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.     

Biorthogonal systems based on {2}-inverse with prescribed range and null space

In this paper, the transformations of generating systems of Euclidean space are examined in connection with the 2-inverseA _T,S ⁽²⁾ , S which has prescribed rangeT and null spaceS of their Gram matrices. The biorthogonal systems of the Moore-Penrose inverse and the Drazin inverse are extended to the {2}-inverseA _T,S ⁽²⁾ .     

The generalized weighted Moore-Penrose inverse

In this paper, we definite a generalized weighted Moore-Penrose inverse A _M,N ⁺ of a given matrixA, and give the necessary and sufficient conditions for its existence. We also prove its uniqueness and give a representation of it. In the end we point out this generalized inverse is also a prescribed rangT and null spaceS of {2}-(or outer) inverse ofA.     

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 .

     

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.     

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

Convolutions of random measures on compact groups

LetG be a compact group andM ₁(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μ _n } _n=?∞ ^+∞ of random measures μ _n =μ _n (ω) inM ₁(G) and the convolutions $$v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n$$ and $$\alpha _n^{( + k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n,n + i} (\omega ),} \alpha _n^{( - k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n - i,n} (\omega )} $$ We describe the setsA _m ⁺ (ω) andA _n ⁺ (ω) of all limit points ofv _m,n(ω) asm→?∞ orn→+∞ and the setA _∞(ω) of its two-sided limit points for typical realizations of {μ _n (ω)} _n=?∞ ^+∞ . Using an appropriate random ergodic theorem we study the limit random measures ρ _n ^(±) (ω)=lim _k→∞ α _n ^(±k) (ω).     

Γ-inverses of Bounded Linear Operators

Let B(H) be the algebra of all the bounded linear operators on a Hilbert space H.For A,P and Q in B(H),if there exists an operator X∈ B(H) such thatAP X QA=A,X QAP X=X,(QAP X)*=QAP X and(X QAP)*=X QAP,then X is said to be the Γ-inverse of A associated with P and Q,and denoted by AP,Q+.In this note,we present some necessary and su?cient conditions for which A+P,Qexists,and give an explicit representation of AP,Q+(if AP,Q+exists).     

On weak solutions of linear coercive integral equations in spacesL 2(X; ℋ k )

Пусть (X,A, μ) - полное про странство с σ-конечно й мерой, и пусть \(\overline {\mu \times \mu } \) . - замык ание меры μ×μ. Пусть далееg: X×X→C - квадратично интегрируемая функц ия по мере \(\overline {\mu \times \mu } \) . Рассматривается лин ейное интегральное у равнение (слабого) типа (1) (1) $$u(t) + A(\mathop \smallint \limits_x g(t,s)u(s)d\mu ) = f(t)\Pi .B.B\,X,$$ гдеА - максимальное р асширение L _k ^′ (в простр анстве ХëрмандераH ₁=B₂к) соотв ествующего линейного (псевдодиф ференциального) опер атораL: S→S; иS обозначает класс Щварца функций Rⁿ→-C. Уст анавливается сущест вование (слабых) решений (1) при н екотором условии коэрпитивно сти на оператор (2) (2) $$(L\Psi )(t) = \Psi (t) + \int\limits_x {g(t,s)L(\Psi (s))d\mu ,} $$ где Ψ принадлежит про странстувуD(Х, S) всех конечно-значных функ ций изX→S. Далее, изучается обобщенна я обратимость максим ального расширения оператора L. Наконец, пр иводится некоторое алгебраическое усло вие, обеспечивающее коэрцитивность L.     

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 $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Polynomial approximation inL p(S) forp>0

For a simple polytopeS inR ^d andp>0 we show that the best polynomial approximationE _n(f)_p≡E_n(f)L_p(S) satisfies $$E_n \left( f \right)_p \leqslant C\omega _S^r \left( {f,\frac{1}{n}} \right)p,$$ where ω _S ^r is a measure of smoothness off. This result is the best possible in the sense that a weak-type converse inequality is shown and a realization of ω _S ^r (f,t)_p via polynomial approximation is proved.     

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.     

Projective characters of the infinite generalized symmetric group

We consider the infinite generalized symmetric group S(∞)?? _m ^∞ , introduce its covering $\tilde B_m $ , and describe all indecomposable characters on the group $\tilde B_m $ .     

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.     

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].

     

An abstract functional differential equation and a related nonlinear Volterra equation

We study the existence, uniqueness, regularity and dependence upon data of solutions of the abstract functional differential equation 1 $$\frac{{du}}{{dt}} + Au \ni G(u) (0 \leqq t \leqq T), u(0) = x,$$ , whereT>0 is arbitrary,A is a givenm-accretive operator in a real Banach spaceX, and \(G:C([0,T]; \overline {D(A)} ) \to L^1 (0, T; X)\) is a given mapping. This study provides simple proofs of generalizations of results by several authors concerning the nonlinear Volterra equation 2 $$u(t) + b * Au(t) \ni F(t) (0 \leqq t \leqq T),$$ , for the case in which X is a real Hilbert space. In (2) the kernelb is real, absolutely continuous on [0,T],b*g(t)=∫ ₀ ¹ (t?s)g(s)ds, andf∈W ^1,1(0,T;X).     

Greedy expansions in Hilbert spaces

We study the rate of convergence of expansions of elements in a Hilbert space H into series with regard to a given dictionary D. The primary goal of this paper is to study representations of an element f ∈ H by a series f ～ ∑ _j=1 ^∞ c _j (f)g _j (f), $g_j \left( f \right) \in \mathcal{D}$ . Such a representation involves two sequences: {g _j (f)} _j=1 ^∞ and {c _j (f) _j=1 ^∞ . In this paper the construction of {g _j (f)} _j=1 ^∞ is based on ideas used in greedy-type nonlinear approximation, hence the use of the term greedy expansion. An interesting open problem questions, “What is the best possible rate of convergence of greedy expansions for f ∈ A ₁(D)?” Previously it was believed that the rate of convergence was slower than $m^{ - \tfrac{1} {4}}$ . The qualitative result of this paper is that the best possible rate of convergence of greedy expansions for $f \in A_1 \left( \mathcal{D} \right)$ is faster than $m^{ - \tfrac{1} {4}}$ . In fact, we prove it is faster than $m^{ - \tfrac{2} {7}}$ .     

Weighted generalized inverses of partitioned matrices in Banachiewicz-Schur form

In this paper the conditions under which the weighted generalized inversesA ^(1,3M), A^(1,4N), A _M,N ^Dg andA ^d,W can be expressed in Banachiewicz-Schur form are considered and some interesting results are established. These results contribute to verify recent results obtained by J. K. Baksalary and G. P. Styan [2] and Y. Wei [15] and these extend their works.     

Algebraic equations on the adèlic closure of a Drinfeld module

Let k be a field of positive characteristic and K = k(V) a function field of a variety V over k and let A _K be the ring of adèles of K with respect to the places on K corresponding to the divisors on V. Given a Drinfeld module $\Phi :\mathbb{F}[t] \to End_K (\mathbb{G}_a )$ over K and a positive integer g we regard both K ^g and A _K ^g as $\Phi \left( {\mathbb{F}_p [t]} \right)$ -modules under the diagonal action induced by Φ. For Γ ? K ^g a finitely generated $\Phi \left( {\mathbb{F}_p [t]} \right)$ -submodule and an affine subvariety $X \subseteq \mathbb{G}_a^g$ defined over K, we study the intersection of X(A _K ), the adèlic points of X, with $\bar \Gamma$ , the closure of Γ with respect to the adèlic topology, showing under various hypotheses that this intersection is no more than X(K) ∩ Γ.