On the vertices of the -additive core

The core of a game v on N, which is the set of additive games φ dominating v such that φ(N)=v(N), is a central notion in cooperative game theory, decision making and in combinatorics, where it is related to submodular functions, matroids and the greedy algorithm. In many cases however, the core is empty, and alternative solutions have to be found. We define the k-additive core by replacing additive games by k-additive games in the definition of the core, where k-additive games are those games whose Möbius transform vanishes for subsets of more than k elements. For a sufficiently high value of k, the k-additive core is nonempty, and is a convex closed polyhedron. Our aim is to establish results similar to the classical results of Shapley and Ichiishi on the core of convex games (corresponds to Edmonds’ theorem for the greedy algorithm), which characterize the vertices of the core.     

On the behavior of the product of independent random variables

For two independent non-negative random variables X and Y, we treat X as the initial variable of major importance and Y as a modifier (such as the interest rate of a portfolio). Stability in the tail behaviors of the product compared with that of the original variable X is of practical interests. In this paper, we study the tail behaviors of the product XY when the distribution of X belongs to the classes L and S, respectively. Under appropriate conditions, we show that the distribution of the product XY is in the same class as X when X belongs to class L or S, in other words, classes L and S are stable under some mild conditions on the distribution of Y. We also show that if the distribution of X is in class L(γ) (γ>0) and continuous, then the product XY is in L if and only if Y is unbounded.     

On the simplicity of the full group of ergodic transformations

LetE denote an invertible, non-singular, ergodic transformation of (0, 1). Then the full group ofE is perfect. IfE preserves the Lebesgue measure, then the full group is simple. IfE preserves no measure equivalent to Lebesgue, then the full group is simple. IfE preserves an infinite measure, then there exists a unique normal subgroup. IfT is any invertible transformation preserving the Lebesgue measure, then the full group is simple if and only ifT is ergodic on its support.     

On the Genus of the Nilpotent Graphs of Finite Groups

The nilpotent graph of a group G is a simple graph whose vertex set is G?nil(G), where nil(G) = {y ∈ G | ? x, y ? is nilpotent ? x ∈ G}, and two distinct vertices x and y are adjacent if ? x, y ? is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.     

An Example of the Effect of Rescaling of the Reflection Coefficient on the Scattering Potential for the One-Dimensional Schrodinger Equation

Let us consider the one-dimensional Schrödinger equation with the scattering potential V(x)=2δ(x) and corresponding reflection coefficient b(k)=?i/(k + i). The potential satisfies a theorem of Deift and Trubowitz which states that non-negative measurable potentials V(x) satisfying a certain range condition have reflection coefficients b(k) such that b(0)=?1. We rescale the reflection coefficient for V(x)=2δ(x) by writing b(k)=?iv/(k + 1) where 0<v<1. It is shown how V(x) changes with v, through the use of the Gelfand-Levitan equation. This example illustrates how sensitive the potential is to rescaling of the reflection coefficient. In particular, the rescaling leads to a negative portion of V(x), as is expected from the Deift-Trubowitz theorem. The example of this paper will be used in a later paper to illustrate the nature of bounds on potentials obtained through the use of variational principles.     

On the upper bounds of the minimum number of rows of disjunct matrices

A 0-1 matrix is d-disjunct if no column is covered by the union of any d other columns. A 0-1 matrix is (d; z)-disjunct if for any column C and any d other columns, there exist at least z rows such that each of them has value 1 at column C and value 0 at all the other d columns. Let t(d, n) and t(d, n; z) denote the minimum number of rows required by a d-disjunct matrix and a (d; z)-disjunct matrix with n columns, respectively. We give a very short proof for the currently best upper bound on t(d, n). We also generalize our method to obtain a new upper bound on t(d, n; z). The work of Y. Cheng and G. Lin is supported by Natural Science and Engineering Research Council (NSERC) of Canada, and the Alberta Ingenuity Center for Machine Learning (AICML) at the University of Alberta. The work of D.-Z. Du is partially supported by National Science Foundation under grant No.CCF0621829.     

On the Decomposition of the Riesz Operator and the Expansion of the Riesz Semigroup

For a Riesz operator T on a reflexive Banach space X with nonzero eigenvalues denote by E(λ_i; T) the eigen-projection corresponding to an eigenvalue λ_i. In this paper we will show that if the operator sequence is uniformly bounded, then the Riesz operator T can be decomposed into the sum of two operators T_p and T_r: T = T_p + T_r, where T_p is the weak limit of T_n and T_r is quasi-nilpotent. The result is used to obtain an expansion of a Riesz semigroup T(t) for t ≥ τ. As an application, we consider the solution of transport equation on a bounded convex body.     

On the Rate of Decay of the Concentration Function of the Sum of Independent Random Variables

Let X₁, ... , X_n be i.i.d. integral valued random variables and S_n their sum. In the case when X₁ has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for max _{k ∊ ℤ} Pr{S_n = k} (which can be expressed in term of the Lévy Doeblin concentration of S_n), under the extra condition that X₁ is not essentially supported by an arithmetic progression. The first aim of the paper is to show that this extra condition cannot be simply ruled out. Secondly, it is shown that if X₁ has a very large tail (larger than a Cauchy-type distribution), then the extra arithmetic condition is not sufficient to guarantee a uniform upper bound for the decay of the concentration of the sum S_n. Proofs are constructive and enhance the connection between additive number theory and probability theory.À Jean-Louis Nicolas, avec amitié et respect2000 Mathematics Subject Classification: Primary—60Fxx, 60Exx, 11Pxx, 11B25     

On the effective computation of the inertia of a non-hermitian matrix

Summary given a complex lower Hessenberg matrixA with unit codiagonal, a hermitian matrixH is constructed such that, ifH is non-singular InA= InH. IfA is real,H is real symmetric. Classical results of Fujiwara on the root-separation problem and of Schwarz on the eigenvalue-separation problem are included as special cases.The authors' research was conducted at the Universidade Estadual de Campinas and supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo, Brasil, under grant n⁰ 78/0490.     

On the difference property of the family of functions with the Baire property

We show that it is consistent with ZFC that the family of functions with the Baire property has the difference property. That is, every function for which f(x + h)-f(x) has the Baire property for every h∈R is of the form f=g + Awhere g has the Baire property and A is additive. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the number of regular vertices of the union of jordan regions

Let \C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in \C cross exactly twice, then their intersection points are called regular vertices of the arrangement \A(\C) . Let R(\C) denote the set of regular vertices on the boundary of the union of \C . We present several bounds on |R(\C)| , depending on the type of the sets of \C . (i) If each set of \C is convex, then |R(\C)|=O(n ^1.5+\eps ) for any \eps>0 . (ii) If no further assumptions are made on the sets of \C , then we show that there is a positive integer r that depends only on s such that |R(\C)|=O(n ^2-1/r ) . (iii) If \C consists of two collections \C ₁ and \C ₂ where \C ₁ is a collection of m convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and \C ₂ is a collection of polygons with a total of n sides, then |R(\C)|=O(m ^2/3 n ^2/3 +m +n) , and this bound is tight in the worst case. Received December 4, 1998, and in revised form June 3, 2000. Online publication Feburary 1, 2001.     

On the centralizer and the commutator subgroup of an automorphism

Let φ be an automorphism of a group G. In this paper, we study the influence of its centralizer on its commutator subgroup when G is polycyclic or metabelian. For instance, when G is metabelian and φ fixed-point-free of prime order p, we prove that is nilpotent of class ≤ p. Also, when G is polycyclic and φ of order 2, we show that if is finite, then so are and .     

A note on the order of the antipode of a pointed Hopf algebra

Let k be a field and let H denote a pointed Hopf k-algebra with antipode S. We are interested in determining the order of S. Building on the work done by Taft and Wilson in [7], we define an invariant for H, denoted m_H, and prove that the value of this invariant is connected to the order of S. In the case where char k?=?0, it is shown that if S has finite order then it is either the identity or has order 2?m_H. If in addition H is assumed to be coradically graded, it is shown that the order of S is finite if and only if m_H is finite. We also consider the case where char k?=?p?>?0, generalizing the results of [7] to the infinite-dimensional setting.     

Bounding the Compression Loss of the FGK Algorithm

An important issue related to coding schemes is their compression loss. A simple measure ε of the compression loss due to a coding scheme different than Huffman coding is defined by ε = A_C − A_H where A_H is the average code length of a static Huffman encoding and A_C is the average code length of an encoding based on the compression scheme C. When the scheme C is the FGK algorithm, Vitter conjectured that ε ≤ K for some real constant K. Here, we use an amortized analysis to prove this conjecture. We show that ε < 2. Furthermore, we show through an example that our bound is asymptotically tight. This result explains the good performance of FGK that many authors have observed through practical experiments.     

The Role of the Legendre Transform in the Study of the Floer Complex of Cotangent Bundles

Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM → R be its Legendre‐dual Lagrangian. In a previous paper we constructed an isomorphism Φ from the Morse complex of the Lagrangian action functional that is associated to L to the Floer complex that is determined by H. In this paper we give an explicit construction of a homotopy inverse Ψ of Φ. Contrary to other previously defined maps going in the same direction, Ψ is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half‐cylinder that on the boundary satisfy half of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a heuristic argument which, independently of any Fredholm and compactness analysis, explains why the spaces of maps that are used in the definition of Φ and Ψ are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our heuristic arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pullback of the second Stiefel‐Whitney class of TM on 2‐tori in M.© 2015 Wiley Periodicals, Inc.     

On the normality of the rings of Schubert varieties

We prove that the cone over a Schubert variety inG/P (P being a maximal parabolic subgroup of classical type) is normal by exhibiting a 2-regular sequence inR(w) (the homogeneous coordinate ring of the Schubert varietyX(w) inG/P under the canonical protective embeddingG/P ⊂→ (p (H° G/P,L)),L being the ample generator of (PicG/P), which vanishes on the singular locus ofX(w). We also prove the surjectivity ofH° (G/Q, L) H° (X(w), L), whereQ is a classical parabolic subgroup (not necessarily maximal) ofG andL is an ample line bundle onG/Q.     

On the hopficity of the polynomial rings

In this paper we prove that if a ringR satisfies the condition that for some integern > 1,a ⁿ =a for everya inR, thenR a hopfian ring implies that the ringR [T] of polynomials is also hopfian. This generalizes a recent result of Varadarajan which states that ifR is a Boolean hopfian ring then the ringR[T] is also hopfian. We show furthermore that there are numerous ringsR satisfying the hypothesis of our theorem which are neither Boolean nor Noetherian.     

An Upper Bound on the Basis Number of the Powers of the Complete Graphs

The basis number of a graph G is defined by Schmeichel to be the least integer h such that G has an h-fold basis for its cycle space. MacLane showed that a graph is planar if and only if its basis number is . Schmeichel proved that the basis number of the complete graph K _n is at most 3>. We generalize the result of Schmeichel by showing that the basis number of the d-th power of K _n is at most 2d+1.     

Estimates of the conformal scalar curvature equation via the method of moving planes

In this paper we derive a local estimate of a positive singular solution u near its singular set Z of the conformal equation where K(x) is a positive continuous function, Z is a compact subset of , and g satisfies that is nonincreasing for t > 0. Assuming that the order of flatness at critical points of K on Z is no less than , we prove that, through the application of the method of moving planes, the inequality holds for any solution of (0.1) with Cap(Z) = 0. By the same method, we also derive a Harnack-type inequality for smooth positive solutions. Let u satisfy Assume that the order of flatness at critical points of K is no less than n - 2; then the inequality holds for R ≤ 1. We also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality (0.4). © 1997 John Wiley & Sons, Inc.