Modified realizability and predicate logic

Semantics of predicate formulas based on the notion of modified realizability for arithmetic formulas and interpretations of the language of arithmetic in all finite types are considered. For a number of natural constructive interpretations, the corresponding predicate logic of modified realizability is proved to be nonarithmetical. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 259–269, February, 1997. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00416 and by the International Science Foundation under grant No. NFQ000. Translated by V. N. Dubrovsky     

Small solutions of quadratic equations with prime variables in arithmetic progressions

A necessary and sufficient solvable condition for diagonal quadratic equation with prime variables in arithmetic progressions is given, and the best qualitative bound for small solutions of the equation is obtained,     

Signomial dual Kuhn-tucker intervals

Signomial programs are a special type of nonlinear programming problems which are especially useful in engineering design. This paper applies interval arithmetic, a generalization of ordinary arithmetic, to a dual equilibrium problem in signomial programming. Two constructive applications are considered. Application I involves uniqueness of local solutions; Application II involves existence and error bounds.The authors are grateful to the National Science Foundation for support through a Graduate Fellowship and Grant No. GK-41301.     

Interval arithmetic in unidimensional signomial programming

This paper applies an interval arithmetic version of Newton's method to unidimensional problems in signomial programming. Unidimensional dual problems occur in engineering design problems formulated as a signomial program with a single degree of difficulty. Unidimensional primal problems are of interest, since many multidimensional search procedures involve unidimensional searches. The interval arithmetic method is guaranteed to generate all the local optima.The authors are grateful to the National Science Foundation for support through a Graduate Fellowship and Grant No. GK-41301.     

The Cauchy Problems and Global Solutions to the Deybe System and Burgers＇ Equations in Pseudomeasure Spaces

In this paper we consider the Cauchy problems of Burgers＇ equations and the Deybe system. Their existence and uniqueness of the time-global solutions for small initial data in some pseudomeasure spaces are obtained. The asymptotic stability of small solutions is proved. As an immediate result the existence and uniqueness of the self-similar solutions are also obtained provided the initial data satisfy the self-similar structures.     

Idempotents in compact semigroups and Ramsey theory

We prove a theorem about idempotents in compact semigroups. This theorem gives a new proof of van der Waerden’s theorem on arithmetic progressions as well as the Hales-Jewett theorem. It also gives an infinitary version of the Hales-Jewett theorem which includes results of T. J. Carlson and S. G. Simpson. Research supported by the National Science Foundation under Grant No. DMS86-05098.     

On Singular Sets of Local Solutions to p-Laplace Equations

The author proves that the on the singular set of a local solution to existence of an optimal control problem. right-hand term of a p-Laplace equation is zero the equation. Such a result is used to study the     

Logarithmic density and measures on semigroups

Davenport and Erdős [3] proved that every setA of integers with the property thata ∈A impliesan ∈A for alln (multiplicative ideal) has a logarithmic density. I generalized [5] this result to sets with the property that if for some numbersa, b, n we havea ∈ A, b ∈ A andan ∈ A, then necessarilybn ∈ A, which I call quasi-ideals. Here a new proof of this theorem is given, applying a result on convolution of measures on discretes semigroups. This leads to further generalizations, including an improvement of a result of Warlimont [8] on ideals in abstract arithmetic semigroups. Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901     

Slope stability and exceptional divisors of high genus

We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse direction we show that slope stability of surfaces can be tested with divisors, and prove that for surfaces with non-negative Kodaira dimension any destabilising divisor must have negative self-intersection and arithmetic genus at least two. We also prove that a destabilising divisor can never be nef, and as an application give an example of a surface that is slope stable but not K-stable. D. Panov was supported by EPSRC grant number EP/E044859/1 and J. Ross was partially supported by the National Science Foundation, Grant No. DMS-0700419.     

Blocking analysis andT(1) theorem

TheT(1) theorem with a weak condition on the distribution kernel is proved by using a new method—blocking analysis. It improves a result of Meyer’s. Project supported by the National Natural Science Foundation of China (Grant No. 19631080) and the Foundation of Advanced Research Center, Zhongshan University.     

A new technique for obtaining diophantine representations via elimination of bounded universal quantifiers

M. Davis proved in the early 1950s that every recursively enumerable set has an arithmetic representation with a unique bounded universal quantifier, known today as the Davis normal form. Davis, H. Putnam, and J. Robinson showed in 1961 how the Davis normal form can be transformed into a purely existential exponential Diophantine representation which uses not only addition and multiplication, but also exponentiation. The present author eliminated the exponentiation in 1970 and thus obtained the unsolvability of Hilbert's tenth problem. The paper presents a new method for transforming the Davis normal form into the exponential Diophantine representation. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 220, 1995, pp. 83–92. Original article Translated by Yu. V. Matiyasevich. The research described in this work was made possible in part by Grant No. 94-01-01030 from the Russsian Foundation for Fundamental Research and Grant No. R43000 from the International Science Foundation.     

An integral estimate of Bessel function and its application

In this paper the authors give a new integral estimate of the Bessel function,which is an extension of Calder(?)n-Zygmund's result.As an application of this result,we prove that the parameterized Marcinkiewicz integralμ_Ω~p with variable kernels is of type (2,2),where the kernel functionΩdoes not have any smoothness on the unit sphere in R~n.     

A mod 2 index theorem for the twisted Signature operator

A mod 2 index theorem for the twisted Signature operator on 4 _q + 1 dimensional manifolds is established. This result generalizes a result of Farber and Turaev, which was proved for the case of orthogonal flat bundles, to arbitrary real vector bundles. It also provides an analytic interpretation of the sign of the Poincare-Reidemeister scalar product defined by Farber and Turaev. Project partially supported by the National Natural Science Foundation of China (Grant No. 19525102), the Fok Ying-Tung Foundation and the Qiu Shi Foundation.     

On Matrix Elements for the Quantized Cat Map Modulo Prime Powers

The quantum cat map is a model for a quantum system with underlying chaotic dynamics. In this paper we study the matrix elements of smooth observables in this model, when taking arithmetic symmetries into account. We give explicit formulas for the matrix elements as certain exponential sums. With these formulas we can show that there are sequences of eigenfunctions for which the matrix elements decay significantly slower then was previously expected. We also prove a limiting distribution for the fluctuation of the normalized matrix elements around their average. Submitted: March 3, 2008., Accepted: August 11, 2008. This material is based upon work supported by the National Science Foundation under agreement No. DMS-0635607. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.     

The Essential Spectrum of Schrödinger, Jacobi, and CMV Operators

We provide a very general result which identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover within a unified framework the HVZ Theorem and Krein's results on orthogonal polynomials with finite essential spectra. Supported in part by The Israel Science Foundation (grant No. 188/02). Supported in part by NSF grant DMS-01 40592. Research supported in part by grant No. 2002068 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.     

Linear independence of refinable distributions supported on [0, 2]×[0, 2]

In this paper a necessary and sufficient condition on symbol is given for linear independence of refinable distributions supported on [0,2]×[0,2]. This project is partially supported by the National Natural Science Foundation of China (No. 69735020), the Tian Tuan Foundation, the Doctoral Bases Promotion Foundation of National Educational Commission of China (No. 97033519) and the Zhejiang Provincial Science Foundation of China (No. 196038).     

Sufficient and Necessary Conditions for Stochastic Comparability of Jump Processes

Abstract This note is devoted to the study of the stochastic comparability of jump processes. On the basis of [2] and [3], it is proved that two jump processes are stochatically comparable if and only if their q-pairs are comparable. Meanwhile, the result concerning the uniqueness given in [6] is also improved upon. Research supported in part by DPFIHE(Grant No.96002704), NNSFC(Grant No.19771008), MCSEC, Ying-Tung Fok Educational Foundation and Youth Science Foundation of BNU     

The Substitution Secant/Finite Difference Method for Large Scale Sparse Unconstrained Optimization

This paper studies a substitution secant/finite difference （SSFD） method for solving large scale sparse unconstrained optimization problems. This method is a combination of a secant method and a finite difference method, which depends on a consistent partition of the columns of the lower triangular part of the Hessian matrix. A q-superlinear convergence result and an r-convergence rate estimate show that this method has good local convergence properties. The numerical results show that this method may be competitive with some currently used algorithms.     

On the proof of the reciprocity law for arithmetic Siegel modular functions

Earlier we obtained a new proof of Shimura’s reciprocity law for the special values of arithmetic Hilbert modular functions. In this note we show how from this result one may derive Shimura’s reciprocity law for special values of arithmetic Siegel modular functions. To achieve this we use Shimura’s classification of the special points of the Siegel space, Satake’s classification of the equivariant holomorphic imbeddings of Hilbert-Siegel modular spaces into a larger Siegel space, and, finally, a corrected version of some of Karel’s results giving an action of the Galois group Gal(Q_ab/Q) on arithmetic Siegel modular forms. Research supported in part by the NSF Grant No. DMS-8601130.     

Fuzzy Topology Based on Residuated Lattice-Valued Logic

We use a semantical method of complete residuated lattice-valued logic to give a generalization of fuzzy topology as a partial answer to a problem by Roser and Turquette. This work is supported by the National Foundation for Distinguished Young Scholars (Grant No: 69725004), Research and Development Project of High-Technology (Grant No: 863-306-ZT06-04-3) and Foundation of Natural Sciences (Grant No: 69823001) of China and Fok Ying-Tung Education Foundation