La más simple. cosa posible: aproximar el término no existente por un bien definida función que se puede cuantificar y la esperanza de el mejor. Como veremos en las próximas secciones, hay más de lo que uno puede hacer. En este punto también hay una decisión importante que debe ser hecho: que la variable q o p debe ser considerada como “des- ¿Cerveza? Una vez que se hace esta elección, entonces implica que la otra variable no existirá: si q se considera como dis- ocre, entonces p no existirá y tenemos que aproximarnos el término cinético p2/2m por otra cosa; si p va a ser la cantidad discreta, entonces q no se definirá y luego tenemos que aproximar el potencial V (q). ¿Qué hap- ¿lápices con potencial periódico? En este caso uno podría ser modelar, por ejemplo, una partícula en una celosía regular como un fonón que vive en un cristal, y luego el natural elección es tener q no bien definido. Por otra parte, la po- tential estará bien definido y no hay aproximación necesario. En la literatura se han considerado ambos escenarios. Por ejemplo, cuando se considera un mecánico cuántico sistema en [2], la posición fue elegida para ser discreta, así que p no existe, y uno está entonces en el tipo A para la polarización del momento (o el tipo B para el q- polarización). Con esta elección, es el término cinético el uno que tiene que ser aproximado, así que una vez que se ha hecho esto, entonces es inmediato considerar cualquier potencial que Por lo tanto, se definirá bien. Por otro lado, cuando con- cosmología cuántica del bucle lateral (LQC), el estándar elección es que la variable de configuración no está definida [4]. Esta elección se hace teniendo en cuenta que LQC se considera como el sector simétrico de la gravedad cuántica del bucle completo donde la conexión (que se considera como la configuración vari- no puede ser promovido a un operador y se puede sólo definir su versión exponencial, a saber, el holón- Omy. En ese caso, la variable canónicamente conjugada, estrechamente relacionado con el volumen, se convierte en «discreto», al igual que en la teoría completa. Este caso es, sin embargo, diferente de la partícula en un ejemplo potencial. Primero podríamos mencionar que la forma funcional de la restricción hamiltoniana que implementa dinámica tiene una estructura diferente, pero la diferencia más importante radica en que el sistema es constreñido. Volvamos al caso de la partícula en una po- tential y para la definición, comencemos con el aux- Marco iliar cinemático en el que: q es discreto, p no pueden ser promovidos y por lo tanto tenemos que aproximarnos el término cinético pÃ3s/2/2m. ¿Cómo se hace esto? El Stan... prescripción dard es definir, en el espacio de configuración C, un «gráfico» regular 0. Esto consiste en un numerable conjunto de puntos, equidistante, y caracterizado por un pa- rameter μ0 que es la separación (constante) entre puntos. El ejemplo más sencillo sería considerar la set 0 = {q R q = nμ0, Esto significa que los kets básicos que se considerarán Se corresponderá precisamente con las etiquetas μn pertenecientes a el gráfico 0, es decir, μn = nμ0. Por lo tanto, sólo considerar los estados de la forma, = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 bn n. (21) Este espacio ‘pequeño’ Hilbert H0, el espacio gráfico Hilbert, es un subespacio del polímero ‘grande’ Hilbert Space Hpoly pero es separable. La condición para un estado de la forma (21) pertenecer al espacio Hilbert H0 es que el co- efficients bn satisfacer: n bn2. Consideremos ahora el término cinético pÃ2/2m. Tenemos para aproximarlo mediante funciones trigonométricas, que se puede construir a partir de las funciones de la forma ei. p/~. Como hemos visto en secciones anteriores, estas funciones pueden ser promovidos a los operadores y actuar como traducción operadores en los kets. Si queremos permanecer en el γ, y no crear ‘nuevos puntos’, entonces uno es a considerar a los operadores que desplazan los kets por la cantidad justa. Es decir, queremos lo básico el operador de turno Vâ ° ° ° ° ° sea tal que mapee el ket con etiqueta nÃ3 al siguiente ket, es decir n+1Ã3. Esto puede... acción realizada mediante la fijación, de una vez por todas, del valor de la se permite que el parámetro  sea  = μ0. Tenemos entonces, Vóz (μ0) · nó = n + μ0ó = n+1ó que es lo que queríamos. Este «operador de turnos» básico ser el bloque de construcción para aproximar cualquier (polinomio) función de p. Con el fin de hacer eso nos damos cuenta de que la función p se puede aproximar por, * * * ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ (μ0 p ~ − e−i donde la aproximación es buena para p • ~/μ0. Por lo tanto, se puede definir un operador regulado p0 que depende de la «escala» μ0 como: = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = [V (μ0) − V (0)] (n+1â − n−1â > ) (22) Para regular el operador, hay (al menos) dos posibilidades, a saber, componer el operador p0 con sí mismo o para definir una nueva aproximación. La operación... ator p0 · p0 tiene la característica que cambia los estados dos pasos en el gráfico a ambos lados. Sin embargo, hay un... otro operador que sólo implica el cambio una vez: 2μ0 · n := [2 − Vâr (μ0) − Vâr (0)] · nâr = (23) lo que corresponde a la aproximación p2 2~ cos(μ0 p/~)), válido también en el régimen p • ~/μ0. Con estas consideraciones, uno puede definir el operador 0, el Hamiltoniano a escala μ0, que en la práctica «vive» en el espacio H0 como, 0 := p+2μ0 + V+ (q), (24) que es un operador bien definido, simétrico en H0. No... que el operador también se define en Hpoly, pero hay su interpretación física es problemática. Por ejemplo, resulta que el valor de expectativa del término cinético calculados en la mayoría de los estados (estados que no están adaptados a al valor exacto del parámetro μ0) es cero. Incluso si uno toma un estado que da expectativas “razonables” valores del término μ0-cinético y lo utiliza para calcular el valor de expectativa del término cinético correspondiente a una ligera perturbación del parámetro μ0 se obtendría cero. Este problema, y otros que surgen cuando se trabaja sobre Hpoly, obliga a uno a asignar una interpretación física a los hamiltonianos 0 sólo cuando su acción está restringida al subespacio H0. Ahora exploremos la forma que toma el Hamiltoniano en las dos posibles polarizaciones. En la q-polarización, la base, etiquetada por n viene dada por las funciones χn(q) = *q,μn. Es decir, las funciones de onda sólo tendrán sup- puerto en el conjunto 0. Alternativamente, se puede pensar en un como completamente caracterizado por el ‘Fourier coeffi- an: فارسى(q) ↔ an, que es el valor que la ola función •(q) toma en el punto q = μn = nμ0. Por lo tanto, el Hamiltoniano toma la forma de una ecuación de diferencia cuando actúa sobre un estado general............................................................................................................................................................................................................................................................ Resolver el tiempo Ecuación independiente de Schrödinger · para resolver la ecuación de diferencia para los coeficientes a. La polarización del impulso tiene una estructura diferente. En este caso, el operador pâ € 2μ0 actúa como una multiplicación operador, = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 1 − cos (μ0 p •(p) (25) El operador correspondiente a q se representará como un operador derivado p): = i~ Łp (p). Para un potencial genérico V (q), tiene que ser definido por medios de teoría espectral definidos ahora en un círculo. ¿Por qué? ¿En un círculo? Por la sencilla razón de que al restringir nosotros mismos a un gráfico regular 0, las funciones de p que preservarlo (cuando actúa como operador de turnos) son de la forma e(i m μ0 p/~) para m entero. Es decir, lo que tenemos son modos Fourier, etiquetados por m, del período 2η ~/μ0 en p. ¿Podemos pretender entonces que la variable de espacio de fase p es ¿Ahora compactado? La respuesta es afirmativa. Los producto interno en las funciones periódicas 0(p) de p que viene del espacio completo Hilbert Hpoly y dado por (p)(p)polio = lim L 7° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° dp (p) (p) (p) es exactamente equivalente al producto interior en el círculo dado por la medida uniforme (p)(p)0 = ∫ /μ0 /μ0 dp (p) (p) (p) con p (/μ0, /μ0). Mientras uno restrinja a... la atención a la gráfica 0, uno puede trabajar en este separable Hilbert espacio H0 de funciones integrables cuadradas en S Inmediatamente, se pueden ver las limitaciones de este descrip- tion. Si el sistema mecánico a cuantificar es tal que sus órbitas tienen valores de los momenta p que son no pequeño en comparación con /μ0 entonces la aproximación tomado será muy pobre, y no esperamos ni el descripción clásica eficaz ni su cuantificación para ser cerca de la estándar. Si, por otro lado, uno es al- en la región en la que la aproximación puede ser Considerado como fiable, entonces tanto clásico como cuántico de- las inscripciones deben aproximarse a la descripción estándar. ¿Qué hace «cerca de la descripción estándar» exactamente necesidades medias, por supuesto, algunas aclaraciones adicionales. In particular está asumiendo la existencia de la habitual Schrödinger representación en la que el sistema tiene un be- havior que también es coherente con las observaciones. Si esto es el caso, la pregunta natural es: ¿cómo podemos ¿Aparear tal descripción de la foto del polímero? ¿Está ahí? un gráfico bastante fino 0 que se aproximará al sistema ¿De tal manera que todas las observaciones sean indistinguibles? O mejor aún, ¿podemos definir un procedimiento, que implica un refinamiento del gráfico 0 tal que uno recupera el ¿Un cuadro estándar? También podría ocurrir que un límite continuo puede ser de- multada, pero no coincide con la «esperada». Pero también podría haber sistemas físicos para los que hay ninguna descripción estándar, o simplemente no tiene sentido. Puede en esos casos la representación del polímero, si ex- ists, proporcionar la descripción física correcta de la sys- ¿Tem en consideración? Por ejemplo, si existe un limitación física de la escala mínima fijada en μ0, como podría ser el caso de una teoría cuántica de la gravedad, entonces la descripción del polímero proporcionaría un verdadero por el valor de determinadas cantidades, como p en nuestro ejemplo. Este podría ser el caso para el lazo cuántico cosmología, cuando exista un valor mínimo para la volumen (proviene de la teoría completa), y el espacio de fase puntos cerca de la “singularidad” se encuentran en la región donde el la aproximación inducida por la escala μ0 se aparta de la descripción clásica estándar. Si en ese caso el poli- sistema cuántico mer se considera más fundamental que el sistema clásico (o su estándar Wheeler-De Witt cuantización), entonces uno interpretaría este dis- crepancias en el comportamiento como señal de la avería de descripción clásica (o su cuantificación ‘naive’). En la siguiente sección presentamos un método para eliminar el regulador μ0 que se introdujo como comieron el paso para construir la dinámica. Más precisamente, nosotros considerará la construcción de un límite continuo de la descripción del polímero mediante una renormalización procedimiento. V. LÍMITE CONTINUO Esta sección consta de dos partes. En el primero motivamos la necesidad de una noción precisa del límite continuo de la representación polimérica, explicando por qué más El enfoque directo e ingenuo no funciona. En la segunda fase: en parte, presentaremos las principales ideas y resultados de el papel [6], donde el hamiltoniano y el físico El espacio de Hilbert en la mecánica cuántica polimérica es... como un continuum límite de teorías eficaces, seguir- Las ideas del grupo de renormalización de Wilson. El resultado El espacio físico Hilbert resulta ser unitariamente isomor- phic a las Hs ordinarias = L2(R, dq) del Schrödinger teoría. Antes de describir los resultados de [6] debemos discutir el significado preciso de llegar a una teoría en el contin- uum. Consideremos, para mayor concreción, la representación del tipo B. sentacion en la q-polarizacion. Es decir, los estados son func... ciones de q y la base ortonormal (q) es dada por funciones características con soporte en q = μ. Déjanos Ahora supongamos que tenemos un estado de Schrödinger L2(R, dq). ¿Cuál es la relación entre Ł(q) y un estado? ¿En Hpoly, X? También estamos interesados en las preguntas opuestas. sión, es decir, nos gustaría saber si hay una preferencia estado en Hs que es aproximado por un estado arbitrario (q) en Hpoly,x. La primera observación obvia es que un Estado Schödinger (q) no pertenece a Hpoly,x ya que tendría una norma infinita. Para ver esa nota que incluso cuando el Estado aspirante puede ser formalmente ampliado en el base como, (q) = (μ) (q) donde la suma es sobre el parámetro μ â € R. Su associ- ated norma en Hpoly,x sería: (q)2polio = (μ)2 → que explota. Tenga en cuenta que para definir una asignación P : Hs → Hpoly, x, hay una gran ambigüedad desde el se necesitan los valores de la función فارسى(q) con el fin de ampliar la función de la onda polimérica. Por lo tanto, sólo podemos definir un mapping en un denso subconjunto D de Hs donde los valores de la funciones están bien definidas (recordemos que en Hs el valor de funciones en un punto dado no tiene significado ya que los estados son clases de equivalencia de funciones). Podríamos, por ejemplo, pedir que la asignación se defina para los representantes de la clases de equivalencia en Hs que son continua por partes. A partir de ahora, cuando nos referimos a un elemento del espacio Nos referiremos a uno de esos representantes. Observe entonces que un elemento de Hs define un elemento de Cyl, el dual al espacio Cylγ, es decir, el espacio de funciones de cilindro con soporte en la celosía (finita) γ = 1, μ2,. .., μN}, de la siguiente manera: (q) : Cylγ C de tal manera que *(q)[(q)] = ( := (μ) - ¡No! - ¡No! - ¡No! - ¡No! - ¡No! • (μi) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Nótese que este mapeo podría ser visto como consistente en dos partes: Primero, una proyección Pγ : Cyl ∗ → Cylγ de tal manera que Pγ() = (q) := i-(μi)i(q)-Cylγ. El Estado se refiere a veces como la «sombra de Ł(q) en la celosía γ». El segundo paso es entonces tomar el interior producto entre la sombra (q) y el estado (q) con respecto al producto interno del polímero poliγ. Ahora este producto interior está bien definido. Note que para cualquier celosía dada γ el proyector correspondiente Pγ puede ser intuitivamente interpretado como una especie de ‘granulación gruesa mapa’ del continuum a la celosía γ. En términos de funciones de q la proyección está reemplazando un continuo función definida en R con una función sobre la celosía γ R, que es un conjunto discreto simplemente restringiendo a γ. Cuanto más fina sea la celosía, más puntos tendremos. en la curva. Como veremos en la segunda parte de este sección, hay de hecho una noción precisa de grano grueso que implementa esta idea intuitiva de una manera concreta. En particular, tendremos que sustituir la celosía γ por una descomposición de la línea real en intervalos puntos de celosía como puntos finales). Consideremos ahora un sistema en el polímero represen- en la que se eligió una celosía particular γ0, por ejemplo con puntos de la forma {qk â R qk = ka0, â k â Z}, es decir, una celosía uniforme con espaciamiento igual a a0. En este caso, cualquier función de onda Schrödinger (del tipo que considerar) tendrá una sombra única en la celosía γ0. Si refinamos la celosía γ 7→ γn dividiendo cada intervalo en 2n nuevos intervalos de longitud a = a0/2 Tenemos una nueva sombra... ows que tienen más y más puntos en la curva. Intu- itativamente, refinando infinitamente el gráfico nos recuperaríamos la función original فارسى(q). Incluso cuando en cada paso finito la sombra correspondiente tiene una norma finita en el poli- mer Hilbert espacio, la norma crece ilimitadamente y el el límite no se puede tomar, precisamente porque no podemos em- cama Hs en Hpoly. Supongamos ahora que estamos interesados en el proceso inverso, es decir, a partir de un polímero teoría sobre una celosía y pidiendo la "onda continua función’ que se aproxima mejor por una función de onda sobre un gráfico. Supongamos, además, que queremos con- sider el límite de la gráfica cada vez más fino. En orden para dar respuestas precisas a estas (y otras) preguntas necesidad de introducir algunas nuevas tecnologías que nos permitirán para superar estas aparentes dificultades. En el resto de esta sección recordaremos estas construcciones para el beneficio del lector. Los detalles se pueden encontrar en [6] (que es una aplicación del formalismo general discutido en [9]). El punto de partida de esta construcción es el concepto de una escala C, que nos permite definir la eficacia de y el concepto de límite continuo. En nuestro caso, un escala es una descomposición de la línea real en la unión de intervalos cerrados-abiertos, que cubren toda la línea y hacen no se intersectan. Intuitivamente, estamos cambiando el énfasis desde los puntos de celosía a los intervalos definidos por el los mismos puntos con el objetivo de aproximar funciones tinuas definidas en R con funciones que son constante en los intervalos definidos por la celosía. Ser precisa, definimos una incrustación, para cada escala Cn de Hpoly a Hs por medio de una función escalonada: •(hombre) χman(q) → *(hombre) m(q)* Hs con n(q) una función característica en el intervalo αm = [hombre, (m + 1)an). Por lo tanto, las sombras (viviendo en la celosía) eran sólo un paso intermedio en el con- estructuración de la función de aproximación; esta función es constante por pieza y se puede escribir como un com- lineal bination de funciones de escalón con los coeficientes proporcionados por las sombras. El desafío ahora es definir en un sentido apropiado cómo se pueden aproximar todos los aspectos de la teoría por medio de esta constante por piezas funciones. Entonces el estrategia es que, para cualquier escala dada, se puede definir un teoría eficaz mediante la aproximación del operador cinético por una combinación de los operadores de traducción que cambian entre los vértices de la descomposición dada, en otros palabras por una función periódica en p. Como resultado uno tiene un conjunto de teorías eficaces a escalas determinadas que son mutuamente relacionados con mapas de granulación gruesa. Este marco era el siguiente: desarrollado en [6]. Para la comodidad del lector nosotros Recordemos brevemente parte de ese marco. Vamos a denotar el espacio cinemático polímero Hilbert en la escala Cn como HCn, y sus elementos de base como eαi,Cn, donde αi = [ian, (i + 1)an) • Cn. Por la construcción de este la base es ortonormal. Los elementos de base en la dualidad Hilbert espacio H*Cn se denotan por i,Cn; también son Ortonormal. Los estados i, Cn tienen una acción simple en Cyl, i,Cn(lx0,q) = i,Cn(lx0). Es decir, si x0 está en el intervalo αi de Cn el resultado es uno y es cero si es No está ahí. Dado cualquier m ≤ n, definimos d*m,n : H*Cn → H como el mapa de ‘granulación gruesa’ entre el doble Hilbert espacios, que envía la parte de los elementos del dual base a cero manteniendo la información del resto: d*m,n(i,Cn) = j,Cm si i = j2 n-m, en el caso contrario d*m,n(i,Cn) = 0. En cada escala la teoría efectiva correspondiente es dado por el hamiltoniano Hn. Estos Hamiltonianos lo harán. ser tratados como formas cuadráticas, hn : HCn → R, dado por hn(­) =  (,Hn), (27) en la que 2Cn es un factor de normalización. Veremos más tarde. que este reescalamiento del producto interior es necesario en para garantizar la convergencia de los renormalizados teoría. La teoría completamente renormalizada a esta escala se obtiene como hrenm := lim - Sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí, sí., sí, sí, sí., sí, sí, sí., sí, sí, sí., sí, sí., sí., sí, sí., sí. (28) y los Hamiltonianos renormalizados son compatibles con el uno al otro, en el sentido de que - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. n = h Con el fin de analizar las condiciones para la convergencia en (28) vamos a expresar el hamiltoniano en términos de su eigen-covectores fin eigenvalues. Trabajaremos con... tiva Hamiltonianos que tienen un espectro puramente discreto (marcado por: · Hn ·, Cn = E/, Cn, Cn. También lo haremos. introducir, como paso intermedio, un corte en la energía niveles. El origen de este corte está en la aproximación del Hamiltoniano de nuestro sistema en una escala dada con a Hamiltoniano de un sistema periódico en un régimen de pequeño energías, como explicamos antes. Por lo tanto, podemos escribir h vcut-offm = vcut-off E/,Cm,Cm ,Cm,(29) donde los covectores eígenos,Cm se normalizan de acuerdo- al producto interior redistribuido por 1 , y el corte... off puede variar hasta una escala dependiente unida, νcut−off ≤ vmax(Cm). El espacio Hilbert de los covectores junto con tal producto interno se llamará H.renCm. En presencia de un corte, la convergencia de la Hamiltonianos microscópicamente corregidos, ecuación (28) es equivalente a la existencia de los dos límites siguientes. El primero es la convergencia de los niveles de energía, E/Cn = E /. (30) Segundo es la existencia de la completamente renormalizada covectores autóctonos, m,n,Cn = - ¿Qué es esto? - ¿Qué es esto? * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 31) Aclaramos que la existencia del límite anterior significa que la letra c) del apartado 1 del artículo 3 del Reglamento (CEE) n° 1408/71 del Consejo, de 17 de diciembre de 1971, relativo a la aplicación de los regímenes de seguridad social a los trabajadores por cuenta ajena, a los trabajadores por cuenta propia, a los trabajadores por cuenta propia y a los trabajadores por cuenta propia, a los trabajadores por cuenta propia y a los trabajadores por cuenta propia, a los trabajadores por cuenta propia y a los trabajadores por cuenta propia, a los trabajadores por cuenta propia y a los trabajadores por cuenta propia, a los trabajadores por cuenta propia y a los trabajadores por cuenta propia, a los trabajadores por cuenta propia y a los trabajadores por cuenta propia, a los trabajadores por cuenta propia y a los trabajadores por cuenta propia, a los trabajadores por cuenta propia y a los trabajadores por cuenta propia, a los trabajadores por cuenta ajena y a los trabajadores por cuenta ajena, a los trabajadores por cuenta ajena y a los trabajadores por cuenta propia, a los trabajadores por cuenta ajena y a los trabajadores por cuenta ajena, a los trabajadores por cuenta ajena, a los trabajadores por cuenta ajena y a los trabajadores por cuenta ajena, a los trabajadores por cuenta ajena y a los trabajadores por cuenta ajena, No... que esta convergencia punto a punto, si se puede llevar a cabo en absoluto, requerirá la afinación de los factores de normalización 2Cn. Pasamos ahora a la cuestión del límite del continuum de los covectores renormalizados. En primer lugar podemos pedir por el existencia del límite El Tribunal de Primera Instancia decidió, en primer lugar, si la Decisión de la Comisión de 17 de diciembre de 1994 (en lo sucesivo, «Decisión impugnada»), que, en el caso de autos, debía interpretarse en el sentido de que la Decisión de la Comisión de 17 de diciembre de 1994 (en lo sucesivo, «Decisión impugnada») no había sido adoptada por el Tribunal de Primera Instancia en el sentido de que la Decisión de la Comisión de 21 de diciembre de 1995 (en lo sucesivo, «Decisión impugnada») no había sido adoptada por el Tribunal de Justicia en el sentido de que la Decisión de la Comisión de 21 de diciembre de 1995 (en lo sucesivo, «Decisión impugnada») no había sido adoptada por el Tribunal de Justicia en el sentido de que la Decisión de la Comisión de 21 de diciembre de 1995 (en lo sucesivo, «Decisión impugnada»). para cualquier فارسىx0,q Cyl. Cuando estos límites existen hay una acción natural de los covectores autóctonos en el continuum límite. A continuación consideramos otra noción del continuum límite de los covectores autóctonos renormalizados. Cuando los covectores autóctonos completamente renormalizados existen, forman una colección que es compatible, - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. = Ren/,Cm. Una secuencia de d - Compatibles ni compatibles. Los covectores maleables definen un elemento de , que es el límite proyectivo de los espacios renormalizados de covec- HerenCn. 33) El producto interior en este espacio está definido por (Cn}, Cn})R := lim (Cn,ΦCn) La inclusión natural de C­0 en es por un antilineal mapa que asigna a cualquier â € â € € â € TM Câ € TM el dâ €-compatible colección shadCn := i(L(αi)) Se le llamará a ShadCn la sombra de # a escala Cn y actúa en Cyl como una función constante a trozos. Claramente otro tipos de funciones de prueba como las funciones de Schwartz son también naturalmente incluidos en . En este contexto una sombra es un estado de la teoría efectiva que se aproxima a un estado en la teoría del continuum. Desde el producto interior en es degenerado, el espacio físico Hilbert se define como Höphys := / ker(·, ·)ren Hphys := Hóphys La naturaleza del espacio físico Hilbert, si es isomórfico al espacio de Schrödinger Hilber, Hs, o no, es determinado por los factores de normalización se obtiene de las condiciones que exigen la compatibil- ity de la dinámica de las teorías eficaces en diferentes básculas. La dinámica del sistema que se examina selecciona el límite del continuum. Volvamos ahora a la definición de la Hamilto- nian en el límite del continuum. En primer lugar considerar la continuación de uum límite del Hamiltoniano (con corte) en el sentido de su convergencia puntual como forma cuadrática. Lo siento. resulta que si el límite de la ecuación (32) existe para todos los covectores autóctonos permitidos por el corte, tenemos vcut-off ren : Hpoly,x → R definido por vcut-off ren (­x0,q) := lim h/cut−off Renn ([lx0,q]Cn). (34) Esta forma cuadrática hamiltoniana en el continuum puede ser de grano grueso a cualquier escala y, como puede ser ex- , produce el Hamilto completamente renormalizado- Nian forma cuadrática a esa escala. Sin embargo, esto no es un límite de continuum completamente satisfactorio porque podemos no retirar el corte auxiliar νcut−off. Si lo intentamos, como incluimos más y más covectores propios en el Hamilto- nian los cálculos hechos a una escala dada divergerían y hacerlos en el continuum es igual de divergente. A continuación exploramos un camino más exitoso. Podemos utilizar el producto interno renormalizado para inducir una acción de los hamiltonianos de corte en vcut-off ren (Cn} := lim h/cut­off renn ((­)Cn, ·)renCn ), donde hemos utilizado el hecho de que (­Cn, ·)renCn • HCn. Los la existencia de este límite es trivial porque el renormalizado Hamiltonianos son sumas finitas y el límite existe término por término. Estos hamiltonianos de corte descienden a lo físico Espacio Hilbert vcut-off ren ([Cn}]):= h vcut-off ren (Cn} para cualquier representante Cn} [Cn}] Hóphys. Por último, podemos abordar la cuestión de la eliminación de la Fuera. El hamiltoniano hren → R se define por la límite := lim & cct−off & cclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclclcl vcut-off ren cuando el límite existe. Su correspondiente forma ermitaña en Hphys se define siempre que exista el límite anterior. Esto concluye nuestra presentación de los principales resultados de [6]. Vamos. ahora consideremos varios ejemplos de sistemas para los que el límite del continuum puede ser investigado. VI. EJEMPLOS En esta sección vamos a desarrollar varios ejemplos de sistemas que han sido tratados con el polímero cuanti- Zation. Estos ejemplos son simples mecánicos cuánticos sistemas, como el oscilador armónico simple y el partículas libres, así como un modelo cosmológico cuántico conocido como cosmología cuántica del bucle. A. El Oscilador Armónico Simple En esta parte, vamos a considerar el ejemplo de un simple Har- Oscilador mónico (SHO) con parámetros m y sicamente descrito por el siguiente Hamiltonian 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 Recuerde que a partir de estos parámetros se puede definir una longitud escala D = - Sí. En el tratamiento estándar se utiliza esta escala para definir una estructura compleja JD (y un r producto de la misma), como hemos descrito en detalle que Selecciona de forma única la representación estándar de Schrödinger. A escala Cn tenemos un Hamiltoniano eficaz para el Oscilador Armónico Simple (SHO) dado por HCn = 1 − como anp má2x2. (35) Si intercambiamos posición e impulso, este Hamilto... nian es exactamente el de un péndulo de masa m, longitud l y sujeto a un campo gravitatorio constante g: Cn = − +mgl(1 − cos ) cuando esas cantidades estén relacionadas con nuestro sistema, m-a-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-d-e-e-d-d-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e-e- , g = ............................................................... Es decir, estamos aproximando, para cada escala Cn el SHO por un péndulo. Hay, sin embargo, un importante diferencia. De nuestro conocimiento del sistema del péndulo, Sabemos que el sistema cuántico tendrá un espectro para la energía que tiene dos behav asintóticos diferentes - iors, el SHO para bajas energías y el rotor planar en el extremo superior, correspondiente a oscilación y rotación soluciones, respectivamente2. A medida que refinamos nuestra escala y ambos la longitud del péndulo y la altura del periódico aumento potencial, esperamos tener un aumento de num- br de estados oscilantes (para un sistema de péndulo dado, sólo hay un número finito de tales estados). Por lo tanto, se justifica considerar el corte en el eigenval de la energía Como se ha dicho en la última sección, dado que sólo esperar un número finito de estados del péndulo a ap- Eigenstatos de SHO próximos. Con estas consideraciones en mente, la pregunta relevante es si las condiciones para el continuum límite a existir está satisfecho. Esta pregunta se ha respondido afirmativamente en [6]. ¿Qué fue? se demostró que los valores propios y eigen func- ciones de los sistemas discretos, que representan un y no degenerados, aproximándose a los de los contin- uum, es decir, del oscilador armónico estándar cuando el producto interior se vuelve a normalizar por un factor 2Cn = 1/2 Esta convergencia implica que existe el límite continuo como lo entendemos. Consideremos ahora la más simple sistema posible, una partícula libre, que tiene sin embargo la particular característica de que el espectro de la energía es Tinuous. 2 Tenga en cuenta que ambos tipos de soluciones están, en el espacio de fase, cerrados. Esta es la razón detrás del espectro puramente discreto. Los la distinción que estamos haciendo es entre esas soluciones dentro de la separatrix, que llamamos oscilante, y aquellos que están por encima de ella que llamamos rotación. B. Partícula libre de polímero En el límite فارسى → 0, el Hamiltoniano de lo Simple El oscilador armónico (35) va al Hamiltoniano de un partícula libre y el tiempo correspondiente independiente La ecuación de Schrödinger, en la p-polarización, está dada por (1 − cos anp ) − CEn (p) = 0 donde ahora tenemos que p â € S1, con p â € ( Por lo tanto, tenemos ECn = 1 − cos ≤ CEn,max. 2 . (36) A cada escala podemos describir la energía de la partícula. está limitado desde arriba y el límite depende de la escala. Nótese que en este caso el espectro es continuo ous, lo que implica que las funciones propias ordinarias de El Hilbert no es normal. Esto impone una limitada en el valor que la energía de la partícula puede tienen, además de los límites en el impulso debido a su “compactación”. Busquemos en primer lugar soluciones eigen a la hora inde- péndulo Schrödinger ecuación, es decir, para la energía eigen- estados. En el caso de la partícula libre ordinaria, estos corresponden a ondas planas de impulso constante de la forma e±( ) y de tal manera que la dispersión ordinaria re- ión p2/2m = E está satisfecho. Estas ondas planas son no cuadrado integrable y no pertenecen a lo ordinario Hilbert espacio de la teoría Schrödinger, pero todavía son útil para extraer información sobre el sistema. Por la partícula libre de polímero que tenemos, Cn(p) = c1♥(p− PCn) + c2/23370/(p+ PCn) donde PCn es una solución de la ecuación anterior consid- , con un valor fijo de ECn. Es decir, PCn = P (ECn) = arccos − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 − El inverso Fourier transforma los rendimientos, en el ‘x represen- dad», Cn(xj) = ∫ /un /un (p) e p j dp = ixjPCn /~ + c2e - ixjPCn /~ .(37) con xj = un j para j â € Z. Tenga en cuenta que las funciones propias son todavía funciones delta (en la representación p) y por lo tanto no (cuadrado) normalizable con respecto al polímero producto interno, que en la polarización p se acaba de dar por la medida ordinaria de Haar en S1, y no hay la cuantificación del impulso (su espectro sigue siendo verdaderamente continuum). Consideremos ahora el tiempo dependiente Schrödinger ecuación, (p, t) = · (p, t). Que ahora toma la forma, (p, t) = (1 − cos (un p/~)) (p, t) que tiene como solución, (p, t) = e− (1−cos (un p/~)) t (p) = e(−iECn /~) t (p) para cualquier función inicial (p), donde la CEn satisface la relación de persión (36). La función de onda (xj, t), la xj-representación de la función de onda, se puede obtener para cualquier tiempo dado t por Fourier transformando con (37) la función de onda (p, t). Con el fin de comprobar la convergencia de la micro- scopicamente corregido Hamiltonians debemos analizar el la convergencia de los niveles de energía y de la ectors. En el límite n → فارسى, ECn → E = p2/2m tan podemos estar seguros de que los valores propios para la energía convergen (al fijar el valor de p). Vamos a escribir el el covector adecuado como Cn = (Cn, ·)ren Cn • H . Entonces nosotros puede traer correcciones microscópicas a escala Cm y mirar para la convergencia de dichas correcciones *RenCm* = lim cn.......................................................................................................................... Es fácil ver que dado cualquier vector de base eαi HCm el límite siguiente: renCm(eαi,Cm) = limCn Cn(dn,m(eαi,Cm)) existe y es igual a (eαi,Cm) = [d Schr](eαi,Cm) = Schr(iam) donde se calcula el valor de la sustancia problema utilizando la partícula libre Hamilto- Nian en la representación de Schrödinger. Esta expresión define el covector adecuado completamente renormalizado en la escala Cm. C. Cosmología cuántica de polímeros En esta sección vamos a presentar una versión de cuántica cosmología que llamamos cosmología cuántica polimérica. Los La idea detrás de este nombre es que la entrada principal en el quan- tización del modelo mini-superespacio correspondiente es el uso de una representación de polímero tal como se entiende aquí. Otra aportación importante es la elección de los elementos fundamentales variables a utilizar y la definición del Hamiltoniano restricción. Distintos grupos de investigación han diferen- ent opciones. Vamos a tomar aquí un modelo simple que tiene recibió mucha atención recientemente, a saber, un isotrópico, cosmología homogénea del FRW con k = 0 y acoplada a un campo escalar sin masa. Como veremos, un el tratamiento del límite continuo de este sistema requiere nuevos instrumentos en desarrollo que están más allá del ámbito de aplicación de este trabajo. Por lo tanto, nos limitaremos a la introducción miento del sistema y de los problemas que deben Resuelto. El sistema a cuantificar corresponde a la fase espacio de espacios cosmológicos que son homogéneos e isotrópico y para los cuales la homogeneidad espacial las rebanadas tienen una geometría intrínseca plana (k = condición 0). El único contenido de materia es un campo escalar sin masa. In este caso la geometría espacio-tiempo es dada por las métricas de el formulario: ds2 = −dt2 + a2(t) (dx2 + dy2 + dz2) donde la función a(t) lleva toda la información y grados de libertad de la parte gravitatoria. En términos de la Coordenadas (a, pa, ­, p­) para el espacio de fase de la Ory, todas las dinámicas son capturadas en el con- strantest C := −3 + 8ηG 2a3 El primer paso es definir la restricción sobre la kine- matical Hilbert espacio para encontrar estados físicos y luego un producto interior físico para construir el Hilbert físico espacio. Primero note que se puede reescribir la ecuación como: p2a a 2 = 8ηG Si, como se hace normalmente, se opta por actuar como un in- tiempo, el lado derecho sería promovido, en la teoría cuántica, a una segunda derivada. La izquierda lado de la mano es, además, simétrico en a y pa. En este punto tenemos la libertad en la elección de la variable que será cuantificado y la variable que no será bien definido en la representación del polímero. El estándar elección es que pa no está bien definido y por lo tanto, a y cualquier cantidad geométrica derivada de ella, se cuantifica. Piel... termorre, tenemos la opción de polarización en la onda función. A este respecto, la elección estándar es seleccionar la a-polarización, en la que una actúa como multiplicación y la aproximación de pa, a saber, sin( diferencia operador en las funciones de onda de a. Para más detalles: esta elección particular véase [5]. En este contexto, adoptaremos la op- posite polarización, es decir, tendremos funciones de onda (pa, فارسى). Al igual que hicimos en los casos anteriores, con el fin de ganar intuición sobre el comportamiento del polímero cuantificado la teoría, es conveniente mirar el prob equivalente- en la teoría clásica, a saber, el sistema clásico Estaríamos aproximándonos a lo no bien definido. servible (pa en nuestro caso actual) por un objeto bien definido (hecho de funciones trigonométricas). Vamos a la simplicidad opte por reemplazar pa 7→ sin( Con esta opción Obtenemos una restricción clásica Hamiltoniana eficaz que depende de : C. := − sin(l pa) - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. + 8ηG 2a3 Ahora podemos calcular ecuaciones efectivas de movimiento por medios de las ecuaciones: := {F, C, para cualquier observable F. C. C. C. C. C., y donde estamos utilizando la eficacia (primero orden) acción: *(pa N C con la opción N = 1. Lo primero que hay que notar es que la cantidad de p.o.p. es una constante de la moción, dado que la variable فارسى es cíclica. La segunda observación es que = 8ηG tiene la misma señal que pÃ3 y nunca desaparece. Por lo tanto, puede ser utilizado como una variable de tiempo (n interna). Los siguiente observación es que la ecuación para , a saber: la ecuación efectiva de Friedman, tendrá un cero para un valor no cero de un dado por 2p2o. Este es el valor en el que se rebotará si el la trayectoria comenzó con un gran valor de a y fue Tracciones. Note que el ‘tamaño’ del universo cuando el rebote se produce depende tanto de la constante dicta la densidad de la materia) y el valor de la celosía tamaño ♥. Aquí es importante subrayar que para cualquier valor (que fija de manera única la trayectoria en el (a, pa) avión), habrá un rebote. En la descripción original en términos de las ecuaciones de Einstein (sin la No hay tal rebote. Si < 0 inicialmente, permanecerá negativo y el universo colapsa, alcanzando la singularidad en un tiempo finito apropiado. ¿Qué sucede dentro de la descripción efectiva si re- afinar la celosía y pasar de ¿N? El único que cambia, para la misma órbita clásica etiquetada por pŁ, es que el rebote se produce en un ‘tiempo posterior’ y para un valor menor de un* pero el cuadro cualitativo sigue siendo Lo mismo. Esta es la principal diferencia con los sistemas considerados antes. En esos casos, uno podría tener trayectoria clásica... rs que quedaron, para una determinada elección de parámetro dentro de la región donde el pecado es una buena Por supuesto, también había trayectorias clásicas. que estaban fuera de esta región, pero entonces podríamos refinar el retícula y encontrar un nuevo valor para el cual el nuevo clas- La trayectoria sical está bien aproximada. En el caso de la cosmología polimérica, este nunca es el caso: Cada clásico la trayectoria pasará de una región donde la sión es buena para una región en la que no lo es; esto es precisamente donde las ‘correcciones cuánticas’ entran en juego y los universos rebotes. Dado que en la descripción clásica, el «original» y las descripciones ‘corregidas’ son tan diferentes que esperamos que, tras la cuantificación, el cuántico correspondiente el- ories, a saber, el polimérico y el Wheeler-DeWitt estar relacionado de manera no trivial (si es que existe). En este caso, con la elección de la polarización y para una particular el orden de los factores que tenemos, sin(lpa) · (pa, ­) = 0 como la ecuación Polymer Wheeler-DeWitt. A fin de abordar el problema del continuo límite de esta teoría cuántica, tenemos que darnos cuenta de que la la tarea es ahora algo diferente que antes. Esto es así. dado que el sistema es ahora un sistema limitado con un operador de restricción en lugar de un no-singular regular sistema con una evolución Hamiltoniana ordinaria. Fortu... nalmente para el sistema que se examina, el hecho de que puede ser considerado como un tiempo interno permite para interpretar la restricción cuántica como una Klein-Gordon ecuación de la forma # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # cuando el operador sea «independiente en el tiempo». Esta al- nos reduce a dividir el espacio de soluciones en ‘positivos y frecuencia negativa», introducir un producto interior físico sobre las soluciones de frecuencia positiva de esta ecuación y un conjunto de observables físicos en función de los cuales de- escriba el sistema. Es decir, se reduce en la práctica la sistema a uno muy similar al caso Schrödinger por tomando la raíz cuadrada positiva de la ecuación anterior: # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # La pregunta que nos interesa es: si el continuum límite de estas teorías (marcado y si se corresponde con el Wheeler- La teoría de DeWitt. Un tratamiento completo de este problema Desgraciadamente, está fuera del ámbito de este trabajo y se informará en otro lugar [12]. VII. DEBATE Resumamos nuestros resultados. En la primera parte de la artículo mostramos que la representación del polímero de la las relaciones canónicas de conmutación se pueden obtener como la el caso limitador de la Fock-Schrödinger ordinario represen- en términos del estado algebraico que define el representación. Estos casos limitantes también pueden ser inter- pretendidos en términos de los estados coherentes definidos naturalmente asociado a cada representación etiquetada por el eter d, cuando se vuelven infinitamente ‘estrujados’. Los dos posibles límites de compresión conducen a dos polímeros diferentes descripciones que, sin embargo, se pueden identificar, como nosotros también han demostrado, con las dos posibles polarizaciones para una representación polímero abstracta. El resultado fue el siguiente: ory tiene, sin embargo, un comportamiento muy diferente como el estándar Uno: El espacio Hilbert no es separable, el representa- es inequivalente unitariamente a la de Schrödinger, y los operadores naturales como pÃ3n ya no están bien definidos. Esta construcción limitante particular del polímero el- Ory puede arrojar algo de luz para sistemas más complicados como las teorías de campo y la gravedad. En los tratamientos regulares de la dinámica dentro de la poli- representación mer, uno necesita introducir algunos extra estructura, como una celosía en el espacio de configuración, a con- construir un Hamiltoniano e implementar la dinámica para el sistema mediante un procedimiento de regularización. ¿Cómo es que esto re- teoría sulting comparar con la teoría del continuum original uno tenía desde el principio? ¿Puede uno esperar eliminar el regulador en la descripción del polímero? Tal como están. no hay relación directa o mapeo del polímero a una teoría de continuum (en caso de que haya una definida). As hemos demostrado, uno puede construir de hecho en un sistema dad, por medio de alguna enmienda apropiada que no se refiera a los derechos humanos, a los derechos humanos, a los derechos humanos y a las libertades fundamentales, y a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos, a los derechos humanos y a las libertades fundamentales, ciones relacionadas con la definición de una escala, a la celosía uno tenía que introducir en la regularización. Con este importante cambio en la perspectiva, y una renormalización priato del producto interior del polímero en cada escala uno puede, sujeto a alguna condición de consistencia- ciones, definir un procedimiento para eliminar el regulador, y llegar a un Hamiltoniano y un espacio Hilbert. Como hemos visto, para algunos ejemplos simples como una partícula libre y el oscilador armónico uno de hecho recupera la descripción de Schrödinger. Para otros sistemas: tems, como los modelos cosmológicos cuánticos, la respuesta no es tan claro, ya que la estructura del espacio de classi- Las soluciones de cal son tales que la «descripción eficaz» producido por la regularización del polímero a diferentes escalas es cualitativamente diferente de la dinámica original. A el tratamiento adecuado de esta clase de sistemas está en marcha y se informará de ello en otro lugar [12]. Tal vez la lección más importante que tenemos En este sentido, se ha aprendido que existe en efecto una riqueza intergubemamental. juego entre la descripción del polímero y el ordinario Representación de Schrödinger. La estructura completa de esta re- dad de la Unión Europea y de los Estados miembros de la Unión Europea, así como de los Estados miembros de la Unión Europea y de los Estados miembros de la Unión Europea. Sólo podemos esperar que una comprensión completa de estas cuestiones arrojará algo de luz en el objetivo final de tratar la dinámica cuántica de los sistemas sobre el terreno independientes de antecedentes, como relatividad. Agradecimientos Agradecemos a A. Ashtekar, G. Hossain, T. Pawlowski y P. Singh para discutir. Este trabajo fue apoyado en parte por subvenciones CONACyT U47857-F y 40035-F, por NSF PHY04-56913, por los Fondos de Investigación Eberly de Penn Estado, por el programa de intercambio AMC-FUMEC y por fondos del CIC-Universidad Michoacana de San Nicolás de Hidalgo. [1] R. Beaume, J. Manuceau, A. Pellet y M. Sirugue, “Estados Invariantes de Traducción en Mecánica Cuántica,” Comun. Matemáticas. Phys. 38, 29 (1974); W. E. Thirring y H. Narnhofer, “Covariante QED sin ric”, Rev. Matemáticas. Phys. 4, 197 (1992); F. Acerbi, G. Mor- chio y F. Strocchi, “Campos singulares infrarrojos y no- representaciones regulares de la conmutación canónica rela- álgebras de tion”, J. Matemáticas. Phys. 34, 899 (1993); F. Cav- allaro, G. Morchio y F. Strocchi, “Una generalización de el teorema de Stone-von Neumann a la representación no regular- sentaciones del CCR-álgebra”, Lett. Matemáticas. Phys. 47 307 (1999); H. Halvorson, "Completaridad de la sentaciones en mecánica cuántica”, Estudios de Historia y Filosofía de la Física Moderna 35 45 (2004). [2] A. Ashtekar, S. Fairhurst y J.L. Willis, “Quantum gravedad, estados de sombra y mecánica cuántica”, Class. Quant. Grav. 20 1031 (2003) [arXiv:gr-qc/0207106]. [3] K. Fredenhagen y F. Reszewski, “Polymer state ap- proximaciones de las funciones de la onda Schrödinger”, Class. Quant. Grav. 23 6577 (2006) [arXiv:gr-qc/0606090]. [4] M. Bojowald, “Loop quantum cosmology”, Living Rev. Rel. 8, 11 (2005) [arXiv:gr-qc/0601085]; A. Ashtekar, M. Bojowald y J. Lewandowski, "Estructura matemática tura de la cosmología cuántica del bucle”, Adv. Teor. Matemáticas. Phys. 7 233 (2003) [arXiv:gr-qc/0304074]; A. Ashtekar, T. Pawlowski y P. Singh, “Naturaleza cualitativa de la big bang: Dinámica mejorada” Phys. Rev. D 74 084003 (2006) [arXiv:gr-qc/0607039] [5] V. Husain y O. Winkler, “Estados semiclásicos para cosmología cuántica” Phys. Rev. D 75 024014 (2007) [arXiv:gr-qc/0607097]; V. Husain V y O. Winkler, “On resolución de singularidad en la gravedad cuántica”, Phys. Rev. D 69 084016 (2004). [arXiv:gr-qc/0312094]. [6] A. Corichi, T. Vukasinac y J.A. Zapata. “Hamil- toniano y espacio físico Hilbert en polímero quan- tum mechanics”, Class. Quant. Grav. 24 1495 (2007) [arXiv:gr-qc/0610072] [7] A. Corichi y J. Cortez, “Cantización canónica de una perspectiva algebraica” (preimpresión) [8] A. Corichi, J. Cortez y H. Quevedo, “Schrödinger y las representaciones de Fock para una teoría de campo sobre Tiempo espacial curvado”, Annals Phys. (NY) 313 446 (2004) [arXiv:hep-th/0202070]. [9] E. Manrique, R. Oeckl, A. Weber y J.A. Zapata, “Loop quantización como un límite continuo” Clase. Quant. Grav. 23 3393 (2006) [arXiv:hep-th/0511222]; E. Manrique, R. Oeckl, A. Weber y J.A. Zapata, “Teo- rios y límite continuo para la cuantificación canónica del bucle” (preimpresión) [10] D.W. Chiou, “Simetrías de Galileo en partículas poliméricas representación”, Class. Quant. Grav. 24, 2603 (2007) [arXiv:gr-qc/0612155]. [11] W. Rudin, análisis de Fourier sobre los grupos, (Interscience, New York, 1962) [12] A. Ashtekar, A. Corichi, P. Singh, “Contrasting LQC y WDW utilizando un modelo exactamente soluble” (preimpresión); A. Corichi, T. Vukasinac y J.A. Zapata, “Continuum límite para el sistema limitado cuántico” (preimpresión). http://arxiv.org/abs/gr-qc/0207106 http://arxiv.org/abs/gr-qc/0606090 http://arxiv.org/abs/gr-qc/0601085 http://arxiv.org/abs/gr-qc/0304074 http://arxiv.org/abs/gr-qc/0607039 http://arxiv.org/abs/gr-qc/0607097 http://arxiv.org/abs/gr-qc/0312094 http://arxiv.org/abs/gr-qc/0610072 http://arxiv.org/abs/hep-th/0202070 http://arxiv.org/abs/hep-th/051122 http://arxiv.org/abs/gr-qc/0612155
704.001
Numerical solution of shock and ramp compression for general material properties
A general formulation was developed to represent material models for applications in dynamic loading. Numerical methods were devised to calculate response to shock and ramp compression, and ramp decompression, generalizing previous solutions for scalar equations of state. The numerical methods were found to be flexible and robust, and matched analytic results to a high accuracy. The basic ramp and shock solution methods were coupled to solve for composite deformation paths, such as shock-induced impacts, and shock interactions with a planar interface between different materials. These calculations capture much of the physics of typical material dynamics experiments, without requiring spatially-resolving simulations. Example calculations were made of loading histories in metals, illustrating the effects of plastic work on the temperatures induced in quasi-isentropic and shock-release experiments, and the effect of a phase transition.
Numerical solution of shock and ramp compression for general material properties Damian C. Swift∗ Materials Science and Technology Division, Lawrence Livermore National Laboratory, 7000, East Avenue, Livermore, CA 94550, U.S.A. (Dated: March 7, 2007; revised April 8, 2008 and July 1, 2008 – LA-UR-07-2051) Abstract A general formulation was developed to represent material models for applications in dynamic loading. Numerical methods were devised to calculate response to shock and ramp compression, and ramp decompression, generalizing previous solutions for scalar equations of state. The numerical methods were found to be flexible and robust, and matched analytic results to a high accuracy. The basic ramp and shock solution methods were coupled to solve for composite deformation paths, such as shock-induced impacts, and shock interactions with a planar interface between different materials. These calculations capture much of the physics of typical material dynamics experiments, without requiring spatially-resolving simulations. Example calculations were made of loading histories in metals, illustrating the effects of plastic work on the temperatures induced in quasi-isentropic and shock-release experiments, and the effect of a phase transition. PACS numbers: 62.50.+p, 47.40.-x, 62.20.-x, 46.35.+z Keywords: material dynamics, shock, isentrope, adiabat, numerical solution, constitutive behavior ∗Electronic address: damian.swift@physics.org http://arxiv.org/abs/0704.0008v3 mailto:damian.swift@physics.org I. INTRODUCTION The continuum representation of matter is widely used for material dynamics in sci- ence and engineering. Spatially-resolved continuum dynamics simulations are the most widespread and familiar, solving the initial value problem by discretizing the spatial domain and integrating the dynamical equations forward in time to predict the motion and defor- mation of components of the system. This type of simulation is used, for instance, to study hypervelocity impact problems such as the vulnerability of armor to projectiles [1, 2], the performance of satellite debris shields [3], and the impact of meteorites with planets, notably the formation of the moon [4]. The problem can be divided into the dynamical equations of the continuum, the state field of the components s(~r), and the inherent properties of the materials. Given the local material state s, the material properties allow the stress τ to be determined. Given the stress field τ(~r) and mass density field ρ(~r), the dynamical equations describe the fields of acceleration, compression, and thermodynamic work done on the materials. The equations of continuum dynamics describe the behavior of a dynamically deforming system of arbitrary complexity. Particular, simpler deformation paths can be described more compactly by different sets of equations, and solved by different techniques than those used for continuum dynamics in general. Simpler deformation paths occur often in experiments designed to develop and calibrate models of material properties. These paths can be regarded as different ways of interrogating the material properties. The principal examples in material dynamics are shock and ramp compression [5, 6]. Typical experiments are designed to induce such loading histories and measure or infer the properties of the material in these states before they are destroyed by release from the edges or by reflected waves. The development of the field of material dynamics was driven by applications in the physics of hypervelocity impact and high explosive systems, including nuclear weapons [7]. In the regimes of interest, typically components with dimensions ranging from millime- ters to meters and pressures from 1GPa to 1TPa, material behavior is dominated by the scalar equation of state (EOS): the relationship between pressure, compression (or mass density), and internal energy. Other components of stress (specifically shear stresses) are much smaller, and chemical explosives react promptly so can be treated by simple mod- els of complete detonation. EOS were developed as fits to experimental data, particularly to series of shock states and to isothermal compression measurements [8]. It is relatively straightforward to construct shock and ramp compression states from an EOS algebraically or numerically depending on the EOS, and to fit an EOS to these measurements. More recently, applications and scientific interest have grown to include a wider range of pressures and time scales, such as laser-driven inertial confinement fusion [9], and experiments are designed to measure other aspects than the EOS, such as the kinetics of phase changes, con- stitutive behavior describing shear stresses, incomplete chemical reactions, and the effects of microstructure, including grain orientation and porosity. Theoretical techniques have also evolved to predict the EOS with ∼1% accuracy [10] and elastic contributions to shear stress with slightly poorer accuracy [11]. A general convention for representing material states is described, and numerical methods are reported for calculating shock and ramp compression states from general representations of material properties. II. CONCEPTUAL STRUCTURE FOR MATERIAL PROPERTIES The desired structure for the description of the material state and properties under dy- namic loading was developed to be as general as possible with respect to the types of material or models to be represented in the same framework, and designed to give the greatest amount of commonality between spatially-resolved simulations and calculations of shock and ramp compressions. In condensed matter on sub-microsecond time scales, heat conduction is often too slow to have a significant effect on the response of the material, and is ignored here. The equations of non-relativistic continuum dynamics are, in Lagrangian form, i.e. along characteristics moving with the local material velocity ~u(~r), Dρ(~r, t) = −ρ(~r, t)div~u(~r, t) (1) D~u(~r, t) ρ(~r, t) div τ(~r, t) (2) De(~r, t) = ||τ(~r, t)grad~u(~r, t)|| (3) where ρ is the mass density and e the specific internal energy. Changes in e can be related to changes in the temperature T through the heat capacity. The inherent properties of each material in the problem are described by its constitutive relation or equation of state τ(s). As well as experiencing compression and work from mechanical deformation, the local material state s(~r, t) can evolve through internal processes such as plastic flow. In general, Ds(~r, t) ≡ ṡ[s(~r, t), U(~r, t)] : U ≡ grad ~u(~r, t) (4) which can also include the equations for ∂ρ/∂t and ∂e/∂t. Thus the material properties must describe at a minimum τ(s) and ṡ[s(~r, t), U(~r, t)] for each material. If they also describe T (s), the conductivity, and ṡ(ė), then heat conduction can be treated. Other functions may be needed for particular numerical methods in continuum dynamics, such as the need for wave speeds (e.g. the longitudinal sound speed), which are needed for time step control in explicit time integration. Internally, within the material properties models, it is desirable to re-use software as much as possible, and other functions of the state are therefore desirable to allow models to be constructed in a modular and hierarchical way. Arithmetic manipulations must be performed on the state during numerical integration, and these can be encoded neatly using operator overloading, so the operator of the appropriate type is invoked automatically without having to include ‘if-then-else’ structures for each operator as is the case in non- object-oriented programming languages such as Fortran-77. For instance, if ṡ is calculated in a forward-time numerical method then changes of state are calculated using numerical evolution equations such as s(t+ δt) = s(t) + δtṡ. (5) Thus for a general state s and its time derivative ṡ, which has an equivalent set of compo- nents, it is necessary to multiply a state by a real number and to add two states together. For a specific software implementation, other operations may be needed, for example to create, copy, or destroy a new instance of a state. The attraction of this approach is that, by choosing a reasonably general form for the constitutive relation and associated operations, it is possible to separate the continuum dynamics part of the problem from the inherent behavior of the material. The relations describing the properties of different types of material can be encapsulated in a library form where the continuum dynamics program need know nothing about the relations for any spe- cific type of material, and vice versa. The continuum dynamics programs and the material properties relations can be developed and maintained independently of each other, provided that the interface remains the same (Table I). This is an efficient way to make complicated material models available for simulations of different types, including Lagrangian and Eule- rian hydrocodes operating on different numbers of dimensions, and calculations of specific loading or heating histories such as shock and ramp loading discussed below. Software in- terfaces have been developed in the past for scalar EOS with a single structure for the state [12], but object-oriented techniques make it practical to extend the concept to much more complicated states, to combinations of models, and to alternative types of model selected when the program is run, without having to find a single super-set state encompassing all possible states as special cases. A very wide range of types of material behavior can be represented with this formalism. At the highest level, different types of behavior are characterized by different structures for the state s (Table II). For each type of state, different specific models can be defined, such as perfect gas, polytropic and Grüneisen EOS. For each specific model, different materials are represented by choosing different values for the parameters in the model, and different local material states are represented through different values for the components of s. In the jargon of object-oriented programming, the ability to define an object whose precise type is undetermined until the program is run is known as polymorphism. For our application, polymorphism is used at several levels in the hierarchy of objects, from the overall type of a material (such as ‘one represented by a pressure-density-energy EOS’ or ‘one represented by a deviatoric stress model’) through the type of relation used to describe the properties of that material type (such as perfect gas, polytropic, or Grüneisen for a pressure-density-energy EOS, or Steinberg-Guinan [13] or Preston-Tonks-Wallace [14] for a deviatoric stress model), to the type of general mathematical function used to represent some of these relations (such as a polynomial or a tabular representation of γ(ρ) in a polytropic EOS) (Table III). States or models may be defined by extending or combining other states or models – this can be implemented using the object-oriented programming concept of inheritance. Thus deviatoric stress models can be defined as an extension to any pressure-density-energy EOS (they are usually written assuming a specific type, such as Steinberg’s cubic Grüneisen form), homo- geneous mixtures can be defined as combinations of any pressure-density-temperature EOS, and heterogeneous mixtures can be defined as combinations of materials each represented by any type of material model. Trial implementations have been made as libraries in the C++ and Java programming languages [15]. The external interface to the material properties was general at the level of representing a generic material type and state. The type of state and model were then selected when programs using the material properties library were run. In C++, objects which were polymorphic at run time had to be represented as pointers, requiring additional software constructions to allocate and free up physical memory associated with each object. It was possible to include general re-usable functions as polymorphic objects when defining models: real functions of one real parameter could be polynomials, transcendentals, tabular with different interpolation schemes, piecewise definitions over different regions of the one dimensional line, sums, products, etc; again defined specifically at run time. Object-oriented polymorphism and inheritance were thus very powerful techniques for increasing software re-use, making the software more compact and more reliable through the greater use of functions which had already been tested. Given conceptual and software structures designed to represent general material proper- ties suitable for use in spatially-resolved continuum dynamics simulations, we now consider the use of these generic material models for calculating idealized loading paths. III. IDEALIZED ONE-DIMENSIONAL LOADING Experiments to investigate the response of materials to dynamic loading, and to calibrate parameters in models of their behavior, are usually designed to apply as simple a loading history as is consistent with the transient state of interest. The simplest canonical types of loading history are shock and ramp [5, 6]. Methods of solution are presented for calculating the result of shock and ramp loading for materials described by generalized material models discussed in the previous section. Such direct solution removes the need to use a time- and space-resolved continuum dynamics simulation, allowing states to be calculated with far greater efficiency and without the need to consider and make allowance for attributes of resolved simulations such as the finite numerical resolution and the effect of numerical and artificial viscosities. A. Ramp compression Ramp compression is taken here to mean compression or decompression. If the material is represented by an inviscid scalar EOS, i.e. ignoring dissipative processes and non-scalar effects from elastic strain, ramp compression follows an isentrope. This is no longer true when dissipative processes such as plastic heating occur. The term ‘quasi-isentropic’ is sometimes used in this context, particularly for shockless compression; here we prefer to refer to the thermodynamic trajectories as adiabats since this is a more appropriate term: no heat is exchanged with the surroundings on the time scales of interest. For adiabatic compression, the state evolves according to the second law of thermody- namics, de = T dS − p dv (6) where T is the temperature and S the specific entropy. Thus ė = T Ṡ − p v̇ = T Ṡ − pdiv~u , (7) or for a more general material whose stress tensor is more complicated than a scalar pressure, de = T dS + τn dv ⇒ ė = T Ṡ + τndiv~u where τn is the component of stress normal to the direction of deformation. The velocity gradient was expressed through a compression factor η ≡ ρ′/ρ and a strain rate ǫ̇. In all ramp experiments used in the development and calibration of accurate material models, the strain has been applied uniaxially. More general strain paths, for instance isotropic or including a shear component, can be treated by the same formalism, and that the working rate is then a full inner product of the stress and strain tensors. The acceleration or deceleration of the material normal to the wave as it is compressed or expanded adiabatically is , (9) from which it can be deduced that where cl is the longitudinal wave speed. As with continuum dynamics, internal evolution of the material state can be calculated simultaneously with the continuum equations, or operator split and calculated periodically at constant compression [16]. The results are the same to second order in the compression increment. Operator-splitting allows calculations to be performed without an explicit en- tropy, if the continuum equations are integrated isentropically and dissipative processes are captured by internal evolution at constant compression. Operator-splitting is desirable when internal evolution can produce highly nonlinear changes, such as reaction from solid to gas: rapid changes in state and properties can make numerical schemes unstable. Operator-splitting is also desirable when the integration time step for internal evolution is much shorter than the continuum dynamics time step. Neither of these considerations is very important for ramp compression without spatial res- olution, but operator-splitting was used as an option in the ramp compression calculations for consistency with continuum dynamics simulations. The ramp compression equations were integrated using forward-time Runge-Kutta nu- merical schemes of second order. The fourth order scheme is a trivial extension. The sequence of operations to calculate an increment of ramp compression is as follows: 1. Time increment: δt = − | ln η| 2. Predictor: s(t + δt/2) = s(t) + ṡm(s(t), ǫ̇) (12) 3. Corrector: s(t+ δt) = s(t) + δtṡm(s(t+ δt/2), ǫ̇) (13) 4. Internal evolution: s(t+ δt) → s(t+ δt) + ∫ t+δt ṡi(s(t ′), ǫ̇) dt′ (14) where ṡm is the model-dependent state evolution from applied strain, and ṡi is internal evolution at constant compression. The independent variable for integration is specific volume v or mass density ρ; for numerical integration finite steps are taken in ρ and v. The step size ∆ρ can be controlled so that the numerical error during integration remains within chosen limits. A tabular adiabat can be calculated by integrating over a range of v or ρ, but when simulating experimental scenarios the upper limit for integration is usually that one of the other thermodynamic quantities reaches a certain value, for example that the normal component of stress reaches zero, which is the case on release from a high pressure state at a free surface. Specific end conditions were found by monitoring the quantity of interest until bracketed by a finite integration step, then bisecting until the stop condition was satisfied to a chosen accuracy. During bisection, each trial calculation was performed as an integration from the first side of the bracket by the trial compression. B. Shock compression Shock compression is the solution of a Riemann problem for the dynamics of a jump in compression moving with constant speed and with a constant thickness. The Rankine- Hugoniot (RH) equations [5] describing the shock compression of matter are derived in the continuum approximation, where the shock is a formal discontinuity in the continuum fields. In reality, matter is composed of atoms, and shocks have a finite width governed by the kinetics of dissipative processes – at a fundamental level, matter does not distinguish between shock compression and ramp compression with a high strain rate – but the RH equations apply as long as the width of the region of matter where unresolved processes occur is constant. Compared with the isentropic states induced by ramp compression in a material represented by an EOS, a shock always increases the entropy and hence the temperature. With dissipative processes included, the distinction between a ramp and a shock may become blurred. The RH equations express the conservation of mass, momentum, and energy across a moving discontinuity in state. They are usually expressed in terms of the pressure, but are readily generalized for materials supporting shear stresses by using the component of stress normal to the shock (i.e., parallel with the direction of propagation of the shock), τn: u2s = −v τn − τn0 v0 − v , (15) ∆up = −(τn − τn0)(v0 − v), (16) e = e0 − (τn + τn0)(v0 − v), (17) where us is the speed of the shock wave with respect to the material, ∆up is the change in material speed normal to the shock wave (i.e., parallel to its direction of propagation), and subscript 0 refers to the initial state. The RH relations can be applied to general material models if a time scale or strain rate is imposed, and an orientation chosen for the material with respect to the shock. Shock compression in continuum dynamics is almost always uniaxial. The RH equations involve only the initial and final states in the material. If a material has properties that depend on the deformation path – such as plastic flow or viscosity – then physically the detailed shock structure may make a difference [17]. This is a limitation of discontinuous shocks in continuum dynamics: it may be addressed as discussed above by including dissipative processes and considering ramp compression, if the dissipative pro- cesses can be represented adequately in the continuum approximation. Spatially-resolved simulations with numerical differentiation to obtain spatial derivatives and forward time differencing are usually not capable of representing shock discontinuities directly, and an artificial viscosity is used to smear shock compression over a few spatial cells [18]. The trajectory followed by the material in thermodynamic space is a smooth adiabat with dissi- pative heating supplied by the artificial viscosity. If plastic work is also included during this adiabatic compression, the overall heating for a given compression is greater than from the RH equations. To be consistent, plastic flow should be neglected while the artificial viscosity is non-zero. This localized disabling of physical processes, particularly time-dependent ones, during the passage of the unphysically smeared shock was previously found necessary for numerically stable simulations of detonation waves by reactive flow [19]. Detonation waves are reactive shock waves. Steady planar detonation (the Chapman- Jouguet state [20]) may be calculated using the RH relations, by imposing the condition that the material state behind the shock is fully reacted. Several numerical methods have been used to solve the RH equations for materials repre- sented by an EOS only [21, 22]. The general RH equations may be solved numerically for a given shock compression ∆ρ by varying the specific internal energy e until the normal stress from the material model equals that from the RH energy equation, Eq. 17. The shock and particle speeds are then calculated from Eqs 15 and 16. This numerical method is particu- larly convenient for EOS of the form p(ρ, e), as e may be varied directly. Solutions may still be found for general material models using ṡ(ė), by which the energy may be varied until the solution is found. Numerically, the solution was found by bracketing and bisection: 1. For given compression ∆ρ, take the low-energy end for bracketing as a nearby state s− (e.g. the previous state, of lower compression, on the Hugoniot), compressed adia- batically (to state s̃), and cooled so the specific internal energy is e(s−). 2. Bracket the desired state: apply successively larger heating increments ∆e to s̃, evolv- ing each trial state internally, until τn(s) from the material model exceeds τn(e − e0) from Eq. 17. 3. Bisect in ∆e, evolving each trial state internally, until τn(s) equals τn(e − e0) to the desired accuracy. As with ramp compression, the independent variable for solution was mass density ρ, and finite steps ∆ρ were taken. Each shock state was calculated independently of the rest, so numerical errors did not accumulate along the shock Hugoniot. The accuracy of the solution was independent of ∆ρ. A tabular Hugoniot can be calculated by solving over a range of ρ, but again when simulating experimental scenarios it is usually more useful to calculate the shock state where one of the other thermodynamic quantities reaches a certain value, often that up and τn match the values from another, simultaneous shock calculation for another material – the situation in impact and shock transmission problems, discussed below. Specific stop conditions were found by monitoring the quantity of interest until bracketed by a finite solution step, then bisecting until the stop condition was satisfied to a chosen accuracy. During bisection, each trial calculation was performed as a shock from the initial conditions to the trial shock compression. C. Accuracy: application to air The accuracy of these numerical schemes was tested by comparing with shock and ramp compression of a material represented by a perfect gas EOS, p = (γ − 1)ρe. (18) The numerical solution requires a value to be chosen for every parameter in the material model, here γ. Air was chosen as an example material, with γ = 1.4. Air at standard tem- perature and pressure has approximately ρ = 10−3 g/cm3 and e = 0.25MJ/kg. Isentropes for the perfect gas EOS have the form pρ−γ = constant, (19) and shock Hugoniots have the form p = (γ − 1) 2e0ρ0ρ+ p0(ρ− ρ0) (γ + 1)ρ0 − (γ − 1)ρ . (20) The numerical solutions reproduced the principal isentrope and Hugoniot to 10−3% and 0.1% respectively, for a compression increment of 1% along the isentrope and a solution tolerance of 10−6GPa for each shock state (Fig. 1). Over most of the range, the error in the Hugoniot was 0.02% or less, only approaching 0.1% near the maximum shock compression. IV. COMPLEX BEHAVIOR OF CONDENSED MATTER The ability to calculate shock and ramp loci in state space, i.e. as a function of vary- ing loading conditions, is particularly convenient for investigating complex aspects of the response of condensed matter to dynamic loading. Each locus can be obtained by a single series of shock or ramp solutions, rather than having to perform a series of time- and space- resolved continuum dynamics simulations, varying the initial or boundary conditions and reducing the solution. We consider the calculation of temperature in the scalar EOS, the effect of material strength and the effect of phase changes. A. Temperature The continuum dynamics equations can be closed using a mechanical EOS relating stress to mass density, strain, and internal energy. For a scalar EOS, the ideal form to close the continuum equations is p(ρ, e), with s = {ρ, e} the natural choice for the primitive state fields. However, the temperature is needed as a parameter in physical descriptions of many contributions to the constitutive response, including plastic flow, phase transitions, and chemical reactions. Here, we discuss the calculation of temperature in different forms of the scalar EOS. 1. Density-temperature equations of state If the scalar EOS is constructed from its underlying physical contributions for continuum dynamics, it may take the form e(ρ, T ), from which p(ρ, T ) can be calculated using the second law of thermodynamics [10]. An example is the ‘SESAME’ form of EOS, based on interpolated tabular relations for {p, e}(ρ, T ) [23]. A pair of relations {p, e}(ρ, T ) can be used as a mechanical EOS by eliminating T , which is equivalent to inverting e(ρ, T ) to find T (ρ, e), then substituting in p(ρ, T ). For a general e(ρ, T ) relation, for example for the SESAME EOS, the inverse can be calculated numerically as required, along an isochore. In this way, a {p, e}(ρ, T ) can be used as a p(ρ, e) EOS. Alternatively, the same p(ρ, T ) relation can be used directly with a primitive state field including temperature instead of energy: s = {ρ, T}. The evolution of the state under mechanical work then involves the calculation of Ṫ (ė), i.e. the reciprocal of the specific heat capacity, which is a derivative of e(ρ, T ). As this calculation does not require e(ρ, T ) to be inverted, it is computationally more efficient to use {p, e}(ρ, T ) EOS with a temperature- based, rather than energy-based, state. The main disadvantage is that it is more difficult to ensure exact energy conservation as the continuum dynamics equations are integrated in time, but any departure from exact conservation is at the level of accuracy of the algorithm used to integrate the heat capacity. Both structures of EOS have been implemented for material property calculations. Taking a SESAME type EOS, thermodynamic loci were calculated with {ρ, e} or {ρ, T} primitive states, for comparison (Fig. 2). For a monotonic EOS, the results were indistinguishable within differences from forward or reverse interpolation of the tabular relations. When the EOS, or the effective surface using a given order of interpolating function, was non- monotonic, the results varied greatly because of non-uniqueness when eliminating T for the {ρ, e} primitive state. 2. Temperature model for mechanical equations of state Mechanical EOS are often available as empirical, algebraic relations p(ρ, e), derived from shock data. Temperature can be calculated without altering the mechanical EOS by adding a relation T (ρ, e). While this relation could take any form in principle, one can also follow the logic of the Grüneisen EOS, in which the pressure is defined in terms of its deviation ∆p(ρ, e − er) from a reference curve {pr, er}(ρ). Thus temperatures can be calculated by reference to a compression curve along which the temperature and specific internal energy are known, {Tr, er}(ρ), and a specific heat capacity defined as a function of density cv(ρ). In the calculations, this augmented EOS was represented as a ‘mechanical-thermal’ form comprising any p(ρ, e) EOS plus the reference curves – an example of software inheritance and polymorphism. One natural reference curve for temperature is the cold curve, Tr = 0K. The cold curve can be estimated from the principal isentrope e(ρ)|s0 using the estimated density variation of the Grüneisen parameter: er(ρ) = e(ρ)|s0 − T0cpe a(1−ρ0/ρ) )γ0−a [24]. In this work, the principal isentrope was calculated in tabular form from the mechanical EOS, using the ramp compression algorithm described above. Empirical EOS are calibrated using experimental data. Shock and adiabatic compression measurements on strong materials inevitably include elastic-plastic contributions as well as the scalar EOS itself. If the elastic-plastic contributions are not taken into account self- consistently, the EOS may implicitly include contributions from the strength. A unique scalar EOS can be constructed to reproduce the normal stress as a function of compression for any unique loading path: shock or adiabat, for a constant or smoothly-varying strain rate. Such an EOS would not generally predict the response to other loading histories. The EOS and constitutive properties for the materials considered here were constructed self- consistently from shock data – this does not mean the models are accurate for other loading paths, as neither the EOS nor the strength model includes all the physical terms that real materials exhibit. This does not in any case matter for the purposes of demonstrating the properties of the numerical schemes. This mechanical-thermal procedure was applied to Al using a Grüneisen EOS fitted to the same shock data used to calculate the {p, e}(ρ, T ) EOS discussed above [24]. Temperatures were in good agreement (Fig. 2). The mechanical-thermal calculations required a similar computational effort to the tabular {p, e}(ρ, T ) EOS with a {ρ, T} primitive states (and were thus much more efficient than the tabular EOS with {ρ, e} states), and described the EOS far more compactly. B. Strength For dynamic compressions to o(10GPa) and above, on microsecond time scales, the flow stress of solids is often treated as a correction or small perturbation to the scalar EOS. However, the flow stress has been observed to be much higher on nanosecond time scales [25], and interactions between elastic and plastic waves may have a significant effect on the compression and wave propagation. The Rankine-Hugoniot equations should be solved self-consistently with strength included. 1. Preferred representation of isotropic strength There is an inconsistency in the standard continuum dynamics treatment of scalar (pres- sure) and tensor (stress) response. The scalar EOS expresses the pressure p(ρ, e) as the dependent quantity, which is the most convenient form for use in the continuum equations. Standard practice is to use sub-Hookean elasticity (hypoelastic form) [16] (Table II), in which the state parameters include the stress deviator σ, evolved by integration σ̇ = G(s)ǫ̇ (22) where G is the shear modulus and ǫ̇ the strain rate deviator. Thus the isotropic and devia- toric contributions to stress are not treated in an equivalent way: the pressure is calculated from a local state involving a strain-like parameter (mass density), whereas the stress de- viator evolves with the time-derivative of strain. This inconsistency causes problems along complicated loading paths because G varies strongly with compression: if a material is sub- jected to a shear strain ǫ, then isotropic compression (increasing the shear modulus from G to G′, leaving ǫ unchanged), then shear unloading to isotropic stress, the true unloading strain is −ǫ, whereas the hypoelastic calculation would require a strain of −ǫG/G′. Using Be and the Steinberg-Guinan strength model as an example of the difference between hy- poelastic and hyperelastic calculations, consider an initial strain to a flow stress of 0.3GPa followed by isothermal, isotropic compression to 100GPa,. the strain to unload to a state of isotropic stress is 0.20% (hyperelastic) and 0.09% (hypoelastic). The discrepancy arises because the hypoelastic model does not increase the deviatoric stress under compression at constant deviatoric strain. The stress can be considered as a direct response of the material to the instantaneous state of elastic strain: σ(ǫ, T ). This relation can be predicted directly with electronic structure calculations of the stress tensor in a solid for a given compression and elastic strain state [11], and is a direct generalization of the scalar equation of state. A more consistent representation of the state parameters is to use the strain deviator ǫ rather than σ, and to calculate σ from scratch when required using σ = G(s)ǫ (23) – a hyperelastic formulation. The state parameters are then {ρ, e, ǫ, ǫ̃p}. The different formulations give different answers when deviatoric strain is accumulated at different compressions, in which case the hyperelastic formulation is correct. If the shear modulus varies with strain deviator – i.e., for nonlinear elasticity – then the definition of G(ǫ) must be adjusted to give the same stress for a given strain. Many isotropic strength models use scalar measures of the strain and stress to parame- terize work hardening and to apply a yield model of flow stress: fǫ||ǫ2||, σ̃ = fσ||σ2||. (24) Inconsistent conventions for equivalent scalar measures have been used by different workers. In the present work, the common shock physics convention was used that the flow stress component of τn is Y where Y is the flow stress. For consistency with published speeds and amplitudes for elastic waves, fǫ = fσ = , in contrast to other values previously used for lower-rate deformation [26]. In principle, the values of fǫ and fσ do not matter as long as the strength parameters were calibrated using the same values then used in any simulations. 2. Beryllium The flow stress measured from laser-driven shock experiments on Be crystals a few tens of micrometers thick is, at around 5-9GPa [25], much greater than the 0.3-1.3GPa mea- sured on microsecond time scales. A time-dependent crystal plasticity model for Be is being developed, and the behavior under dynamic loading depends on the detailed time depen- dence of plasticity. Calculations were performed with the Steinberg-Guinan strength model developed for microsecond scale data [24], and, for the purposes of rough comparison, with elastic-perfectly plastic response with a flow stress of 10GPa. The elastic-perfectly plastic model neglected pressure- and work- hardening. Calculations were made of the principal adiabat and shock Hugoniot, and of a release adiabat from a state on the principal Hugoniot. Calculations were made with and without strength. Considering the state trajectories in stress-volume space, it is interesting to note that heating from plastic flow may push the adiabat above the Hugoniot, because of the greater heating obtained by integrating along the adiabat compared with jumping from the initial to the final state on the Hugoniot (Fig. 3). Even with an elastic-perfectly plastic strength model, the with-strength curves do not lie exactly 2 Y above the strengthless curves, because heating from plastic flow contributes an increasing amount of internal energy to the EOS as compression increases. An important characteristic for the seeding of instabilities by microstructural variations in shock response is the shock stress at which an elastic wave does not run ahead of the shock. In Be with the high flow stress of nanosecond response, the relation between shock and particle speeds is significantly different from the relation for low flow stress (Fig. 4). For low flow stress, the elastic wave travels at 13.2 km/s. A plastic shock travels faster than this for pressures greater than 110GPa, independent of the constitutive model. The speed of a plastic shock following the initial elastic wave is similar to the low strength case, because the material is already at its flow stress, but the speed of a single plastic shock is appreciably higher. For compression to a given normal stress, the temperature is significantly higher with plastic flow included. The additional heating is particularly striking on the principal adi- abat: the temperature departs significantly from the principal isentrope. Thus ramp-wave compression of strong materials may lead to significant levels of heating, contrary to com- mon assumptions of small temperature increases [27]. Plastic flow is largely irreversible, so heating occurs on unloading as well as loading. Thus, on adiabatic release from a shock- compressed state, additional heating occurs compared with the no-strength case. These levels of heating are important as shock or release melting may occur at a significantly lower shock pressure than would be expected ignoring the effect of strength. (Fig. 5.) C. Phase changes An important property of condensed matter is phase changes, including solid-solid poly- morphism and solid-liquid. An equilibrium phase diagram can be represented as a single overall EOS surface as before. Multiple, competing phases with kinetics for each phase trans- formation can be represented conveniently using the structure described above for general material properties, for example by describing the local state as a set of volume fractions fi of each possible simple-EOS phase, with transition rates and equilibration among them. This model is described in more detail elsewhere [19]. However, it is interesting to investi- gate the robustness of the numerical scheme for calculating shock Hugoniots when the EOS has the discontinuities in value and gradient associated with phase changes. The EOS of molten metal, and the solid-liquid phase transition, can be represented to a reasonable approximation as an adjustment to the EOS of the solid: ptwo-phase(ρ, e) = psolid(ρ, ẽ) (25) where e : T (ρ, e) < Tm(ρ) e−∆h̃m : ∆h̃m ≡ cv(ρ, e) [T (ρ, e)− Tm(ρ)] < ∆hm e−∆hm : otherwise and ∆hm is the specific latent heat of fusion. Taking the EOS and a modified Lindemann melting curve for Al [24], and using ∆hm = 0.397MJ/kg, the shock Hugoniot algorithm was found to operate stably across the phase transition (Fig. 6). V. COMPOSITE LOADING PATHS Given methods to calculate shock and adiabatic loading paths from arbitrary initial states, a considerable variety of experimental scenarios can be treated from the interaction of loading or unloading waves with interfaces between different materials, in planar geometry for uniaxial compression. The key physical constraint is that, if two dissimilar materials are to remain in contact after an interaction such as an impact or the passage of a shock, the normal stress τn and particle speed up in both materials must be equal on either side of the interface. The change in particle speed and stress normal to the waves were calculated above for compression waves running in the direction of increasing spatial ordinate (left to right). Across an interface, the sense is reversed for the material at the left. Thus a projectile impacting a stationary target to the right is decelerated from its initial speed by the shock induced by impact. The general problem at an interface can be analyzed by considering the states at the instant of first contact – on impact, or when a shock traveling through a sandwich of ma- terials first reaches the interface. The initial states are {ul, sl; ur, sr}. The final states are {uj, s l; uj, r r} where uj is the joint particle speed, τn(s l) = τn(s r), and s i is connected to si by either a shock or an adiabat, starting at the appropriate initial velocity and stress, and with orientation given by the side of the system each material occurs on. Each type of wave is considered in turn, looking for an intersection in the up − τn plane. Examples of these wave interactions are the impact of a projectile with a stationary target (Fig. 7), release of a shock state at a free surface or a material (e.g. a window) of lower shock impedance (hence reflecting a release wave into the shocked material – Fig. 8), reshocking at a surface with a material of higher shock impedance (Fig. 8), or tension induced as materials try to separate in opposite directions when joined by a bonded interface (Fig. 9). Each of these scenarios may occur in turn following the impact of a projectile with a target: if the target is layered then a shock is transmitted across each interface with a release or a reshock reflected back, depending on the materials; release ultimately occurs at the rear of the projectile and the far end of the target, and the oppositely-moving release waves subject the projectile and target to tensile stresses when they interact (Fig. 10). As an illustration of combining shock and ramp loading calculations, consider the problem of an Al projectile, initially traveling at 3.6 km/s, impacting a stationary, composite target comprising a Mo sample and a LiF release window [28, 29]. The shock and release states were calculated using published material properties [24]. The initial shock state was calculated to have a normal stress of 63.9GPa. On reaching the LiF, the shock was calculated to transmit at 27.1GPa, reflecting as a release in the Mo. These stresses match the continuum dynamics simulation to within 0.1GPa in the Mo and 0.3GPa in the LiF, using the same material properties (Fig. 11). The associated wave and particle speeds match to a similar accuracy; wave speeds are much more difficult to extract from the continuum dynamics simulation. An extension of this analysis can be used to calculate the interaction of oblique shocks with an interface [30]. VI. CONCLUSIONS A general formulation was developed to represent material models for applications in dynamic loading, suitable for software implementation in object-oriented programming lan- guages. Numerical methods were devised to calculate the response of matter represented by the general material models to shock and ramp compression, and ramp decompression, by direct evaluation of the thermodynamic pathways for these compressions rather than spatially-resolved simulations. This approach is a generalization of earlier work on solutions for materials represented by a scalar equation of state. The numerical methods were found to be flexible and robust: capable of application to materials with very different properties. The numerical solutions matched analytic results to a high accuracy. Care was needed with the interpretation of some types of physical response, such as plas- tic flow, when applied to deformation at high strain rates. The underlying time-dependence of processes occurring during deformation should be taken into account. The actual history of loading and heating experienced by material during the passage of a shock may influence the final state – this history is not captured in the continuum approximation to material dynamics, where shocks are treated as discontinuities. Thus care is also needed in spa- tially resolved simulations when shocks are modeled using artificial viscosity to smear them unphysically over a finite thickness. Calculations were shown to demonstrate the operation of the algorithms for shock and ramp compression with material models representative of complex solids including strength and phase transformations. The basic ramp and shock solution methods were coupled to solve for composite defor- mation paths, such as shock-induced impacts, and shock interactions with a planar interface between different materials. Such calculations capture much of the physics of typical ma- terial dynamics experiments, without requiring spatially-resolving simulations. The results of direct solution of the relevant shock and ramp loading conditions were compared with hydrocode simulations, showing complete consistency. Acknowledgments Ian Gray introduced the author to the concept of multi-model material properties soft- ware. Lee Markland developed a prototype Hugoniot-calculating computer program for equations of state while working for the author as an undergraduate summer student. Evolutionary work on material properties libraries was supported by the U.K. Atomic Weapons Establishment, Fluid Gravity Engineering Ltd, andWessex Scientific and Technical Services Ltd. Refinements to the technique and applications to the problems described were undertaken at Los Alamos National Laboratory (LANL) and Lawrence Livermore National Laboratory (LLNL). The work was performed partially in support of, and funded by, the National Nuclear Se- curity Agency’s Inertial Confinement Fusion program at LANL (managed by Steven Batha), and LLNL’s Laboratory-Directed Research and Development project 06-SI-004 (Principal Investigator: Hector Lorenzana). The work was performed under the auspices of the U.S. Department of Energy under contracts W-7405-ENG-36, DE-AC52-06NA25396, and DE- AC52-07NA27344. References [1] J.K. Dienes, J.M. Walsh, in R. Kinslow (Ed), “High-Velocity Impact Phenomena” (Academic Press, New York, 1970). [2] D.J. Benson, Comp. Mech. 15, 6, pp 558-571 (1995). [3] J.W. Gehring, Jr, in R. Kinslow (Ed), “High-Velocity Impact Phenomena” (Academic Press, New York, 1970). [4] R.M. Canup, E. Asphaug, Nature 412, pp 708-712 (2001). [5] For a recent review and introduction, see e.g. M.R. Boslough and J.R. Asay, in J.R. Asay, M. Shahinpoor (Eds), “High-Pressure Shock Compression of Solids” (Springer-Verlag, New York, 1992). [6] For example, C.A. Hall, J.R. Asay, M.D. Knudson, W.A. Stygar, R.B. Spielman, T.D. Pointon, D.B. Reisman, A. Toor, and R.C. Cauble, Rev. Sci. Instrum. 72, 3587 (2001). [7] M.A. Meyers, “Dynamic Behavior of Materials” (Wiley, New York, 1994). [8] G. McQueen, S.P. March, J.W. Taylor, J.N. Fritz, W.J. Carter, in R. Kinslow (Ed), “High- Velocity Impact Phenomena” (Academic Press, New York, 1970). [9] J.D. Lindl, “Inertial Confinement Fusion” (Springer-Verlag, New York, 1998). [10] D.C. Swift, G.J. Ackland, A. Hauer, G.A. Kyrala, Phys. Rev. B 64, 214107 (2001). [11] J.P. Poirier, G.D. Price, Phys. of the Earth and Planetary Interiors 110, pp 147-56 (1999). [12] I.N. Gray, P.C. Thompson, B.J. Parker, D.C. Swift, J.R. Maw, A. Giles and others (AWE Aldermaston), unpublished. [13] D.J. Steinberg, S.G. Cochran, M.W. Guinan, J. Appl. Phys. 51, 1498 (1980). [14] D.L. Preston, D.L. Tonks, and D.C. Wallace, J. Appl. Phys. 93, 211 (2003). [15] A version of the software, including representative parts of the material model library and the algorithms for calculating the ramp adiabat and shock Hugoniot, is available as a supplemen- tary file provided with the preprint of this manuscript, arXiv:0704.0008. Software support, and versions with additional models, are available commercially from Wessex Scientific and Technical Services Ltd (http://wxres.com). [16] D. Benson, Computer Methods in Appl. Mechanics and Eng. 99, 235 (1992). http://arxiv.org/abs/0704.0008 http://wxres.com [17] J.L. Ding, J. Mech. and Phys. of Solids 54, pp 237-265 (2006). [18] J. von Neumann, R.D. Richtmyer, J. Appl. Phys. 21, 3, pp 232-237 (1950). [19] R.M. Mulford, D.C. Swift, in preparation. [20] W. Fickett, W.C. Davis, “Detonation” (University of California Press, Berkeley, 1979). [21] R. Menikoff, B.J. Plohr, Rev. Mod. Phys. 61, pp 75-130 (1989). [22] A. Majda, Mem. Amer. Math. Soc., 41, 275 (1983). [23] K.S. Holian (Ed.), T-4 Handbook of Material Property Data Bases, Vol 1c: Equations of State, Los Alamos National Laboratory report LA-10160-MS (1984). [24] D.J. Steinberg, Equation of State and Strength Properties of Selected Materials, Lawrence Livermore National Laboratory report UCRL-MA-106439 change 1 (1996). [25] D.C. Swift, T.E. Tierney, S.-N. Luo, D.L. Paisley, G.A. Kyrala, A. Hauer, S.R. Greenfield, A.C. Koskelo, K.J. McClellan, H.E. Lorenzana, D. Kalantar, B.A. Remington, P. Peralta, E. Loomis, Phys.Plasmas 12, 056308 (2005). [26] R. Hill, “The Mathematical Theory of Plasticity” (Clarendon Press, Oxford, 1950). [27] C.A. Hall, Phys. Plasmas 7, 5, pp 2069-2075 (2000). [28] D.C. Swift, A. Seifter, D.B. Holtkamp, and D.A. Clark, Phys. Rev. B 76, 054122 (2007). [29] A. Seifter and D.C. Swift, Phys. Rev. B 77, 134104 (2008). [30] E. Loomis, D.C. Swift, J. Appl. Phys. 103, 023518 (2008). TABLE I: Interface to material models required for explicit forward-time continuum dynamics simulations. purpose interface calls program set-up read/write material data continuum dynamics equations stress(state) time step control sound speed(state) evolution of state (deformation) d(state)/dt(state,grad ~u) evolution of state (heating) d(state)/dt(state,ė) internal evolution of state d(state)/dt manipulation of states create and delete add states multiply state by a scalar check for self-consistency Parentheses in the interface calls denote functions, e.g. “stress(state)” for “stress as a function of the instantaneous, local state.” The evolution functions are shown in the operator-split structure that is most robust for explicit, forward-time numerical solutions and can also be used for calculations of the shock Hugoniot and ramp compression. Checks for self-consistency include that mass density is positive, volume or mass fractions of components of a mixture add up to one, TABLE II: Examples of types of material model, distinguished by different structures in the state vector. model state vector effect of mechanical strain s ṡm(s, gradu) mechanical equation of state ρ, e −ρdiv~u,−pdiv~u/ρ thermal equation of state ρ, T −ρdiv~u,−pdiv~u/ρcv heterogeneous mixture {ρ, e, fv}i {−ρdiv~u,−pdiv~u/ρ, 0}i homogeneous mixture ρ, T, {fm}i {−ρdiv~u,−pdiv~u/ρcv , 0i traditional deviatoric strength ρ, e, σ, ǫ̃p −ρdiv~u, −pdiv~u+fp||σǫ̇p|| , Gǫ̇e, fǫ||ǫ̇ The symbols are ρ: mass density; e: specific internal energy, T : temperature, fv: volume fraction, fm: mass fraction, σ: stress deviator, fp: fraction of plastic work converted to heat, gradup: plastic part of velocity gradient, G: shear modulus, ǫ̇e,p: elastic and plastic parts of strain rate deviator, ǫ̃p: scalar equivalent plastic strain, fǫ: factor in effective strain magnitude. Reacting solid explosives can be represented as heterogeneous mixtures, one component being the reacted products; reaction, a process of internal evolution, transfers material from unreacted to reacted components. Gas-phase reaction can be represented as a homogeneous mixture, reactions transferring mass between components representing different types of molecule. Symmetric tensors such as the stress deviator are represented more compactly by their 6 unique upper triangular components, e.g. using Voigt notation. TABLE III: Outline hierarchy of material models, illustrating the use of polymorphism (in the object-oriented programming sense). material (or state) type model type mechanical equation of state polytropic, Grüneisen, energy-based Jones-Wilkins-Lee, (ρ, T ) table, etc thermal equation of state temperature-based Jones-Wilkins- Lee, quasiharmonic, (ρ, T ) table, reactive equation of state modified polytropic, reactive Jones- Wilkins-Lee spall Cochran-Banner deviatoric stress elastic-plastic, Steinberg-Guinan, Steinberg-Lund, Preston-Tonks- Wallace, etc homogeneous mixture mixing and reaction models heterogeneous mixture equilibration and reaction models Continuum dynamics programs can refer to material properties as an abstract ‘material type’ with an abstract material state. The actual type of a material (e.g. mechanical equation of state), the specific model type (e.g. polytropic), and the state of material of that type are all handled transparently by the object-oriented software structure. The reactive equation of state has an additional state parameter λ, and the software operations are defined by extending those of the mechanical equation of state. Spalling materials can be represented by a solid state plus a void fraction fv, with operations defined by extending those of the solid material. Homogeneous mixtures are defined as a set of thermal equations of state, and the state is the set of states and mass fractions for each. Heterogeneous mixtures are defined as a set of ‘pure’ material properties of any type, and the state is the set of states for each component plus its volume fraction. 0.0001 0.001 0.01 0.001 0.01 mass density (g/cm3) isentrope Hugoniot 0.0001 0.001 0.01 0.001 0.01 mass density (g/cm3) isentrope Hugoniot FIG. 1: Principal isentrope and shock Hugoniot for air (perfect gas): numerical calculations for general material models, compared with analytic solutions. 0 1000 2000 3000 4000 5000 temperature (K) solid: Grueneisen dashed: SESAME 3716 FIG. 2: Shock Hugoniot for Al in pressure-temperature space, for different representations of the equation of state. 0.7 0.75 0.8 0.85 0.9 0.95 1 volume compression each pair of lines: upper is Hugoniot, lower is adiabat FIG. 3: Principal adiabat and shock Hugoniot for Be in normal stress-compression space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly plastic with Y = 10GPa (dotted). 0 20 40 60 80 100 120 140 normal stress (GPa) elastic wave plastic shock FIG. 4: Principal adiabat and shock Hugoniot for Be in shock speed-normal stress space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly plastic with Y = 10GPa (dotted). 0 1000 2000 3000 4000 5000 temperature (K) principal adiabat principal Hugoniot release adiabat FIG. 5: Principal adiabat, shock Hugoniot, and release adiabat for Be in normal stress-temperature space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly plastic with Y = 10GPa (dotted). 0 1000 2000 3000 4000 5000 temperature (K) melt locus solid Hugoniot FIG. 6: Demonstration of shock Hugoniot solution across a phase boundary: shock-melting of Al, for different initial porosities. initial state particle speed initial state of projectile principal Hugoniot of target principal Hugoniot of projectile shock state: intersection of target FIG. 7: Wave interactions for the impact of a flat projectile moving from left to right with a stationary target. Dashed arrows are a guide to the sequence of states. For a projectile moving from right to left, the construction is the mirror image reflected in the normal stress axis. states particle speed secondary Hugoniot of target initial shock state in target principal Hugoniot: high impedance window low impedance window target release isentrope target release at free surface window release FIG. 8: Wave interactions for the release of a shocked state (shock moving from left to right) into a stationary ‘window’ material to its right. The release state depends whether the window has a higher or lower shock impedance than the shocked material. Dashed arrows are a guide to the sequence of states. For a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis. projectile release in projectile and target final tensile state in projectile and target particle speed target release target release projectile release initial shock state FIG. 9: Wave interactions for the release of a shocked state by tension induced as materials try to separate in opposite directions when joined by a bonded interface. Material damage, spall, and separation are neglected: the construction shows the maximum tensile stress possible. For general material properties, e.g. if plastic flow is included, the state of maximum tensile stress is not just the negative of the initial shock state. Dashed arrows are a guide to the sequence of states. The graph shows the initial state after an impact by a projectile moving from right to left; for a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis. tension ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� target impact shocks transmitted shock; reflected wave free surface release release interactions: FIG. 10: Schematic of uniaxial wave interactions induced by the impact of a flat projectile with a composite target. 0 5 10 15 20 position (mm) LiFAl Mo reflected transmitted release shock original shock state FIG. 11: Hydrocode simulation of Al projectile at 3.6 km/s impacting a Mo target with a LiF release window, 1.1µs after impact. Structures on the waves are elastic precursors. List of figures 1. Principal isentrope and shock Hugoniot for air (perfect gas): numerical calculations for general material models, compared with analytic solutions. 2. Shock Hugoniot for Al in pressure-temperature space, for different representations of the equation of state. 3. Principal adiabat and shock Hugoniot for Be in normal stress-compression space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic- perfectly plastic with Y = 10GPa (dotted). 4. Principal adiabat and shock Hugoniot for Be in shock speed-normal stress space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic- perfectly plastic with Y = 10GPa (dotted). 5. Principal adiabat, shock Hugoniot, and release adiabat for Be in normal stress- temperature space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly plastic with Y = 10GPa (dotted). 6. Demonstration of shock Hugoniot solution across a phase boundary: shock-melting of Al, for different initial porosities. 7. Wave interactions for the impact of a flat projectile moving from left to right with a stationary target. Dashed arrows are a guide to the sequence of states. For a projectile moving from right to left, the construction is the mirror image reflected in the normal stress axis. 8. Wave interactions for the release of a shocked state (shock moving from left to right) into a stationary ‘window’ material to its right. The release state depends whether the window has a higher or lower shock impedance than the shocked material. Dashed arrows are a guide to the sequence of states. For a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis. 9. Wave interactions for the release of a shocked state by tension induced as materials try to separate in opposite directions when joined by a bonded interface. Material damage, spall, and separation are neglected: the construction shows the maximum tensile stress possible. For general material properties, e.g. if plastic flow is included, the state of maximum tensile stress is not just the negative of the initial shock state. Dashed arrows are a guide to the sequence of states. The graph shows the initial state after an impact by a projectile moving from right to left; for a shock moving from right to left, the construction is the mirror image reflected in the normal stress axis. 10. Schematic of uniaxial wave interactions induced by the impact of a flat projectile with a composite target. 11. Hydrocode simulation of Al projectile at 3.6 km/s impacting a Mo target with a LiF release window, 1.1µs after impact. Structures on the waves are elastic precursors. Introduction Conceptual structure for material properties Idealized one-dimensional loading Ramp compression Shock compression Accuracy: application to air Complex behavior of condensed matter Temperature Density-temperature equations of state Temperature model for mechanical equations of state Strength Preferred representation of isotropic strength Beryllium Phase changes Composite loading paths Conclusions Acknowledgments References References List of figures
Introduction Conceptual structure for material properties Idealized one-dimensional loading Ramp compression Shock compression Accuracy: application to air Complex behavior of condensed matter Temperature Density-temperature equations of state Temperature model for mechanical equations of state Strength Preferred representation of isotropic strength Beryllium Phase changes Composite loading paths Conclusions Acknowledgments References References List of figures
Solución numérica de choque y compresión de rampa para propiedades materiales generales Damian C. Swift* División de Ciencia y Tecnología de Materiales, Laboratorio Nacional Lawrence Livermore, 7000, East Avenue, Livermore, CA 94550, U.S.A. (Fecha: 7 de marzo de 2007; versión revisada el 8 de abril de 2008 y el 1 de julio de 2008 – LA-UR-07-2051) Resumen Se elaboró una formulación general para representar modelos de materiales para aplicaciones en dinámica cargando. Se diseñaron métodos numéricos para calcular la respuesta a la compresión de golpes y rampas, y descompresión de rampa, generalizando soluciones previas para ecuaciones escalares de estado. El número se encontró que los métodos eran flexibles y robustos, y que los resultados analíticos se ajustaban a una alta precisión. Los métodos básicos de la rampa y la solución de choque se acoplaron para resolver la deformación compuesta rutas, tales como impactos inducidos por choque, e interacciones de choque con una interfaz planar entre diferentes materiales. Estos cálculos captan gran parte de la física de las dinámicas materiales típicas experimentos, sin requerir simulaciones de resolución espacial. Se hicieron cálculos de ejemplo de historia de la carga en metales, ilustrando los efectos del trabajo plástico en las temperaturas inducidas en experimentos cuasi-isentrópicos y de liberación de choque, y el efecto de una transición de fase. Números PACS: 62.50.+p, 47.40.-x, 62.20.-x, 46.35.+z Palabras clave: dinámica material, choque, isentropo, adiabat, solución numérica, comportamiento constitutivo * Dirección electrónica: damian.swift@physics.org http://arxiv.org/abs/0704.0008v3 mailto:damian.swift@physics.org I. INTRODUCCIÓN La representación continua de la materia se utiliza ampliamente para la dinámica material de la ciencia y la tecnología. Ence e ingeniería. Las simulaciones de dinámica de continuum espacialmente resueltas son las más generalizado y familiar, resolviendo el problema de valor inicial discretizando el dominio espacial e integrar las ecuaciones dinámicas hacia adelante en el tiempo para predecir el movimiento y defor- nes de los componentes del sistema. Este tipo de simulación se utiliza, por ejemplo, para estudiar problemas de impacto hipervelocidad tales como la vulnerabilidad de la armadura a los proyectiles [1, 2], el rendimiento de los escudos de desechos de satélites [3], y el impacto de los meteoritos con los planetas, en particular la formación de la luna [4]. El problema se puede dividir en las ecuaciones dinámicas del continuum, el campo de estado de los componentes s(~r), y las propiedades inherentes de los materiales. Teniendo en cuenta el estado material local s, las propiedades materiales permiten el estrés por determinar. Teniendo en cuenta el campo de tensión (~r) y el campo de densidad de masa (~r), la dinámica ecuaciones describen los campos de aceleración, compresión y trabajo termodinámico realizado sobre los materiales. Las ecuaciones de la dinámica del continuum describen el comportamiento de una deformación dinámica sistema de complejidad arbitraria. Trayectorias de deformación particulares y más simples se pueden describir más compactamente por diferentes conjuntos de ecuaciones, y resuelto por diferentes técnicas que las utilizadas para la dinámica del continuum en general. Caminos de deformación más simples ocurren a menudo en experimentos diseñado para desarrollar y calibrar modelos de propiedades del material. Estos caminos pueden ser considerados como diferentes formas de interrogar las propiedades materiales. Los principales ejemplos en material la dinámica son la compresión de choque y rampa [5, 6]. Los experimentos típicos están diseñados para inducir tales historias de carga y medir o inferir las propiedades del material en estos estados antes de ser destruidos por liberación de los bordes o por ondas reflejadas. El desarrollo del campo de la dinámica material fue impulsado por aplicaciones en el física de los impactos a hipervelocidad y de los sistemas de explosivos elevados, incluidas las armas nucleares [7]. En los regímenes de interés, por lo general los componentes con dimensiones que van desde ters a metros y presiones de 1GPa a 1TPa, el comportamiento material está dominado por el Ecuación escalar del estado (EOS): la relación entre presión, compresión (o masa) densidad), y la energía interna. Otros componentes del estrés (específicamente esfuerzos de corte) son: mucho más pequeño, y los explosivos químicos reaccionan rápidamente por lo que puede ser tratado por Els de detonación completa. EOS se desarrollaron como ajuste a los datos experimentales, en particular a series de estados de choque y a mediciones de compresión isotérmica [8]. Es relativamente directo para construir estados de compresión de choque y rampa de un EOS algebraicamente o numéricamente dependiendo del EOS, y para ajustar un EOS a estas mediciones. Más recientemente, las aplicaciones y el interés científico han crecido para incluir una gama más amplia de presiones y escalas de tiempo, tales como la fusión inercial de confinamiento impulsado por láser [9], y los experimentos son El objetivo de este estudio es medir otros aspectos distintos de la EOS, tales como la cinética de los cambios de fase, Comportamiento estetutivo que describe tensiones de corte, reacciones químicas incompletas y los efectos de microestructura, incluyendo orientación de grano y porosidad. Las técnicas teóricas también tienen evolucionó para predecir el EOS con una precisión de +1% [10] y contribuciones elásticas al estrés por cizallamiento con una precisión ligeramente inferior [11]. Se describe una convención general para representar estados materiales, y métodos numéricos se informan para calcular los estados de compresión de choque y rampa de las representaciones generales de propiedades materiales. II. ESTRUCTURA CONCEPTUAL PARA PROPIEDADES MATERIALES La estructura deseada para la descripción del estado del material y las propiedades bajo dy- la carga namic se desarrolló para ser lo más general posible con respecto a los tipos de material o modelos que deben estar representados en el mismo marco, y diseñados para dar la mayor cantidad de similitud entre simulaciones espacialmente resueltas y cálculos de choque y rampa compresiones. En la materia condensada en escalas de tiempo sub-microsegundo, la conducción de calor es a menudo demasiado lenta para tienen un efecto significativo en la respuesta del material, y es ignorado aquí. Las ecuaciones de la dinámica del continuum no relativista son, en forma lagrangiana, es decir. a lo largo de las características movimiento con la velocidad de material local ~u(~r), (~r, t) = (~r, t)div~u(~r, t) (1) D~u(~r, t) (~r, t) (~r, t) (2) De(~r, t) = (~r, t)grad~u(~r, t) (3) donde la densidad de masa y la energía interna específica. Los cambios en e pueden estar relacionados a los cambios en la temperatura T a través de la capacidad de calor. Las propiedades inherentes de cada material en el problema se describen por su relación constitutiva o ecuación de estado (s). Además de experimentar la compresión y el trabajo de la deformación mecánica, el local El estado del material s(~r, t) puede evolucionar a través de procesos internos como el flujo de plástico. En general, Ds(~r, t) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * que también puede incluir las ecuaciones para /­t y ­e/­t. Por lo tanto, las propiedades del material deben Describa como mínimo los siguientes elementos para cada material: Si también describen T (s), la conductividad, y ė(ė), entonces la conducción del calor puede ser tratada. Otras funciones pueden ser: necesarios para determinados métodos numéricos en la dinámica del continuum, como la necesidad de velocidades (por ejemplo, la velocidad de sonido longitudinal), que se necesitan para el control del paso del tiempo en forma explícita integración del tiempo. Internamente, dentro de los modelos de propiedades materiales, es deseable reutilizar software tanto como sea posible, y otras funciones del estado son por lo tanto deseables para permitir modelos que se construirán de forma modular y jerárquica. Las manipulaciones aritméticas deben se realiza en el estado durante la integración numérica, y estos se pueden codificar cuidadosamente utilizando sobrecarga del operador, por lo que el operador del tipo apropiado se invoca automáticamente sin tener que incluir estructuras «si-entonces-else» para cada operador, como es el caso en lenguajes de programación orientados a objetos como Fortran-77. Por ejemplo, si se calcula en un método numérico de tiempo de avance entonces los cambios de estado se calculan utilizando numérico ecuaciones de evolución tales como s(t+ ­t) = s(t) + ­t». 5) Por lo tanto, para un estado general s y su derivado de tiempo, que tiene un conjunto equivalente de compo- nents, es necesario multiplicar un estado por un número real y agregar dos estados juntos. Para una implementación de software específica, pueden ser necesarias otras operaciones, por ejemplo: crear, copiar o destruir una nueva instancia de un estado. La atracción de este enfoque es que, al elegir una forma razonablemente general para el relación constitutiva y operaciones asociadas, es posible separar el continuum dinámica parte del problema del comportamiento inherente del material. Relaciones describiendo las propiedades de diferentes tipos de material se puede encapsular en una forma de biblioteca donde el programa de dinámica continua no necesita saber nada sobre las relaciones para tipo cífico de material, y viceversa. Los programas de dinámica continua y el material las relaciones de propiedades pueden ser desarrolladas y mantenidas independientemente unas de otras, siempre y cuando que la interfaz sigue siendo la misma (cuadro I). Esta es una manera eficiente de complicar modelos de materiales disponibles para simulaciones de diferentes tipos, incluyendo Lagrangian y Eule- rios que funcionan en diferentes números de dimensiones, y cálculos de historial de carga o calefacción, como el choque y la carga en rampa que se examina a continuación. Programas informáticos en- terfaces se han desarrollado en el pasado para escalar EOS con una sola estructura para el estado [12], pero las técnicas orientadas a objetos hacen práctico extender el concepto a mucho más estados complicados, a combinaciones de modelos, y a tipos alternativos de modelos seleccionados cuando se ejecuta el programa, sin tener que encontrar un solo estado super-set que abarque todo posibles estados como casos especiales. Una gama muy amplia de tipos de comportamiento material se puede representar con este formalismo. En el nivel más alto, diferentes tipos de comportamiento se caracterizan por diferentes estructuras para el estado s (cuadro II). Para cada tipo de estado, se pueden definir diferentes modelos específicos, tales como: como gas perfecto, politrópico y Grüneisen EOS. Para cada modelo específico, diferentes materiales se representan eligiendo diferentes valores para los parámetros en el modelo, y diferentes los estados materiales locales se representan a través de diferentes valores para los componentes de s. jerga de programación orientada a objetos, la capacidad de definir un objeto cuyo tipo preciso no está determinado hasta que el programa se ejecuta se conoce como polimorfismo. Para nuestra aplicación, polimorfismo se utiliza en varios niveles en la jerarquía de objetos, desde el tipo general de un material (como «uno representado por un EOS de presión-densidad-energía» o «uno representado por un modelo de estrés desviatorio») a través del tipo de relación utilizado para describir las propiedades de tipo de material (como gas perfecto, politrópico, o Grüneisen para una densidad de presión-energía EOS, o Steinberg-Guinan [13] o Preston-Tonks-Wallace [14] para un modelo de estrés desviatorio, al tipo de función matemática general utilizada para representar algunas de estas relaciones (como como polinomio o representación tabular de γ(l) en un EOS politrópico) (Tabla III). Estados o modelos pueden definirse extendiendo o combinando otros estados o modelos - esto puede ser se aplica utilizando el concepto de herencia basado en la programación orientada a los objetos. Así desviatoria modelos de estrés pueden definirse como una extensión a cualquier presión-densidad-energía EOS (que son Por lo general escrito asumiendo un tipo específico, como la forma cúbica Grüneisen de Steinberg), homo- las mezclas genéticas pueden definirse como combinaciones de cualquier EOS a presión-densidad-temperatura, y mezclas heterogéneas pueden definirse como combinaciones de materiales representados cada uno por cualquier tipo de modelo de material. Las implementaciones de prueba se han hecho como bibliotecas en la programación C++ y Java idiomas [15]. La interfaz externa con las propiedades del material era general a nivel de representar un tipo y estado de material genérico. El tipo de estado y modelo eran entonces seleccionado cuando los programas que utilizan la biblioteca de propiedades de material se ejecutaron. En C++, objetos que eran polimórficos en el tiempo de ejecución tuvo que ser representado como punteros, lo que requiere adicional construcciones de software para asignar y liberar la memoria física asociada con cada objeto. Era posible incluir funciones reutilizables generales como objetos polimórficos al definir modelos: funciones reales de un parámetro real podrían ser polinomios, trascendentales, tabular con diferentes sistemas de interpolación, definiciones por partes en diferentes regiones de la línea dimensional, sumas, productos, etc; otra vez definido específicamente en el tiempo de ejecución. Orientado a los objetos El polimorfismo y la herencia eran por lo tanto técnicas muy poderosas para aumentar el software reutilizar, haciendo el software más compacto y más fiable a través de un mayor uso de funciones que ya se habían puesto a prueba. Dadas las estructuras conceptuales y de software diseñadas para representar lazos adecuados para su uso en simulaciones de dinámica de continuum espacialmente resueltos, ahora consideramos el uso de estos modelos genéricos de material para calcular las rutas de carga idealizadas. III. CARGA DE UN DIMENSIONAL IDEALIZADA Experimentos para investigar la respuesta de los materiales a la carga dinámica, y para calibrar los parámetros en los modelos de su comportamiento, son generalmente diseñados para aplicar como simple una carga la historia como es consistente con el estado transitorio de interés. Los tipos canónicos más simples de el historial de carga son el choque y la rampa [5, 6]. Los métodos de solución se presentan para el cálculo el resultado del choque y la carga en rampa para los materiales descritos por los modelos de materiales generalizados se examina en la sección anterior. Tal solución directa elimina la necesidad de utilizar un tiempo- y simulación de la dinámica del continuum resuelta desde el espacio, que permite calcular los estados con mucho mayor eficiencia y sin la necesidad de tener en cuenta y tener en cuenta los atributos de simulaciones resueltas como la resolución numérica finita y el efecto de la resolución numérica y viscosidades artificiales. A. Compresión de rampas La compresión de la rampa se toma aquí para significar compresión o descompresión. Si el material está representado por un EOS escalar invisible, es decir. ignorando procesos disipativos y no escalares efectos de la tensión elástica, la compresión de la rampa sigue un isentrope. Esto ya no es cierto. cuando se producen procesos disipativos como el calentamiento de plástico. El término «cuasi-isentrópico» es a veces utilizados en este contexto, especialmente para la compresión sin golpes; aquí preferimos se refieren a las trayectorias termodinámicas como adiabats, ya que se trata de un término más adecuado: ningún calor se intercambia con el entorno en las escalas de tiempo de interés. Para la compresión adiabática, el estado evoluciona de acuerdo con la segunda ley de termo- los namics, de = T dS − p dv (6) donde T es la temperatura y S la entropía específica. Por lo tanto ė = T − p v = T − pdiv~u , (7) o para un material más general cuyo tensor de tensión sea más complicado que una presión escalar, de = T dS + n dv  ė = T + * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * en la que el componente del estrés es normal a la dirección de la deformación. La velocidad El gradiente se expresó a través de un factor de compresión η ° ° ° ° ° ° y una tasa de deformación. En total experimentos en rampa utilizados en el desarrollo y calibración de modelos de materiales precisos, la cepa se ha aplicado uniaxialmente. Vías de deformación más generales, por ejemplo isotrópicas o incluyendo un componente de corte, puede ser tratado por el mismo formalismo, y que el trabajo tasa es entonces un producto interno completo de los tensores de tensión y tensión. La aceleración o desaceleración del material normal a la onda a medida que se comprime o expandido adiabáticamente es , (9) de la cual puede deducirse que donde cl es la velocidad de onda longitudinal. Como con la dinámica del continuum, la evolución interna del estado material se puede calcular simultáneamente con las ecuaciones de continuum, o operador dividido y calculado periódicamente a compresión constante [16]. Los resultados son iguales al segundo orden en la compresión incremento. El fraccionamiento del operador permite realizar los cálculos sin un explici- tropy, si las ecuaciones del continuum están integradas isentrópicamente y los procesos disipativos son capturado por la evolución interna en constante compresión. La división del operador es deseable cuando la evolución interna puede producir altamente no lineal cambios, como la reacción del sólido al gas: cambios rápidos en el estado y las propiedades pueden que los esquemas numéricos sean inestables. El reparto de operadores también es deseable cuando la integración el paso del tiempo para la evolución interna es mucho más corto que el paso del tiempo de la dinámica continua. Ninguna de estas consideraciones es muy importante para la compresión de rampas sin res- olución, pero el operador-splitting se utilizó como una opción en los cálculos de compresión de rampa para la coherencia con las simulaciones de dinámica continuum. Las ecuaciones de compresión de rampa se integraron usando Runge-Kutta nu- esquemas mericos de segundo orden. El esquema del cuarto orden es una extensión trivial. Los la secuencia de operaciones para calcular un incremento de compresión de rampa es la siguiente: 1. Incremento de tiempo: T = − ln 2. Predictor: s(t + t/2) = s(t) + (m(s)(t), ) (12) 3. Corrector: s(t+ ­t) = s(t) + ­t­óm(s(t+ ­t/2), ) (13) 4. Evolución interna: s(t+ t) → s(t+ t) + ∫ tÃ3 °t •i(s)(t) ′), ) dt′ (14) donde m es la evolución del estado dependiente del modelo a partir de la cepa aplicada, y i es interna evolución en constante compresión. La variable independiente para la integración es volumen específico v o densidad de masa los pasos finitos de integración numérica son tomados en ♥ y v. El tamaño del paso se puede controlar así que el error numérico durante la integración permanece dentro de los límites elegidos. Un adiabat tabular se puede calcular mediante la integración en un rango de v o ♥, pero al simular experimental escenarios el límite superior para la integración suele ser que uno de los otros termodinámicos las cantidades alcanzan un valor determinado, por ejemplo, que el componente normal del estrés alcanza cero, que es el caso en la liberación de un estado de alta presión en una superficie libre. Específico condiciones finales se encontraron mediante el seguimiento de la cantidad de interés hasta que entre corchetes por un finito paso de integración, a continuación, biseccionar hasta que la condición de parada se satisfizo a una precisión elegida. Durante la bisección, cada cálculo de ensayo se realizó como una integración desde el primer lado del soporte por la compresión del ensayo. B. Compresión por choque La compresión de choque es la solución de un problema de Riemann para la dinámica de un salto en compresión moviéndose con velocidad constante y con un espesor constante. El Rankine... Las ecuaciones de Hugoniot (RH) [5] que describen la compresión de choque de la materia se derivan en la aproximación del continuum, donde el choque es una discontinuidad formal en el continuum campos. En realidad, la materia está compuesta de átomos, y los choques tienen un ancho finito gobernado por la cinética de los procesos disipativos – a un nivel fundamental, la materia no distingue entre compresión de choque y compresión de rampa con una alta tasa de deformación, pero el RH las ecuaciones se aplican siempre y cuando la anchura de la región de la materia donde los procesos no resueltos ocurren es constante. En comparación con los estados isentrópicos inducidos por la compresión de rampa en un material representado por un EOS, un choque siempre aumenta la entropía y por lo tanto la temperatura. Con procesos disipativos incluidos, la distinción entre una rampa y una El shock se puede desdibujar. Las ecuaciones RH expresan la conservación de la masa, el impulso y la energía a través de un la discontinuidad en movimiento en estado. Por lo general se expresan en términos de la presión, pero son fácilmente generalizado para materiales que soportan tensiones de cizallamiento mediante el uso del componente de estrés normal al choque (es decir, paralelo con la dirección de propagación del choque), u2s = −v N-N-N-N-O-N-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O v0 − v , (15) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • — (ln) — (l) — (l) — (l) — (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) (l) e = e0 − (ln + ln0)(v0 − v), (17) donde nosotros es la velocidad de la onda de choque con respecto al material, arriba es el cambio en velocidad del material normal a la onda de choque (es decir, paralela a su dirección de propagación), y El subíndice 0 se refiere al estado inicial. Las relaciones RH se pueden aplicar a los modelos de material general si una escala de tiempo o velocidad de deformación se impone, y una orientación elegida para el material con respecto al choque. Shock la compresión en la dinámica del continuum es casi siempre uniaxial. Las ecuaciones RH implican sólo los estados inicial y final en el material. Si un material tiene propiedades que dependen de la trayectoria de deformación – tales como flujo de plástico o viscosidad – entonces físicamente la estructura de choque detallada puede hacer una diferencia [17]. Esto es una limitación. de choques discontinuos en la dinámica del continuum: puede abordarse como se ha señalado anteriormente mediante la inclusión de procesos disipativos y la consideración de la compresión en rampa, si la Los procesos pueden representarse adecuadamente en la aproximación del continuum. Resuelto desde el punto de vista espacial simulaciones con diferenciación numérica para obtener derivados espaciales y tiempo de avance las diferencias no suelen ser capaces de representar las discontinuidades de choque directamente, y un la viscosidad artificial se utiliza para la compresión de choque de frotis en unas pocas células espaciales [18]. Los trayectoria seguida por el material en el espacio termodinámico es un adiabat suave con dissi- calefacción pative suministrada por la viscosidad artificial. Si el trabajo de plástico también se incluye durante este compresión adiabática, el calentamiento total para una compresión dada es mayor que desde el Ecuaciones RH. Para ser consistente, el flujo de plástico debe ser descuidado mientras la viscosidad artificial no es cero. Esta desactivación localizada de los procesos físicos, en particular los que dependen del tiempo, durante el paso de la conmoción no físicamente manchada se encontró previamente necesario para simulaciones numéricamente estables de ondas de detonación por flujo reactivo [19]. Las ondas de detonación son ondas de choque reactivas. Detonación plana constante (el Chapman- Estado de Jouguet [20]) puede calcularse utilizando las relaciones RH, mediante la imposición de la condición que el estado material detrás del shock es totalmente reaccionado. Varios métodos numéricos se han utilizado para resolver las ecuaciones RH para los materiales repré- enviado por un EOS únicamente [21, 22]. Las ecuaciones generales de RH pueden ser resueltas numéricamente para un compresión de choque dada variando la energía interna específica e hasta el estrés normal del modelo material es igual a la de la ecuación de energía RH, Eq. 17. El shock y Las velocidades de partículas se calculan a partir de Eqs 15 y 16. Este método numérico es particu- larly conveniente para EOS de la forma p(l, e), ya que e puede variar directamente. Las soluciones todavía pueden se encuentran para los modelos de material general que utilizan (ė), por lo que la energía puede ser variada hasta se encuentra la solución. Numéricamente, la solución se encontró por soporte y bisección: 1. Para la compresión dada, tomar el extremo de baja energía para el soporte como un estado cercano s− (por ejemplo: el estado anterior, de compresión inferior, en el Hugoniot), adia comprimida baticamente (estado s"), y se enfría por lo que la energía interna específica es e(s-). 2. Bracket el estado deseado: aplicar incrementos de calefacción sucesivamente más grandes en cada estado de ensayo internamente, hasta que el (los) n(s) del modelo de material supere el (e − e0) de Eq. 17. 3. Bisecte en la e, evolucionando cada estado de prueba internamente, hasta que el n(s) es igual al n(e − e0) a la precisión deseada. Al igual que con la compresión en rampa, la variable independiente para la solución fue la densidad de masa y pasos finitos fueron tomados. Cada estado de shock fue calculado independientemente del resto, así que los errores numéricos no se acumularon a lo largo del shock Hugoniot. La exactitud de la la solución era independiente de. Un Hugoniot tabular se puede calcular resolviendo sobre un rango de............................................................................................................................ calcular el estado de choque en el que una de las otras cantidades termodinámicas alcanza una determinada valor, a menudo que hasta y Łn coinciden con los valores de otro, cálculo de choque simultáneo para otro material – la situación en los problemas de transmisión de impactos y choques, discutido abajo. Las condiciones específicas de parada se comprobaron mediante el control de la cantidad de interés hasta entre corchetes por un paso de solución finita, luego bisecar hasta que la condición de parada se satisfizo a un Precisión elegida. Durante la bisección, cada cálculo de ensayo se realizó como un shock de la condiciones iniciales para la compresión de choque del ensayo. C. Precisión: aplicación al aire La precisión de estos esquemas numéricos fue probada comparando con el choque y la rampa compresión de un material representado por un EOS de gas perfecto, p = (γ − 1) (18) La solución numérica requiere que se elija un valor para cada parámetro del material modelo, aquí γ. El aire fue elegido como material de ejemplo, con γ = 1.4. Aire en el tem- la peratura y la presión tienen unas dimensiones aproximadas de 10 a 3 g/cm3 y de 0,25 MJ/kg. Isentropos para el gas perfecto EOS tienen la forma p = constante, (19) y el shock Hugoniots tienen la forma p = (γ − 1) 2e0-0 p0( ­0) (γ + 1)}0 − (γ − 1) . (20) Las soluciones numéricas reprodujeron el isentrope principal y Hugoniot al 10-3% y al 0,1% respectivamente, para un incremento de compresión del 1% a lo largo del isentrope y una tolerancia a la solución de 10-6GPa por cada estado de shock (fig. 1). Sobre la mayor parte del rango, el error en el Hugoniot fue igual o inferior al 0,02%, aproximándose sólo al 0,1% cerca de la compresión máxima de choque. IV. COMPLEJO COMPATIBILIDAD DE LA IMPORTACIÓN CONDENADA La capacidad de calcular choque y loci rampa en el espacio de estado, es decir. en función de la diversidad de las condiciones de carga, es particularmente conveniente para investigar aspectos complejos de la respuesta de la materia condensada a la carga dinámica. Cada locus puede ser obtenido por un solo serie de soluciones de choque o rampa, en lugar de tener que realizar una serie de tiempo y espacio- simulaciones de dinámica continua resueltas, variando las condiciones iniciales o de frontera, y reducir la solución. Consideramos el cálculo de la temperatura en el escalar EOS, el efecto de la fuerza material y el efecto de los cambios de fase. A. Temperatura Las ecuaciones de dinámica continua se pueden cerrar usando un EOS mecánico relacionado con el estrés a la densidad de masa, la tensión y la energía interna. Para un EOS escalar, la forma ideal para cerrar el ecuaciones continuum es p(l, e), con s =, e} la elección natural para el estado primitivo campos. Sin embargo, la temperatura es necesaria como parámetro en las descripciones físicas de muchos contribuciones a la respuesta constitutiva, incluidos el flujo de plástico, las transiciones de fase, y reacciones químicas. Aquí, discutimos el cálculo de la temperatura en diferentes formas de la escalar EOS. 1. Ecuaciones de densidad-temperatura del estado Si el EOS escalar se construye a partir de sus contribuciones físicas subyacentes para el continuum la dinámica, puede tomar la forma e(l, T ), a partir de la cual p(l, T ) se puede calcular utilizando la segunda ley de la termodinámica [10]. Un ejemplo es la forma ‘SESAME’ de EOS, basada en relaciones tabulares interpoladas para {p, e}(l, T ) [23]. Un par de relaciones {p, e}(l, T ) puede ser utilizado como un EOS mecánico mediante la eliminación de T, que es equivalente a invertir e(l, T ) para encontrar T (l, e), sustituyéndolo en p(l, T ). Para una relación general e(l, T ), por ejemplo para la SESAME EOS, el inverso se puede calcular numéricamente según sea necesario, a lo largo de un isochore. In de esta manera, un {p, e}(l, T ) puede ser utilizado como un p(l, e) EOS. Alternativamente, la misma relación p(l, T ) se puede utilizar directamente con un campo de estado primitivo incluyendo la temperatura en lugar de la energía: s =, T}. La evolución del estado bajo El trabajo mecánico implica entonces el cálculo de (ė), es decir. el recíproco del calor específico capacidad, que es un derivado de e(l, T ). Dado que este cálculo no requiere que e(l, T ) sea invertido, es computacionalmente más eficiente para utilizar {p, e}(l, T ) EOS con una temperatura- Estado basado, en lugar de basado en la energía. La principal desventaja es que es más difícil para garantizar la conservación exacta de la energía a medida que las ecuaciones de dinámica continua se integran en tiempo, pero cualquier desviación de la conservación exacta está en el nivel de precisión del algoritmo utilizado para integrar la capacidad de calor. Ambas estructuras de EOS han sido implementadas para cálculos de propiedades materiales. Tomando a SESAME tipo EOS, los loci termodinámicos fueron calculados con, e} o, T} primitivos los estados, para la comparación (Fig. 2). Para un EOS monotónico, los resultados fueron indistinguibles dentro de las diferencias de interpolación hacia adelante o hacia atrás de las relaciones tabulares. Cuándo el EOS, o la superficie efectiva utilizando un orden dado de función de interpolación, no fue monotónicos, los resultados variaron mucho debido a la no-unidad al eliminar T para el , e} estado primitivo. 2. Modelo de temperatura para ecuaciones mecánicas de estado EOS mecánicos a menudo están disponibles como empíricas, algebraicas relaciones p(l, e), derivadas de Datos de choque. La temperatura se puede calcular sin alterar el EOS mecánico añadiendo a relación T (l, e). Si bien esta relación podría adoptar cualquier forma en principio, también se puede seguir la lógica del Grüneisen EOS, en la que la presión se define en términos de su desviación (p, e-er) de una curva de referencia {pr, er}(l). Por lo tanto, las temperaturas se pueden calcular por referencia a una curva de compresión a lo largo de la cual la temperatura y la energía interna específica son conocidos, {Tr, er}(l), y una capacidad calorífica específica definida como función de la densidad cv(l). En los cálculos, esta EOS aumentada se representó como una forma «mecánica-térmica» que comprende cualquier p(e), e) EOS más las curvas de referencia – un ejemplo de herencia de software y polimorfismo. Una curva de referencia natural para la temperatura es la curva de frío, Tr = 0K. La curva de frío puede estimarse a partir del isentrope principal e(l)s0 utilizando la variación de densidad estimada del parámetro Grüneisen: er(l) = e(l)s0 − T0cpe a(10/l) )γ0−a [24]. En este trabajo, el isentropo principal se calculó en forma tabular a partir de la mecánica EOS, usando el algoritmo de compresión de rampa descrito anteriormente. El EOS empírico se calibra con datos experimentales. Amortiguación y compresión adiabática medidas en materiales fuertes inevitablemente incluyen contribuciones elásticas-plásticas, así como el EOS escalar en sí mismo. Si las contribuciones elásticas-plásticas no se tienen en cuenta sistemáticamente, la EOS puede incluir implícitamente contribuciones de la fuerza. Un único EOS escalar se puede construir para reproducir el estrés normal en función de la compresión para cualquier trayectoria de carga única: choque o adiabat, para una tensión constante o que varíe suavemente tasa. Tal EOS generalmente no predeciría la respuesta a otras historias de carga. Los EOS y propiedades constitutivas de los materiales considerados aquí fueron construidos auto- consistentemente a partir de datos de choque – esto no significa que los modelos son precisos para otras cargas caminos, ya que ni el EOS ni el modelo de fuerza incluye todos los términos físicos que real exposición de materiales. Esto no importa en ningún caso a los efectos de demostrar propiedades de los esquemas numéricos. Este procedimiento mecánico-térmico se aplicó a Al utilizando un Grüneisen EOS instalado en el los mismos datos de choque utilizados para calcular el {p, e}(l, T ) EOS analizado anteriormente [24]. Temperaturas estaban de acuerdo (Fig. 2). Los cálculos mecánicos-térmicos requirieron un similar esfuerzo computacional para el tabular {p, e}(l, T ) EOS con un, T} estados primitivos (y eran por lo tanto mucho más eficiente que el EOS tabular con, e} estados), y describió el EOS mucho más compacto. B. Dosis Para compresiones dinámicas a o(10GPa) y superiores, en escalas de tiempo de microsegundo, el flujo El estrés de los sólidos a menudo se trata como una corrección o una pequeña perturbación del EOS escalar. Sin embargo, se ha observado que el estrés de flujo es mucho mayor en las escalas de tiempo de nanosegundos [25], y las interacciones entre ondas elásticas y plásticas pueden tener un efecto significativo sobre la compresión y la propagación de ondas. Las ecuaciones de Rankine-Hugoniot deben ser resueltas auto-consistente con la fuerza incluida. 1. Representación preferida de la fuerza isotrópica Existe una inconsistencia en el tratamiento de la dinámica continua estándar de escalar (pres- respuesta del tensor (estrés). El EOS escalar expresa la presión p(l, e) como la cantidad dependiente, que es la forma más conveniente para su uso en las ecuaciones de continuum. La práctica habitual consiste en utilizar la elasticidad subhookea (forma hipoelástica) [16] (cuadro II), en que los parámetros de estado incluyen el desviador de estrés = G(s) (22) donde G es el módulo de corte y el desviador de la tasa de deformación. Por lo tanto, el isotrópico y el devia- las contribuciones técnicas al estrés no se tratan de manera equivalente: la presión se calcula de un estado local que implica un parámetro similar a la deformación (densidad de masa), mientras que el estrés de- viator evoluciona con el tiempo-derivado de la cepa. Esta inconsistencia causa problemas a lo largo de rutas de carga complicadas porque G varía fuertemente con la compresión: si un material es sub- se inyecta a una cepa de cizallamiento, a continuación, compresión isotrópica (aumento del módulo de cizallamiento de G a G′, dejando sin cambios ), después la descarga de cizallamiento a la tensión isotrópica, la descarga verdadera la cepa es, mientras que el cálculo hipoelástico requeriría una cepa de G/G′. Uso Ser y el modelo de fuerza Steinberg-Guinan como ejemplo de la diferencia entre cálculos poelásticos e hiperelásticos, considerar una cepa inicial a un esfuerzo de flujo de 0.3GPa seguido de compresión isotrópica isotrópica a 100GPa,. la cepa a descargar a un estado de estrés isotrópico es 0,20% (hiperelástico) y 0,09% (hipoelástico). La discrepancia surge porque el modelo hipoelástico no aumenta el estrés desviatorio en la compresión en tensión desviatoria constante. El estrés puede ser considerado como una respuesta directa del material al estado instantáneo de cepa elástica: (, T ). Esta relación puede predecirse directamente con la estructura electrónica cálculos del tensor de tensión en un sólido para un determinado estado de compresión y tensión elástica [11], y es una generalización directa de la ecuación escalar del estado. Una representación más coherente de los parámetros de estado es utilizar el desviador de la deformación más bien que el desviador de la deformación, y calcular a partir de rascar cuando sea necesario utilizando  = G(s)® (23) – una formulación hiperelástica. Los parámetros de estado son entonces, e,, p}. Las diferentes formulaciones dan diferentes respuestas cuando se acumula tensión desviatoria a diferentes compresiones, en cuyo caso la formulación hiperelástica es correcta. Si la cizalla El módulo varía con el desviador de deformación – es decir, para la elasticidad no lineal – a continuación, la definición de G(­) debe ajustarse para dar el mismo estrés para una determinada cepa. Muchos modelos de resistencia isotrópica utilizan medidas escalares de la tensión y el estrés para terilizar el trabajo endurecimiento y aplicar un modelo de rendimiento de tensión de flujo: F2, = - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. (24) Los diferentes trabajadores han utilizado convenios inconsistentes para medidas escalares equivalentes. En el presente trabajo, se utilizó la convención de física de choque común que el estrés de flujo componente de la Y donde Y es el estrés de flujo. Por coherencia con las velocidades publicadas y amplitudes para ondas elásticas, , en contraste con otros valores utilizados anteriormente para una deformación de menor velocidad [26]. En principio, los valores de las letras f) y f) no importan mientras los parámetros de resistencia fueron calibrados utilizando los mismos valores utilizados en cualquier simulación. 2. Berilio La tensión de flujo medida a partir de los experimentos de choque impulsados por láser en los cristales de Be unas pocas decenas de micrómetros de espesor es, en torno a 5-9GPa [25], mucho mayor que el seguro en escalas de tiempo de microsegundos. Un modelo de plasticidad cristalina dependiente del tiempo para Be está siendo desarrollado, y el comportamiento bajo carga dinámica depende del tiempo detallado depen- Dence de la plasticidad. Los cálculos se realizaron con el modelo de resistencia Steinberg-Guinan desarrollados para datos a escala de microsegundos [24], y, a efectos de comparación aproximada, con respuesta elástica-perfectamente plástica con un esfuerzo de flujo de 10GPa. El plástico elástico perfectamente modelo descuidado presión- y trabajo- endurecimiento. Se hicieron cálculos del principal adiabat y el shock Hugoniot, y de una liberación adiabat de un estado en el principal Hugoniot. Los cálculos se hicieron con y sin fuerza. Teniendo en cuenta las trayectorias del estado en el espacio de volumen de estrés, es interesante notar que el calentamiento del flujo de plástico puede empujar el adiabat por encima del Hugoniot, debido a la mayor calentamiento obtenido mediante la integración a lo largo del adiabat en comparación con el salto de el estado inicial al final en el Hugoniot (Fig. 3). Incluso con un plástico elástico perfecto modelo de fuerza, las curvas con fuerza no mienten exactamente 2 Y por encima de las curvas sin fuerza, porque la calefacción a partir del flujo de plástico contribuye a aumentar la cantidad de energía interna a la EOS a medida que aumenta la compresión. Una característica importante para la siembra de inestabilidades por variaciones microestructurales en respuesta de choque es el estrés de choque en el que una onda elástica no se ejecuta por delante de la Choque. En Be con el alto estrés de flujo de la respuesta nanosegundo, la relación entre el choque y la velocidad de las partículas es significativamente diferente de la relación para el bajo esfuerzo de flujo (Fig. 4). Por bajo esfuerzo de flujo, la onda elástica viaja a 13,2 km/s. Un choque de plástico viaja más rápido que esto para presiones superiores a 110GPa, independientemente del modelo constitutivo. La velocidad de un choque de plástico después de la onda elástica inicial es similar a la caja de baja resistencia, porque el el material ya está en su tensión de flujo, pero la velocidad de un solo choque de plástico es sensiblemente Más alto. Para la compresión a una tensión normal dada, la temperatura es significativamente más alta con flujo de plástico incluido. La calefacción adicional es particularmente llama- abat: la temperatura se aparta significativamente del isentropo principal. Así la onda de la rampa la compresión de materiales fuertes puede conducir a niveles significativos de calefacción, contrariamente a la hipótesis de pequeñas subidas de temperatura [27]. El flujo de plástico es en gran parte irreversible, así que la calefacción se produce tanto en la descarga como en la carga. Por lo tanto, en la liberación adiabática de un shock- estado comprimido, se produce calefacción adicional en comparación con el caso sin resistencia. Estos los niveles de calentamiento son importantes ya que el choque o el derretimiento de la liberación pueden ocurrir a una presión de choque de lo que cabría esperar ignorando el efecto de la fuerza. (Fig. 5.) C. Cambios de fase Una propiedad importante de la materia condensada son los cambios de fase, incluyendo polisólidos sólidos Morfismo y líquido sólido. Un diagrama de fase de equilibrio se puede representar como un solo superficie total de EOS como antes. Múltiples fases competidoras con cinética para cada fase trans- formación se puede representar convenientemente utilizando la estructura descrita anteriormente para general propiedades materiales, por ejemplo, al describir el estado local como un conjunto de fracciones de volumen fi de cada posible fase de EOS simple, con tasas de transición y equilibrio entre ellas. Este modelo se describe con más detalle en otras partes [19]. Sin embargo, es interesante investigar puerta la robustez del esquema numérico para calcular el choque Hugoniots cuando el EOS tiene las discontinuidades en valor y gradiente asociadas con los cambios de fase. El EOS de metal fundido, y la transición de fase sólido-líquido, se puede representar a un aproximación razonable como ajuste a la EOS del sólido: pdosfase(l, e) = psólido(l, ) (25) donde e : T (l, e) < Tm(l) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ehm : de lo contrario es el calor específico latente de la fusión. Tomando el EOS y un Lindemann modificado la curva de fusión para Al [24], y utilizando el algoritmo de choque Hugoniot fue se encontró que funciona de forma estable a lo largo de la transición de fase (Fig. 6). V. PATOS COMPUESTOS DE CARGO Dados los métodos para calcular el choque y las rutas de carga adiabáticas desde el inicial arbitrario los estados, una variedad considerable de escenarios experimentales se pueden tratar a partir de la interacción de ondas de carga o descarga con interfaces entre diferentes materiales, en geometría plana para la compresión uniaxial. La restricción física clave es que, si dos materiales diferentes son para permanecer en contacto después de una interacción como un impacto o el paso de un shock, el la velocidad normal de la tensión y la velocidad de las partículas en ambos materiales deben ser iguales a ambos lados de la interfaz. El cambio en la velocidad de las partículas y el estrés normal a las ondas fueron calculados arriba para ondas de compresión que corren en la dirección de aumentar la ordenación espacial (de izquierda a derecha). A través de una interfaz, el sentido se invierte para el material a la izquierda. Por lo tanto, un proyectil impacto en un objetivo estacionario a la derecha se desacelera a partir de su velocidad inicial por el choque inducidos por el impacto. El problema general en una interfaz se puede analizar considerando los estados en el instantánea del primer contacto – en el impacto, o cuando un shock viaja a través de un sandwich de ma- terials primero llega a la interfaz. Los estados iniciales son {ul, sl; ur, sr}. Los estados finales son {uj, s l; uj, r r} donde uj es la velocidad de la partícula de la junta, ­n(s) l) = n(s) r), y s i está conectado a si por un shock o un adiabat, comenzando con la velocidad inicial y el estrés adecuados, y con la orientación dada por el lado del sistema cada material se produce en. Cada tipo de onda se considera a su vez, en busca de una intersección en el plano ascendente. Ejemplos de ello Las interacciones de ondas son el impacto de un proyectil con un objetivo estacionario (Fig. 7), liberación de un estado de choque en una superficie libre o un material (por ejemplo, a ventana) de menor impedancia de choque (de ahí reflejando una onda de liberación en el material conmocionado – Fig. 8), que chocan en una superficie con una material de mayor impedancia al choque (fig. 8), o tensión inducida como materiales tratar de separar en direcciones opuestas cuando se une una interfaz enlazada (Fig. 9). Cada uno de estos escenarios puede ocurrir a su vez después del impacto de un proyectil con un objetivo: si el objetivo está en capas entonces un choque se transmite a través de cada interfaz con una liberación o una réplica reflejada en la espalda, dependiendo de los materiales; la liberación se produce en última instancia en la parte trasera del proyectil y el el extremo lejano del objetivo, y las ondas de liberación que se mueven opuestamente sujetan el proyectil y objetivo a tensiones de tracción cuando interactúan (Fig. 10). Como ilustración de la combinación de cálculos de choque y carga en rampa, considere el problema de un proyectil Al, viajando inicialmente a 3,6 km/s, impactando en un objetivo compuesto estacionario que comprende una muestra de Mo y una ventana de liberación de LiF [28, 29]. Los estados de shock y liberación fueron calculados utilizando propiedades de materiales publicados [24]. El estado de shock inicial se calculó para tiene un estrés normal de 63.9GPa. Al llegar a la LiF, el shock fue calculado para transmitir a 27.1GPa, reflejándose como un lanzamiento en el Mo. Estos esfuerzos coinciden con la dinámica del continuum simulación a dentro de 0,1GPa en el Mo y 0,3GPa en el LiF, utilizando el mismo material propiedades (Fig. 11). Las velocidades de onda y partícula asociadas coinciden con una precisión similar; Las velocidades de onda son mucho más difíciles de extraer de la simulación de la dinámica de continuum. Una extensión de este análisis se puede utilizar para calcular la interacción de choques oblicuos con una interfaz [30]. VI. CONCLUSIONES Se elaboró una formulación general para representar modelos de materiales para aplicaciones en carga dinámica, adecuada para la implementación de software en la programación orientada a objetos lan- ¡Guages! Se diseñaron métodos numéricos para calcular la respuesta de la materia representada por los modelos de material general para la compresión de golpes y rampas, y la descompresión de rampas, mediante la evaluación directa de las vías termodinámicas para estas compresiones en lugar de simulaciones espacialmente resueltas. Este enfoque es una generalización de la labor anterior sobre soluciones para los materiales representados por una ecuación escalar de estado. Los métodos numéricos fueron encontrados ser flexible y robusto: capaz de aplicarse a materiales con propiedades muy diferentes. Las soluciones numéricas combinaban los resultados analíticos con una alta precisión. Se necesita atención con la interpretación de algunos tipos de respuesta física, como por ejemplo: flujo tic, cuando se aplica a la deformación a altas tasas de deformación. La dependencia temporal subyacente deben tenerse en cuenta los procesos que se produzcan durante la deformación. La historia real la carga y el calentamiento experimentados por el material durante el paso de un choque puede influir el estado final – esta historia no se captura en la aproximación continuum al material dinámica, donde los choques se tratan como discontinuidades. Por lo tanto, la atención también es necesaria en el spa. simulaciones resueltas cuando los choques se modelan utilizando la viscosidad artificial para untarlos unphysically sobre un espesor finito. Se demostró que los cálculos demuestran el funcionamiento de los algoritmos de choque y compresión de rampa con modelos de material representativos de sólidos complejos, incluida la resistencia y las transformaciones de fase. Los métodos básicos de la rampa y la solución de choque se acoplaron para resolver Vías de comunicación, tales como impactos inducidos por choque, e interacciones de choque con una interfaz planar entre diferentes materiales. Tales cálculos captan gran parte de la física de la experimentos de dinámica terial, sin requerir simulaciones de resolución espacial. Resultados de la solución directa de las condiciones de choque y de carga en rampa pertinentes se compararon con simulaciones de hidrocódigo, mostrando consistencia completa. Agradecimientos Ian Gray presentó al autor el concepto de propiedades materiales multimodelo ware. Lee Markland desarrolló un prototipo de programa informático de cálculo de Hugoniot para ecuaciones de estado mientras trabaja para el autor como estudiante de verano de pregrado. El trabajo evolutivo sobre las bibliotecas de propiedades materiales fue apoyado por el U.K. Atomic Establecimiento de Armas, Fluid Gravity Engineering Ltd, y Wessex Científico y Técnico Services Ltd. Los refinamientos de la técnica y las aplicaciones a los problemas descritos fueron: realizado en el Laboratorio Nacional de Los Alamos (LANL) y Lawrence Livermore National Laboratorio (LLNL). El trabajo se llevó a cabo parcialmente en apoyo de, y financiado por, programa de fusión de confinamiento inercial de la Agencia de curidad en LANL (gestionado por Steven Batha), Proyecto de Investigación y Desarrollo Dirigido a Laboratorios y LLNL 06-SI-004 (Principal Investigador: Héctor Lorenzana). El trabajo se llevó a cabo bajo los auspicios de los Estados Unidos. Departamento de Energía en virtud de los contratos W-7405-ENG-36, DE-AC52-06NA25396 y DE- AC52-07NA27344. Bibliografía [1] J.K. Dienes, J.M. Walsh, en R. Kinslow (Ed), “High-Velocity Impact Phenomenas” (Academic Press, Nueva York, 1970). [2] D.J. Benson, Comp. Mech. 15, 6, pp 558-571 (1995). [3] J.W. Gehring, Jr, en R. Kinslow (Ed), “High-Velocity Impact Phenomenas” (Prensa Académica, Nueva York, 1970). [4] R.M. Canup, E. Asphaug, Nature 412, pp 708-712 (2001). [5] Para una revisión e introducción recientes, véase, por ejemplo: M.R. Boslough y J.R. Asay, en J.R. Asay, M. Shahinpoor (Eds), “Compresión de choque de alta presión de sólidos” (Springer-Verlag, New York, 1992). [6] Por ejemplo, C.A. Hall, J.R. Asay, M.D. Knudson, W.A. Stygar, R.B. Spielman, T.D. Pointon, D.B. Reisman, A. Toor y R.C. Cauble, Rev. Sci. Instrum. 72, 3587 (2001). [7] M.A. Meyers, “Comportamiento dinámico de los materiales” (Wiley, Nueva York, 1994). [8] G. McQueen, S.P. Marzo, J.W. Taylor, J.N. Fritz, W.J. Carter, en R. Kinslow (Ed), “High- Fenómenos de Impacto de Velocidad” (Prensa Académica, Nueva York, 1970). [9] J.D. Lindl, “Inertial Confinament Fusion” (Springer-Verlag, Nueva York, 1998). [10] D.C. Swift, G.J. Ackland, A. Hauer, G.A. Kyrala, Phys. Rev. B 64, 214107 (2001). [11] J.P. Poirier, G.D. Price, Phys. de la Tierra y los Interiores Planetarios 110, págs. 147 y 56 (1999). [12] I.N. Gray, P.C. Thompson, B.J. Parker, D.C. Swift, J.R. Maw, A. Giles y otros (AWE Aldermaston), inédito. [13] D.J. Steinberg, S.G. Cochran, M.W. Guinan, J. Appl. Phys. 51, 1498 (1980). [14] D.L. Preston, D.L. Tonks, y D.C. Wallace, J. Appl. Phys. 93, 211 (2003). [15] Una versión del software, incluyendo partes representativas de la biblioteca de modelos de material y la Los algoritmos para calcular el adiabat rampa y el choque Hugoniot, está disponible como un supplemen- archivo tary proporcionado con la preimpresión de este manuscrito, arXiv:0704.008. Apoyo a programas informáticos, y versiones con modelos adicionales, están disponibles comercialmente de Wessex Scientific y Technical Services Ltd (http://wxres.com). [16] D. Benson, Métodos informáticos en Appl. Mecánica e Ing. 99, 235 (1992). http://arxiv.org/abs/0704.0008 http://wxres.com [17] J.L. Ding, J. Mech. y Phys. de Solids 54, págs. 237 y 265 (2006). [18] J. von Neumann, R.D. Richtmyer, J. Appl. Phys. 21, 3, págs. 232 a 237 (1950). [19] R.M. Mulford, D.C. Swift, en preparación. [20] W. Fickett, W.C. Davis, “Detonación” (Universidad de California Press, Berkeley, 1979). [21] R. Menikoff, B.J. Plohr, Rev. Mod. Phys. 61, págs. 75 y 130 (1989). [22] A. Majda, Mem. Amer. Matemáticas. Soc., 41, 275 (1983). [23] K.S. Holian (Ed.), T-4 Manual de bases de datos de bienes materiales, Vol 1c: Ecuaciones de Estado, Informe del Laboratorio Nacional de Los Alamos LA-10160-MS (1984). [24] D.J. Steinberg, Ecuación de Estado y Propiedades de Fuerza de Materiales Seleccionados, Lawrence Informe del Laboratorio Nacional Livermore UCRL-MA-106439 cambio 1 (1996). [25] D.C. Swift, T.E. Tierney, S.-N. Luo, D.L. Paisley, G.A. Kyrala, A. Hauer, S.R. Greenfield, A.C. Koskelo, K.J. McClellan, H.E. Lorenzana, D. Kalantar, B.A. Remington, P. Peralta, E. Loomis, Phys.Plasmas 12, 056308 (2005). [26] R. Hill, “La teoría matemática de la plasticidad” (Clarendon Press, Oxford, 1950). [27] C.A. Hall, Phys. Plasmas 7, 5, pp 2069-2075 (2000). [28] D.C. Swift, A. Seifter, D.B. Holtkamp, y D.A. Clark, Phys. Rev. B 76, 054122 (2007). [29] A. Seifter y D.C. Swift, Phys. Rev. B 77, 134104 (2008). [30] E. Loomis, D.C. Swift, J. Appl. Phys. 103, 023518 (2008). CUADRO I: Interfaz con los modelos materiales necesarios para una dinámica de continuidad explícita en el futuro simulaciones. llamadas de interfaz de propósito configuración del programa lectura/escritura de datos de material continuo dinámica ecuaciones estrés (estado) tiempo paso control de la velocidad del sonido (estado) evolución del estado (deformación) d(estado)/dt(estado,grado ~u) Evolución del estado (calentamiento) d(estado)/dt(estado,ė) evolución interna del estado d(estado)/dt manipulación de estados crear y eliminar añadir estados multiplicar el estado por un escalar comprobar la autocoherencia Los paréntesis en las llamadas de la interfaz denotan funciones, por ejemplo. “estrés (estado)” para “estrés en función de las funciones de evolución se muestran en la estructura de operador-dividido. que es más robusto para soluciones numéricas explícitas de tiempo de avance y también se puede utilizar para cálculos del choque Hugoniot y compresión de rampa. Los cheques de auto-coherencia incluyen que la densidad de masa es positiva, el volumen o las fracciones de masa de los componentes de una mezcla se suman a una, CUADRO II: Ejemplos de tipos de modelo material, distinguidos por diferentes estructuras en el estado vector. modelo de estado efecto vectorial de la tensión mecánica s(s), gradú Ecuación mecánica del estado, e div~u,-pdiv~u/ Ecuación térmica del estado, T div~u,-pdiv~u/lcv mezcla heterogénea, e, fv}i div~u,−pdiv~u/l, 0}i mezcla homogénea, T, {fm}i div~u,−pdiv~u/Ćcv, 0i la fuerza desviatoria tradicional, e, , p div~u, −pdiv~u+fpp , Ge, F Los símbolos son: densidad de masa; e: energía interna específica, T: temperatura, fv: fracción de volumen, fm: fracción de masa, : desviador de tensión, fp: fracción de trabajo de plástico convertido al calor, gradup: Parte plástica del gradiente de velocidad, G: módulo de corte, e,p: partes elásticas y plásticas de la tasa de deformación desviador, p: cepa plástica equivalente escalar, f: factor en la magnitud efectiva de la cepa. Reaccionando Los explosivos sólidos pueden representarse como mezclas heterogéneas, siendo un componente la reacción productos; reacción, un proceso de evolución interna, transfiere material de no reaccionado a reaccionado componentes. La reacción en fase gaseosa puede representarse como una mezcla homogénea, reacciones transferencia de masa entre componentes que representan diferentes tipos de molécula. Simétrico tensores como el desviador de tensión se representan más compactamente por su 6 superior único componentes triangulares, por ejemplo: utilizando la notación Voigt. CUADRO III: Esquema de la jerarquía de los modelos materiales, que ilustra el uso del polimorfismo (en el sentido de programación orientado a objetos). Tipo de modelo de material (o estado) ecuación mecánica del estado politrópico, Grüneisen, basado en energía Jones-Wilkins-Lee, (­, T ) mesa, etc. ecuación térmica del estado basado en la temperatura Jones-Wilkins- Lee, mesa cuasiarmoníaca, ecuación reactiva de estado politrópico modificado, reactivo Jones- Wilkins-Lee spall Cochran-Banner estrés desviatorio elástico-plástico, Steinberg-Guinan, Steinberg-Lund, Preston-Tonks... Wallace, etc. modelos homogéneos de mezcla y reacción modelos heterogéneos de equilibrio y reacción de la mezcla Los programas de dinámica continua pueden referirse a las propiedades materiales como un ‘tipo material’ abstracto con un estado material abstracto. El tipo real de un material (e.g. ecuación mecánica de state), el tipo de modelo específico (por ejemplo, politrópico), y el estado del material de ese tipo son todos manejado transparentemente por la estructura de software orientada a objetos. La ecuación reactiva de estado tiene un parámetro de estado adicional ♥, y las operaciones de software se definen extendiendo los de la ecuación mecánica de estado. Los materiales de espaciado pueden ser representado por un estado sólido más una fracción de vacío fv, con operaciones definidas mediante la extensión de las de el material sólido. Las mezclas homogéneas se definen como un conjunto de ecuaciones térmicas de estado, y el estado es el conjunto de estados y fracciones de masa para cada uno. Las mezclas heterogéneas se definen como conjunto de propiedades de material puro de cualquier tipo, y el estado es el conjunto de estados para cada componente más su fracción de volumen. 0,0001 0,001 0,01 0,001 0,01 densidad de masa (g/cm3) isentrope Hugoniot 0,0001 0,001 0,01 0,001 0,01 densidad de masa (g/cm3) isentrope Hugoniot FIG. 1: Principal isentrope y choque Hugoniot para el aire (gas perfecto): cálculos numéricos para modelos de materiales generales, en comparación con soluciones analíticas. 0 1000 2000 3000 4000 5000 temperatura (K) Sólido: Grueneisen chasquido: SESAME 3716 FIG. 2: Shock Hugoniot para Al en el espacio a presión-temperatura, para las diferentes representaciones de la ecuación de estado. 0,7 0,75 0,80,85 0,90,95 1 compresión de volumen cada par de líneas: la parte superior es Hugoniot, inferior es adiabat FIG. 3: Principal adiabat y choque Hugoniot para Estar en el espacio normal de la compresión del estrés, descuidando resistencia (dashed), para la resistencia Steinberg-Guinan (sólida), y para el plástico elástico-perfectamente con Y = 10GPa (punto). 0 20 40 60 80 100 120 140 estrés normal (GPa) onda elástica choque de plástico FIG. 4: principal adiabat y choque Hugoniot para Estar en choque velocidad-espacio de estrés normal, descuidando resistencia (dashed), para la resistencia Steinberg-Guinan (sólida), y para el plástico elástico-perfectamente con Y = 10GPa (punto). 0 1000 2000 3000 4000 5000 temperatura (K) principal adiabat principal Hugoniot liberación adiabat FIG. 5: principal adiabat, choque Hugoniot, y lanzamiento de adiabat para Be en la temperatura de estrés normal espacio, despreocupando la fuerza (dashed), para Steinberg-Guinan fuerza (sólida), y para el elástico-perfectamente plástico con Y = 10GPa (punto). 0 1000 2000 3000 4000 5000 temperatura (K) locus de fusión Hugoniot sólido FIG. 6: Demostración de la solución de choque Hugoniot a través de un límite de fase: la fusión de choque de Al, para diferentes porosidades iniciales. estado inicial velocidad de las partículas estado inicial de proyectil director Hugoniot del objetivo principal Hugoniot de proyectil Estado de shock: intersección del objetivo FIG. 7: Interacciones de onda para el impacto de un proyectil plano que se mueve de izquierda a derecha con una objetivo estacionario. Las flechas estrujadas son una guía de la secuencia de estados. Para un proyectil en movimiento de derecha a izquierda, la construcción es la imagen del espejo reflejada en el eje de tensión normal. estados velocidad de las partículas secundaria Hugoniot del objetivo estado de choque inicial en el objetivo Director Hugoniot: ventana de alta impedancia baja impedancia ventana isentrope de liberación objetivo liberación objetivo en superficie libre ventana liberación FIG. 8: Interacciones de onda para la liberación de un estado de choque (choque que se mueve de izquierda a derecha) en un material estacionario de «ventana» a su derecha. El estado de liberación depende de si la ventana tiene una impedancia de choque mayor o menor que el material conmocionado. Las flechas estrujadas son una guía para el secuencia de estados. Para un choque que se mueve de derecha a izquierda, la construcción es la imagen del espejo reflejado en el eje de tensión normal. Liberación de proyectil en proyectil y objetivo Estado de tracción final en proyectil y objetivo velocidad de las partículas liberación de objetivo liberación de objetivo Liberación de proyectil estado de choque inicial FIG. 9: Interacciones de ondas para la liberación de un estado escandalizado por la tensión inducida por los materiales para separarse en direcciones opuestas cuando se une una interfaz enlazada. Daños materiales, escalofríos y la separación se descuidan: la construcción muestra el esfuerzo de tracción máximo posible. Por cuestiones generales propiedades del material, por ejemplo: si se incluye el flujo de plástico, el estado de tensión máxima no es sólo el negativo del estado de choque inicial. Las flechas estrujadas son una guía de la secuencia de estados. Los el gráfico muestra el estado inicial después de un impacto por un proyectil que se mueve de derecha a izquierda; para un choque moviéndose de derecha a izquierda, la construcción es la imagen del espejo reflejada en el eje de tensión normal. tensión - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. - No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. objetivo choques de impacto choque transmitido; onda reflejada superficie libre liberación interacciones de liberación: FIG. 10: Esquema de las interacciones de ondas uniaxiales inducidas por el impacto de un proyectil plano con una objetivo compuesto. 0 5 10 15 20 posición (mm) LiFAl Mo reflejados transmitidos liberación shock original estado de shock FIG. 11: Simulación de hidrocódigo del proyectil Al a 3,6 km/s impactando un objetivo Mo con un LiF ventana de liberación, 1,1μs después del impacto. Las estructuras en las ondas son precursores elásticos. Lista de cifras 1. Principal isentrope y choque Hugoniot para el aire (gas perfecto): cálculos numéricos para modelos de material general, en comparación con soluciones analíticas. 2. Shock Hugoniot para Al en el espacio a presión-temperatura, para diferentes representaciones de la ecuación del estado. 3. Principal adiabat y choque Hugoniot para estar en el espacio normal de la compresión del estrés, Abandonar la fuerza (dashed), para la resistencia Steinberg-Guinan (sólida), y para el elástico- perfectamente plástico con Y = 10GPa (punto). 4. Principal adiabat y choque Hugoniot para estar en choque velocidad-espacio de estrés normal, Abandonar la fuerza (dashed), para la resistencia Steinberg-Guinan (sólida), y para el elástico- perfectamente plástico con Y = 10GPa (punto). 5. Principal adiabat, choque Hugoniot, y liberar a adiabat para estar en el estrés normal- espacio de temperatura, resistencia despreocupante (dashed), para la fuerza Steinberg-Guinan (sólido), y para plástico elástico-perfectamente con Y = 10GPa (punto). 6. Demostración de la solución de choque Hugoniot a través de un límite de fase: Al, para diferentes porosidades iniciales. 7. Interacciones de onda para el impacto de un proyectil plano que se mueve de izquierda a derecha con una objetivo estacionario. Las flechas estrujadas son una guía de la secuencia de estados. Para un proyectil moviéndose de derecha a izquierda, la construcción es la imagen del espejo reflejada en el normal eje de esfuerzo. 8. Interacciones de onda para la liberación de un estado de choque (choque que se mueve de izquierda a derecha) en un material estacionario de «ventana» a su derecha. El estado de liberación depende de si la ventana tiene una impedancia de choque mayor o menor que el material conmocionado. Dashed Las flechas son una guía para la secuencia de estados. Para un choque que se mueve de derecha a izquierda, la construcción es la imagen del espejo reflejada en el eje de tensión normal. 9. Interacciones de ondas para la liberación de un estado de choque por la tensión inducida como materiales tratar de separarse en direcciones opuestas cuando se une con una interfaz enlazada. Material se descuidan los daños, la caída y la separación: la construcción muestra el máximo Es posible el estrés por tracción. En el caso de las propiedades materiales generales, por ejemplo: si se incluye el flujo de plástico, el estado de tensión de tracción máxima no es sólo el negativo del estado de choque inicial. Las flechas estrujadas son una guía de la secuencia de estados. El gráfico muestra el estado inicial después de un impacto por un proyectil que se mueve de derecha a izquierda; para un choque que se mueve de de derecha a izquierda, la construcción es la imagen del espejo reflejada en el eje de tensión normal. 10. Esquema de interacciones de ondas uniaxiales inducidas por el impacto de un proyectil plano con un objetivo compuesto. 11. Simulación de hidrocódigo del proyectil Al a 3,6 km/s impactando un objetivo Mo con un LiF ventana de liberación, 1,1μs después del impacto. Las estructuras en las ondas son precursores elásticos. Introducción Estructura conceptual para propiedades del material Carga unidimensional idealizada Compresión de rampas Compresión por choque Precisión: aplicación al aire Comportamiento complejo de la materia condensada Temperatura Ecuaciones de densidad-temperatura del estado Modelo de temperatura para ecuaciones mecánicas de estado Dosis Representación preferida de la fuerza isotrópica Berilio Cambios de fase Vías de carga compuestas Conclusiones Agradecimientos Bibliografía Bibliografía Lista de cifras
704.001
Partial cubes: structures, characterizations, and constructions
" Partial cubes are isometric subgraphs of hypercubes. Structures on a graph\ndefined by means of s (...TRUNCATED)
"Partial cubes: structures, characterizations, and\nconstructions\nSergei Ovchinnikov\nMathematics D (...TRUNCATED)
"Introduction\nA hypercube H(X) on a set X is a graph which vertices are the finite subsets\nof X ; (...TRUNCATED)
"Cubos parciales: estructuras, caracterizaciones, y\nconstrucciones\nSergei Ovchinnikov\nDepartament (...TRUNCATED)
704.001
"Computing genus 2 Hilbert-Siegel modular forms over $\\Q(\\sqrt{5})$ via\n the Jacquet-Langlands c (...TRUNCATED)
" In this paper we present an algorithm for computing Hecke eigensystems of\nHilbert-Siegel cusp fo (...TRUNCATED)
"COMPUTING GENUS 2 HILBERT-SIEGEL MODULAR\nFORMS OVER Q(\n5) VIA THE JACQUET-LANGLANDS\nCORRESPONDEN (...TRUNCATED)
"Introduction\nLet F be a real quadratic field of narrow class number one and let B be the\nunique ( (...TRUNCATED)
"COMPUTANDO GENUS 2 HILBERT-SIEGEL MODULAR\nFORMULARIOS SOBRE Q(\n5) VIA LAS JACQUETAS-LANGLANDAS\nC (...TRUNCATED)
704.001
"Distribution of integral Fourier Coefficients of a Modular Form of Half\n Integral Weight Modulo P (...TRUNCATED)
" Recently, Bruinier and Ono classified cusp forms $f(z) := \\sum_{n=0}^{\\infty}\na_f(n)q ^n \\in (...TRUNCATED)
"DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS OF A\nMODULAR FORM OF HALF INTEGRAL WEIGHT MODULO\nPR (...TRUNCATED)
"Introduction and Results\nLet Mλ+ 1\n(Γ0(N), χ) and Sλ+ 1\n(Γ0(N), χ) be the spaces, respecti (...TRUNCATED)
"DISTRIBUCIÓN DE CUARTOS COEFICIENTES INTEGRALES DE UNA\nFORMA MODULAR DE MÓDULO DE PESO INTEGRAL\ (...TRUNCATED)
704.001
"$p$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral\n Weight" (...TRUNCATED)
" Serre obtained the p-adic limit of the integral Fourier coefficient of\nmodular forms on $SL_2(\\ (...TRUNCATED)
"p-ADIC LIMIT OF THE FOURIER COEFFICIENTS OF WEAKLY\nHOLOMORPHIC MODULAR FORMS OF HALF INTEGRAL WEIG (...TRUNCATED)
"Introduction and Statement of Main Results\nSerre obtained the p-adic limits of the integral Fourie (...TRUNCATED)
"LÍMITE PÁDICO DE LOS CUARTOS COEFICIENCIAS DE LA DEBILIDAD\nFORMAS MODULARES HOLOMÓRFICAS DE PES (...TRUNCATED)

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