pinecone/core-2020-05-10-deduplication huggingface dataset:
Partitions of Points into Simplices withk-dimensional Intersection. Part II: Proof of Reay’s Conjecture in Dimensions 4 and 5

AbstractA longstanding conjecture of Reay asserts that every set X of (m− 1)(d+ 1) +k+ 1 points in general position in Rd has a partition X1, X2,⋯ , Xmsuch that∩i=1mconvXiis at least k -dimensional. Using the tools developed in and oriented matroid theory, we prove this conjecture for d= 4 and d= 5. How about, to that end, we introduce the notion of a k -lopsided oriented matroid and we characterize these combinatorial objects for certain values of k. Divisibility properties for subsets of Rd with other independence conditions are also obtained, thus settling several particular cases of a generalization of Reay’s conjecture
Realizable but not Strongly Euclidean Oriented Matroids†

AbstractThe extension space conjecture of oriented matroid theory claims that the space of all (non-zero, non-trivial, single-element) extensions of a realizable oriented matroid of rank r is homotopy equivalent to an (r− 1)-sphere. In 1993, Sturmfels and Ziegler proved the conjecture for the class of strongly Euclidean oriented matroids, which includes those of rank at most 3 or corank at most 2. They did not provide any example of a realizable but not strongly Euclidean oriented matroid. Here we produce two such examples for the first time, one with rank 4 and one with corank 3. Both have 12 elements
The Terwilliger Algebra of the Hypercube

AbstractWe give an introduction to the Terwilliger algebra of a distance-regular graph, focusing on the hypercube QDof dimension D. Let X denote the vertex set ofQD . Fix a vertex x∈X, and letT=T(x) denote the associated Terwilliger algebra. We show thatT is the subalgebra of MatX(C) generated by the adjacency matrixA and a diagonal matrix A*=A* (x), where A*has yy entryD− 2 ∂(x, y) for all y∈X , and where ∂ denotes the path-length distance function. We show that A andA* satisfy A2A*− 2AA*A+A*A2&=& 4A* , A*2A− 2A*AA*+AA*2&=& 4 A. Using the above equations, we find the irreducible T -modules. For each irreducible T -module W, we display two orthogonal bases, which we call the standard basis and the dual standard basis. We describe the action of A andA* on each of these bases. We give the transition matrix from the standard basis to the dual standard basis for W. We compute the multiplicity with which each irreducible T -module W appears inCX . We give an elementary proof that QDhas the Q -polynomial property. We show that T is a homomorphic image of the universal enveloping algebra of the Lie algebrasl2 (C). We obtain an element φ of T that generates the center ofT . We obtain the central primitive idempotents of T as polynomials in φ
On Traces ofd-stresses in the Skeletons of Lower Dimensions of Piecewise-lineard-manifolds

AbstractWe show how a d -stress on a piecewise-linear realization of an oriented (non-simplicial, in general)d -manifold in Rdnaturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of self-stresses in spatial frameworks. The mappings we construct are not linear, but polynomial. In the 1860–70s J. C. Maxwell described an interesting relationship between self-stresses in planar frameworks and vertical projections of polyhedral 2-surfaces. We offer a spatial analog of Maxwell’s correspondence based on our polynomial mappings. By applying our main result we derive a class of three-dimensional spider webs similar to the two-dimensional spider webs described by Maxwell. We also conjecture an important property of our mappings that is based on the lower bound theorem (g2(d+ 1) =dim Stress2≥ 0) for d -pseudomanifolds generically realized in Rd+1
Cutting a Bunch of Grapes by a Plane

AbstractLet Dndenote a family of disjoint n disks in the plane. The max–min ratio λ of Dnis the ratio (the maximum radius)/(the minimum radius) among the disks in Dn. We prove that (1) If logλ=o(n), then there is a line both sides of which contain n/2 −o(n) intact disks (such a line is called an almost-halving line), (2) for any constant c> 0, there is a family of disjoint n disks with logλ=cn that has no almost-halving line. The max–min ratio λ of a family Bnof disjoint n balls in R3is defined similarly. We also prove that (3) for any n≥ 3, there is a family of disjoint n balls in R3such that every plane H in R3has a side that contains at most 2 intact balls of the family, and (4) if logλ=o((n/logn)1/3) for a family Bnof n disjoint balls in R3, then there is a plane both sides of which contain n/2 −o(n) intact balls of Bn
Cocircuit Graphs and Efficient Orientation Reconstruction in Oriented Matroids

AbstractWe consider the cocircuit graph GMof an oriented matroid M, which is the 1-skeleton of the cell complex formed by the span of the cocircuits of M. As a result of Cordovil, Fukuda, and Guedes de Oliveira, the isomorphism class of M is not determined by GM, but it is determined if M is uniform and the vertices in GMare paired if they are associated to negative cocircuits; furthermore the reorientation class of an oriented matroid M with rank(M) ≥ 2 is determined by GMif every vertex in GMis labeled by the zero support of the associated cocircuit. In this paper we show that the isomorphism class of a uniform oriented matroid is determined by the cocircuit graph, and we present polynomial algorithms which provide constructive proofs to all these results. Furthermore it is shown that the correctness of the input of the algorithms can be verified in polynomial time
Orlik–Solomon Type Algebras

AbstractWe introduce χ -algebras, and show that a χ -algebra has the NBC basis property. We also show that a certain ideal used in the construction has the so-called BC basis property. The Orlik–Solomon algebra of a matroid, the Orlik–Terao algebra of a set of vectors, and the Cordovil algebra of an oriented matroid areχ -algebras. We define a new χ -algebra from a set of vectors, close to the Orlik–Terao, Orlik–Terao, and Cordovil algebras, but nevertheless different. Our proof provides a unified short and elementary proof of the NBC basis property for these algebras
Active Orders for Matroid Bases

AbstractWe introduce three orderings of the basis set of an ordered matroid, defined in terms of basis activities. We show that the active orders have lattice structures. This property contains both the neighboring property of the lexicographic ordering of bases, and the maximality property of the product ordering, a form of the Greedy Algorithm Theorem. The active orders are closely related to the Tutte polynomial and the Orlik–Solomon algebra of a matroid. In particular, we show that, in a simple ordered matroid, a basis is the join in the external lattice of its components in the NBC basis of the Orlik–Solomon algebra of the matroid
Enumeration of Hypergraphs

AbstractIn this paper, we will study enumeration of hypergraphs. Let Spbe a symmetric group acting on ap-setX. It induces the permutation group Sp*acting on the set of all subsets ofX. Our problem is reduced to finding a good formula for the cycle index ofSp* . A crucial point is to calculate the cycle inventory of the permutation induced by a single cycle. Their formulas are given inductively. We will give the cycle index ofSp* explicitly and concretely. We will find a relation between the cycle inventory of the induced permutation and the number of convexk-gons of a given regularp-gon and obtain the formula for its cycle inventory
A Result on Spreads of the Generalized QuadrangleT2(O), withOan Oval Arising from a Flock, and Applications

AbstractIf F is a flock of the quadratic cone K of PG (3, q), q even, then the corresponding generalized quadrangle S(F) of order (q2,q ) has subquadrangles T2(O), with O an oval, of order q. We prove in a geometrical way that any such T2(O) has spreads S consisting of an element y∈O and the q2lines not in the plane PG (2,q ) of O of q quadratic cones Kx,x∈O− { y }, of the space PG (3, q) containing T2(O), where Kxhas vertex x, is tangent to PG (2, q) at xy and has nucleus line xn, with n the nucleus of O. We also show how the oval O can be directly constructed from the flock F
New Distance Regular Graphs Arising from Dimensional Dual Hyperovals

AbstractIn 4 we have studied the semibiplanes Σm,he=Af(Sm,he) obtained as affine expansions of the d -dimensional dual hyperovals of Yoshiara 6. We continue that investigation here, but from a graph theoretic point of view. Denoting byΓm, he the incidence graph of (the point-block system of)Σm, he, we prove that Γm,heis distance regular if and only if eitherm+h=e or (m+h,e ) = 1. In the latter case, Γm,hehas the same array as the coset graph Kheof the extended binary Kasami code K(2e, 2h) but, as we prove in this paper, we have Γm, he ∼ =Kheif and only if m=h. Finally, by exploiting some information obtained on Γm, he, we prove that if e≤ 13 and m≠=h with (m+h, e) = 1, then Σm,heis simply connected
On a New Family of Flag-transitive Semibiplanes

AbstractEach of the d -dimensional dual hyperovalsSmh discovered by Yoshiara 20 gives rise, via affine expansion, to a flag-transitive semibiplane Af(Smh). We prove that, if m+h=d+ 1, thenAf (Smh) is an elation semibiplane. In the other cases, ifd> 2 then Af(Smh) is not isomorphic to any of the examples we are aware of, except possibly for certain semibiplanes obtained fromDn -buildings defined over GF(2). However, many semibiplanes live hidden as quotients inside halved hypercubes. It is thus quite natural to ask whether any of our semibiplanes are like that. We prove thatAf (Smh) is a quotient of a halved hypercube if and only ifh=m
The Pseudo-geometric Graphs for Generalized Quadrangles of Order (3, t)

AbstractThe values t= 1, 3, 5, 6, 9 satisfy the standard necessary conditions for existence of a generalized quadrangle of order (3, t). This gives the following possible parameter sets for strongly regular graphs that are pseudo-geometric for such a generalized quadrangle: (v, k,λ , μ) = (16, 6, 2, 2), (40, 12, 2, 4), (64, 18, 2, 6), (76, 21, 2, 7) and (112, 30, 2, 10). It is well known that there are two graphs with the first parameter set and that there is just one graph with the last set of parameters. Recently, the second author has shown that there are precisely 28 strongly regular graphs with the second parameter set. Non-existence of a strongly regular graph with the fourth set of parameters was proved by the first author. Here we complete the classification by announcing that there are exactly 167 non-isomorphic strongly regular graphs with parameters (64, 18, 2, 6)
On Permutation Groups of Finite Type

AbstractA permutation group G is said to be a group of finite type { k }, k a positive integer, if each non-identity element of G has exactly k fixed points. We show that a group G can be faithfully represented as an irredundant permutation group of finite type if and only if G has a non-trivial normal partition such that each component has finite bounded index in its normalizer. An asymptotic structure theorem for locally (soluble-by-finite) groups of finite type is proved. Finite sharp irredundant permutation groups of finite type, notp -groups, are determined
Non-rigidity Degree of a Lattice and Rigid Lattices

AbstractVoronoi defines a partition of the cone of positive semidefinite n -ary formsPn into L -type domains. Each L -type domain is an open polyhedral cone of dimensionk , 1 ≤k≤n(n+ 1)2, where n is the number of variables and dimension of the corresponding lattice. We define a non-rigidity degree of a lattice as the dimension of the L -type domain containing the lattice. We prove that the non-rigidity degree of a lattice equals the corank of a system of equalities connecting norms of minimal vectors of cosets of 2 L in L. A lattice of non-rigidity degree 1 is called rigid. A lattice is rigid if any of its sufficiently small deformations distinct from a homothety changes its L -type. Using the list of 84 zone-contracted Voronoi polytopes inR5given by Engel [8], we give a complete list of seven five-dimensional rigid lattices
Holey Segre Varieties

AbstractHoley Segre varieties are introduced, which generalize classical Segre varieties and whose existence is suggested by the fact that any non-prime finite field contains proper subfields. More precisely, a holey Segre variety is the tensor product PG(n, q) ⊗PG(m,qk ). An algebraic description of such an object is given and its structure is investigated. Furthermore, special choices of the parameters are considered, in which cases a holey Segre variety admits a partition into classical Segre varieties. Also, some holey Segre varieties are isomorphic to non-canonical subgeometries embedded in suitable Segre varieties
Generalized Pattern Avoidance

AbstractRecently, Babson and Steingrı&#x0301;msson have introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We will consider pattern avoidance for such patterns, and give a complete solution for the number of permutations avoiding any single pattern of length three with exactly one adjacent pair of letters. For eight of these 12 patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. We also give some results for the number of permutations avoiding two different patterns. These results relate the permutations in question to Motzkin paths, involutions and non-overlapping partitions. Furthermore, we define a new class of set partitions, called monotone partitions, and show that these partitions are in one-to-one correspondence with non-overlapping partitions
Base-sortable Matroids and Koszulness of Semigroup Rings

AbstractMotivated by a question in commutative algebra and inspired by the work of Sturmfels, we introduce the class of base-sortable matroids and show that it is closed under several matroid operations. All matroids of rank 2 are base-sortable and we give a characterization of base-sortability by excluded minors in the case of graphic matroids and rank 3 matroids. Transversal matroids with certain presentations are also base-sortable. For a base-sortable matroid M, the basis monomial ring RMis shown to be Koszul, by proving that the toric ideal of this ring has a quadratic Gröbner basis. Extending the concept of combinatorial pure subrings considered by Herzog, Hibi and Ohsugi we define the matroid operations deletion, contraction and duality for homogeneous semigroup rings with square-free monomial generators, and observe that Koszulness is preserved under these operations
Answers to Questions by Dénes on Latin Power Sets

AbstractThe i th power, Li, of a Latin square L is that matrix obtained by replacing each row permutation in L by its i th power. A Latin power set of cardinalitym≥ 2 is a set of Latin squares { A, A2, A3, . . . ,Am }. We prove some basic properties of Latin power sets and use them to resolve questions asked by Dénes and his various collaborators. Dénes has used Latin power sets in an attempt to settle a conjecture by Hall and Paige on complete mappings in groups. Dénes suggested three generalisations of the Hall–Paige conjecture. We refute all three with counterexamples. Elsewhere, Dénes et al. unsuccessfully tried to construct three mutually orthogonal Latin squares of order 10 based on a Latin power set. We confirm as a result of an exhaustive computer search that there is no Latin power set of the kind sought. However we do find a set of four mutually orthogonal 9 × 10 Latin rectangles. We also show the non-existence of a 2-fold perfect (10, 9, 1)-Mendelsohn design which was conjectured to exist by Dénes. Finally, we prove a conjecture originally due to Dénes and Keedwell and show that two others of Dénes and Owens are false
The Quotients ofG4,5,120

AbstractThe groups G4,5, mare significantly useful because they are symmetric groups of the map {4, 5}n, which is constructed from the tessellation {4, 5} of the hyperbolic plane. It is now known (Geom. Ded., 2, 48 (1993), 231–242) that PSL(2, q) is a factor group of the Coxeter groups Gk, l, mif −1 is a quadratic residue in the Galois field GF(q), where q is a prime power. By using a diagrammatic argument, propounded by Higman, we show that all but finitely many Anand Snare factor groups ofG4,5,120
A Remark on Non-Uniform Hyperplanes of Finite Thick Dual Polar Spaces

AbstractLet Δ be a finite thick dual polar space, and let H be a hyperplane ofΔ . Calling the elements of Δ of type 2 quads, we call a quadα≠⊂H singular (respectively subquadrangular or ovoidal) ifH meets α in the perp of a point (respectively in a full subquadrangle or in an ovoid). A hyperplane is said to be uniform if all quads not contained in it are either singular or subquadrangular or ovoidal. We prove that each non-uniform hyperplane H of a finite thick dual polar space meets some quad in the perp of a point
Some Properties of the Yoshiara Family of Planes

AbstractIn [4], Yoshiara constructed a family of q+ 3 planes inPG (5, q), q even, satisfying certain intersection properties. This is then used to construct extended generalized quadrangles of order (q+ 1, q− 1). In this note we examine the properties of these planes and show that the existence of a set of q+ 2 planes satisfying the intersection properties necessarily implies the existence of the (q+ 3)th plane, and that two such families meet in at most (2 q+ 6)/3 planes. We also pose some open completeness problems
The Cyclic Model forPG(n, q) and a Construction of Arcs

AbstractThe n -dimensional finite projective space, PG(n, q), admits a cyclic model, in which the set of points of PG(n, q) is identified with the elements of the group Zqn+qn−1+⋯+q+1. It was proved by Hall (1974, Math. Centre Tracts, 57, 1–26) that in the cyclic model of PG(2, q), the additive inverse of a line is a conic. The following generalization of this result is proved:In the cyclic model of PG(n, q), the additive inverse of a line is a (q+ 1)-arc if n+ 1 is a prime and q+ 1 >n.It is also shown that the additive inverse of a line is always a normal rational curve in some subspace PG(m, q), where m+ 1| n+ 1
MacMahon’s Partition Analysis: The Omega Package

AbstractIn his famous book ‘Combinatory Analysis’ MacMahon introduced Partition Analysis (‘Omega Calculus’) as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations. The object of this paper is to show that partition analysis is ideally suited for being implemented in computer algebra. To this end we have developed the computer algebra packageOmega. In addition to an introduction to basic facts of ‘Omega Calculus’, we present a number of applications that illustrate the usage of the package
Parker Vectors for Infinite Groups

AbstractWe take the first step towards establishing a theory of Parker vectors for infinite permutation groups, with an emphasis towards oligomorphic groups. We show that, on the one hand, many results for finite groups extend naturally to the infinite case (Parker’s Lemma, multiplicative properties, etc.), while on the other, in the infinite case some genuinely new phenomena arise. We also note that calculating Parker vectors of oligomorphic groups is akin to counting circulant combinatorial objects, mirroring in a sense the combinatorial meaning of the orbit-counting sequence of an oligomorphic group. Finally we explicitly find the Parker vectors for some groups, one of which being the automorphism group of the Rado graph
Enumeration of 2 -connected Loopless 4 -regular Maps on the Plane

AbstractIn this paper rooted (near-) 4-regular maps on the plane are counted with respect to the root-valency, the number of edges, the number of inner faces, the number of non-root vertex loops, the number of non-root vertex blocks, and the number of multi-edges. As special cases, formulae of several types of rooted 4-regular maps such as 2-connected 4-regular planar maps, rooted 2-connected (connected) 4-regular planar maps without loops are also presented. Several known results on 4-regular maps and trees of Tutte are also concluded. Finally, asymptotic formulae for the numbers of those types of maps are given
On the Geometry of Hermitian Matrices of Order Three Over Finite Fields

AbstractSome geometry of Hermitian matrices of order three over GF(q2) is studied. The variety coming from rank 2 matrices is a cubic hypersurface M73of PG(8,q ) whose singular points form a variety H corresponding to all rank 1 Hermitian matrices. BesideM73 turns out to be the secant variety of H. We also define the Hermitian embedding of the point-set of PG(2, q2) whose image is exactly the variety H. It is a cap and it is proved that PGL(3, q2) is a subgroup of all linear automorphisms of H. Further, the Hermitian lifting of a collineation of PG(2, q2) is defined. By looking at the point orbits of such lifting of a Singer cycle of PG(2, q2) new mixed partitions of PG(8,q ) into caps and linear subspaces are given
Automorphism Groups of Generalized Quadrangles via an Unusual Action of PΓL (2, 2Sph)

AbstractWe introduce a new, simpler method for the calculation of automorphism groups of flock quadrangles in characteristic 2, and illustrate the power of this new perspective by obtaining new characterizations of three of the four known families of flock quadrangles. Underlying this perspective is an unusual action ofPΓL(2, 2Sph) which leads to new insights about the groups of ovals in PG(2, 2Sph), and is of considerable interest independent of the connection with flock quadrangles, which arises via the herd model
A Characterization ofQ(5, q) Using One SubquadrangleQ(4,q )

AbstractLet Γ be a finite generalized quadrangle of order (q,q2 ), and suppose that it has a subquadrangle Δ isomorphic to Q(4,q ). We show that Γ is isomorphic to the classical generalized quadrangle Q(5,q ) if at least one of the following holds: (1) all linear collineations of Δ extend toΓ ; (2) all subtended ovoids are classical (and we present a uniform proof independent of the characteristic). Further, for q odd, we prove that if every triad { x, y, z } ofΔ is 3-regular in Γ and { x, y, z }⊥⊥⊂Δ, then Γ is classical. We also show that, if for every centric triad { x, y, z } of an ovoid O of the quadrangleΔ ∼ =Q(4, q), q odd, all points of { x,y , z }⊥⊥belong to O, then O is classical
On the Natural Imprint Function of a Graph

AbstractIn this paper, some characterizations of median and quasi-median graphs are extended to general isometric subgraphs of Cartesian products using the concept of an imprint function as introduced by Tardif. This extends the well known concepts of medians in median graphs as well as imprints in quasi-median graphs. We introduce absolute C-median graphs in analogy to absolute retracts, and derive a connection with the canonical isometric embedding of graphs into Cartesian products. Absolute C-median graphs strictly include classes of irreducible graphs and absolute (weak) retracts as well as many median-like classes, such as weakly median graphs, pre-median graphs, and weakly modular graphs. New characterizations of quasi-median graphs and of median graphs are obtained along the way. Finally, we propose a conjecture on the amalgamation procedure for absolute C-median graphs, and prove the fixed box theorem for this class modulo the conjecture
Antipodal Metrics and Split Systems

AbstractRecall that a metric d on a finite set X is called antipodal if there exists a map σ: X→X: x∣→ __x so that d(x, __ x) =d(x,y ) +d(y, __ x) holds for all x,y∈X. Antipodal metrics canonically arise as metrics induced on specific weighted graphs, although their abundance becomes clearer in light of the fact that any finite metric space can be isometrically embedded in a more or less canonical way into an antipodal metric space called its full antipodal extension.In this paper, we examine in some detail antipodal metrics that are, in addition, totally split decomposable. In particular, we give an explicit characterization of such metrics, and prove that—somewhat surprisingly—the full antipodal extension of a proper metricd on a finite set X is totally split decomposable if and only if d is linear or#X= 3 holds