Best approximation by splines on classes of periodic functions in the metric of L

We have obtained the exact value of the upper bound on the best approximations in the metric of L on the classes W^rH of functionsf C ₂ ^r for which ¦f ^(r) (x)-f ^(r) (x)) ¦ <(¦ x-xf) [ (t) is the upwards-convex modulus of continuity] by subspaces of r-th order polynomial splines of defect 1 with respect to the partitioning k/n.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 655–664, November, 1976.     

Behavior on the real line of entire functions represented by Dirichlet series

Conditions are found under which for an entire function f represented by a Dirichlet series with finite Ritt order on some sequence (x_k), 0 < x_k , as k one has ¦f(x_k)¦=M_t((1 + 0(1) x_k), M_f(x)=sup {¦ f (z) ¦:Re z x}.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 265–269, February, 1991.     

Einige Aspekte in der Zuordnungstheorie

Zusammenfassung Gegeben seien endliche MengenX, Y undZ X × Y, Z _x ={y¦(x,y) Z},Z _y ={x¦(x,y) Z}.Man nenntA X (bzw.B Y)zuordenbar, wenn es eine Injektion:A Y (bzw.: B X) mit(x) Z _x (bzw.(y) Z _y ) gibt, und (A, B) mit #A=#B > 0 einZuordnungspaar, wenn eine Bijektionf:A B mitf(x)Z _x B (bzw.f ^–1 (y) Z _y A) existiert. Die Bijektionf heißtZuordnungsplan fürA, B.In der vorliegenden Arbeit werden Fragen nach der Existenz von optimal zuordenbaren Mengen und optimalen Zuordnungspaaren behandelt, wenn man auf den MengenX undY Ordnungen vorgibt, wobei auch Nebenbedingungen berücksichtigt werden. In manchen Fällen lassen sich anhand der Beweise Zuordnungspläne oder ihre Berechnungsvorschrift explizit angeben.Zum Schluß werden die Aussagen an konkreten, dem Bereich der Wirtschaftswissenschaften entnommenen Beispielen erläutert.

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Summary LetX, Y be finite sets andZ X × Y, Z _x ={y¦(x,y) Z},Z _y ={x¦(x,y)Z}. A X (resp.B Y) is calledassignable if there is an injection: A Y (resp.: B X) with (x) Z _x (resp.(y) Z _y ), (A, B) with #A=#B > 0 anassigned pair if there is a bijection f:A B withf (x) Z _x B (resp.f ^–1(y) Z _y A). The bijectionf is called aplan forA andB.In this paper problems are discussed concerning the existence of optimal assignable sets and optimal assigned pairs ifX andY are totally ordered, additional constraints are also considered. In some cases the proofs give explicit constructions of plans. The results are illustrated by application to problems occurring in Operations Research.

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Diese Arbeit ist mit Unterstützung des Sonderforschungsbereiches 72 an der Universität Bonn entstanden.     

Defining a metric in a linear space by means of a family of subsets

Necessary and sufficient conditions are given on a familyA _r ^r>0 of subsets of a real linear space X under which infr > 0 x A _r is a quasinorm [l] on X. It is shown that for any symmetric (about zero) closed set A in a normed space X containing the ball {x X: x l there exists a quasinorm ¦·¦ on X such that A = {x X ¦x¦ 1}. Examples are constructed of linear metric spaces in which there exists a Chebyshev line which is not an approximately compact set.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 237–246, February, 1976.     

The best one-sided approximation of the classes WrHω

In this paper we calculate the upper bounds of the best one-sided approximations, by trigonometric polynomials and splines of minimal defect in the metric of the space L, of the classes W^rH (r = 2, 4, 6, ...) of all 2-periodic functions f(x) that are continuous together with their r-th derivative f^r(x) and such that for any points x and x we have ¦f ^r (x) f^r (x) ¦ (x–x¦), where (t) is a modulus of continuity that is convex upwards.Translated from Matematicheskie Zametki, Vol. 21, No. 3, 313–327, March, 1977.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

α-Convexity of Schlicht functions

Upper and lower bounds are obtained for the radius of-convexity, R, of the schlicht within ¦z¦< 1 functions g(z), g(0)=0, and g(0)=1, for values ranging from 0 to 0.313.... The exact value of R is determined for 0.313... < 1. The results constitute the solution to a problem recently posed by the Roumanianmathematician P. T. Mocanu [1].Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 227–232, February, 1972.     

The C*-algebra of the N-dimensional harmonic oscillator

In this paper we study the Fredholm thoery of a C*-algebraOl of o-order pseudo-differential operators on L²(ⁿ). IfK denotes the ideal of all compact operators of L², the algebraOl will be generated by (i) the idealK, (ii) a function algebra CS(ⁿ) and (iii) by the bounded operators x_j, D_j, j=1,...,n, = H^–1/2, H=1+¦x¦²–. We show thatOl/K is a commutative C*-algebra with identity and obtain its Gelfany space M. This provides Fredholm criterion and index formula for a graded algebra of partial differential operators including all oeprators with polynomial coefficients. We also give Fredholm criterion and index formula for systems of such operators.     

Projections on Lp-spaces of polyanalytic functions

The fundamental result: for an arbitrary bounded, simply connected domain in , the subspace L_n,m ^p() of the space L^p(, ) ( is the plane Lebesgue measure, p 1), consisting of the (m, n)-analytic functions in , is complemented in LP(, ) (a function f is said to be (m, n)-analytic if (^m+n/¯Z^mZⁿ)f=0 in ). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczyski, the space L_n,m ^P() is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic L^P-functions in is complemented in L^P(, ). This result has been known earlier only for smooth domains.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 15–33, 1991.     

Kato's equality and spectral decomposition for positive C0-groups

Let A be the generator of a C₀-semigroup {T(t); t0} defined on a Banach lattice E. It is shown that T(t) is a lattice homomorphism for all t>0 if and only if A satisfies <¦x¦, Ax>= (xD(A), x D(A)) (where q: EE is the evaluation mapping). This equality is used to obtain a spectral decomposition for generators of positive groups.     

Random Logistic Maps II. The Critical Case

Let (X _n ) ₀ be a Markov chain with state space S=[0,1] generated by the iteration of i.i.d. random logistic maps, i.e., X _n+1=C _n+1 X _n (1–X _n ),n0, where (C _n ) ₁ are i.i.d. random variables with values in [0, 4] and independent of X ₀. In the critical case, i.e., when E(log C ₁)=0, Athreya and Dai⁽²⁾ have shown that X _n ^P 0. In this paper it is shown that if P(C ₁=1)<1 and E(log C ₁)=0 then(i) X _n does not go to zero with probability one (w.p.1) and in fact, there exists a 0<<1 and a countable set (0,1) such that for all xA(0,1), P _x (X _n for infinitely many n1)=1, where P _x stands for the probability distribution of (X _n ) ₀ with X ₀=x w.p.1. A is a closed set for (X _n ) ₀.(ii) If is the supremum of the support of the distribution of C ₁, then for all xA (a)

Simplices by point-sliding and the Yamnitsky-Levin algorithm

Yamnitsky and Levin proposed a variant of Khachiyan's ellopsoid method for testing feasibility of systems of linear inequalities that also runs in polynomial time but uses simplices instead of ellipsoids. Starting with then-simplexS and the half-space {x¦a ^Tx }, the algorithm finds a simplexS _YL of small volume that enclosesS {x¦a ^Tx }. We interpretS _YL as a simplex obtainable by point-sliding and show that the smallest such simplex can be determined by minimizing a simple strictly convex function. We furthermore discuss some numerical results. The results suggest that the number of iterations used by our method may be considerably less than that of the standard ellipsoid method.     

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.     

Diameters of classes of continuous functions in the Lp space

In the L_p(a, b) space the exact values of n-diameters (n=1, 2, ...) are found of the class H[a, b] of the functions f(x) such that ¦f(x)-f(x)¦ (¦x-x¦), where (t) is a given continuity module which is convex upwards.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 493–500, November, 1971.     

Computations of K- and L-Theory of Cocompact Planar Groups

The verification of the isomorphism conjectures of Baum and Connes and Farrell and Jones for certain classes of groups is used to compute the algebraic K- and L-theory and the topological K-theory of cocompact planar groups (=cocompact N.E.C-groups) and of groups G appearing in an extension where is a finite group and the conjugation -action on ⁿ is free outside . These computations apply, for instance, to two-dimensional crystallographic groups and cocompact Fuchsian groups.     

On the formula for the distribution of the eigenvalues of singular differential operators

In this note we construct exampLes of a function q(x), which grows arbitrarily rapidly, and a function q(x) (c₁¦x¦ q (x) c₂¦x¦ , > >0) such that for a Sturm-Liouville operator with the constructed potential functions q (x), the classical formula for the number of eigenvalues of the operator that do not exceed is not true.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 361–368, September, 1973.     

Quasimonotonicity,regularity and duality for nonlinear systems of partial differential equations

Summary We prove partial regularity for the vector-valued differential forms solving the system (A(x, ))=0, d=0, and for the gradient of the vector-valued functions solving the system div A(x, Du)=B(x, u, Du). Here the mapping A, with A(x, w) (1+ + ¦¦²)^{(p – 2)/2} (p2), satisfies a quasimonotonicity condition which, when applied to the gradient A(x, )=Df(x, ) of a real-valued functionf, is analogous to but stronger than quasiconvexity for f. The case 1相似文献   

The Lattice of Interpretability Types of Cantor Varieties

For integers 1 m < n, a Cantor variety with m basic n-ary operations _i and n basic m-ary operations _k is a variety of algebras defined by identities _k(₁( ), ... , _m( )) = _k and i(₁( ), ... ,_n( )) = y _i, where = (x ₁., ... , x _n) and = (y ₁, ... , y _m). We prove that interpretability types of Cantor varieties form a distributive lattice, , which is dual to the direct product ₁ × ₂ of a lattice, ₁, of positive integers respecting the natural linear ordering and a lattice, ₂, of positive integers with divisibility. The lattice is an upper subsemilattice of the lattice of all interpretability types of varieties of algebras.     

Brownian motions on infinite dimensional quadric hypersurfaces

Summary A potential theory on an infinite dimensional quadric hypersurfaceS is developed following Lévy's limiting procedure. For a given real sequence { _n } _n=1 a quadratic fromh(x) on an infinite dimensional real sequence spaceE is defined by and a quadric hypersurfaceS is defined byS:={xE;h(x)=c}, and the Laplacian onS is introduced by the limiting procedure. Instead of a direct use of , the Brownian motion(t)=( ₁(t)), ₂(t),...), the diffusion process ((t),P ^x ) onS with the generator is constructed by solving a system of stochastic differential equations according to . The law of large numbers forX _n (t:=( _n , _n (t)) is proved, and ergodic properties are discussed.     

Some Conditions for a C0-Semigroup to Be Asymptotically Finite-Dimensional

We study the class of bounded C ₀-semigroups T=(T _t ) _t0 on a Banach space X satisfying the asymptotic finite dimensionality condition: codim X ₀(T)<, where X ₀(T):={x X:lim_t T _t x=0}. We prove a theorem which provides some necessary and sufficient conditions for asymptotic finite dimensionality.