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.     

Large deviations of the first passage time for a random walk with semiexponentially distributed jumps

Suppose that ξ, ξ(1), ξ(2), ... are independent identically distributed random variables such that ?ξ is semiexponential; i.e., $P( - \xi \geqslant t) = e^{ - t^\beta L(t)} $ is a slowly varying function as t → ∞ possessing some smoothness properties. Let E ξ = 0, D ξ = 1, and S(k) = ξ(1) + ? + ξ(k). Given d > 0, define the first upcrossing time η ₊(u) = inf{k ≥ 1: S(k) + kd > u} at nonnegative level u ≥ 0 of the walk S(k) + kd with positive drift d > 0. We prove that, under general conditions, the following relation is valid for $u = (n) \in \left[ {0, dn - N_n \sqrt n } \right]$ : 0.1 $P(\eta + (u) > n) \sim \frac{{E\eta + (u)}}{n}P(S(n) \leqslant x) as n \to \infty $ , where x = u ? nd < 0 and an arbitrary fixed sequence N _n not exceeding $d\sqrt n $ tends to ∞. The conditions under which we prove (0.1) coincide exactly with the conditions under which the asymptotic behavior of the probability P(S(n) ≤ x) for $x \leqslant - \sqrt n $ was found in [1] (for $x \in \left[ { - \sqrt n ,0} \right]$ it follows from the central limit theorem).     

Behavior of solutions of quasilinear elliptic inequalities in an unbounded domain

We consider the solutions of the inequalityLu≤?(¦gradu¦), whereL is a uniformly elliptic homogeneous operator and ? is a function increasing faster than any linear function but not faster thanξ lnξ, in the unbounded domain $$\left\{ {x \in \mathbb{R}^n |\sum\limits_{i = 2}^n {x_i^2< (\psi (x_1 ))^2 ,} {\text{ }} - \infty< x_1< \infty } \right\},$$ , whereψ is a bounded function with bounded derivative. We estimate the growth of the solutions in terms of $\int_0^{x_1 } {(1/\psi (r))dr}$ . For the special case in which?(ξ)=aξ lnξ+C, the solutionsu(x ₁,x ₂,...,x _n ) grow as $\left( {\int_0^{x_1 } {(1/\psi (r))dr} } \right)^N$ , whereN is any given number anda=a(N).     

Group pursuit in Pontryagin’s nonstationary example

On the interval [t ₀, ∞), we consider the following group pursuit problem with one evader: 1 $$ z_i^{(l)} + a_1 (t)z_i^{(l - 1)} + a_2 (t)z_i^{(l - 2)} + \cdots + a_l (t)z_i = u_i - v, u_i ,v \in V, z_i^{(q)} (t_0 ) = z_i^q , $$ where z _i , u _i , v ∈ R ^v , (v ≥ 2), V is a strictly convex compact set in R ^v , the functions a ₁(t), a ₂(t), …, a _l (t) are continuous, i = 1, 2, …, n and q = 0, 1, …, l ? 1. Let ? _q (t, s) be the solution of the Cauchy problem $$ \begin{gathered} \omega ^{(l)} + a_1 (t)\omega ^{(l - 1)} + a_2 (t)\omega ^{(l - 2)} + \cdots + a_l (t)\omega = 0, \omega ^{(q)} (s) = 1, \hfill \\ \omega ^{(r)} (s) = 0, r = 0, \ldots q - 1,q + 1, \ldots ,l - 1, \hfill \\ \end{gathered} $$ and let $$ \xi _\iota (t) = \varphi _0 (t,t_0 )Z_i^0 + \varphi _1 (t,t_0 )Z_i^1 + \cdots + \varphi _{l - 1} (t,t_0 )Z_i^{l - 1} . $$ We prove that if there exist continuous functions α _i (t) and ξ _i ¹ (t) such that the ξ _i ¹ (t) are Bohr almost periodic on [t ₀, ∞), α _i (t) > 0 for all t ≥ t ₀, lim _t→∞(ξ _i ¹ (t) ? α _i (t)ξ _i (t)) = 0, lim _t→∞(min _i α _i (t) ∝ _t0 ^t |? _l?1(t, s)| ds) = ∞, and there exist points h _i ⁰ ∈ H _i ¹ = {ξ _i ¹ (t), t ∈ [0, ∞)} such that 0 ∈ Int co{h _i ⁰ }, then the pursuit problem with evader discrimination is solvable.     

Multi-point boundary value problems for one-dimensionalp-Laplacian at resonance

In this paper, we consider the multi-point boundary value problems for one-dimensional p-Laplacian at resonance: $(\phi _p (x'(t)))' = f(t,x(t),x'(t))$ subject to the boundary value conditions: $(\phi _p (x'(0)) = \sum\limits_{i = 1}^{n - 2} {\alpha _i \phi _p (x'(\xi _i ))} $ , $(\phi _p (x'(1)) = \sum\limits_{j = 1}^{m - 2} {\beta _j \phi _p (x'(\eta _i ))} $ where ? _p (s)=|s|^p-2 s, p>1,α_i(1≤i≤n-2)∈R,β_{jit}(1≤j≤m-2)∈R, 0<ξ₁<ξ₂<...<ξ_n-2<1, 0<η₁<η₂<...<η_m-2<1, By applying the extension of Mawhin’s continuation theorem, we prove the existence of at least one solution. Our result is new.     

The semigroup approach to first order quasilinear equations in several space variables

The Cauchy problem for u _t + Σ _{i = 1} ⁿ (φ _i (u)) _xi = 0 is treated via the theory of semigroups of nonlinear transformations. This treatment requires the development of results concerning the time-independent equation u + Σ _{i = 1} ⁿ (φ _i (u)) _xi = h for h∈L ¹(Rⁿ), which in turn is studied via the regularized equation $$ u + \sum\nolimits_{i = 1}^n {\left( {\phi _i \left( u \right)} \right)} _{xi} - \varepsilon \Delta u = h $$ .     

Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type

In this paper, we examine, in a systematic fashion, some ill-posed problems arising in the theory of heat conduction. In abstract terms, letH be a Hilbert space andA: D (A)?H→H be an unbounded normal operator, we consider the boundary value problemü(t)=Au(t), 0<t<∞,u(0)=u ₀∈D(A), \(\mathop {\lim }\limits_{t \to 0} \left\| {u\left( t \right)} \right\| = 0\) . The problem of recoveringu ₀ whenu(T) is known for someT>0 is not well-posed. Suppose we are given approximationsx ₁,x ₂,…,x _N tou(T ₁),…,u(T _N) with 0<T, <…<T _N and positive weightsP _i,i=1,…,n, \(\sum\limits_{i = 1}^N {P_i = 1} \) such that \(Q_2 \left( {u_0 } \right) = \sum\limits_{i = 1}^N {P_i } \left\| {u\left( {T_i } \right) - x_i } \right\|^2 \leqslant \varepsilon ^2 \) . If ‖u _t(0)‖≤E for some a priori constantE, we construct a regularized solution ν(t) such that \(Q\left( {\nu \left( 0 \right)} \right) \leqslant \varepsilon ^2 \) while \(\left\| {u\left( 0 \right) - \nu \left( 0 \right)} \right\| = 0\left( {ln \left( {E/\varepsilon } \right)} \right)^{ - 1} \) and \(\left\| {u\left( t \right) - \nu \left( t \right)} \right\| = 0\left( {\varepsilon ^{\beta \left( t \right)} } \right)\) where 0<β(t)<1 and the constant in the order symbol depends uponE. The function β(t) is larger thant/m whent _k andk is the largest integer such that \((\sum\limits_{k = 1}^N {P_i (T_i )} )< (\sum\limits_{k = 1}^N {P_i (T_i )} = m\) , which β(t)=t/m on [T _k, m] and β(t)=1 on [m, ∞). Similar results are obtained if the measurement is made in the maximum norm, i.e.,Q _∞(u ₀)=max{‖u(T _i)?x _i‖, 1≤i≤N}.     

Summation of weakly dependent sequences by Cesàro methods

Пусть (X, A, u) — пространст во с конечной мерой, (ξ_k) ₁ ^∞ — последовательност ь функций, \(\xi _k \varepsilon L_{2r} (X), r > 1, \int\limits_X {\xi _k d\mu = 0} \) . Изучаются условия, п ри которых справедли вgа - у. з. б.ч., т. e. (ξ _k) суммируется к ну лю почти всюду методо м (С, а),а > 0. Приведем два резу льтата. 1) Если (ξ _k) — слабо мульт ипликативная систем а (в частности, мартингал-разности или независимая сист ема), то условие $$\mathop \sum \limits_1^\infty \mathop {\smallint }\limits_X \left| {\xi _k } \right|^{2r} d\mu \cdot c_r (k,\alpha )< \infty $$ влечетβ - у.з.б.ч. Здесьc _r(k,α)=k ^-2rα при 0<α<(r+1)/2r, c_r=k^?(r+1) In3r-1 k приа=(r+1)/2r, с_r=k^?(r+1) при а >(r+1)/2r. 2) Если (ξ _k) независимы,Mξ _k=0, (r+1)/2r<α=1, то условия $$\mathop \sum \limits_{k = 1}^\infty \frac{{(M\xi _k^2 )^r }}{{k^{r + 1} }}< \infty ,\mathop \sum \limits_{k = 1}^\infty \frac{{M|\xi _k |^{2r} }}{{k^{2r\alpha } }}< \infty $$ влекут за собой а - у. з. б. ч.     

On global relations for hypergeometric series

Let [K:Q]=k, f_i εK[[z]], ξ εK, Q εK[y₁,...,y_m]. A relation Q(f₁(ξ),..., f_m(ξ))=0 is called global if it holds in any local field where all f_i(ξ) exist. The paper establishes that for series of the form $$\sum\limits_{n = 0}^\infty {\frac{{(\mu _1 )_n \ldots (\mu _p )_n }}{{(\lambda _1 )_n \ldots (\lambda _{q - 1} )_n n!}}\left( {\frac{{z^{p - q} }}{{q - p}}} \right)^n , p > q,} $$ with some natural hypotheses on parameters global relations do not exist. Bibliography: 9 titles.     

On the regularity of solutions of model nonlinear elliptic systems with the oblique derivative type boundary condition

Properties of generalized solutions of model nonlinear elliptic systems of second order are studied in the semiball $B_1^ + = B_1 (0) \cap \{ x_n > 0\} \subset $ ? ⁿ , with the oblique derivative type boundary condition on $\Gamma _1 = B_1 (0) \cap \{ x_n = 0\} $ . For solutionsu∈H ¹(B ₁ ⁺ ) of systems of the form $\frac{d}{{dx_\alpha }}a_\alpha ^k (u_x ) = 0, k \leqslant {\rm N}$ , it is proved that the derivatives u_x are Hölder in $B_1^ + \cup \Gamma _1 )\backslash \Sigma $ , where H_n?p(σ)=0,p>2. It is shown for continuous solutions u from H¹(B₁/⁺) of systems $\frac{d}{{dx_\alpha }}a_\alpha ^k (u,u_x ) = 0$ that the derivatives u_x are Hölder on the set $(B_1^ + \cup \Gamma _1 )\backslash \Sigma , dim_\kappa \Sigma \leqslant n - 2$ . Bibliography: 13 titles.     

Asymptotics for the minimization of a Ginzburg-Landau functional

LetΩ ? ?² be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classH _g ¹ ={u εH ¹(Ω; ?);u=g on ?Ω} whereg:?Ω? → ? is a prescribed smooth map with ¦g¦=1 on ?Ω? and deg(g, ?Ω)=0. Let uu _ε be a minimizer for E_ε onH _g ¹ . We prove that u_ε → u₀ in \(C^{1,\alpha } (\bar \Omega )\) as ε → 0, where u₀ is identified. Moreover \(\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 \) .     

On some boundary value problems with integral conditions for functional differential equations

For the functional differential equationu ⁽ⁿ⁾ (t)=f(u)(t) we have established the sufficient conditions for solvability and unique solvability of the boundary value problems $$u^{(i)} (0) = c_i (i = 0,...,m - 1), \smallint _0^{ + \infty } |u^{(m)} (t)|^2 dt< + \infty $$ and $$\begin{gathered} u^{(i)} (0) = c_i (i = 0),...,m - 1, \hfill \\ \smallint _0^{ + \infty } t^{2j} |u^{(j)} (t)|^2 dt< + \infty (j = 0,...,m), \hfill \\ \end{gathered} $$ wheren≥2,m is the integer part of $\tfrac{n}{2}$ ,c _i ∈R, andf is the continuous operator acting from the space of (n?1)-times continuously differentiable functions given on an interval [0,+∞] into the space of locally Lebesgue integrable functions.     

Instability of Standing Wave for the Klein–Gordon–Hartree Equation

The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein–Gordon– Hartree equation     

Spectral asymptotics and trace formulas for differential operators with unbounded coefficients

In the space L ²[0, π], we consider the operators $$ L = L_0 + V, L_0 = - y'' + (\nu ^2 - 1/4)r^{ - 2} y (\nu \geqslant 1/2) $$ with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L ²[0, π] satisfying the condition $$ \int\limits_0^\pi {r^\varepsilon } (\pi - r)^\varepsilon |V(r)|dr < \infty , \varepsilon \in [0,1] $$ . We prove the trace formula Σ _n=1 ^∞ [µ _n ? λ _n ? Σ _k=1 ^m α _k ⁽ⁿ⁾ ] = 0.     

On the Intersection of the Normalizers of the \boldsymbol{\cal{F}}-Residuals of Subgroups of a Finite Group

In this paper we give criteria for a finite group to belong to a formation. As applications, recent theorems of Li, Shen, Shi and Qian are generalized. Let G  be a finite group, $\cal F$ a formation and p  a prime. Let $D_{\mathcal {F}}(G)$ be the intersection of the normalizers of the $\cal F$ -residuals of all subgroups of G, and let $D_{\mathcal {F}}^{p}(G)$ be the intersection of the normalizers of $(H^{\cal F}O_{p'}(G))$ for all subgroups H of G. We then define $D_{\mathcal F}^{0}(G)=D_{\mathcal F, p}^{~0}(G)=1$ and $D_{\mathcal F}^{i+1}(G)/D_{\mathcal F}^{i}(G)=D_{\mathcal F}(G/D_{\mathcal F}^{i}(G))$ , $D_{\mathcal F, p}^{i+1}(G)/D_{\mathcal F, p}^{~i}(G)=D_{\mathcal F, p}(G/D_{\mathcal F, p}^{~i}(G))$ . Let $D_{\mathcal {F}}^{\infty}(G)$ and $D_{\mathcal {F}, p}^{~\infty}(G)$ denote the terminal member of the ascending series of $D_{\mathcal F}^{i}(G)$ and $D_{\mathcal F, p}^{~i}(G)$ respectively. In this paper we prove that under certain hypotheses, the the $\cal F$ -residual $G^{\cal F}$ is nilpotent (respectively,p-nilpotent) if and only if $G=D_{\mathcal {F}}^{\infty}(G)$ (respectively, $G=D_{\mathcal {F}, p}^{~\infty}(G)$ ). Further more, if the formation $\cal F$ is either the class of all nilpotent groups or the class of all abelian groups, then $G^{\cal F}$ is p-nilpotent if and only if and only if every cyclic subgroup of G order p and 4 (if p?=?2) is contained in $D_{\mathcal {F}, p}^{~\infty}(G)$ .     

On some new nonlocal solvability theorems for various classes of nonlinear differential equations

We study nonlinear boundary value problems of the form $$ [\Psi u']' + F(x;u',u) = g, u(0) = u(1) = 0 $$ , where Φ is a coercive continuous operator from L _p to L _q , and $$ F(x;u'',u',u) = g, u(0) = u(1) = 0 $$ ; first- and second-order partial differential equations $$ \Phi (x_1 ,x_2 ;u'_1 ,u'_2 ,u) = 0, \sum\limits_{i = 1}^\infty {[\Psi _i (u'_{x_i } )]'_{x_i } + F(x; \ldots ,u'_{x_i } , \ldots ,u) = g_i } $$ ; and general equations F(x; ..., u″ _ii , ...., ...., u′ _i , ...; u) = g(x) of elliptic type. We consider the corresponding boundary value problems of parabolic and hyperbolic type. The proof is based on various a priori estimates obtained in the paper and a nonlocal implicit function theorem.     

Power Majorization and Majorization of Sequences

Let \(\bar x\) , \(\bar y\ \in\ R_n\) be vectors which satisfy x₁ ≥ x₂ ≥ … ≥ x_n and y₁ ≥ y₂ >- … ≥ y_n and Σx_i = Σy_i. We say that \(\bar x\) is power majorized by \(\bar y\) if Σx_i ^p ≤ Σy_i ^p for all real p ? [0, 1] and Σx_i ^p ≥ Σy_i ^p for p ∈ [0, 1]. In this paper we give a classification of functions ? (which includes all possible positive polynomials) for which \(\bar\phi(\bar x) \leq \bar\phi(\bar y)\) (see definition below) when \(\bar x\) is power majorized \(\bar y\) . We also answer a question posed by Clausing by showing that there are vectors \(\bar x\) , \(\bar y\ \in\ R^n\) of any dimension n ≥ 4 for which there is a convex function ? such that \(\bar x\) is power majorized by \(\bar y\) and \(\bar\phi(\bar x)\ >\ \bar\phi(\bar y)\) .     

Existence of multiple positive solutions for one-dimensional p-Laplacian operator

In this paper, we consider the multipoint boundary value problem for one-dimensional p-Laplacian $$(\phi_{p}(u'))'+f(t,u,u')=0,\quad t\in [0,1],$$ subject to the boundary value conditions: $$u'(0)=\sum_{i=1}^{n-2}\alpha_{i}u'(\xi_{i}),\qquad u(1)=\sum_{i=1}^{n-2}\beta_{i}u(\xi_{i}),$$ where φ _p (s)=|s| ^p?2?s,p>1;ξ _i ∈(0,1) with 0<ξ ₁<ξ ₂<???<ξ _n?2<1 and α _i ,β _i satisfy α _i ,β _i ∈[0,∞),0≤∑ _i=1 ^n?2 α _i <1 and 0≤∑ _i=1 ^n?2 β _i <1. Using a fixed point theorem for operators in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.     

Classification of rotational special Weingarten surfaces of minimal type in {\mathbb{S}^2 \times \mathbb{R}} and {\mathbb{H}^2 \times \mathbb{R}}

In this paper we classify the complete rotational special Weingarten surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ ; i.e. rotational surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ whose mean curvature H and extrinsic curvature K _e satisfy H = f(H ² ? K _e ), for some function ${f \in \mathcal{C}^1([0,+\infty))}$ such that f(0) = 0 and 4x(f′(x))² < 1 for any x ≥ 0. Furthermore we show the existence of non-complete examples of such surfaces.     

Existence of multiple positive solutions for m-point fractional boundary value problems with p-Laplacian operator on infinite interval

In this paper we consider the following m-point fractional boundary value problem with p-Laplacian operator on infinite interval where 0<????1, 2<????3, $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative, ?? _p (s)=|s| ^p?2 s,p>1, (?? _p )^?1=?? _q , $\frac{1}{p}+\frac{1}{q}=1$ . 0<?? ₁<?? ₂<?<?? _m?2<+??, ?? _i ??0, i=1,2,??,m?2 satisfies $0 <\sum_{i=1}^{m-2}\beta_{i}\xi_{i}^{\alpha-1} < \Gamma(\alpha)$ . We establish solvability of the above fractional boundary value problems by means of the properties of the Green function and some fixed-point theorems.