On relatively short and long sides of convex pentagons

By the relative distance of pointsa andb of a convex bodyC we mean the ratio of the Euclidean distance ofa andb to the half of the Euclidean distance ofa, b C such thatab is a longest chord ofC parallel to the segmentab. We say that a sideab of a convexn-gon is relatively short (respectively: relatively long) if the relative distance ofa andb is at most (respectively: at least) the relative distance of two consecutive vertices of the regularn-gon. We show that every convexn-gon, wheren 5, has a relatively short side and a relatively long side, and that it is affine-regular if and only if all its sides are of equal relative lengths.Research supported in part by Komitet Bada Naukowych (Committee of Scientific Research), grant number 2 2005 92 03.     

Non-Uniformly Elliptic Equations with Natural Growth in the Gradient

We prove the existence of bounded solutions for a class of nonlinear elliptic problems of type–div(a(x,u,Du))=H(x,u,Du)+f, uW ^1,p ₀()L (),where a(x,,)b(||)|| ^p , b is a continuous monotone decreasing function and |H(x,,)| k()|| ^p , k is a continuous monotone increasing function.     

C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

We prove that if aC ¹ smooth change of variable : generates a bounded composition operatorff° in the spaceA _p()=L ^p ,p2, then is linear (affine).We also prove that for a nonlinearC ¹ mapping , the norms of exponentialse ⁱ as Fourier multipliers inL ^p () tend to infinity (,||). In both results the condition C ¹ is sharp, it cannot be replaced by the Lipschitz condition.     

Nonlocal problems in the theory of the motion equations of Kelvin-Voight fluids

For the motion equations of Kelvin-Voight fluids one proves: 1) a global theorem for the existence and uniqueness of a solution (v;{u_e}) of the initial-boundary value problem on the semiaxis t R⁺ from the class W ¹ (R⁺); W ₂ ² () H()) with initial condition v_o(x) W ₂ ² () H() when the right-hand side f(x, t) L(R ⁺; L₂()); 2) a global theorem for the existence and uniqueness of a solution (v; {u_l}) on the entire axisR from the classW ¹ (R; W ₂ ² () H()) when the right-hand side f(x, t) L(R; L₂()); 3) a global theorem for the existence of at least one solution (v; {u_l}), periodic with respect to t with period , from the class W ¹ (R ⁺; W ₂ ² () H()) when the right-hand side f(x, t) L(R ⁺; L₂()) is periodic with respect to t with period , and a local uniqueness theorem for such a solution; 4) a theorem for the existence and uniqueness in the small of a solution (v; {u_l}), almost periodic with respect to t R, from V. V. Stepanov's class S ¹ (R; W ₂ ² ()H()) when the right-hand side f(x, t) S(R; L₂()) is almost periodic with respect to t; 5) the linearization principle (Lyapunov's first method) is justified in the theory of the exponential stability of the solutions of an initial-boundary value problem in the space H() and conditions are given for the exponential stability of a stationary and periodic solution, with respect to t R, of the system (1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 146–185, 1990.     

Subgroups of a finite group whose algebra of invariants is a complete intersection

Let G be a finite subgroup of GL(V), where V is a finite-dimensional vector space over the field K and char KG. We show that if the algebra of invariants K(V)^G of the symmetric algebra of V is a complete intersection then K(V)^H is also a complete intersection for all subgroups H of G such that H={ Gv (v)=v for all v V^H}.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 63–67, 1982.     

Some function classes closed under infimal convolution

It is shown that the following classes of functionsf, each defined on some subsetD of a fixed Hausdorff topological vector spaceE, are closed under the binary operation of infimal convolution: (a) the class of functionsf:D[–, ) having connectedstrict lower level sets; (b) the class of functionsf:D having compact connected lower level sets; and (c) the class of functionsf:D having compact lower level sets and for which every local minimizer is global. In (a),E need not be Hausdorff; while in (a) and (b), the word connected may be replaced by the word path-connected.     

Green's theorem from the viewpoint of applications

Making use of a line integral defined without use of the partition of unity, Green's theorem is proved in the case of two-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces W^1,p () H^1,p () (1 p < ).     

Quasigraded lie algebras,Kostant-Adler scheme,and integrable hierarchies

Using special anisotropic quasigraded Lie algebras, we obtain a number of new hierarchies of integrable nonlinear equations in partial derivatives admitting zero-curvature representations. Among them are an anisotropic deformation of the Heisenberg magnet hierarchy, a matrix and vector generalization of the Landau—Lifshitz hierarchies, new types of matrix and vector anisotropic chiral-field hierarchies, and other types of anisotropic hierarchies.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 329–345, February, 2005.     

Generation of alternating groups by pairs of conjugates

Considering the conjugacy classes of the alternating group of degreen, those classes that contain a pair of generators are in the majority. In fact, the proportion of such classes is 1 –(n), and(n) 0 asn .     

An M/G/1 queue with second optional service

We study an M/G/1 queue with second optional service. Poisson arrivals with mean arrival rate (>0) all demand the first essential service, whereas only some of them demand the second optional service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B(v) and that of the second optional service are exponential with mean service time 1/₂ (₂>0). The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly. The well-known Pollaczec–Khinchine formula and some other known results including M/D/1, M/E_k/1 and M/M/1 have been derived as particular cases.     

Averaging in parabolic systems subject to weakly dependent random actions. The L2-approach

The first initial-boundary problem for a parabolic equation with a small parameter under external action described by some random process satisfying an arbitrary condition of weak dependence is considered. Averaging of the coefficients over a time variable is carried out. The existence of a generalized solution for the initial stochastic problem as well as for the problem with an averaged equation which turns out to be deterministic is assumed. Exponential bounds of the type of the well-known Bernstein inequalities for a sum of independent random variables are established for the probability of the deviation of the solution of the initial equation from the solution of the averaged problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 315–322, March, 1991.     

Stationary regenerative phenomena

Summary Corresponding to anyp-functionp, a stationary version of the associated regenerative phenomenon is constructed for which the underlying probability measure may have infinite total mass (though it will always be -finite ifp is standard). As a trivial consequence,p is a positive-definite function. The construction is generalised to quasi-Markov chains.     

On a method of K. Uhlenbeck for proving partial regularity for solutions of certain nonlinear elliptic systems

We extend a method of K. Uhlenbeck for proving the partial regularity of solutions of nonlinear elliptic systems. We do not employ reverse Hölder type inequalities.     

Renormalization-group study of correlation functions and composite operators in the stochastic magnetohydrodynamic model

By means of the renormalization-group approach, the scaling behavior of the correlation functions in stochastic magnetohydrodynamics is studied for the case of developed turbulence. To complete the results obtained previously, we investigate the infrared asymptotics of the Green's functions with an arbitrary number of external fields in both the kinetic and magnetic regimes. Special consideration is given to the problem of dynamic scaling behavior in the magnetic regime. The critical dimensions of the composite operators of the energy functional are calculated. Using the short-distance expansion, we find the relation between the correlation functions within the inertial range and the integral turbulence scale.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 1, pp. 142–154, April, 1996.Translated by M. V. Checkhova.     

The state density of elliptic operators with random potentials

Summary The existence of the state density N() is established for certain linear elliptic coercive selfadjoint operators subject to perturbation by a unbounded random potential and its asymptotic properties discussed as (and for certain Gaussian perturbations as — ). The estimates depend on recent worlc by Métivier on the approximation numbers of the classical Sobolev spaces.The research of the author was supported by F.I.N.E.P. at U.F.R.G.S. under contract B/31/79/183/00/00 and by the CNPq.     

Elementary modifications and line configurations in ℙ<Superscript>2</Superscript>

Associated to a projective arrangement of hyperplanes ${\mathcal A}$ ⁿ is the module D$({\mathcal A})$, which consists of derivations tangent to ${\mathcal A}$. We study D$({\mathcal A})$ when ${\mathcal A}$ is a configuration of lines in ². In this setting, we relate the deletion/restriction construction used in the study of hyperplane arrangements to elementary modifications of bundles. This allows us to obtain bounds on the Castelnuovo-Mumford regularity of D$({\mathcal A})$. We also give simple combinatorial conditions for the associated bundle to be stable, and describe its jump lines. These regularity bounds and stability considerations impose constraints on Teraos conjecture.     

Homogeneous grouping of nuclear fuel cans through simulated annealing and tabu search

Collections of cans containing nuclear fuel have to be grouped in batches that are as homogeneous as possible with respect to several criteria. This highly combinatorial problem, which can be described as grouping or clustering, is tackled using simulated annealing and tabu search. Both approaches are submitted to extensive experimentation on a real data set and several artificial ones. Two variants of the basic approaches called Locally optimized simulated annealing and Tabu search with variable offset are also tested. Sensitivity to parameter choice and to problem size are investigated. All four algorithms outperform a local search heuristic previously proposed in the literature; on the class of instances dealt with, a remarkably stable ranking of the four algorithms emerges.     

Justification of an asymptotic formula for solutions of the perturbed Klein-Fock-Gordon equation

A problem with periodic boundary conditions is considered for the Fock-Klein-Gordon equation perturbed by a small nonlinear operator R[t, u, u_x, u_xx]:. It is shown that under certain conditions the solution exists and is close to the known asymptotic solution on the interval 0 t l/.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 84–92, 1981.In conclusion, the author wishes to thank I. A. Molotkov, V. M. Babich, Ya. V. Kurylev, M. M. Popov, V. B. Filippov, and all the participants in the seminar on wave diffraction (LOMI) for valuable advice, pointers, and discussions.     

Quadrature formulas with positive weights

Recent developments in the theory of stability or contractivity of numerical methods for solving ordinary differential equations (see for instance [4], [5], [8]) have renewed the interest for the study of quadrature formulas with positive weights. Nørsett-Wanner [8] and Burrage [2], [3] have given characterisation of such quadrature formulas of order 2m–2 or 2m–3. In this paper we extend these investigations to the case of formulas of order 2m–4 and then to the case where the order is 2m–7. Finally we use these results to characterise the algebraically stable methods out of a 12-parameter family of implicit Runge-Kutta methods of order 2m–4.     

Partial Orders on Transformation Semigroups

In 1986, Kowol and Mitsch studied properties of the so-called natural partial order on T(X), the total transformation semigroup defined on a set X. In particular, they determined when two total transformations are related under this order, and they described the minimal and maximal elements of (T(X), ). In this paper, we extend that work to the semigroup P(X) of all partial transformations of X, compare with another natural partial order on P(X), characterise the meet and join of these two orders, and determine the minimal and maximal elements of P(X) with respect to each order.This author gratefully acknowledges the generous support of Centro de Matematica, Universidade do Minho, Portugal during his visit in May–June 2001.Received May 27, 2002; in revised form November 27, 2002 Published online May 16, 2003