On the Gap between the First Eigenvalues of the Laplacian on Functions and p-Forms

We study the first positive eigenvalue ^(p) ₁(g) of the Laplacian on p-forms for a connected oriented closed Riemannianmanifold (M, g) of dimension m. We show that for 2 p m – 2 a connected oriented closed manifold M admits three metrics g _i (i = 1, 2, 3) such that ^(p) ₁(g ₁)> ⁽⁰⁾ ₁(g ₁),^(p) ₁(g ₂) < ⁽⁰⁾ ₁(g ₂) and^(p) ₁(g ₃)= ⁽⁰⁾ ₁(g ₃).Furthermore, if (M, g) admits a nontrivial parallel p-form,then ^(p) ₁ ⁽⁰⁾ ₁ always holds.     

Even diameters of the classes W(r)H ω in the space C2π

For even values of n we find the exact values of the diameters d_n(W^(r)H) of the classes of 2-periodic functions ((t) is an arbitrary convex upwards modulus of continuity) in the space C₂. We find that d_2n(W^(r)H)=d_2n–1(W^(r)H) (n=1, 2, ... r=0, 1, 2, ...).Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 387–392, March, 1974.The author expresses his thanks to N. P. Korneichuk for his interest in my work.     

Conjugation for polynomial mappings

We consider Keller's functions, namely polynomial functionsf:C ⁿ C ⁿ with detf(x)=1 at allx C ⁿ. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open.Without loss of generality assumef(0)=0 andf'(0)=I. We study the existence of certain mappingsh , > 1, defined by power series in a ball with center at the origin, such thath(0)=I andh (f(x))=h (x). So eachh conjugates f to its linear part I in a ball where it is injective.We conjecture that for Keller's functionsf of the homogeneous formf(x)=x +g(x),g(sx)=s ^dg(x),g(x)ⁿ=0,xC ⁿ,sC the conjugationh for f is anentire function.     

Growth of length of sequential derivation transformed into natural one

We consider the (&, )-fragment of the intuitionistic propositional calculus. It is proved that under the standard transformation of a Gentzen derivation into a natural derivation(), the length of (())2^2·length( ). There is constructed a sequence of Gentzen derivations of length _i, for which the length of (( _i))2^{1/3·length(i}), which shows that the upper bound obtained is not too weak.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 192–196, 1979.     

On rationalized H- and co-H-spaces with an appendix on decomposable H- and co-H-spaces

Let R be a subring of the rationals with 1/2, 1/3R; let S _R ⁿ denote the R-local n-sphere and define _R ⁿ :=S _R ⁿ for n odd, _R ⁿ :=S _R ⁿ for n>0 even. An H-space (resp. a 1-conn. co-H-space) is decomposable over R, if it is homotopy equivalent to a weak product of spaces _R ⁿ (resp. to a wedge of R-local spheres). We prove that, if E is grouplike decomposable of finite type over R, the functor [-,E] is determined on finite dim. complexes by the Hopf algebra M^*(E;R); here M^* denotes the unstable cohomotopy functor of H.J. Baues. If C is cogrouplike decomposable over R, the functor [C,-] is determined on 1-conn. R-local spaces by _*(C) as a cogroup in the category of M-Lie algebras. For R = the functor [-,E] is also determined by the Lie algebra _*(E) and [C,-] by the Berstein coalgebra associated to the comultiplication of C.     

An embedding theorem for a limiting exponent

We consider the function space B _p ^l () of functionsf(x), defined on the domain of a certain class and characterized by specific differential-difference properties in L_p(). We prove a theorem on the embedding B _p,q ^l () L_q in the case whenl=n/p –n/q >0 and its generalization for vectorl, p, q.Translated from Matematicheski Zametki, Vol. 6, No. 2, pp. 129–138, August, 1969.     

Computational complexity of the integration problem for anisotropic classes

We determine the exact order of -complexity of the numerical integration problem for the anisotropic class W^r(I^d) and H^r(I^d) with respect to the worst case randomized methods and the average case deterministic methods. We prove this result by developing a decomposition technique of Borel measure on unit cube of d-dimensional Euclidean space. Moreover by the imbedding relationship between function classes we extend our results to the classes of functions W_p(I^d) and H_p(I^d). By the way we highlight some typical results and stress the importance of some open problems related to the complexity of numerical integration. Project supported by the fund of Personnel Division of Nankai University and the Program of One Hundred Distinguished Chinese Scientists of the Chinese Academy of Sciences.     

Codings of graphs with binary edge labels

LetG(V,E) be a graph. A mappingf:E{0,1} ^m is called a (binary) coding ofG, if the induced mapping , assigns different vectors to the vertices. For the Boolean sum,f is called aB-code, and for the mod 2 sum anM-code. Letm _B (G) resp.m _M (G) be the smallest lengthm for whichB-codes resp.M-codes are possible. Trivially,m _B (G),m _M (G) log₂|V|. Improving results of Z. Tuza we showm _B (G)log₂|V| + 1,m _M (G)log₂|V|+4.     

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.     

Smoothness of solutions of a nonlinear ode

Smoothness of aC -functionf is measured by (Carleman) sequence {M _k} ₀ ; we sayfC _M [0, 1] if|f ^(k) (t)|CR ^k M _k,k=0, 1, ... withC, R>0. A typical statement proven in this paper isTHEOREM: Let u, b be two C -functions on [0, 1]such that (a) u=u ²+b, (b) |b ^(k) (t)|CR ^k (k!) , >1,k_–.Then |u^(k)(t)|C₁R^k((k–1)!),k_–.The first author acknowledges the hospitality of Mathematical Research Institute of the Ohio State University during his one month visit there in the spring of 1999     

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.     

On the postulation of a general projection of a curve inP 3

Summary We say that a curve C in P ³ has maximal rank if for every integer k the restriction map r_c(k):H ⁰(P ³, O_P3(k)) H⁰ (C, O_C(k))has maximal rank. Here we prove the following results. Theorem 1Fix integers g, d with 0g3,dg+3.Fix a curve X of genus g and L Pic^d (X).If g=3and X is hyperelliptic, assume d8. Let _L(X)be the image of X by the complete linear system H ⁰(X, L). Then a general projection of _L(X)into P ³ has maximal rank. Theorem 2For every integer g0,there exists an integer d(g, 3)such that for every dd(g, 3),for every smooth curve X of genus g and every LPic^d (X) the general projection of _L(X)into P ³ has maximal rank.     

Fields with vanishing K2. Torsion in H1(X,K2) and Ch2(x)

This paper describes fields F of nonzero characteristic with the property that for all finite extensions E/F K₂E=0. We consider a somewhat wider class of fields which includes finite and separably closed fields. For smooth projective varieties X over such a field we show that the groups H¹(X, K₂){} and H²(X_et, (2)), NH³(X_et, (2)) and Ch²(X){} are isomorphic. These results are applied to describe the groups SK₁ of a smooth affine curve over such a field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 108–118, 1982.     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

The dependence on parameters of the modulus problem for families of several classes of curves

Let, where A={a₁,..., a_n} and B={b₁,...,b_m} are systems of distinguished points, and let H be a family of homotopic classes H_i, i=1, ..., j + m, of closed Jordan curves in C, where the classes H_j+, =1, ..., m, consist of curves that are homotopic to a point curve in b. Let ={₁,...,_j+m} be a system of positive numbers. By P=P(,A,B) we denote the extremal-metric problem for the family H and the numbers : for the modulusU=U(,A,B) of this problem we have the equality , whereD ^*={D ₁ ^* ,...,D _j+m ^* } is a system of domains realizinga maximum for the indicated sum in the family of all systemsD={D ₁,...,D _j+m } of domains, associated with the family H (byU(D _i )) we denote the modulus of the domain D_i, associated with the class H_i). In the present paper we investigate the manner in whichU=U(,A,B) and the moduliU=(D ₁ ^* ) depend on the parameters _i, a_k, b; moreover, we consider the conditions under which some of the doubly connected domains D _i ^* ,i=1,...,j, from the system D^* turn out to be degenerate (Theorems 1–3). In particular, one obtains an expression for the gradient of the function M, as function of the parameter a=a_k (Theorem 4). One gives some applications of the obtained results (Theorem 5).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 136–148, 1985.     

Compact groups having almost discrete orbit hypergroups

LetG be a compact group of automorphism acting continuously on a compact groupH. Then the orbit spaceH ^G is a compact hypergroup. We characterize, all solvable groupsH and compact automorphism groupsG for whichH ^G is almost discrete, i.e.,H ^G is homeomorphic to the one-point-compactification of . It turns out that thenH is isomorphic either to the infinite direct product (p) of the cyclic groups (p) or to _p ⁿ ( _p the group of allp-adic numbers) for some primep and some . The almost discrete orbit hypergroupsH ^G are determined explicitly for some examples.     

The summation of some series with variable coefficients by approximate analytical expressions

In a series of the form _k ^=1/ a _k f _k (x),a _k , which need be given only numerically, is approximated by an analytical functiong(k). If then _k ^=1/ g(k)f _k (x) can be summed exactly, it yields an approximation to the given series.Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation.     

Convex programs with an additional reverse convex constraint

A method is presented for solving a class of global optimization problems of the form (P): minimizef(x), subject toxD,g(x)0, whereD is a closed convex subset ofR ⁿ andf,g are convex finite functionsR ⁿ . Under suitable stability hypotheses, it is shown that a feasible point is optimal if and only if 0=max{g(x):xD,f(x)f( )}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblemsQ _k ,k=1, 2, ..., each of which consists in maximizing the convex functiong(x) over some polyhedronS _k . The method is similar to the outer approximation method for maximizing a convex function over a compact convex set.     

On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains

We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W _m ¹( _s ) [W _m ¹( _s )]^* in a sequence of perforated domains _s . Under a certain condition imposed on the local capacity of the set \ _s , we prove the following principle of compensated compactness: , where r _s(x) and z _s(x) are sequences weakly convergent in W _m ¹() and such that r _s(x) is an analog of a corrector for a homogenization problem and z _s(x) is an arbitrary sequence from whose weak limit is equal to zero.     

An algorithm for constructing an optimal structural schedule

The article is devoted to the problem of finding an optimal schedule for a class of functionals ƒ which allows for the existence of a structural set of activities. The functionalƒ(R), where, is defined in the following way: where {_i(t)} is a structural set of functions, and the function F is defined on any finite set of arguments and satisfies the following conditions: 1)F(x)=(x); 2) F(x₁,x₂)=(x₁,x₂), F(x₁,x₂,...x₃)= (x₁, F(x₂,...,x_s)), S2; 3) and do not decrease in each of their arguments, and moreover, 3a) strictly increases with the increase of both arguments, 3b) if (x₁,x₂)>(x₁, x₂ (x₂, x₃)> (x₂,x₃), then F(x₁,x₂,x₃)>F(x₁,x₂,x₃).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 5–20, 1983.