bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the Integral Extensions of Quadratic Forms Over Local Fields
Authors: Band Melvin 
Published: December 2010

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.
Eigenfunctions of Operator-Valued Analytic Functions
Authors: Sherman Malcolm J. 
Published: December 2010

This paper is a sequel to [2], whose primary purposes are to clarify and generalize the concept introduced there of an eigenfunction of an inner function, and to answer questions raised there concerning the equivalence of several possible forms of the definition. A new definition, proposed here, leads to a complete characterization of the eigenfunctions of Potapov inner functions of normal operators, and the result is more satisfactory than [2, Theorem 3.4], although the latter is used strongly in the proof.Let be an inner function in the sense of Lax; i.e., is almost everywhere (a.e.) a unitary operator on a separable Hilbert space and belongs weakly to the Hardy class H2. An analytic function q (which will have to be a scalar inner function) was defined to be an eigenfunction of if the set of z in the disk {z: |z| ≦ 1} for which is invertible is a set of linear measure 0 on the circle {z: |z| = 1}.
Existence Theorems for Some Non-Linear Equations of Evolution
Authors: Clements John C. 
Published: December 2010

In recent years considerable attention has been focused on non-linear hyperbolic differential equations with the object of establishing the existence of global solutions. It is our aim here to establish the existence of weak solutions of boundary value problems for non-linear equations of the form(1-1)where d is a real constant called the damping coefficient, u(t) is a vector-valued function defined on a subinterval of the real line into a space of complex-valued functions u(x) defined on a bounded domain Ω in the real Euclidean space EN of N dimensions, ut(t) ≡ du(t)/dt, and A(t) is the family of partial differential operators of order 2m (m = 1, 2, …) on Ω given in generalized divergence form by(1-2)with
Linear Transformations on Matrices: the Invariance of the Third Elementary Symmetric Function
Authors: Beasley Leroy B. 
Published: December 2010

Let T be a linear transformation on Mn the set of all n × n matrices over the field of complex numbers, . Let A ∈ Mn have eigenvalues λ1, …, λn and let Er(A) denote the rth elementary symmetric function of the eigenvalues of A :Equivalently, Er(A) is the sum of all the principal r × r subdeterminants of A. T is said to preserve Er if Er[T(A)] = Er(A) for all A ∈ Mn. Marcus and Purves [3, Theorem 3.1] showed that for r ≧ 4, if T preserves Er then T is essentially a similarity transformation; that is, either T: A → UAV for all A ∈ Mn or T: A → UAtV for all A ∈ Mn, where UV = eiθIn, rθ ≡ 0 (mod 2π).
Ordered Groups Satisfying the Maximal Condition Locally
Authors: Hursey R. J. , Rhemtulla A. H. 
Published: December 2010

Let denote the class of all (fully) ordered groups satisfying the maximal condition on subgroups, and let denote the class of all locally groups. In this paper we investigate the family of convex subgroups of groups.It is well known (see [1, pp. 51, 54]) that every convex subgroup of an is normal in G, and for any jump D –< C in the family of convex subgroups, [G′, C] ⊆ D. We observe that these properties are also true for any group and record, without proof, the following.THEOREM 1. Any convex subgroup of angroup G is normal in G, and for any jump D –< C in the family of convex subgroups, [G′, C] ⊆ D.
Group Representations and Cardinal Algebras
Authors: Deliyannis Platon C. 
Published: December 2010

Our aim in this work is to show that the global theory of group representations (as presented in [4] for example) is actually and naturally a part of the theory of cardinal algebras. In this sense it is analogous to the work of Kaplansky [2] and Loomis [3] on operator rings. In his paper Loomis proposed an abstract scheme for representation theory; it appears, however, that the idea of abstracting the equivalence class of a representation is more suitable. It is the set of all these classes that forms the cardinal algebra which we study.The connection between cardinal algebras and operator theory has been made explicit by Fillmore [1], where he worked out a dimension theory for a class of cardinal algebras satisfying certain conditions. In the next section we shall discuss the relation between the two systems; we could say that the main difference lies in our introducing such axioms as to make possible the study of type III cases.
On the Rank Numbers of an Arc
Authors: Turgeon J. 
Published: December 2010

The kth rank number, rankkB, of a differentiable arc B in real projective n-space is the least upper bound of the number of osculating k-spaces of B which meet an (n – k – l)-flat, k = 0, 1, …, n – 1. The number rank0B is called the order of B; cf. 1.1-1.3. It has been conjectured by Peter Scherk that(0.1)equality holding if and only if B has the order n; cf. [2, p. 396]. In this paper we prove the following results.THEOREM 1. If B is a differentiable elementary arc, then (0.1) holds for k = 0, 1, …, n – 1.THEOREM 2. If B is a differentiable elementary arc and order B > n, then rankkB > (k + 1) (n – k) for k = 1, …, n – 2.By a theorem of Park [3, p. 38], every differentiable arc contains a subarc of order n. This eliminates the assumption that B is elementary from Theorem 1. We do not know whether it can be dropped from Theorem 2.
The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution
Authors: Gauthier Paul 
Published: December 2010

Let f(z) be a function meromorphic in the unit disc D = (|z| < 1). We consider the maximum modulusand the minimum modulusWhen no confusion is likely, we shall write M(r) and m(r) in place of M(r,f) and m(r,f).Since every normal holomorphic function belongs to an invariant normal family, a theorem of Hayman [6, Theorem 6.8] yields the following result.THEOREM 1. If f(z) is a normal holomorphic function in the unit disc D, then(1)This means that for normal holomorphic functions, M(r) cannot grow too rapidly. The main result of this paper (Theorem 5, also due to Hayman, but unpublished) is that a similar situation holds for normal meromorphic functions.
Simple Quotients of Euclidean Lie Algebras
Authors: Moody Robert V. 
Published: December 2010

In [2], we considered a class of Lie algebras generalizing the classical simple Lie algebras. Using a field Φ of characteristic zero and a square matrix (Aij) of integers with the properties (1) Aii = 2, (2) Aij ≦ 0 if i ≠ j, (3) Aij = 0 if and only if Ajt = 0, and (4) is symmetric for some appropriate non-zero rational a Lie algebra E = E((Aij)) over Φ can be constructed, together with the usual accoutrements: a root system, invariant bilinear form, and Weyl group.For indecomposable (A ij), E is simple except when (Aij) is singular and removal of any row and corresponding column of (Aij) leaves a Cartan matrix. The non-simple Es, Euclidean Lie algebras, were our object of study in [3] as well as in the present paper. They are infinite-dimensional, have ascending chain condition on ideals, and proper ideals are of finite codimension.
On the Classification of Lie Pseudo-Algebras
Authors: van Quê Ngö 
Published: December 2010

For every ( differentiable) bundle E over a manifold M, Jk(E) denotes the set of all k-jets of local (differentiable) sections of the bundle E. Jk(E) is a bundle over M such that if X is a section of E, thenis a (differentiable) section of Jk(E). If E is a vector bundle, Jk(E) is a vector bundle and we have the canonical exact sequence of vector bundleswhere Sk(T*) is the symmetric Whitney tensor product of the cotangent vector bundle T* of M. and π is the canonical morphism which associates to each k-jet of section its jet of inferior order.
Minimal Relations for Certain Wreath Products of Groups
Authors: Johnson D. L. 
Published: December 2010

Let p be a rational prime, G a non-trivial finite p group, and K the field of p elements, regarded as a trivial G-module according to context; then we define:d(G) = dimKH1(G, K), the minimal number of generators of G,r(G) = dimKH2(G, K),r′(G) = the minimal number of relations required to define G,where, in the last equation, it is sufficient to take the minimum over those presentations of G with d(G) generators. It is well known (see § 2) that the following inequalities hold:We shall consider only finite p-groups, so that the class of groups with r = d coincides with that consisting of those groups whose Schur multiplicator is trivial.
k-Degenerate Graphs
Authors: Lick Don R. , White Arthur T. 
Published: December 2010

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graphFor the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.
Algebraic Extensions of Commutative Regular
Rings
Authors: Raphael R. M. 
Published: December 2010

In this paper we study algebraic closures for commutative semiprime rings.
The main interest, however, is with rings which are regular in the sense of
von Neumann. These play the same role with respect to semiprime rings as
fields do with respect to integral domains. Two generally distinct notions
are defined: “algebraic” and “weak-algebraic” extensions. Each has the
transitivity property and yields a closure which is unique up to isomorphism
and is “universal”. Both coincide in fields.The extensions here called “algebraic” were studied independently by Enochs
[5] and myself. Our results on these extensions proceed from
a different point of view, and allow us to answer a question posed by
Enochs. Furthermore, these results are required (and were developed) in
order to obtain the weak-algebraic closure, which was the original closure
sought. The motivation for the weak-algebraic extensions is found in the
work of Shoda [14, p. 134, no. 1].
On Quasi-Essential Subgroups of Primary Abelian
Groups
Authors: Benabdallah Khalid , Irwin John M. 
Published: December 2010

All groups considered in this paper are abelian. A subgroup
N of a group G is said to be a
quasi-essential subgroup of G if G =
〈H, K〉 whenever H is an
N-high subgroup of G and
K is a pure subgroup of G containing
N. We started the study of such subgroups in
[5]; in particular, we characterized subsocles of a primary
group which were both quasi-essential and centres of purity. In this paper
we show that quasi-essential subsocles of a primary group are necessarily
centres of purity answering thus in the affirmative a question raised in
[5].We obtain the following theorem: A subsocle S of a p-group G is
quasi-essential if and only if either S ⊂ G1or (pnG)[p] ⊃ S ⊃
(pn+1G)[p] for some non-negative integer n. The
notation is that of [1]. If G is a group,
thenwhere p is a prime integer.
A Difference-Differential Basis Theorem
Authors: Cohn Richard M. 
Published: December 2010

Our aim in this paper is to extend to difference-differential rings the
beautiful theorem of Kolchin [5, Theorem 3] for the
differential case. The necessity portion of Kolchin's result is not
obtained.What might well be called the Ritt basis theorem states that if a
commutative ring R with identity is finitely generated over
a subring R0, then the ascending chain condition for radical ideals of
R0 implies the ascending chain condition for radical ideals of
R. (This is indeed a basis theorem. If we define a basis
for a radical ideal A to be a finite set B
such that then every
radical ideal of a ring R has a basis if and only if the
ascending chain condition for radical ideals holds in R.)
It is the Ritt basis theorem rather than the Hilbert basis theorem which has
appropriate generalizations in differential and difference algebra, where in
fact it originated.
Higher Monotonicity Properties of Certain Sturm-Liouville
Functions. III
Authors: Lorch Lee , Muldoon M. E. , Szego Peter 
Published: December 2010

A Sturm-Liouville function is simply a non-trivial solution of the
Sturm-Liouville differential equation(1.1)considered, together with everything else in this study, in the real domain.
The associated quantities whose higher monotonicity properties are
determined here are defined, for fixed λ > –1, to be(1.2)where y(x) is an arbitrary (non-trivial) solution of (1.1)
and x1, x2, … is any finite or infinite sequence of consecutive zeros of
any non-trivial solution z(x) of (1.1) which may or may not
be linearly independent of y(x). The condition
λ > –1 is required to assure convergence of the
integral defining Mk, and the function W(x) is taken subject to the
same restriction.
Vertex Subgroups of Irreducible Representations of Solvable
Groups
Authors: Malzan J. 
Published: December 2010

If ρ(G) is a finite, real, orthogonal
group of matrices acting on the real vector space V, then
there is defined [5], by the action of
ρ(G), a convex subset of the unit
sphere in V called a fundamental region.
When the unit sphere is covered by the images under
ρ(G) of a fundamental region, we obtain
a semi-regular figure.The group-theoretical problem in this kind of geometry is to find when the
fundamental region is unique. In this paper we examine the subgroups,
ρ(H), of
ρ(G) with a view of finding what
subspace, W of V consists of vectors held
fixed by all the matrices of ρ(H). Any
such subspace lies between two copies of a fundamental region and so
contributes to a boundary of both. If enough of these boundaries might be
found, the fundamental region would be completely described.
Pentagon-Generated Trivalent Graphs with Girth 5
Authors: Robertson Neil 
Published: December 2010

The terminology of [1] will be assumed in what follows. LetPb(G) stand for the set of pentagons in the graphG.Call a graphpentagongeneratedwhen it is the union of its contained pentagons. LetP5,3be the class of connected trivalent pentagon-generated graphs with girth 5. These graphs form a family including the Petersen graph and the graph of the dodecahedron. They are studied here and completely classified in terms of a decomposition which all but some specifically determined indecomposable graphs admit.Assume henceforth thatH∈P5,3. LetEk(H) be the set of edges in exactlyk∈ 0 pentagons ofH. ClearlyEk(H) = 0 ifk≠ 1, 2, 3, 4 and |E1(H) ∩E(P)r ≦ 2, for allP∈P5(H).P∈P5(H) issingularwhen |E1(H) ∩E(P)r = 2,.
Commutative Endomorphism Rings
Authors: Zelmanowitz J. 
Published: December 2010

The problem of classifying the torsion-free abelian groups with commutative
endomorphism rings appears as Fuchs’ problems in [4, Problems
46 and 47]. They are far from solved, and the obstacles to a solution appear
formidable (see [4; 5]). It is, however, easy to see that the
only dualizable abelian group with a commutative endomorphism ring is the
infinite cyclic group. (An R-module Miscalled dualizable if HomR(M, R) ≠ 0.) Motivated by this, we study the class
of prime rings R which possess a dualizable module
M with a commutative endomorphism ring. A
characterization of such rings is obtained in § 6, which as would be
expected, places stringent restrictions on the ring and the module.Throughout we will write homomorphisms of modules on the side opposite to
the scalar action. Rings will not be assumed to contain identity elements
unless otherwise indicated.
Finite-to-One Open Mappings
Authors: Duda Edwin , Haynsworth W. Hugh 
Published: December 2010

The class of finite-to-one open mappings on manifolds contains some
important subclasses. Any non-constant analytic function from a bounded
region in its domain of definition is finite-to-one. Church [2]
showed that any light strongly open Cn map f: Rn → Rn is discrete. A number of papers concerning discrete open mappings on
manifolds have been published; see [1-6; 8-9; 11-14].A result of Černavskiĭ [1] (see also [13]) shows
that for any discrete strongly open mapping f :
Mn → Nn of an n-manifold into an
n-manifold, the branch set of f has
dimension less than n – 1. If f is also a
closed map, then N(f) is finite and the set of points
x for which N(x, f) =
N(f) is an open dense connected subset of
Mn.
Transitive Orientation of Graphs and Identification of
Permutation Graphs
Authors: Pnueli A. , Lempel A. , Even S. 
Published: December 2010

The graphs considered in this paper are assumed to be finite, with no edge
joining a vertex to itself and with no two distinct edges joining the same
pair of vertices. An undirected graph will be denoted by G
or (V, E), where V is the set of vertices
and E is the set of edges. An edge joining the vertices
i,j ∊ V will be denoted by the
unordered pair (i,j).An orientation of G = (V,
E) is an assignment of a unique direction i →
j or j → i to every
edge (i,j) ∊ E. The resulting directed
image of G will be denoted by G→ or (V, E→), where
E→ is now a set of ordered pairs E→ =
{[i,j]| (i,j) ∊ E and
i → j}. Notice the difference in
notation (brackets versus parentheses) for ordered and unordered pairs.
Approximation by Unimodular Functions
Authors: Fisher Stephen 
Published: December 2010

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.
On the Point-Arboricity of a Graph and its Complement
Authors: Mitchem John 
Published: December 2010

The point-arboricity ρ (G) of a graph G is defined as the minimum number of subsets into which the point set V(G) of G may be partitioned so that each subset induces an acyclic subgraph. Equivalently, the point-arboricity of G may be defined as the least number of colours needed to colour the points of G so that no cycle of G has all of its points coloured the same. This term was introduced by Chartrand, Geller, and Hedetniemi [1], although the concept was first considered by Motzkin [4].As with the chromatic number of a graph G, which we denote by χ(G), there is no explicit formula for the point-arboricity of a graph. However, Nordhaus and Gaddum [5] have shown that if G is a graph with p points, then
On Certain Functional Identities in EN
Authors: Mercer A. McD. 
Published: December 2010

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity1.1in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to1.2In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of1.3to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.
Actions of a Locally Compact Group with Zero
Authors: Hanson T. H. McH. 
Published: December 2010

In [2] we find the definition of a locally compact group with zero as a locally compact Hausdorff topological semigroup, S, which contains a non-isolated point, 0, such that G = S – {0} is a group. Hofmann shows in [2] that 0 is indeed a zero for S, G is a locally compact topological group, and the unit, 1, of G is the unit of S. We are to study actions of S and G on spaces, and the reader is referred to [4] for the terminology of actions.If X is a space (all are assumed Hausdorff) and A ⊂ X, A* denotes the closure of A. If {xρ} is a net in X, we say limρxρ = ∞ in X if {xρ} has no subnet which converges in X.
Some Imbedding Theorems for Sobolev Spaces
Authors: Adams R. A. , Fournier John 
Published: December 2010

We shall be concerned throughout this paper with the Sobolev space Wm,p(G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En. For each positive integer m and each real p ≧ 1 the space Wm,p(G) consists of all u in LP(G) whose distributional partial derivatives of all orders up to and including m are also in LP(G). With respect to the norm1.1Wm,p(G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm(G) which, together with their partial derivatives of orders up to and including m, are in LP(G) forms a dense subspace of Wm,p(G).
On the Generality of the AP-Integral
Authors: Cross G. E. 
Published: December 2010

In 1955 Taylor [6] constructed an AP-integral sufficiently strong to integrate Abel summable series with coefficients o(n). He showed that the AP-integral includes the special Denjoy integral and further that, when applied to trigonometric series, the AP-integral is more powerful than the SCP-integral of Burkill [1] and the P2-integral of James [3]. The present paper shows that the AP-integral includes the SCP-integral, and, under natural assumptions, the P2-integral.After completing this manuscript I was advised by Skvorcov that he had shown [5] under more general conditions that the P2-integral is included in the AP-integral. The proof in the present paper seems to have some value in its own right and is considerably shorter.Since the definition of the AP-integral is essentially for a function defined in (0, 2π] and elsewhere by 2π-periodicity, we shall consider SCP-integrable and P2-integrable functions defined similarly.
On Stable Diffeomorphism of Exotic Spheres in the Metastable Range
Authors: Antonelli P. L. 
Published: December 2010

Let ⊖np+ 1denote the subgroup of the Kervaire-Milnor groupθnconsisting of thosen-spheres which imbed with trivial normal bundle in Euclidean(n + p+ 1)-space,n<2p.It is known that such imbeddings always exist [6], and that the normal bundle is independent of the imbedding[10].Following [2], we write Ωn,pfor the quotientθn/⊖np+ 1.The order of Ωn,pafter identifying each element with its inverse, is equal to the number of diffeomorphically distinct (orientation preserved) Σn×Sp[2; 5]. Indeed,Ωn,pis closely linked to the problem of determining the number of smooth structuresα(n, p)onSn×Sp.For instance, ifΩn,p= 0 thenα(n, p)equals the order ofθn+p[5]. Specific results are easily read off Table I and Theorem 2.1.