The degree of maps between certain 6-manifolds

For manifolds M,M of the form S² e⁴ e⁶ we compute the homomorphisms H_*M H_*M between homology groups which are realizable by a map F: M M.     

Quasianapole Interaction of Charged Fermions

We obtain the analytic expression for the total cross section of the reaction e ^– e ⁺l ^– l ⁺ (l=,) taking possible quasianapole interaction effects into account. We find numerical restrictions on the interaction parameter value from data for the reaction e ^– e ^+–+ in the energy domain below the Z ⁰ peak.     

Some remarks on the modulus of continuity of a conformal mapping of the disk onto a Jordan domain

Letd(;z, t) be the smallest diameter of the arcs of a Jordan curve with endsz andt. Consider the rapidity of decreasing ofd(;)=sup{d(;z, t):z, t , ¦z–t¦} (as 0,0) as a measure of nicety of . Letg(x) (x0) be a continuous and nondecreasing function such thatg(x)x,g(0)=0. Put¯g(x)=g(x)+x, h(x)=(¯g(x))². LetH(x) be an arbitrary primitive of 1/h ^–1(x). Note that the functionH ^–1 x is positive and increasing on (–, +),H ^–1 0 asx– andH ^–1+ asx +. The following statement is proved in the paper.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 176–184, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236 and by the International Science Foundation under grant No. NCF000.     

Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

Let A be a set, and let E be the Banach space of bounded functions : A R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,) : R × E E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to . Then there exists a solution u: [a,b] × [0,T] E of the problem ut(x,t) – uxx(x,t) = F(x,t,u(x,t)) ((x,t) R), u(x,t) = 0 ((x,t) R R).     

Turán's extremal problem in random graphs: Forbidding odd cycles

For 0<1 and graphsG andH, we writeGH if any -proportion of the edges ofG span at least one copy ofH inG. As customary, we writeC ^k for a cycle of lengthk. We show that, for every fixed integerl1 and real >0, there exists a real constantC=C(l, ), such that almost every random graphG _{n, p} withp=p(n)Cn ^–1+1/2l satisfiesG _n,p1/2+ C ^2l+1. In particular, for any fixedl1 and >0, this result implies the existence of very sparse graphsG withG _1/2+ C ^2l+1.The first author was partially supported by NSERC. The second author was partially supported by FAPESP (Proc. 93/0603-1) and by CNP_q (Proc. 300334/93-1). The third author was partially sopported by KBN grant 2 1087 91 01.     

Estimates for lebesgue constants in dimension two

Summary Let D denote the interior of a piecewise regular curve of R² having a point with Gauss curvature different from zero. We show that the Lebesgue constants L ^D relative to D behave like ^1/2 as .     

Weakly isomorphic transformations that are not isomorphic

Summary A new method for construction of transformations T _i: (X _i, B _i, _i) , i=1,2, that are factors of each other but that are not measuretheoretically isomorphic is provided. This method uses ergodic product cocycles of the form S ⁱ ₁xS ⁱ ₂x...,, where : XZ ₂ is a cocycle, S belongs to the centralizer of T and T is an ergodic translation on a compact, monothetic group X.     

Computation of Almost Split Sequences with Applications to Relatively Projective and Prinjective Modules

Let be an Artin algebra, let mod be the category of finitely generated -modules, and let Amod be a contravariantly finite and extension closed subcategory. For an indecomposable and not Ext-projective module CA, we compute the almost split sequence 0ABC0 in A from the almost split sequence 0DTrCEC0 in mod. Since the computation is particularly simple if the minimal right A-approximation of DTrC is indecomposable for all indecomposable and not Ext-projective CA, we manufacture subcategories A with the desired property using orthogonal subcategories. The method of orthogonal subcategories is applied to compute almost split sequences for relatively projective and prinjective modules.     

Isomorphism of one-place functors Ext

Let be an associative ring with identity. One considers the category of left (unitary) -modules m and also the contravariant and the covariant functors Ext ¹ ( ,A) and Ext ¹ (A, ): M_z M. One proves the following results: (1) If the homomorphism of -modules A B induces an isomorphism Ext ¹ ( ,A)Ext ¹ ( ,B), then there exist injective -modules J₁ and J₂ such that AJ₁BJ₂. (2) Every functorial morphism Ext ¹ ( ,A)Ext ¹ ( ,B) induces a certain homomorphism of -modules AB. One also obtains a dual result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 71–74, 1981.     

Arcsine Law and One Generalization

The limit distribution as T of the functional ₀ ^T f(X(t)) d t / T is found for one-dimensional null-recurrent Markov process X(t) with the diffusion a(x) and zero drift for regular in rather weak sense behavior of a(x) and f(x) as x ± .     

275-01275-01275-01Regularity of the spatial derivatives of the solutions to parabolic systems in divergence form

Sunto Si dimostra che, nelle ipotesi: fⁱ L^2,(Q, ^N), 0<< n+2, i=0,1,..., n, uL^2,(Q,^N)H _–T ^*0,1/2() (Q,^N), D_iuL^2,(Q,^N),i=1,2,...,n, la soluzione v: Q ^N del problema di Cauchy-Dirichlet: ha derivate spaziali D_iv appartenenti a L^2,(Q,^N) e che sussiste la maggiorazione:.

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Lavoro eseguito con contributo finanziario del M.U.R.S.T. e nell'ambito del G.N.A.F.A. del C.N.R.     

Groups acting simply transitively on the vertices of a building of typeà 2, II: The casesq=2 andq=3

If (P, L) is a projective plane and is a triangle presentation compatible with a point-line correspondence :P L, then gives rise to a group and a thick building of typeà ₂ on the vertices of which acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence :P L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these , is isomorphic to the building associated withG=PGL(3,K) whereK is a local field with discrete valuation and residual field of orderq. We identify the for which this is the case, and in these cases, find embeddings of intoG. We also describe the arithmetic nature of these groups.     

Asymptotic behavior of the solution of an elliptic boundary value problem in a thin domain

The asymptotic behavior as h0 of the solution of a mixed boundary value problem is investigated for an elliptic (in the sense of Petrovskii) system of second-order differential equations in the n-dimensional cylinder Q _h =×(–h/2, h/2) of small altitude h; is a domain in R ⁿ –1. The limit problem in contains a small parameter =h , (0, 1), for higher-order derivatives and degenerates regularly, as 0, into an elliptic problem of a lower order. It is shown that the limit problem and its corresponding degenerate problem (=0) are uniquely solvable. An estimate for the difference of solutions of the original and the limit problem in the energy norm is established. As an example, a problem on the deformation of a thin plate in the framework of the Cosserat continuum is considered.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 191–208, 1990.     

Quasiclassical asymptotic behavior of the scattering cross section for asymptotically homogeneous potentials

One considers the total scattering cross section on the potential gV(x), x^m, m3, for large values of the coupling constant g and of the wave number k. One assumes that V(x)(x/|1x|)|x|^–, 2>m+1, as ¦x¦. It is shown that for gk^–1 , g^3–ak^2(a–2) the scattering cross section is equal asymptotically to _a(gk^–1), x=(m–1)(–1)^–1. Here the coefficient _a is determined only by the function and the number . Under the additional conditions >0, V>0, the indicated asymptotic behavior holds in the large domain gk^–1 , gk^a–z c(gk^–1), >0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 105–136, 1986.     

The survival of the large dimensional basic contact process

Summary We consider the d-dimensional basic contact process obtaining the limit value of the probability of survival when d+, and showing that the finite dimensional distributions of the upper invariant measure become of the product form as d+.     

Functions with a finite dirichlet integral in a domain with cusp points on the boundary

Let be a domain in ⁿ, n >2, the boundary of which has a cusp point, pointing inside or outside the domain. The purpose of the paper is to characterize the traces on of the elements of the space H¹() of functions with a finite Dirichlet integral. As a consequence one establishes the existence of a linear continuous extension operator H¹ () H¹(ⁿ) under the presence of an interior cusp point on . Theorems on domains with cusps are proved with the aid of results on cylindrical domains. In the space of functions with a finite Dirichlet integral in the exterior or the interior of the cylinder one introduces the norm, depending on a small parameter and generating a norm of the trace on as an element of the quotient space. The latter is placed in correspondence with an explicitly described norm of functions on the boundary, uniformly equivalent relative to . One constructs an operator of extension of functions from the exterior of the cylinder to Rⁿ, preserving H¹, whose norm is uniformly bounded relative to . For the optimal operator of extension from the inside of the cylinder one finds the asymptotic behavior of the norm as 0. From these results there follow similar theorems on functions with a finite Dirichlet integral inside and outside a thin closed tube (of width ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 117–137, 1983.     

On the problem of two linearized wells

We study the behaviour of sequences of elastic deformationsy ^{n n} whose gradients approach two linearized wells, and give an application to magnetostriction.This article was processed by the author using the style filepljour1m from Springer-Verlag.     

On depth first search strees inm-out digraphs

We consider depth first search (DFS for short) trees in a class of random digraphs: am-out model. Let _i be thei ^th vertex encountered by DFS andL(i, m, n) be the height of _i in the corresponding DFS tree. We show that ifi/n asn, then there exists a constanta(,m), to be defined later, such thatL(i, m, n)/n converges in probability toa(,m) asn. We also obtain results concerning the number of vertices and the number of leaves in a DFS tree.     

Homotopy Structures for Algebras over a Monad

Let T be a monad over a category A. Then a homotopy structure for A, defined by a cocylinder P : A A, or path-endofunctor, can be lifted to the category A ^T of Eilenberg–Moore algebras over T, provided that P is consistent with T in a natural sense, i.e. equipped with a natural transformation : T P P T satisfying some obvious axioms. In this way, homotopy can be lifted from well-known, basic situations to various categories of algebras for instance, from topological spaces to topological semigroups, or spaces over a fixed space (fibrewise homotopy), or actions of a fixed topological group (equivariant homotopy); from categories to strict monoidal categories; from chain complexes to associative chain algebras. The interest is given by the possibility of lifting the homotopy operations (as faces, degeneracy, connections, reversion, interchange, vertical composition, etc.) and their axioms from A to A ^T , just by verifying the consistency between these operations and : T P P T. When this holds, the structure we obtain on our category of algebras is sufficiently powerful to ensure the main general properties of homotopy.     

Number of quasimodes of “bouncing-ball” type

A new two-scale expansion is proposed for eigenfunctions of bouncing-ball type and the corresponding eigenvalues of the Laplace operator with the Dirichlet condition in a domain of the plane. The eigenfunctions are concentrated in a neighborhood of a stable diameter of the domain and are numbered with two indices (p, q) where p is the number of longitudinal nodes and q the number of nodes in a direction orthogonal to the diameter. The validity of the asymptotic expansions is guaranteed for 0 q const p^–1 and >0 as p +.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 172–182, 1981.