Jackson-type inequalities and widths of function classes in L 2

The sharp Jackson-type inequalities obtained by Taikov in the space L ₂ and containing the best approximation and the modulus of continuity of first order are generalized to moduli of continuity of kth order (k = 2, 3, ... ). We also obtain exact values of the n-widths of the function classes F(k, r, Φ) and F _k ^r (h), which are a generalization of the classes F(1, r, Φ) and F _k ^r (h) studied by Taikov.     

ON possible Turán densities

The Turán density π(F) of a family F of k-graphs is the limit as n → ∞ of the maximum edge density of an F-free k-graph on n vertices. Let Π _∞ ^(k) consist of all possible Turán densities and let Π _fin ^(k) ? Π _∞ ^(k) be the set of Turán densities of finite k-graph families. Here we prove that Π _fin ^(k) contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that Π _fin ^(k) contains an irrational number for each k ≥ 3. Also, we show that Π _∞ ^(k) has cardinality of the continuum. In particular, Π _∞ ^(k) ≠ Π _fin ^(k) .     

On the Rabinowitz Floer homology of twisted cotangent bundles

Let (M, g) be a closed connected orientable Riemannian manifold of dimension n????2. Let ??:?=??? ₀?+??? ^* ?? denote a twisted symplectic form on T ^* M, where ${\sigma\in\Omega^{2}(M)}$ is a closed 2-form and ?? ₀ is the canonical symplectic structure ${dp\wedge dq}$ on T ^* M. Suppose that ?? is weakly exact and its pullback to the universal cover ${\widetilde{M}}$ admits a bounded primitive. Let ${H:T^{*}M\rightarrow\mathbb{R}}$ be a Hamiltonian of the form ${(q,p)\mapsto\frac{1}{2}\left|p\right|^{2}+U(q)}$ for ${U\in C^{\infty}(M,\mathbb{R})}$ . Let ?? _k :?=?H ^?1(k), and suppose that k?>?c(g, ??, U), where c(g, ??, U) denotes the Ma?é critical value. In this paper we compute the Rabinowitz Floer homology of such hypersurfaces. Under the stronger condition that k?>?c ₀(g, ??, U), where c ₀(g, ??, U) denotes the strict Ma?é critical value, Abbondandolo and Schwarz (J Topol Anal 1:307?C405, 2009) recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k?>?c(g, ??, U), thus covering cases where ?? is not exact. As a consequence, we deduce that the hypersurface ?? _k is never (stably) displaceable for any k?>?c(g, ??, U). This removes the hypothesis of negative curvature in Cieliebak et?al. (Geom Topol 14:1765?C1870, 2010, Theorem 1.3) and thus answers a conjecture of Cieliebak, Frauenfelder and Paternain raised in Cieliebak et?al. (2010). Moreover, following Albers and Frauenfelder (2009; J Topol Anal 2:77?C98, 2010) we prove that for k?>?c(g, ??, U), any ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ has a leaf-wise intersection point in ?? _k , and that if in addition ${\dim\, H_{*}(\Lambda M;\mathbb{Z}_{2})=\infty}$ , dim M????2, and the metric g is chosen generically, then for a generic ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ there exist infinitely many such leaf-wise intersection points.     

On ≤k-Edges, Crossings, and Halving Lines of Geometric Drawings of K n

Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by $\operatorname {cr}(P)$ , is the rectilinear crossing number of P. A halving line of P is a line passing through two points of P that divides the rest of the points of P in (almost) half. The number of halving lines of P is denoted by h(P). Similarly, a k-edge, 0??k??n/2?1, is a line passing through two points of P and leaving exactly k points of P on one side. The number of ??k-edges of P is denoted by E _??k (P). Let $\overline {\mathrm {cr}}(n)$ , h(n), and E _??k (n) denote the minimum of $\operatorname {cr}(P)$ , the maximum of h(P), and the minimum of E _??k (P), respectively, over all sets P of n points in general position in the plane. We show that the previously best known lower bound on E _??k (n) is tight for k<?(4n?2)/9? and improve it for all k???(4n?2)/9?. This in turn improves the lower bound on $\overline {\mathrm {cr}}(n)$ from $0.37968\binom{n}{4}+\varTheta (n^{3})$ to $\frac{277}{729}\binom{n}{4}+\varTheta (n^{3})\geq 0.37997\binom{n}{4}+\varTheta (n^{3})$ . We also give the exact values of $\overline {\mathrm {cr}}(n)$ and h(n) for all n??27. Exact values were known only for n??18 and odd n??21 for the crossing number, and for n??14 and odd n??21 for halving lines.     

On the order of magnitude of Fourier coefficients with respect to uniformly bounded orthonormal systems

пУсть {Φ _k (x) ∶k>0} — РАВНО МЕРНО ОгРАНИЧЕННАь ОРтОНОРМИРОВАННАь с ИстЕМА НА пРОстРАНстВЕX с кО НЕЧНОИ пОлОжИтЕльНО И МЕРОИ. Дль ФУНкцИИf ?L ^p (X) Дль НЕкОтОРОгО 1≤p≤2 ВВОДИМ ЕЕ кОЁФФИцИЕН ты ФУРьЕc _k (f) пО сИстЕМЕΦ _k (x). Дль жАДАННОгОα>0 М ы РАссМАтРИВАЕМ пОслЕДОВАтЕльНОсть {λ _k ^α :k ≥ 0} пОлОжИтЕльНых ЧИсЕл тАкИх, ЧтО т.Е. тАкИх, ЧтО гДЕA _n ^α — тАк НАжыВАЕМыЕ ЧИслА ЧЕжАРО. Мы ИжУЧАЕМ ЁтУ жАДАЧУ Дль ДВОИНых ОРтОгОНА льНых сИстЕМ, А тАкжЕ Дль НЕпРЕРыВНых ФУНкцИИ. пРИ ДОкАжАтЕльстВЕ М ы кОМБИНИРУЕМ МЕтОДы т ЕОРИИ сУММИРОВАНИь И ФУНкцИОНАльНОгО АН АлИжА.     

Asymptotic Results for Tail Probabilities of Sums of Dependent and Heavy-Tailed Random Variables

Abstract Let X1,X2,...be a sequence of dependent and heavy-tailed random variables with distributions F1,F2,…. on (-∞,∞),and let т be a nonnegative integer-valued random variable independent of the seq...     

On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W ₂ ^θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s _2k (q) = λ _k ^1/2 (q) ? k, s _2k?1(q) = μ _k ^1/2 (q) ? k ? 1/2, where {λ _k } ₁ ^∞ and {μ _k } ₁ ^∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t ₂ ^θ such that the mapping F:W ₂ ^θ?1 →t ₂ ^θ defined by the equality F(q) = {s _n } ₁ ^∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W ₂ ^θ?1 and t ₂ ^θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥_τ ≤ C∥q∥_θ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥q∥_θ? ≤ R, but is independent of the function q varying in this ball.     

A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

A partial orthomorphism of ${\mathbb{Z}_{n}}$ is an injective map ${\sigma : S \rightarrow \mathbb{Z}_{n}}$ such that ${S \subseteq \mathbb{Z}_{n}}$ and ??(i)?Ci ? ??(j)? j (mod n) for distinct ${i, j \in S}$ . We say ?? has deficit d if ${|S| = n - d}$ . Let ??(n, d) be the number of partial orthomorphisms of ${\mathbb{Z}_{n}}$ of deficit d. Let ??(n, d) be the number of partial orthomorphisms ?? of ${\mathbb{Z}_n}$ of deficit d such that ??(i) ? {0, i} for all ${i \in S}$ . Then ??(n, d) =???(n, d)n ²/d ² when ${1\,\leqslant\,d < n}$ . Let R _{k, n} be the number of reduced k ×?n Latin rectangles. We show that $$R_{k, n} \equiv \chi (p, n - p)\frac{(n - p)!(n - p - 1)!^{2}}{(n - k)!}R_{k-p,\,n-p}\,\,\,\,(\rm {mod}\,p)$$ when p is a prime and ${n\,\geqslant\,k\,\geqslant\,p + 1}$ . In particular, this enables us to calculate some previously unknown congruences for R _{n, n} . We also develop techniques for computing ??(n, d) exactly. We show that for each a there exists??? _a such that, on each congruence class modulo??? _a , ??(n, n-a) is determined by a polynomial of degree 2a in n. We give these polynomials for ${1\,\leqslant\,a\,\leqslant 6}$ , and find an asymptotic formula for ??(n, n-a) as n ?? ??, for arbitrary fixed a.     

Quasikonvexe Multiplikatoren starker Konvergenz

пУсть {f _k; f _k ^* ?X×X^* — пОлНАь БИОРтОгОНАльНАь сИс тЕМА В БАНАхОВОМ пРОстРАН стВЕ X (X^* — сОпРьжЕННОЕ пРОст РАНстВО). пУсть (?→+0) $$\begin{gathered} S_n f = \sum\limits_{k = 0}^n {f_k^* (f)f_k ,} K(f,t) = \mathop {\inf }\limits_{g \in Z} (\left\| {f - g} \right\|_x + t\left| g \right|_z ), \hfill \\ X_0 = \{ f \in X:\mathop {\lim }\limits_{n \to \infty } \left\| {S_n f - f} \right\|_x = 0\} ,X_\omega = \{ f \in X:K(f,t) = 0(\omega (t))\} , \hfill \\ \end{gathered} $$ гДЕZ?X — НЕкОтОРОЕ пОД пРОстРАНстВО с пОлУН ОРМОИ ¦·¦ И Ω — МОДУль НЕпРЕРыВНО стИ УДОВлЕтВОРьУЩИИ Усл ОВИУ sup Ω(t)/t=∞. пОслЕДОВАтЕ льНОстьΤ={Τ _k} кОМплЕксНых ЧИ сЕл НАжыВАЕтсь МНОжИтЕл ЕМ сИльНОИ схОДИМОст И ДльX _Τ, жАпИсьΤ?М[X _Τ,X _Τ], ЕслИ Д ль кАжДОгО ЁлЕМЕНтАf?X _Τ сУЩЕстВ УЕт тАкОИ ЁлЕМЕНтf ^τ?х₀, ЧтОf _k ^* (f ^τ)=Τ_kf _k ^* (f) Дль ВсЕхk. ДОкА жАНО сРЕДИ ДРУгИх слЕДУУЩ ЕЕ УтВЕРжДЕНИЕ. тЕОРЕМА. пУсmь {f_k; f _k ^* } —Н ЕкОтОРыИ (с, 1)-БАжИс тАк ОИ, ЧтО ВыпОлНьУтсь НЕРАВЕН стВА тИпА НЕРАВЕНстВА ДжЕ ксОНА с пОРьДкОМ O(?_n) u тИ пА НЕРАВЕНстВА БЕРНшmЕИ НА с пОРьДкОМ O(1/?_n). ЕслИ пОслЕДОВАтЕл ьНОсть Τ кВАжИВыпУкл А И ОгРАНИЧЕНА, тО $$\tau \in M[X_{\omega ,} X_0 ] \Leftrightarrow \omega (\varphi _n )\tau _n \left\| {S_n } \right\|_{[X,X]} = o(1).$$ ЁтОт ОБЩИИ пОДхОД НЕМ ЕДлЕННО ДАЕт клАссИЧ ЕскИЕ РЕжУльтАты, ОтНОсьЩИ Есь к ОДНОМЕРНыМ тРИгОНОМЕтРИЧЕскИМ РьДАМ. НО тЕпЕРь ВОжМО жНы ДАльНЕИшИЕ пРИлОжЕН Иь, НАпРИМЕР, к РАжлОжЕНИьМ пО пОлИ НОМАМ лЕжАНДРА, лАгЕР РА ИлИ ЁРМИтА.     

Tail Behavior of Sums and Maxima of Sums of Dependent Subexponential Random Variables

In this paper, we consider dependent random variables X _k , k=1,2,?? with supports on [?b _k ,??), respectively, where the b _k ??0 are some finite constants. We derive asymptotic results on the tail probabilities of the quantities $S_{n}=\sum_{k=1}^{n} X_{k}$ , X _(n)=max?_1??k??n X _k and S _(n)=max?_1??k??n S _k , n??1 in the case where the random variables are dependent with heavy-tailed (subexponential) distributions, which substantially generalize the results of Ko and Tang (J. Appl. Probab. 45, 85?C94, 2008).     

Best trigonometric and bilinear approximations of classes of functions of several variables

Order-sharp estimates of the best orthogonal trigonometric approximations of the Nikol’skii-Besov classes B _p,θ ^r of periodic functions of several variables in the space L _q are obtained. Also the orders of the best approximations of functions of 2d variables of the form g(x, y) = f(x?y), x, y ∈ $\mathbb{T}$ ^d = Π _j=1 ^d [?π, π], f(x) ∈ B _p,θ ^r , by linear combinations of products of functions of d variables are established.     

Comparison Inequalities for One Sided Normal Probabilities

Several sharp upper and lower bounds for the ratio of two normal probabilities $\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(1)}_i\leq \mu_i\bigr\}\Biggr)\Big/\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(0)}_i\leq \mu_i\bigr\}\Biggr)$ are given in this paper for various cases, where (ξ ₁ ⁽⁰⁾ ,ξ ₂ ⁽⁰⁾ ,…,ξ _n ⁽⁰⁾ ) and (ξ ₁ ⁽¹⁾ ,ξ ₂ ⁽¹⁾ , …,ξ _n ⁽¹⁾ ) are standard normal random variables with covariance matrices R ⁰=(r _ij ⁰ ) and R ¹=(r _ij ¹ ), respectively.     

Solutions of second order degenerate integro-differential equations in vector-valued function spaces

We study the well-posedness of the second order degenerate integro-differential equations(P2):(Mu)(t)+α(Mu)(t) = Au(t)+ft-∞ a(ts)Au(s)ds + f(t),0t2π,with periodic boundary conditions M u(0)=Mu(2π),(Mu)(0) =(M u)(2π),in periodic Lebesgue-Bochner spaces Lp(T,X),periodic Besov spaces B s p,q(T,X) and periodic Triebel-Lizorkin spaces F s p,q(T,X),where A and M are closed linear operators on a Banach space X satisfying D(A) D(M),a∈L1(R+) and α is a scalar number.Using known operatorvalued Fourier multiplier theorems,we completely characterize the well-posedness of(P2) in the above three function spaces.     

Multipliers for the Weyl transform and Laguerre expansions

Let P_k denote the projection of L²(R ^R ) onto the kth eigenspace of the operator (-δ+?x?² andS _N ^α =(1/A _N ^α )Σ _k ^N =0A _N?k ^α P _k . We study the multiplier transformT _N ^α for the Weyl transform W defined byW(T _N ^αf )=S _n ^αW(f) . Applications to Laguerre expansions are given.     

Inversion complexity of self-correcting circuits for a certain sequence of boolean functions

The asymptotics L _k ^? (f ₂ ⁿ ) ?? n min{k+1, p} is obtained for the sequence of Boolean functions $f_2^n \left( {x_1 , \ldots ,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n}$ for any fixed k, p ?? 1 and growing n, here L _k ^? (f ₂ ⁿ ) is the inversion complexity of realization of the function f ₂ ⁿ by k-self-correcting circuits of functional elements in the basis B = {&, ?}, p is the weight of a reliable invertor.     

On vector bundles over surfaces and Hilbert schemes

Let X be an irreducible smooth projective surface over ${{\mathbb{C}}}$ and Hilb ^d (X) the Hilbert scheme parametrizing the zero-dimensional subschemes of X of length d. Given a vector bundle E on X, there is a naturally associated vector bundle ${{\mathcal{F}}_d(E)}$ over Hilb ^d (X). If E and V are semistable vector bundles on X such that ${{\mathcal{F}}_d(E)}$ and ${{\mathcal{F}}_d(V)}$ are isomorphic, we prove that E is isomorphic to V. A key input in the proof is provided by Biswas and Nagaraj (see [1]).     

An initial value problem for a singular parabolic equation of even order

At first Cauchy-problem for the equation: \(L[u(X,t)] \equiv \sum\limits_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_1^2 }} + \frac{{2v}}{{\left| X \right|^2 }}} \sum\limits_{i = 1}^n {x_i \frac{{\partial u}}{{\partial x_i }} - \frac{{\partial u}}{{\partial t}} = 0} \) wheren≥1,v—an arbitrary constant,t>0,X=(x ₁, …, x_n)∈E ⁿ/{0}, |X|= =(x ₁ ² +…+x _n ² )^1/2, with 0 being a centre of coordinate system, is studied. Basing on the above, the solution of Cauchy-Nicolescu problem is given which consist in finding a solution of the equationL ^p [u (X, t)]=0, withp∈N subject the initial conditions \(\mathop {\lim }\limits_{t \to \infty } L^k [u(X,t)] = \varphi _k (X)\) ,k=0, 1,…,p?1 and ?_k(X) are given functions.     

Bipartite graphs with five eigenvalues and pseudo designs

A pseudo (v,k,??)-design is a pair $(X, \mathcal{B})$ , where X is a v-set, and $\mathcal{B}=\{B_{1},\ldots,B_{v-1}\}$ is a collection of k-subsets (blocks) of X such that any two distinct B _i ,B _j intersect in ?? elements, and 0????<k??v?1. We use the notion of pseudo designs to characterize graphs of order n whose (adjacency) spectrum contains zero and ±?? with multiplicity (n?3)/2 where $0<\theta\le\sqrt{2}$ . Meanwhile, partial results confirming a conjecture of?O. Marrero on a characterization of pseudo (v,k,??)-designs are obtained.     

Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

A Representation of Hyperbolic Motions Including the Infinite-Dimensional Case

Let X be a real inner product space of (finite or infinite) dimension ???2, O(X) be its group of all surjective (hence bijective) orthogonal transformations of X, T(X) be the set of all hyperbolic translations of X and M(X, hyp) be the group of all hyperbolic motions of X. The following theorem will be proved in this note. Every ${\mu\in M(X,{\mbox hyp})}$ has a representation ?? = T · ?? with uniquely determined ${T\in T(X)}$ and uniquely determined ${\omega\in O(X)}$ .