Extensions of hereditary algebras

Let be a triangular matrix algebra, uhere k is an algebraically closed field, B is the path algebra of an oriented Dynkin diagram of type E₆ or E₇ or E₈ and M is a finite dimensional k-B-bimodule. The aim of this paper is to determine the representation type of A for any orientation of the Dynkin diagram and for any indecomposable B-module M. This classification is obtained by comparing the representation types of the algebras and using the theory of tilting modules.     

Counting the number of isotone selfmappings of crowns

Letn>1. The number of all strictly increasing selfmappings of a 2n-element crown is . The number of all order-preserving selfmappings of a 2n-element crown is

On theq-log-concavity of Gaussian binomial coefficients

We give a combinatorial proof that is a polynomial inq with nonnegative coefficients for nonnegative integersa, b, k, l withab andlk. In particular, fora=b=n andl=k, this implies theq-log-concavity of the Gaussian binomial coefficients , which was conjectured byButler (Proc. Amer. Math. Soc. 101 (1987), 771–775).     

Ancestor tree for arbitrary multi-terminal cut functions

In many applications, a function is defined on the cuts of a network. In the max-flow min-cut theorem, the function on a cut is simply the sum of all capacities of edges across the cut, and we want the minimum value of a cut separating a given pair of nodes. To find the minimum cuts separating pairs of nodes, we only needn – 1 computations to construct the cut-tree. In general, we can define arbitrary values associated with all cuts in a network, and assume that there is a routine which gives the minimum cut separating a pair of nodes. To find the minimum cuts separating pairs of nodes, we also only needn – 1 routine calls to construct a binary tree which gives all minimum partitions. The binary tree is analogous to the cut-tree of Gomory and Hu.     

Inclusion-exclusion inequalities

Ifμ is a positive measure, andA ₂, ...,A _n are measurable sets, the sequencesS ₀, ...,S _n andP _[0], ...,P _[n] are related by the inclusion-exclusion equalities. Inequalities among theS _i are based on the obviousP _[k]≧0. Letting =the average average measure of the intersection ofk of the setsA _i , it is shown that (−1) ^k Δ ^k M _i ≧0 fori+k≦n. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS ₀=1, whenS ₁≧N−1, and for 1≦k<N≦n andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN, for all sequencesM ₀, ...,M _n of sufficiently large length if and only if for 0<t<1.     

Gauss polynomials and the rotation algebra

Newton's binomial theorem is extended to an interesting noncommutative setting as follows: If, in a ring,ba=ab with commuting witha andb, then the (generalized) binomial coefficient arising in the expansion

Коэффициенты рядов Ф урье и класс Липшица

We generalize and sharpen certain results concerning Fourier series from the Lipschitz class. In particular, for sinnx we prove the following: Let ¦b_n¦n^–2L(n) where L(x) is a continuous and slowly oscillating function. Then

On the number of faces of centrally-symmetric simplicial polytopes

I. Bárány and L. Lovász [Acta Math. Acad. Sci. Hung.40, 323–329 (1982)] showed that ad-dimensional centrally-symmetric simplicial polytopeP has at least 2 ^d facets, and conjectured a lower bound for the numberf _i ofi-dimensional faces ofP in terms ofd and the numberf ₀ =2n of vertices. Define integers A. Björner conjectured (unpublished) that (which generalizes the result of Bárány-Lovász sincef _d–1 = h _i ), and more strongly that , which is easily seen to imply the conjecture of Bárány-Lovász. In this paper the conjectures of Björner are proved.Partially supported by NSF grant MCS-8104855. The research was performed when the author was a Sherman Fairchild Distinguished Scholar at Caltech.     

A 2-extension of the field ℚ of rational numbersof rational numbers

It is proved that the rational number field has one, and only one, normal 2-extension (_{2, t8)}/with group isomorphic to .If is the maximal subfield of a real-closed field, which does not contain ,then the algebraic closure of is isomorphic to the field .Bibliography: 7titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 192–196.     

How to find many counterfeit coins?

We propose an algorithm for findingm defective coins, that uses at most + 15m weighings on a balance scale, wheren is the number of all coins.     

Rate-of-convergence bound for difference schemes for the system of equilibrium equations of a rigidly clamped nonhomogeneous anisotropic elastic solid

Exact difference scheme operators are used to construct a difference scheme for a system of equilibrium equations of a rigidly clamped nonhomogeneous anisotropic elastic solid. The rate-of-convergence bound .Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 14–19, 1987.     

Interlineation of functions of 2 variables on M (M≥2) lines with the highest algebraic precision

A general algorithm is proposed for constructing interlineation operators , x=(x₁, x₂) with the properties

Über lineare Komplexe von Räumen höchster Dimension einer nicht entarteten Quadrik

Sunto Conoscendo il modello minimo per la totalità degli S _k ^I d'una schiera giacenti su di una quadrica Q_2k di S_2k+1 si definisce il ? complesso lineare di S _k ^I ? come imagine d'una sezione iperpiana di . Il problema della classificazione projettiva di tali complessi lineari è posto e risolto per k≤5.      

Analytic semigroups,degenerate elliptic operators and applications to nonlinear cauchy problems

In this paper we prove L^p () and -norm estimates for the solution of the elliptic equation:

Solutions of minimal period for a semilinear wave equation

In questo lavoro si considera il problema

On the existence of components of the Noether-Lefschetz locus with given codimension

It is known that the codimension c of a component of the Noether-Lefschetz locus NL(d) satisfies . We prove that ford≥47 and for every integer there exists a component of NL(d) with codimension c. This is done with families of surfaces of degree d in ℙ³ containing a curve lying on a cubic or on a quartic surface or a curve with general moduli. Moreover we produce an explicit example, for everyd≥4, of components of maximum codimension , thus giving a new proof of the fact that these components are dense in the locus of smooth surfaces (density theorem).     

A Minimal-Automaton-Based Algorithm for the Reliability of Con(d,k, n) Systems

A d-within-consecutive-k-out-of-n system, abbreviated as Con(d, k, n), is a linear system of n components in a line which fails if and only if there exists a set of k consecutive components containing at least d failed ones. So far the fastest algorithm to compute the reliability of Con(d, k, n) is Hwang and Wright's algorithm published in 1997, where . In this paper we use automata theory to reduce to . For d small or close to k, we have reduced from exponentially many (in k) to polynomially many. The computational complexity of our final algorithm is , where .     

Decomposition Method for a Class of Monotone Variational Inequality Problems

In the solution of the monotone variational inequality problem VI(, F), with

A proof of Shelah's partition theorem

A self contained proof of Shelah's theorem is presented: If is a strong limit singular cardinal of uncountable cofinality and 2 > ⁺ then .     

A theorem on the average number of subfaces in arrangements and oriented matroids

It is known that for simple arrangements in thed-dimensional Euclidean spaceR ^d The average number ofj-dimensional subfaces of ak-dimensional face is less than . In this paper, we show that this is also true for all arrangements inR ^d and for all oriented matroids, and we give combinatorial proofs.