Topologically invariant Chern numbers of projective varieties

We prove that a rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three we prove that only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation-preserving. These results solve a long-standing problem of Hirzebruch's. We also determine the linear combinations of Chern numbers that can be bounded in terms of Betti numbers.     

Reduction of Silverman-like games to games on bounded sets

A two person zero sum game is regarded as Silverman-like if the strategy sets are sets of real numbers bounded below, the payoff function is bounded, the maximum payoff is achieved whenever the second player's numbery exceeds the first player's numberx by “too much”, and the minimum is achieved wheneverx exceedsy by “too much”. Explicit upper bounds are obtained for pure strategies to be included in an optimal mixed strategy in such games. In particular, if the strategy sets are discrete, the games may be reduced to games on specified finite sets.     

Decidability and Specker sequences in intuitionistic mathematics

A bounded monotone sequence of reals without a limit is called a Specker sequence. In Russian constructive analysis, Church's Thesis permits the existence of a Specker sequence. In intuitionistic mathematics, Brouwer's Continuity Principle implies it is false that every bounded monotone sequence of real numbers has a limit. We claim that the existence of Specker sequences crucially depends on the properties of intuitionistic decidable sets. We propose a schema (which we call ED ) about intuitionistic decidability that asserts “there exists an intuitionistic enumerable set that is not intuitionistic decidable” and show that the existence of a Specker sequence is equivalent to ED . We show that ED is consistent with some certain well known axioms of intuitionistic analysis as Weak Continuity Principle, bar induction, and Kripke Schema. Thus, the assumption of the existence of a Specker sequence is conceivable in intuitionistic analysis. We will also introduce the notion of double Specker sequence and study the existence of them (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A proof of the bounded graph conjecture

Summary An infinite graph is called bounded if for every labelling of its vertices with natural numbers there exists a sequence of natural numbers which eventually exceeds the labelling along any ray in the graph. We prove an old conjecture of Halin, which characterizes the bounded graphs in terms of four forbidden topological subgraphs.Oblatum 17-IV-1991 & 25-X-1991     

Specker sequences revisited

Specker sequences are constructive, increasing, bounded sequences of rationals that do not converge to any constructive real. A sequence is said to be a strong Specker sequence if it is Specker and eventually bounded away from every constructive real. Within Bishop's constructive mathematics we investigate non‐decreasing, bounded sequences of rationals that eventually avoid sets that are unions of (countable) sequences of intervals with rational endpoints. This yields surprisingly straightforward proofs of certain basic results fromconstructive mathematics. Within Russian constructivism, we show how to use this general method to generate Specker sequences. Furthermore, we show that any nonvoid subset of the constructive reals that has no isolated points contains a strictly increasing sequence that is eventually bounded away from every constructive real. If every neighborhood of every point in the subset contains a rational number different from that point, the subset contains a strong Specker sequence. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the regularization of mixed complementarity problems

A variational inequality problem (VIP) satisfying a constraint qualification can be reduced to a mixed complementarity problem (MOP). Monotonicity of the VIP implies that the MOP is also monotone. Introducing regularizing perturbations, a sequence of strictly monotone mixed complementarity problems is generated. It is shown that, if the original problem is solvable, the sequence of computable inexact solutions of the strictly monotone MCP's is bounded and every accumulation point is a solution. Under an additional condition on the precision used for solving each subproblem, the sequence converges to the minimum norm solution of the MCP.     

On Conservative Matrix Methods for Double Sequence Spaces

We consider two types of summability methods of double sequences defined by 4-dimensional matrices. Our concern is with methods preserving the regular convergence (in Hardy"s sense) and the -convergence of double sequences defined as follows: every column is convergent and the sequence of column limits converges as well. Of primary interest is whether the following well-known facts of summability concerning the summation of bounded sequences by conservative matrices can be extended to 4-dimensional matrices: (1) if the space of all convergent sequences is closed in a summability domain then the matrix does not sum bounded divergent sequences, (2) if a conservative matrix sums a divergent sequence then it sums an unbounded sequence and (3) two regular matrices are consistent on bounded sequences. We emphasize that the negative answer to (2), in the case of regular convergence, shows that the main result in [8] fails in general.     

On characterizations of the ordered field of real numbers

This self-contained note could find classroom use in an introductory course on analysis. It is proved that an ordered field F is complete (that is, order-isomorphic to the field of real numbers) if and only if each bounded monotonic sequence in F converges in F. Also established is the key tool that an ordered field is complete if and only if it is Archimedean and Cauchy-complete, along with a number of characterizations of Archimedean fields.     

On balanced games with infinitely many players: Revisiting Schmeidler's result

We consider transferable utility cooperative games with infinitely many players and the core understood in the space of bounded additive set functions. We show that, if a game is bounded below, then its core is non-empty if and only if the game is balanced. This finding generalizes Schmeidler (1967) “On Balanced Games with Infinitely Many Players”, where the game is assumed to be non-negative. We also generalize Schmeidler's (1967) result to the case of restricted cooperation too.     

Weakly Computable Real Numbers

A real number x is recursively approximable if it is a limit of a computable sequence of rational numbers. If, moreover, the sequence is increasing (decreasing or simply monotonic), then x is called left computable (right computable or semi-computable). x is called weakly computable if it is a difference of two left computable real numbers. We show that a real number is weakly computable if and only if there is a computable sequence (x_s)_s of rational numbers which converges to x weakly effectively, namely the sum of jumps of the sequence is bounded. It is also shown that the class of weakly computable real numbers extends properly the class of semi-computable real numbers and the class of recursively approximable real numbers extends properly the class of weakly computable real numbers.     

Interior estimates for Monge-Ampère equation in terms of modulus of continuity

We investigate the Monge-Ampère equation subject to zero boundary value and with a positive right-hand side function assumed to be continuous or essentially bounded. Interior estimates of the solution's first and second derivatives are obtained in terms of moduli of continuity. We explicate how the estimates depend on various quantities but have them independent of the solution's modulus of convexity. Our main theorem has many useful consequences. One of them is the nonlinear dependence between the Hölder seminorms of the solution and of the right-side function, which confirms the results of Figalli, Jhaveri & Mooney in [7]. Our technique is in part inspired by Jian & Wang in [11] which includes using a sequence of so-called sections.     

A characterization of conjugate WCG banach spaces

It is proved that a WCG Banach space X is isomorphic to a conjugate Banach space if and only if there exists a closed subspace V of its conjugate space X' with positive characteristic such that X possesses the following summability property with respect to V: For every bounded sequence in X there exists a regular essentially positive summability method A such that the A-means of the sequence are σ(X,V)-convergent in X. This extends a well-known theorem of Nishiura-Waterman [8] and yields an analogous characterization of quasi-reflexive spaces. Conjugate spaces of smooth Banach spaces can also be characterized by the above summability condition.     

Trace-positive polynomials and the quartic tracial moment problem

The tracial analog of Hilbert's classical result on positive binary quartics is presented: a trace-positive bivariate noncommutative polynomial of degree at most four is a sum of hermitian squares and commutators. This is applied via duality to investigate the truncated tracial moment problem: a sequence of real numbers indexed by words of degree four in two noncommuting variables with values invariant under cyclic permutations of the indexes, can be represented with tracial moments of matrices if the corresponding moment matrix is positive definite. Understanding trace-positive polynomials and the tracial moment problem is one of the approaches to Connes' embedding conjecture.     

Convergence of a large time step generalization of Godunov's method for conservation laws

A natural generalization of Godunov's method for Courant numbers larger than 1 is obtained by handling interactions between neighboring Riemann problems linearly, i.e., by allowing waves to pass through one another with no change in strength or speed. This method is well defined for arbitrarily large Courant numbers and can be written in conservation form. It follows that if a sequence of approximations converges to a limit u(x,t) as the mesh is refined, then u is a weak solution to the system of conservation laws. For scalar problems the method is total variation diminishing and every sequence contains a convergent subsequence. It is conjectured that in fact every sequence converges to the (unique) entropy solution provided the correct entropy solution is used for each Riemann problem. If the true Riemann solutions are replaced by approximate Riemann solutions which are consistent with the conservation law, then the above convergence results for general systems continue to hold.     

A splitting method for quadratic programming problem

1. IntroductionThe quadratic programming (QP) problem is the most simple one in nonlinear pro-gramming and plays a very important role in optimization theory and applications.It is well known that matriX splitting teChniques are widely used for solving large-scalelinear system of equations very successfully. These algorithms generate an infinite sequence,in contrast to the direct algorithms which terminate in a finite number of steps. However,iterative algorithms are considerable simpler tha…     

随机变量序列的大数定律及常返性定理

对一类有界独立或相依的随机变量序列|ξn|,获得了它的伯努利大数定律、波雷尔强大数定律及常返性定理.作为应用,得出了Loève专著[1]中的推广的伯努利大数定律、常返性定理,改进了[1]中的推广的波雷尔强大数定律.     

一般型三维簇的三典范映射

研究了一般型三维射影代数簇的三典范映射性质.当几何亏格p_g≥2时,整数m≥4的多典范映射结构已经比较清晰,但是三典范映射的结构还有很多未知问题.如果三典范映射φ_3不是双有理的,本文证明了φ_3的映射次数(如果φ_3是一般有限映射)或者一般纤维中一般不可约分支的几何亏格(如果φ_3是纤维型)有一个上界.     

On multiplicities of cocharacters for algebras with superinvolution

In this paper we deal with finitely generated superalgebras with superinvolution, satisfying a non-trivial identity, whose multiplicities of the cocharacters are bounded by a constant. Along the way, we prove that the codimension sequence of such algebras is polynomially bounded if and only if their colength sequence is bounded by a constant.     

Problems of Linear Discrete Games of Pursuit

In this paper, we consider two problems of linear discrete games of pursuit. In each of them, the terms of the sequence defining the pursuer’s control are bounded by some positive number. In the first problem, the terms of the sequence defining the quarry’s control are bounded by some positive number and, in the second problem, the sum of the pth powers of the terms of this sequence is bounded by a given number. For each problem, we obtain a necessary and sufficient condition for the termination of pursuit from all points in space.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 707–718.Original Russian Text Copyright ©2005 by G. I. Ibragimov.     

Register computations on ordinals

We generalize ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. Ordinal register machines are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets of ordinals in a model of the Zermelo-Fraenkel axioms ZFC. This allows the following characterization of computable sets: a set of ordinals is ordinal register computable if and only if it is an element of Gödel’s constructible universe L.