Hypersurfaces of semi-C-reducible and S4-like Finsler spaces

The notions of semi C-reducible and S4-like Finsler spaces have been introduced by Matsumoto and Shibata ([6]). The object of the present paper is to study some properties of the hypersurfaces immersed in semi-C-reducible and S4-like Finsler spaces. It has been proved that a hypersurface of semi-C-reducible Finsler space is a semi-C-reducible while the condition, under with a hypersurface of S4-like Finsler space will be a S-4like space, has been obtained. The condition under which a hypersurface of semi-C-reducible Landsberg space will be a Landsberg space has also been obtained. After using the so called “T-condition” (Matsumoto [5]) we have discussed the condition under which a hypersurface of a semi-C-reducible Finsler spaceF _n satisfying T-condition will also satisfy T-condition.     

The γ-vector of a barycentric subdivision

We prove that the γ-vector of the barycentric subdivision of a simplicial sphere is the f-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the h-vector of the barycentric subdivision of a boolean complex.     

Cubical Subdivisions and Local h-Vectors

Face numbers of triangulations of simplicial complexes were studied by Stanley by use of his concept of a local h-vector. It is shown that a parallel theory exists for cubical subdivisions of cubical complexes, in which the role of the h-vector of a simplicial complex is played by the (short or long) cubical h-vector of a cubical complex, defined by Adin, and the role of the local h-vector of a triangulation of a simplex is played by the (short or long) cubical local h-vector of a cubical subdivision of a cube. The cubical local h-vectors are defined in this paper and are shown to share many of the properties of their simplicial counterparts. Generalizations to subdivisions of locally Eulerian posets are also discussed.     

On the M?bius function of a lower Eulerian Cohen–Macaulay poset

A certain inequality is shown to hold for the values of the M?bius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen–Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley’s nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen–Macaulay meet-semilattice and Adin’s nonnegativity conjecture for the cubical h-vector of a Cohen–Macaulay cubical complex.     

On subdivisions of simplicial complexes: Characterizing localh-vectors

A well-known combinatorial invariant of simplicial complexes is theh-vector, which has been the subject of much combinatorial research. This paper deals withlocal h-vectors, recently defined by Stanley as a tool for studyingh-vectors of simplicial subdivisions. The face-vector of any simplicial complex can only increase when the complex is subdivided; how does theh-vector change? Motivated by this question, Stanley derived certain useful properties of localh-vectors. In this paper we use mainly geometric arguments to show that these properties characterize localh-vectors, andregular localh-vectors.     

h-Vectors of Gorenstein polytopes

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP, c∈N, can be written as the sum of c lattice points in P. (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies.     

Equivalences and the complete hierarchy of intersection graphs of paths in a tree

An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex in G, such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T. The class of graphs that have an (h,s,t)-representation is denoted by [h,s,t]. It is well known that the class of chordal graphs corresponds to the class [3, 3, 1]. Moreover, it was proved by Jamison and Mulder that chordal graphs correspond to orthodox-[3, 3, 1] graphs defined below.In this paper, we investigate the class of [h,2,t] graphs, i.e., the intersection graphs of paths in a tree. The [h,2,1] graphs are also known as path graphs [F. Gavril, A recognition algorithm for the intersection graphs of paths in trees, Discrete Math. 23 (1978) 211-227] or VPT graphs [M.C. Golumbic, R.E. Jamison, Edge and vertex intersection of paths in a tree, Discrete Math. 55 (1985) 151-159], and [h,2,2] graphs are known as the EPT graphs. We consider variations of [h,2,t] by three main parameters: h, t and whether the graph has an orthodox representation. We give the complete hierarchy of relationships between the classes of weakly chordal, chordal, [h,2,t] and orthodox-[h,2,t] graphs for varied values of h and t.     

On the Gorensteinness of broken circuit complexes and Orlik–Terao ideals

It is proved that the broken circuit complex of an ordered matroid is Gorenstein if and only if it is a complete intersection. Several characterizations for a matroid that admits such an order are then given, with particular interest in the h-vector of broken circuit complexes of the matroid. As an application, we prove that the Orlik–Terao algebra of a hyperplane arrangement is Gorenstein if and only if it is a complete intersection. Interestingly, our result shows that the complete intersection property (and hence the Gorensteinness as well) of the Orlik–Terao algebra can be determined from the last two nonzero entries of its h-vector.     

Completely unimodal numberings of a simple polytope

A completely unimodal numbering of the m vertices of a simple d-dimensional polytope is a numbering 0, 1, …,m−1 of the vertices such that on every k-dimensional face (2≤k≤d) there is exactly one local minimum (a vertex with no lower-numbered neighbors on that face). Such numberings are abstract objective functions in the sense of Adler and Saigal [1]. It is shown that a completely unimodal numbering of the vertices of a simple polytope induces a shelling of the facets of the dual simplicial polytope. The h-vector of the dual simplicial polytope is interpreted in terms of the numbering (with respect to using a local-improvement algorithm to locate the vertex numbered 0). In the case that the polytope is combinatorially equivalent to a d-dimensional cube, a ‘successor-tuple’ for each vertex is defined which carries the crucial information of the numbering for local-improvement algorithms. Combinatorial properties of these d-tuples are studied. Finally the running time of one particular local-improvement algorithm, the Random Algorithm, is studied for completely unimodal numberings of the d-cube. It is shown that for a certain class of numberings (which includes the example of Klee and Minty [8] showing that the simplex algorithm is not polynomial and all Hamiltonian saddle-free injective pseudo-Boolean functions [6]) this algorithm has expected running time that is at worst quadratic in the dimension d.     

Modification of a Finsler space by a normalized semi-parallel vector field

LetF _n be a Finsler space with metric functionF(x, y). M. Matsumoto [6] has defined a modified Finsler spaceF _n ^* whose metric functionF ^*(x, y) is given byF ^*2 = = F² + (X_i(x)yⁱ)², whereX _i are the components of a covariant vector which is a function of coordintae only. Since a concurrent vector is a function of coordinate only, Matsumoto and Eguchi [9] have studied various properties of the modified Finsler spaceF _n ^* under the assumption thatX _i are the components of a concurrent vector field inF _n. In this paper we shall introduce the concept of semi-parallel vector field inF _n and study the properties of modified Finsler spaceF _n ^* .     

Improving the bounds of the multiplicity conjecture: The codimension 3 level case

The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen-Macaulay algebra A in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of A. All the examples studied so far have lead to conjecture (see [J. Herzog, X. Zheng, Notes on the multiplicity conjecture. Collect. Math. 57 (2006) 211-226] and [J. Migliore, U. Nagel, T. Römer, Extensions of the multiplicity conjecture, Trans. Amer. Math. Soc. (preprint: math.AC/0505229) (in press)]) that, moreover, the bounds of the MC are sharp if and only if A has a pure MFR. Therefore, it seems a reasonable-and useful-idea to seek better, if possibly ad hoc, bounds for particular classes of Cohen-Macaulay algebras.In this work we will only consider the codimension 3 case. In the first part we will stick to the bounds of the MC, and show that they hold for those algebras whose h-vector is that of a compressed algebra.In the second part, we will (mainly) focus on the level case: we will construct new conjectural upper and lower bounds for the multiplicity of a codimension 3 level algebra A, which can be expressed exclusively in terms of the h-vector of A, and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of A is not pure.Even though proving our bounds still appears too difficult a task in general, we are already able to show them for some interesting classes of codimension 3 level algebras A: namely, when A is compressed, or when its h-vector h(A) ends with (…,3,2). Also, we will prove our lower bound when h(A) begins with (1,3,h₂,…), where h₂≤4, and our upper bound when h(A) ends with (…,h_c−1,h_c), where h_c−1≤h_c+1.     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Construction of additional variables conforming to a common factor model

It is shown that if a random p-vector x conforms to an r-common factor model and an external variable Z is given, then there exists a family of linear combinations X of x and Z such that [x′: X]′ conforms to an r-common factor model     

On the f- and h-triangle of the barycentric subdivision of a simplicial complex

For a simplicial complex Δ we study the behavior of its f- and h-triangle under the action of barycentric subdivision. In particular we describe the f- and h-triangle of its barycentric subdivision sd(Δ). The same has been done for f- and h-vector of sd(Δ) by F. Brenti, V. Welker (2008). As a consequence we show that if the entries of the h-triangle of Δ are nonnegative, then the entries of the h-triangle of sd(Δ) are also nonnegative. We conclude with a few properties of the h-triangle of sd(Δ).     

A type-B associahedron

The (type-A) associahedron is a polytope related to polygon dissections which arises in several mathematical subjects. We propose a B-analogue of the associahedron. Our original motivation was to extend the analogies between type-A and type-B noncrossing partitions, by exhibiting a simplicial polytope whose h-vector is given by the rank-sizes of the type-B noncrossing partition lattice, just as the h-vector of the (simplicial type-A) associahedron is given by the Narayana numbers. The desired polytope Q^B_n is constructed via stellar subdivisions of a simplex, similarly to Lee's construction of the associahedron. As in the case of the (type-A) associahedron, the faces of Q^B_n can be described in terms of dissections of a convex polygon, and the f-vector can be computed from lattice path enumeration. Properties of the simple dual $\text{Q}{}^{\mathrm{<mtext>B</mtext><mtext>1</mtext>}}{}_{\mathrm{n}}$ are also discussed and the construction of a space tessellated by $\text{Q}{}^{\mathrm{<mtext>B</mtext><mtext>1</mtext>}}{}_{\mathrm{n}}$ is given. Additional analogies and relations with type A and further questions are also discussed.     

An elementary bound for the number of points of a hypersurface over a finite field (Summary)

This short note is a summary of our paper with the same title [M. Homma and S. J. Kim, An elementary bound for the number of points of a hypersurface over a finite field, preprint 2012]. We establish an upper bound for the number of points of a hypersurface without a linear component over a finite field, which is analogous to the Sziklai bound for a plane curve.     

On the Flag f-vector of a Graded Lattice with Nontrivial Homology

It is proved that the Boolean algebra of rank n minimizes the flag f-vector among all graded lattices of rank n, whose proper part has nontrivial top-dimensional homology. The analogous statement for the flag h-vector is conjectured in the Cohen-Macaulay case.     

Möbius Isoparametric Hypersurfaces in S n+1 with Two Distinct Principal Curvatures

A hypersurface x : M → S ⁿ⁺¹ without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = ?ρ ^?2∑ _i (e _i (H) + ∑ _j (h _ij ?Hδ _ij )e _j (log ρ))θ _i vanishes and its Möbius shape operator $ {\Bbb {S}}A hypersurface x : M → S ^{n +1} without umbilic point is called a M?bius isoparametric hypersurface if its M?bius form Φ = −ρ⁻²∑ _i (e _i (H) + ∑ _j (h _ij −Hδ _ij )e _j (log ρ))θ _i vanishes and its M?bius shape operator ? = ρ⁻¹(S−Hid) has constant eigenvalues. Here {e _i } is a local orthonormal basis for I = dx·dx with dual basis {θ _i }, II = ∑ _ij h _ij θ _i ⊗θ _i is the second fundamental form, and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S ^{n +1} is a M?bius isoparametric hypersurface, but the converse is not true. In this paper we classify all M?bius isoparametric hypersurfaces in S ^{n +1} with two distinct principal curvatures up to M?bius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact M?bius isoparametric hypersurface embedded in S ^{n +1} can take only the values 2, 3, 4, 6. Received September 7, 2001, Accepted January 30, 2002     

Universal inequalities in Ehrhart theory

In this paper, we show the existence of universal inequalities for the h*-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the h*-polynomial that are independent of both the dimension and the degree of P. More precisely, we prove that the coefficients h* ₁ and h* ₂ of the h*-vector (h* ₀, h* ₁,..., h* _d) of a lattice polytope of any degree satisfy Scott’s inequality if h* ₃ = 0.     

A Bound of the Finslerian Ricci Scalar

In the Riemannian as well as in the Finslerian geometry, certain conditions on the Ricci scalar or the Ricci tensor provide obstructions on the topology of the base manifold and so on the configuration of cut points by limitations of the injectivity radius, see the Bonnet–Myers theorem and its variants and generalizations. In this paper, we show that conversely, prescribing the injectivity radius of a Finsler manifold, some limitations of the Ricci scalar are obtained. Some consequences of the condition that the Ricci tensor is h-parallel with respect to the Chern–Rund connection are found. In addition, some classes of examples are provided.