On the best quadrature formulas for some classes of continuous functions

We obtain the best quadrature formulas for classes of continuous functions defined by various restrictions on the moduli of continuity with respect to increase and decrease. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1284–1288, September, 1998.     

Two decidable classes of formulas of the modal logic S5

With the use of S. Yu. Maslov's inverse method, we prove the decidability by derivability of two classes of formulas of the quantifier modal logic S5 containing only one-place predicate variables. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 350–360, July–September, 2000. Translated by Remigijus Lapinskas     

Trigonometric solutions of nonlinear first-order ordinary differential equations over a Banach algebra

In this paper we continue the study initiated in the recent paper (see Russian Mathematics (Iz. VUZ) 50 (8), 5–17 (2006)). We determine differentiation formulas for basic trigonometric functions and describe classes of nonlinear first-order ordinary differential equations (ODE.1) over a finite-dimensional Banach algebra whose solutions are the mentioned functions.     

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.     

The power of some forms of the induction axiom in the multiplicative arithmetic

In this paper, the sequential variants of the multiplicative arithmetic with the below-defined formsAIO,I ⁺, andI ⁻ of the induction axiom are investigated, and some relations between the classes of theorems in these systems are determined. In addition, the multiplicative systems with these induction axioms, for which the classes of derivable formulas are equivalent, are presented. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 36–47, January–March, 2000.     

Separating signs in the propositional satisfiability problem

In 1980, Monien and Speckenmeyer and (independently) Dantsin proved that the satisfiability of a propositional formula in CNF can be checked in less than 2^N steps (N is the number of variables). Later, many other upper bounds for SAT and its subproblems were proved. A formula in CNF is in CNF-(1, ∞) if each positive literal occurs in it at most once. In 1984, Luckhardt studied formulas in CNF-(1, ∞). In this paper, we prove several a new upper bounds for formulas in CNF-(1, ∞) by introducing new signs separation principle. Namely, we present algorithms working in time of order 1.1939^K and 1.0644^L for a formula consisting of K clauses containing L literal occurrences. We also present an algorithm for formulas in CNF-(1, ∞) whose clauses are bounded in length. Bibliography: 14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 241, 1997, pp. 30–71 Partially supported by the INTAS-RFBR, grant 95-0095, and by the Pro Mathematica foundation (France). Translated by E. A. Hirsch.     

Enumeration of quadrics of projective spaces over local rings

Applying the methods of integral representations of combinatorial sums, G. P. Egorychev and E. V. Zima (Comm. Algebra 36, 1426–1436 (2008)) have found simple formulas for the number of classes of projectively congruent quadrics and posed the problem of finding an algebraic proof and an interpretation of the obtained formulas. The aimof the present paper is to give a solution to this problem.     

Asymptotic distribution of classes of positive binary quadratic forms with conditions of divisibility of coefficients

Asymptotic formulas for the number of classes of positive binary quadratic forms with conditions of divisibility of extreme coefficients are obtained by the discrete ergodic method. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 123–130, 2005.     

Satisfiability of bright formulas

We investigate one solvable subclass of quantified formulas in pure predicate calculus and obtain a necessary and sufficient condition for the satisfiability of formulas from this subclass. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1432–1435, October, 2007.     

Fast computation of incomplete elliptic integral of first kind by half argument transformation

We developed a new method to calculate the incomplete elliptic integral of the first kind, F(j|m){F(\varphi|m)} , by using the half argument formulas of Jacobian elliptic functions. The method reduces the magnitude of j{\varphi} by repeated usage of the formulas while fixing m. The method is sufficiently precise in the sense that the maximum relative error is 3–5 machine epsilons at most. Thanks to the simplicity of the half argument formulas, the new procedure is significantly faster than the existing procedures. For example, it runs 20–60% faster than Bulirsch’ function, el1, and 1.9–2.2 times faster than the method using Carlson’s function, R _F .     

A family of A-stable Runge Kutta collocation methods of higher order for initial-value problems

^{We consider the construction of a special family of Runge–Kutta(RK) collocation methods based on intra-step nodal points ofChebyshev–Gauss–Lobatto type, with A-stability andstiffly accurate characteristics. This feature with its inherentimplicitness makes them suitable for solving stiff initial-valueproblems. In fact, the two simplest cases consist in the well-knowntrapezoidal rule and the fourth-order Runge–Kutta–LobattoIIIA method. We will present here the coefficients up to eighthorder, but we provide the formulas to obtain methods of higherorder. When the number of stages is odd, we have considereda new strategy for changing the step size based on the use ofa pair of methods: the given RK method and a linear multistepone. Some numerical experiments are considered in order to checkthe behaviour of the methods when applied to a variety of initial-valueproblems.}     

Interpolatory product quadratures for Cauchy principal value integrals with Freud weights

Summary. We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value integrals on the real line with Freud–type weight functions. The formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration. As a by–product, we obtain new bounds for the derivative of the functions of the second kind for these weight functions. Received July 15, 1997 / Revised version received August 25, 1998     

Chern classes in Deligne cohomology for coherent analytic sheaves

In this article, we construct Chern classes in rational Deligne cohomology for coherent sheaves on a smooth complex compact manifold. We prove that these classes satisfy the functoriality property under pullbacks, the Whitney formula and the Grothendieck–Riemann–Roch theorem for projective morphisms between smooth complex compact manifolds.     

Modification systems and integration in their Chow groups

We introduce a notion of integration on the category of proper birational maps to a given variety X, with value in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers of nonsingular birational varieties; ‘stringy’ Chern classes of singular varieties; and a zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern–Schwartz–MacPherson classes of possibly singular varieties, placing them into a context in which a ‘change-of-variable’ formula holds.     

Coherent Tangent Bundles and Gauss–Bonnet Formulas for Wave Fronts

We give a definition of ‘coherent tangent bundles’, which is an intrinsic formulation of wave fronts. In our application of coherent tangent bundles for wave fronts, the first fundamental forms and the third fundamental forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisfy the completely same conditions, and so can reverse roles with each other. For a given wave front of a 2-manifold, there are two Gauss–Bonnet formulas. By exchanging the roles of the fundamental forms, we get two new additional Gauss–Bonnet formulas for the third fundamental form. Surprisingly, these are different from those for the first fundamental form, and using these four formulas, we get several new results on the topology and geometry of wave fronts.     

Chern invariants of some flat bundles in the arithmetic Deligne cohomology

In this note, we investigate the cycle class map between the rational Chow groups and the arithmetic Deligne cohomology, introduced by Green–Griffiths and Asakura–Saito. We show nontriviality of the Chern classes of flat bundles in the arithmetic Deligne Cohomology in some cases and our proofs also indicate that generic flat bundles can be expected to have nontrivial classes. This provides examples of non-zero classes in the arithmetic Deligne cohomology which become zero in the usual rational Deligne cohomology.     

Eigenvectors of the Baxter-Bazhanov-Stroganov τ<Superscript>(2)</Superscript>(<Emphasis Type="Italic">t</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>) model with fixed-spin boundary conditions

We give explicit formulas for the eigenvectors of the transfer matrix of the Baxter-Bazhanov-Stroganov (BBS) model (N-state spin model) with fixed-spin boundary conditions. We obtain these formulas from the formulas for the eigenvectors of the periodic BBS model by a limit procedure. The latter formulas were derived in the framework of Sklyanin’s method of separation of variables. In the case of fixed-spin boundaries, we solve the corresponding T-Q Baxter equations for the functions of separated variables explicitly. As a particular case, we obtain the eigenvectors of the Hamiltonian of the Ising-like ℤ_N quantum chain model. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 94–108, April, 2008.     

Talbot quadratures and rational approximations

Many computational problems can be solved with the aid of contour integrals containing e ^z in the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly O(3^-N ), where N is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, O(9.28903^-N ), can be achieved by using a different approach: best supremum-norm rational approximants to e ^z for z∈(–∞,0], following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody–Meinardus–Varga approximants by the method of Carathéodory–Fejér approximation. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65D30, 41A20     

Some formulas related to Hurwitz–Lerch zeta functions

In this paper, we consider multiplication formulas and their inversion formulas for Hurwitz–Lerch zeta functions. Inversion formulas give simple proofs of known results, and also show generalizations of those results. Next, we give a generalization of digamma and gamma functions in terms of Hurwitz–Lerch zeta functions, and consider its properties. In all the sections, various results are always proved by multiplication and inversion formulas.