The f-Vector of the Descent Polytope

On formal power series defined by infinite linear systems

Formal analysis is the study of formal power series, formal Laurent series, formal root series, and other formal series or formal functionals. This book is the first comprehensive presentation of the topic that systematically introduces formal analysis, including its algebraic, analytic, and topological structure, along with various applications.

Studying the extreme kernel face complexes of a given dimension, we obtain some lower estimates of the number of shortest face complexes in the n-dimensional unit cube. The number of shortest complexes of k-dimensional faces is shown to be of the same logarithm order as the number of complexes consisting of at most 2n−1 different k-dimensional faces if 1 ≤ k ≤ c · n and c < 0.5. This implies similar lower bounds for the maximum length of the kernel DNFs and the number of the shortest DNFs of Boolean functions.

Extended formulations for convex hulls of some bilinear functions

Preface. On Discrete Approximations of Continuous Functions with Bounded Second Derivative G.G. Amanzhaev. On a Property of Perfect Binary Codes S.V. Avgustinovich. Edge Neighborhoods in Normal Maps S.V. Avgustinovich, O.V. Borodin. An Upper Bound for the Length of a Snake in the n-Dimensional Unit Cube P.G. Emel'yanov. Operations and Isometric Embeddings of Graphs Related to the Metric Prolongation Property T.I. Fedoryaeva. Effective Algorithms for Solving Multi-Level Plant Location Problem E.Kh. Gimadi. On the Complexity of Realization of Boolean Functions by Circuits over an Infinite Basis O.M. Kasim-Zade. Dynamic Problems of Choosing Optimal Composition of a System of Technical Tools Yu.A. Kochetov, M.G. Pashchenko. Lower Bounds in the Problem of Choosing an Optimal Composition of a Two-Level System of Technical Tools Yu.A. Kochetov, M.G. Pashchenko. On Schedules of a Single Machine Jobs with Processing Times Nonlinear in Time A.V. Kononov. Problems of Reconstructing Words from Their Fragments and Applications V.K. Leont'ev. A Vertex Variant of the Kleitman-West Problem B. List. Regular Graphs Whose Every Edge Belongs to Many Triangles A.A. Makhnev. On Closed Classes in k-Valued Logic Which Contain a Homogeneous Switching Function S.S. Marchenkov. Strongly Essential Variables of the Functions of k-Valued Logic I. Mirchev. On Comparison Between the Sizes of Read-k-Times Branching Programs E.A. Okol'nishnikova. Some Optimization Problems of Scheduling the Transmission of Messages in a Local Communication Network A.V. Pyatkin. The Maximum-Weight Traveling Salesman Problem in Finite-Dimensional Real Spaces A.I. Serdyukov. Nonstrict Vector Summation in the Plane and Its Applications to Scheduling Problems S.V. Sevast'yanov.Bounds for Approximate Solution of a Scheduling Problem P.I. Sharygin. On Certain Bounds for the Weight of l- Balanced Boolean Functions Yu.V. Tarannikov. Spectral Properties of Perfect Binary (n,3)-Codes A.Yu. Vasile'va. Distributive Coloring of Graph Vertices V.G. Vizing. Deciphering Threshold Functions of k-Valued Logic N.Y. Zolotykh, V.N. Shevchenko. Index.

Helicon wave (whistler waves bounded transversely by a magnetic field surface) propagation is investigated for a cylindrically symmetric curved (flaring) boundary using a finite-element method where the propagation region is divided into a sequence of truncated cones. In each conical segment, a local spherical coordinate system is used with the origin at the apex of the cone. A vector wave equation for the fields is formulated for a cold plasma and reduced, in spherical coordinates, to a pair of coupled partial differential equations (PDEs) for two scalar functions. The PDEs are separable for the azimuthal eigenvalue m = 0. The -dependence is a Legendre function, with non-integer eigenvalues determined by the cone angle. The dependence on the radial coordinate, x, satisfies a fourth-order ordinary differential equation (ODE). A straightforward numerical integration of the equation from to fails for large values of , because of the existence of an exponentially growing solution to the equation. Consequently, a different approach is needed. Four independent solutions, valid for , in the form of power series in x (PS) are obtained which are each asymptotically proportional to as . For x large, four asymptotic expansions (AE) are obtained in the form of series in 1/x times or . The problem then is to find four linear combinations of the four PSs, each of which approaches one of the AEs in the limit by first deriving a double-integral representation of each PS, valid for all x, and then using its limit to match each AE to a linear combination of the PSs. For special values of or 4N, N an integer) closed-form solutions (polynomials times exponentials) result that are exact representations of the solution for all x. Solutions are computed and illustrated for an outgoing wave using the closed-form solution for special values of , and as a function of and x using the PS and AE, which have a large range of overlap. The propagating solutions of the ODE are also obtained using a WKB method and are used to calculate the propagation in a slowly diverging parabola of revolution.

We give a very short survey of the results on placing of points into the unit n-dimensional cube with mutual distances at least one. The main result is that into the 5-dimensional unit cube there can be placed no more than 40 points.

Formal power series and regular operations on fuzzy languages

We investigate the translation equation (T) , and the cocycle equations (Co1) , (Co2) in [ x ], the ring of formal power series over . Here we restrict ourselves to iteration groups of type I, i. e. to solutions of (T) of the form , where c 1 ? 1 is a generalized exponential function. It is easy to prove that the coefficient functions c n (s), a n (s), s n (s) of are polynomials, c 1(s), polynomials or rational functions in one more generalized exponential function and, in a special case, polynomials in an additive function. For example, we obtain . For this we do not need any detailed information about the polynomials P n . Under some conditions on the exponential and additive functions it is possible to replace the exponential and additive functions by independent indeterminates. In this way we obtain formal versions of the translation equation and the cocycle equations in rings of the form [ x ], [ x ], [ x ], [ x ], and [ x ]. We solve these equations in a completely algebraic way, by deriving formal differential equations or Aczel�Jabotinsky type equations for these problems. It is possible to get the structure of the coefficients in great detail which are now polynomials or rational functions. We prove the universal character (depending on certain parameters) of these polynomials or rational functions. And we deduce the canonical form S -1(yS(x)) for iteration groups of type I. This approach seems to be more clear and more general than the original one. Some simple substitutions allow us to solve these problems in rings of the form ( [ u ]) [ x ], i. e. where the coefficient functions are formal series.

A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.

In this paper, we develop a theory of free holomorphic functions on noncommutative Reinhardt domains , generated by positive regular free holomorphic functions f in n noncommuting variables and by positive integers m ≥ 1, where is the algebra of all bounded linear operators on a Hilbert space . Noncommutative Berezin transforms are used to study Hardy algebras H ∞ (D f, rad m ) and domain algebras A(D f, rad m ) associated with and compositions of free holomorphic functions. We obtain noncommutative Cartan type results for formal power series, in several noncommuting indeterminates, which leave invariant the nilpotent parts of the corresponding domains. As a consequence, we characterize the set of all free biholomorphic functions with F(0) = 0. We show that the free biholomorphic classification of the domains is the same as the classification, up to unital completely isometric isomorphisms having completely contractive hereditary extension, of the corresponding noncommutative domain algebras A(D f, rad m ). In particular, we prove that Ψ : A(D f, rad 1 ) → A(D g, rad 1 ) is a unital completely isometric isomorphism if and only if there is a free biholomorphic map φ ∈ Bih(D g 1 , D f 1 ) such that This implies that the noncommutative domains and are free biholomorphic equivalent if and only if the domain algebras A(D f, rad 1 ) and A(D g, rad 1 ) are completely isometrically isomorphic. Using the interaction between the theory of functions in several complex variables and our noncommutative theory, we provide several results concerning the free biholomorphic classification of the noncommutative domains and the classification, up to completely isometric isomorphisms, of the associated noncommutative domain (resp. Hardy) algebras. In particular, we characterize the unit ball of among the noncommutative domains , up to free biholomorphisms. We also obtain characterizations for the unitarily implemented isomorphisms of noncommutative Hardy (resp. domain) algebras in terms of free biholomorphic functions between the corresponding noncommutative domains.

We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system ePdy/dx = A(x,e)y with A holomorphic at (x0,0), suppose that x0 is formally not a turning point in the sense that no singularities appear at x0 during the standard formal solution procedure with formal fractional power series in e. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x0. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses.

Substituting non-unit formal power series into formal power series is a well-understood concept. This article describes a sense in which one can define substitution of unit formal power series, implemented in Maple’s MultivariatePowerSeries package.

Let S n be the symmetric group, T d be the subspace of C[x 1 , x 2 ,…,x n ] which contains all of the homogeneous polynomials of degree d. The representation of S n determined by T d is denoted by λ d , and the irreducible representation characterized by partition a is denoted by π(a). We denote the multiplicity of π(a) entries into λ d by N a d . As an extension of [4], this note simplifies the recursion formula of the formal power series (?)N a d t d , and obtains the explicity expressions of generating function...

Inspired by the skein-valued open Gromov–Witten theory of Ekholm and Shende and the Gopakumar–Vafa formula, we associate to each pair of non-negative integers (g,l) a formal power series with values in the HOMFLY-PT skein of a disjoint union of l solid tori. The formal power series can be thought of as open BPS-states of genus g with l boundary components and reduces to the contribution of a single BPS-state of genus g for l=0 . Using skein theoretic methods, we show that the formal power series satisfy gluing identities and multi-cover skein relations corresponding to an elliptic boundary node of the underlying curves. For (g,l)=(0,1) , we prove a crossing formula which is the multi-cover skein relation corresponding to a hyperbolic boundary node, also known as the pentagon identity.