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.     

On coefficients of null-series and on sets of uniqueness of trigonometric and Walsh systems

В работе доказываютс я следующие утвержде ния. Теорема I.Пусть ? _n ↓0u \(\sum\limits_{n = 0}^\infty {\varepsilon _n^2 = + \infty } \) .Тогд а существует множест во Е?[0, 1]с μЕ=0 такое что:1. Существует ряд \(\sum\limits_{n = 0}^\infty {a_n W_n } (t)\) с к оеффициентами ¦а _n ¦≦{in¦n¦, который сх одится к нулю всюду вне E и ε∥a_n∥>0.2. Если b _n ¦=о(ε _n )и ряд \(\sum\limits_{n = 0}^\infty {b_n W_n (t)} \) сх одится к нулю всюду вн е E за исключением быть может некоторого сче тного множества точе к, то b _n =0для всех п. Теорема 3.Пусть ? _n ↓0u \(\mathop {\lim \sup }\limits_{n \to \infty } \frac{{\varepsilon _n }}{{\varepsilon _{2n} }}< \sqrt 2 \) Тогд а существует множест во E?[0, 1] с υ E=0 такое, что:

1. Существует ряд \(\sum\limits_{n = - \infty }^{ + \infty } {a_n e^{inx} ,} \sum\limits_{n = - \infty }^{ + \infty } {\left| {a_n } \right|} > 0,\) кот орый сходится к нулю в сюду вне E и ¦a_n≦¦n¦ для n=±1, ±2, ...
2. Если ряд \(\sum\limits_{n = - \infty }^{ + \infty } {b_n e^{inx} } \) сходится к нулю всюду вне E и ¦b_v¦=о(ε ¦n¦), то b_n=0 для всех я. Теорема 5. Пусть послед овательности S⁽¹⁾={ε ₀ ⁽¹⁾ , ε ₁ ⁽¹⁾ , ε ₂ ⁽¹⁾ , ...} u S²=ε ₀ ⁽²⁾ , ε ₁ ⁽²⁾ . ε ₂ ⁽²⁾ монотонно стремятся к нулю, \(\mathop {\lim \sup }\limits_{n \to \infty } \varepsilon ^{(i)} /\varepsilon _{2n}^{(i)}< 2,i = 1,2\) , причем \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n^{(2)} /\varepsilon _n^{(i)} = + \infty \) . Тогда для каждого ε>O н айдется множество Е? [-π,π], μE >2π — ε, которое является U(S¹), но не U(S¹) — множеством для тригонометричес кой системы. Аналог теоремы 5 для си стемы Уолша был устан овлен в [7].

     

On certain estimates for orthogonal polynomials depending on parameters

Рассматривается сис тема ортогональных м ногочленов {P _n (z)} ₀ ^∞ , удовлетворяющ их условиям $$\frac{1}{{2\pi }}\int\limits_0^{2\pi } {P_m (z)\overline {P_n (z)} d\sigma (\theta ) = \left\{ {\begin{array}{*{20}c} {0,m \ne n,P_n (z) = z^n + ...,z = \exp (i\theta ),} \\ {h_n > 0,m = n(n = 0,1,...),} \\ \end{array} } \right.} $$ где σ (θ) — ограниченная неу бывающая на отрезке [0,2π] функция с бесчисленным множе ством точек роста. Вводится последовательность параметров {а_{n 0} ^∞ , независимых дру г от друга и подчиненных единств енному ограничению { ¦а_n¦<1} ₀ ^∞ ; все многочлены {Р _n (z)} 0/∞ можно найти по формуле $$P_0 = 1,P_{k + 1(z)} = zP_k (z) - a_k P_k^ * (z),P_k^ * (z) = z^k \bar P_k \left( {\frac{1}{z}} \right)(k = 0,1,...)$$ . Многие свойства и оце нки для {P _n (z)} ₀ ^∞ и (θ) можн о найти в зависимости от этих параметров; например, условие \(\mathop \Sigma \limits_{n = 0}^\infty \left| {a_n } \right|^2< \infty \) , бо лее общее, чем условие Г. Cerë, необходимо и достато чно для справедливости а симптотической форм улы в области ¦z¦>1. Пользуясь этим ме тодом, можно найти также реш ение задачи В. А. Стекло ва.     

Bounds for the spectral abscissa of an element in a Banach algebra

For an arbitrary element x with spectrum sp(x) in a Banach algebra with identity e ≠ 0 we define the upper (lower) spectral abscissa \(\mathop {\sigma + (x)}\limits_{( - )} = \mathop {\max }\limits_{(\min )} \operatorname{Re} \lambda ,\lambda \in sp(x)\) . With the aid of the spectral radius \(\rho (x) = \mathop {\max }\limits_{\lambda \in sp(x)} \left| \lambda \right| = \mathop {\lim }\limits_{n \to + \infty } \parallel x^n {{1 - } \mathord{\left/ {\vphantom {{1 - } n}} \right. \kern-0em} n}\) we prove the following bounds: γ_?(x)?σ_?(x)?Γ_?(x)?₊(x)?σ₊(x)?γ₊(x), Γ_(±)(x)=(2δ_(±))^?1 (ρ _δ ² )_(±)?δ _(±) ² ?ρ ₀ ² )(δ_(±)≠0), γ_(±)(x)= (±)ρ_δ(±)?δ_(±), δ₊?0, δ_??0 ρ _(±) ^δ = ρ(x+eδ_(±)). We mention a case where equality is achieved, some corollaries,and discuss the sharpness of the bounds: for every ? > 0 there is a δ: ¦δ¦ ≥ρ ₀ ² /2?, such that Δ: = ¦γ_(±) x?Γ_(±) x¦?ε and conversely, if the bounds are computed for some δ ≠ 0, then △ ≤ρ ₀ ² /2 ¦δ¦. An example is considered.     

Spacings-ratio empirical processes

We consider an empirical process based upon ratios of selected pairs of spacings, generated by independent samples of arbitrary sizes. As a main result, we show that when both samples are uniformly distributed on (possibly shifted) intervals of equal lengths, this empirical process converges to a mean-centered Brownian bridge of the form B _C (u) = B(u)?6Cu(1?u) Σ ₀ ¹ B(s)ds, where B(·) denotes a Brownian bridge, and C, a constant. The investigation of the class of Gaussian processes {B _C (·): C ∈ ?} leads to some unexpected distributional identities such as B ₂(·) $ \underline{\underline d} $ B(·). We discuss this and similar results in an extended framework.     

A quadratic approximation for protein sequence to structure mapping

The existence and representations of some generalized inverses, includingA _{T, *} ⁽²⁾ ,A _{T, *} ^(1,2) ,A _{T, *} ^(2,3) ,A _*,S ⁽²⁾ ,A _*,S ^(1,2) andA _*,S ^(2,4) , are showed. As applications, the perturbation theory for the generalized inverseA _T,S ⁽²⁾ and the perturbation bound for unique solution of the general restricted systemAx=b (dim (AT)=dimT,b∈AT andx∈T) are studied. Moreover, a characterization and representation of the generalized inverseA _{T, *} ^Emphasis>(2) is obtained.     

Singular integrals along Lipschitz curves with holomorphic kernels

If γ(x)=x+iA(x),tan ^?1‖A′‖_∞<ω<π/2,S _ω ⁰ ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S _ω ⁰ and \(\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}\) , denotingT f (z)= ∫?(z-ζ)f(ζ)dζ, ?f∈C ₀(γ), ?z∈suppf, where C_c(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent:

1. T can be extended to be a bounded operator on L²(γ);
2. there exists a function ?₁ ∈H ^∞(S _ω ⁰ ) such that ?′₁(z)=?(z)+?(-z), ?z∈S _ω ⁰ ?z∈S _w ⁰ .

     

A constructive description of Hölder classes on closed Jordan curves

Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that ¦f(z) ? R_n(z)¦ ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ ₁ ⁺ /n^(z) and ρ ₁ ^- /n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) .     

The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.     

Rings Whose Nilpotent Elements form a Levitzki Radical Ring

In this article, we first discuss the relations among JH^r = 0, JH^r?1·H = 0, and JH^r·x = 0. Then we give a counterexample to the question mentioned in the Remarks of [3 Cheng, C. C., Sakkalis, T., Wang, S. S. S. (1994). A case of the Jacobian conjecture. J. Pure and Applied Algebra 96:15–18.[Crossref], [Web of Science ®] , [Google Scholar]] and prove the equivalence among JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r)) = 0, JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r?1))·H(x^(r)) = 0, and JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r))·x^(r) = 0. Finally, we give partial answer to Conjecture 2 in [4 Connell, E., Zweibel, J. (1991). Exact and coexact matrices. J. Algebra 142:110–117.[Crossref], [Web of Science ®] , [Google Scholar]].     

Equilibrium constants of UO 2 2+ , Cu2+ with dicarboxylic acids in dioxane-water mixtures

The interactions of uranyl and copper ions with dicarboxylic acids such as succinic, itaconic, adipic and malonic acids have been investigated in 20, 40, 60 and 80% (v/v) dioxane-water mixtures at 30°C and 0·06 ionic strength by Calvin-Bjerrum titration technique. UO ₂ ²⁺ , Cu²⁺ form only 1:1 complexes with succinic, itaconic and adipic acids and both 1:1 and 1:2 with malonic acid in the pH range 2·5–5. The plots of pK/log K vs mole fraction gave linear while those with 1/D gave non-linear. The complexation equilibria: are proposed. The radius of anionic ligand is calculated from Born equation and attempts are made to understand the medium effect on the pK values.     

Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

For the equation $$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$ whereh ₀ ^o =0,h ₀ ¹ =0 (t) ≡ 0,h _j ^o = const > 0,h ₁ ^j (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation $$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$ connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.     

Tensor products of power Köthe spaces of different types

The multirectangular characteristics µ _m ^(λ,c) are applied to the isomorphic classification of tensor products of the form $ E_0 (a)\widehat \otimes E_\infty (b) $ . We single out a subclass of tensor products such that the two-rectangular characteristic µ ₂ ^(λ,c) is a complete invariant on this class.     

On copositive approximation by algebraic polynomials

Для функцииf ∈C[?1, 1] с ог раниченным числом пе ремен знака строится последовательность многочленовр _п , коположительных сf (т.е.f(x)p _n (x)≥0, ?1≤х<1) и таких, что $$\left\| {f - p_n } \right\|_\infty \leqslant C\omega _\varphi ^3 (f,n^{ - 1} ),$$ гдеω _? ³ (f, δ) — модуль непр ерывности Дитциана-Т отика третьего порядка. Изв естно, чтоω _? ³ нельзя заменить ни наω _? ⁴ , ни на ω⁴. Таким образом, приведенная оценка точна в некотором смы сле. В качестве следст вия установлена эквивал ентность соотношений $$E_n (f) = O(n^{ - \alpha } )\user2{}E_n^{(0)} (f,r) = O(n^{ - \alpha } )\user2{}0< \alpha< 3.$$      

Connection between random curves,changes of time,and regenerative times of stochastic processes

The product of spaces Φ × D is considered, where Φ is the set of all continuous, nondecreasing functions ?:[0,∞)→(0,∞), ?(0)=0, ?(t)→∞(t→∞), and D is the set of all right continuous functions ξ:(0,∞)→X; here X is some metric space. Two mappings are defined: the first is the projection q(?,ξ)=ξ, and the second is the change of time U(?,ξ)=ξº?. The following equivalence relation is defined on D: $$\xi _1 \sim \xi _2 \Leftrightarrow \exists _{\varphi _1 , \varphi _1 } \in \Phi :\xi _1 ^\circ \varphi _1 = \xi _2 ^\circ \varphi _2 $$ . Let? be the set of all equivalence classes, and let L be the mapping ξ₄～ξ₂, Lξ is called the curve corresponding to ξ. The following theorem is proved: two stochastic processes with probability measures P¹ and P² on D possess identical random curves (i.e.,P¹ºL^?1=P²ºL^?1) if and only if there exist two changes of time (i.e., probability measures Q¹ and Q² on ?×D for which P¹=Q¹ºq^?1, P²=Q²ºq^?1 which take these two processes into a process with measure \(\tilde P\) (i.e., Q¹ºu^?1=Q²ºu^?1,=～P) If (P _x ¹ )_x∈X and (P _x ² )_x∈X are two families of probability measures for which P _x ¹ ºL^?1=P _x ² ºL^?1?x∈X then for each x ε X the corresponding measures Q _X ¹ andQ _X ² can be found in the following manner. The set of regenerative times of the family \(\left( {\tilde P_x } \right)_{x \in X} \) contains all stopping times which are simultaneously regenerative times of the families (p _x ¹ )_x∈X and (P _x ² )_x∈X and possess a certain special property of first intersection.     

On positive linear interpolation operators

В этой работе мы даем о бобщение понятия нор мальной системы точек, введен ного Фейером [3]. Наше определ ение включает и случа й бесконечного интерв ала (0, ∞). Доказано, в частности, что систе ма точек 0<x ₁ ⁽ⁿ⁾ /(n)<... _n ⁽ⁿ⁾ <∞ является нормальной в смысле нашего определения тогда и т олько тогда, когда вып олняются оценки — фиксированное чис ло, 0≦?<1. Мы доказываем, что есл и точкиx _k ⁽ⁿ⁾ /(n) являются ну лями многочлена ЛагерраL _n ^(α) (x), то они образуют норма льную систему в том и т олько том случае, когда ?1<α≦0. Мы получаем, таким обр азом, положительный интерполяционный пр оцесс для каждой нормальной системы т очек и устанавливаем теорему сходимости для того с лучая, когда эти точки являются ну лямиL _n ^(α) (x) при — 1相似文献   

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Relative average widths of Sobolev spaces in L 2(ℝ d )

In this paper, in order to consider the problems of relative width on ? ^d , we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality $$ \overline K _\sigma (U(W_2^\alpha ),M(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) $$ hold, where U(W ₂ ^α ) is the Riesz potential or Bessel potential of the unit ball in L ²(? ^k ) and the notations $ \overline K _\sigma $ (·, ·,L ₂(? ^d )) and $ \overline d _\sigma $ (·, L ₂(? ^d )) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that $$ \overline K _\sigma (U(W_2^\alpha ) \cap B(L_2 (\mathbb{R}^d )),U(W_2^\beta ) \cap B(L_2 (\mathbb{R}^d ))L_2 (\mathbb{R}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 (\mathbb{R}^d )), $$ where 0 × β × α.     

A method of constructing integrable linear equations and its application to Hill's equation

Starting with a given equation of the form $$\ddot x + [\lambda + \varepsilon f(t)] x = 0$$ , where λ > 0 and ? ? l is a small parameter [heref(t) may be periodic, and so Hill's equation is included], we construct an equation of the form y + [λ + ?f (t) + ?² _g (t)]y = 0, integrable by quadratures, close in a certain sense to the original equation. For x₀ = y₀ and x ₀ ^′ = y ₀ ^′ , an upper bound is obtained for ¦y—x¦ on an interval of length Δt.     

A weight space invariant with respect to a singular linear operator

For the singular operator $$Su = \int_a^b {\frac{{K(x, s) u (s)}}{{s - x}}} ds$$ invariant weight spacesλ _α ^β , _p (u(x)∈λ _α ^β , _p if 1⁰,u (x) ρ (x)∈ H _β ⁰ , 2⁰.‖u‖L _p(ρ₀)<∞, ρ (x) = (x?a) (b ?x)^1+β, ρ₀(x)=(b?x)^α(p?1), 0<α, β<1,p>1H ₀ ^β is a Hölder space. Multiplicative inequalities of the type of Kh. Sh. Mukhtarov are also obtained.