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"... Richard P. Stanley is well known for his fundamental and important contributions to combinatorics and its relationship to algebra and geometry, in particular in the theory of simplicial complexes. Two kinds of simplicial complexes play central roles in combinatorics: partitionable complexes and Coh ..."
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Richard P. Stanley is well known for his fundamental and important contributions to combinatorics and its relationship to algebra and geometry, in particular in the theory of simplicial complexes. Two kinds of simplicial complexes play central roles in combinatorics: partitionable complexes and CohenMacaulay complexes. Stanley posed a central conjecture relating these two notions: Are all CohenMacaulay simplicial complexes partitionable? In a 1982 Inventiones Mathematicae paper [4], Stanley defined what is now called the Stanley depth of a graded module over a graded commutative ring. Stanley depth is a geometric invariant of a module that, by a conjecture of Stanley, relates to an algebraic invariant of the module, called simply the depth. It is shown in [2] that this conjecture implies his conjecture about partitionable CohenMacaulay simplicial complexes. Our aim here is to introduce the notion of the Stanley depth. Let K be a field and S = K[x1,..., xn] the Kalgebra of polynomials over K in n indeterminates x1,..., xn. We may write x = {x1,..., xn} and denote S by K[x] for convenience. A monomial in S is a product xa = xa11... xann for a vector a = (a1,..., an) ∈ωn of nonnegative integers. The M. R. Pournaki is associate professor of mathematics at
WHAT IS... STANLEY DEPTH?
"... Richard P. Stanley is well known for his fundamental and important contributions to combinatorics and its relationship to algebra and geometry, in particular in the theory of simplicial complexes. Two kinds of simplicial complexes play central roles in combinatorics: partitionable complexes and Cohe ..."
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Richard P. Stanley is well known for his fundamental and important contributions to combinatorics and its relationship to algebra and geometry, in particular in the theory of simplicial complexes. Two kinds of simplicial complexes play central roles in combinatorics: partitionable complexes and Cohen–Macaulay complexes. Stanley posed a central conjecture relating these two notions: Are all Cohen–Macaulay simplicial complexes partitionable? In a 1982 Inventiones Mathematicae paper [4], Stanley defined what is now called the Stanley depth of a graded module over a graded commutative ring. Stanley depth is a geometric invariant of a module that, by a conjecture of Stanley, relates to an algebraic invariant of the module, called simply the depth. It is shown in [2] that this conjecture implies his conjecture about partitionable Cohen–Macaulay simplicial complexes. Our aim here is to introduce the notion of the Stanley depth. Let K be a field and S = K[x1,..., xn] the Kalgebra of polynomials over K in n indeterminates x1,..., xn. We may write x = {x1,..., xn} and denote S by K[x] for convenience. A monomial in S is a product xa = xa11... x
Hidden Connections Between General Relativity and Finsler
 Geometry, Nuovo. Cim. B118 (2003) 345–351, gr–qc/0312053
"... Modern formulation of Finsler geometry of a manifold M utilizes the equivalence between this geometry and the Riemannian geometry of V TM, the vertical bundle over the tangent bundle of M, treating TM as the base space. We argue that this approach is unsatisfactory when there is an indefinite metric ..."
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Cited by 2 (0 self)
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Modern formulation of Finsler geometry of a manifold M utilizes the equivalence between this geometry and the Riemannian geometry of V TM, the vertical bundle over the tangent bundle of M, treating TM as the base space. We argue that this approach is unsatisfactory when there is an indefinite metric on M because the corresponding Finsler fundamental function would not be differentiable over TM (even without its zero section) and therefore TM cannot serve as the base space. We then make the simple observation that for any differentiable Lorentzian metric on a smooth spacetime, the corresponding Finsler fundamental function is differentiable exactly on a proper subbundle of TM. This subbundle is then used, in place of TM, to provide a satisfactory basis for modern Finsler geometry of manifolds with Lorentzian metrics. Interestingly, this Finslerian property of Lorentzian metrics does not seem to exist for general indefinite Finsler metrics and thus, Lorentzian metrics appear to be of special relevance to Finsler geometry. We note that, in contrast to the traditional formulation of Finsler geometry, having a Lorentzian metric in the modern setting does not imply reduction to pseudoRiemannian geometry because metric and connection are entirely disentangled in the modern formulation and there is a new indispensable nonlinear connection, necessary for construction of Finsler tensor bundles. It is concluded that general relativity—without any modification—has a close bearing on Finsler geometry and a modern Finsler formulation of the theory is an appealing idea. Furthermore, in any such attempt, the metric should probably be left unchanged (not generalized) or the newly discovered property, which provides a satisfactory basis for the corresponding Finsler geometry, would be lost.
Sequentially Sr simplicial complexes and sequentially S2 graphs
, 2010
"... We introduce sequentially Sr modules over a commutative graded ring and sequentially Sr simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially CohenMacaulay, and satisfying Serre’s condition Sr. In analogy with the sequentially CohenMacaulay ..."
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We introduce sequentially Sr modules over a commutative graded ring and sequentially Sr simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially CohenMacaulay, and satisfying Serre’s condition Sr. In analogy with the sequentially CohenMacaulay property, we show that a simplicial complex is sequentially Sr if and only if its pure iskeleton is Sr for all i. Forr = 2, we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially Sr if and only if the minimal free resolution of the ideal of its Alexander dual is componentwise linear in the first r steps. We apply these results for a graph, i.e., for the simplicial complex of the independent sets of vertices of a graph. We characterize sequentially Sr cycles showing that the only sequentially S2 cycles are odd cycles and, for r ≥ 3, no cycle is sequentially Sr with the exception of cycles of length 3 and 5. We extend certain known results on sequentially CohenMacaulay graphs to the case of sequentially Sr graphs. We prove that a bipartite graph is vertex decomposable if and only if it is sequentially S2. We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially S2. Finally, we propose some questions.
Some Properties Of Finite Morphisms On Double Points
"... Abstract. For a finite morphism f: X → Y of smooth varieties such that f maps X birationally onto X ′ = f(X), the local equations of f are obtained at the double points which are not triple. If C is the conductor of X over X ′ , and D = Sing(X ′ ) ⊂ X ′ , ∆ ⊂ X are the subschemes defined by C, t ..."
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Abstract. For a finite morphism f: X → Y of smooth varieties such that f maps X birationally onto X ′ = f(X), the local equations of f are obtained at the double points which are not triple. If C is the conductor of X over X ′ , and D = Sing(X ′ ) ⊂ X ′ , ∆ ⊂ X are the subschemes defined by C, then D and ∆ are shown to be complete intersections at these points, provided that C has ”the expected ” codimension. This leads one to determine the depth of local rings of X ′ at these double points. On the other hand, when C is reduced in X, it is proved that X ′ is weakly normal at these points, and some global results are given. For the case of affine spaces, the local equations of X ′ at these points are computed.
A Basic Family Of Iteration Functions For Polynomial Root Finding And Its Characterizations
 J. of Comp. and Appl. Math
, 1997
"... Let p(x) be a polynomial of degree n 2 with coefficients in a subfield K of the complex numbers. For each natural number m 2, let Lm (x) be the m2m lower triangular matrix whose diagonal entries are p(x) and for each j = 1; : : : ; m 0 1, its jth subdiagonal entries are p (j) (x)=j!. For i = 1; ..."
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Cited by 17 (12 self)
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Let p(x) be a polynomial of degree n 2 with coefficients in a subfield K of the complex numbers. For each natural number m 2, let Lm (x) be the m2m lower triangular matrix whose diagonal entries are p(x) and for each j = 1; : : : ; m 0 1, its jth subdiagonal entries are p (j) (x)=j!. For i = 1; 2, let L (i) m (x) be the matrix obtained from Lm (x) by deleting its first i rows and its last i columns. L (1) 1 (x) j 1. Then, the function Bm (x) = x 0 p(x) det(L (1) m01 (x))=det(L (1) m (x)) is a member of S(m; m + n 0 2), where for any M m, S(m; M) is the set of all rational iteration functions such that for all roots ` of p(x) , g(x) = ` + P M i=m fl i (x)(` 0 x) i , with fl i (x)'s also rational and welldefined at `. Given g 2 S(m; M), and a simple root ` of p(x), g (i) (`) = 0, i = 1; : : : ; m0 1, and fl m (`) = (01) m g (m) (`)=m!. For Bm (x) we obtain fl m (`) = (01) m det(L (2) m+1 (`))=det(L (1) m (`)). For m = 2 and 3, Bm (x) coincides with Newton'...
Betti numbers of transversal monomial ideals
, 809
"... In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of tminors of a generic pluric ..."
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In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of tminors of a generic pluricirculant matrix is a transversal monomial ideal. Using a Gröbner basis for this ideal, it is shown that the initial ideal of a generic pluricirculant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the EliahouKervair resolution, the Betti numbers of this initial ideal is computed and it is proved that, for some significant values of t, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper, naturally arise in the study of generic singularities of algebraic varieties. Key Words: Betti numbers; Pluricirculant matrix; Stable monomial ideal; Transversal monomial ideal.
ON GRADED CLASSICAL PRIME AND GRADED PRIME SUBMODULES
"... Abstract. Let G be a group with identity e: Let R be a Ggraded commutative ring and M a graded Rmodule. In this paper, we introduce several results concerning graded classical prime submodules. For example, we give a characterization of graded classical prime submodules. Also, the relations betw ..."
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Abstract. Let G be a group with identity e: Let R be a Ggraded commutative ring and M a graded Rmodule. In this paper, we introduce several results concerning graded classical prime submodules. For example, we give a characterization of graded classical prime submodules. Also, the relations between graded classical prime and graded prime submodules of M are studied.
ON THE DEFINING NUMBER OF (2n − 2)VERTEX COLORINGS OF Kn × Kn
"... Abstract. In a given graph G = (V, E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is call ..."
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Abstract. In a given graph G = (V, E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G, c). In this note we study d(G = Kn × Kn, 2n − 2). We determine an upper bound for d(G = Kn × Kn, 2n − 2) for all n and its exact value for some n. 1.
© Electronic Publishing House DIRAC STRUCTURES ON HILBERT SPACES
, 1997
"... Abstract. For a real Hilbert space (H, 〈,〉), a subspace L ⊂ H ⊕H is said to be a Dirac structure on H if it is maximally isotropic with respect to the pairing 〈(x,y),(x′,y′)〉+ = (1/2)(〈x,y′〉+ 〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures on H ..."
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Abstract. For a real Hilbert space (H, 〈,〉), a subspace L ⊂ H ⊕H is said to be a Dirac structure on H if it is maximally isotropic with respect to the pairing 〈(x,y),(x′,y′)〉+ = (1/2)(〈x,y′〉+ 〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures on H are in onetoone correspondence with isometries on H, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structure L on H, every z ∈H is uniquely decomposed as z = p1(l)+p2(l) for some l ∈ L, where p1 and p2 are projections. When p1(L) is closed, for any Hilbert subspace W ⊂ H, an induced Dirac structure on W is introduced. The latter concept has also been generalized.
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