On holomorphic solutions of some inhomogeneous linear differential equations in a banach space over a non-archimedean field

Let A be a closed linear operator on a Banach space $ \mathfrak{B} $ \mathfrak{B} over the field Ω of complex p-adic numbers having an inverse operator defined on the whole $ \mathfrak{B} $ \mathfrak{B} , and f be a locally holomorphic at 0 $ \mathfrak{B} $ \mathfrak{B} -valued vector function. The problem of existence and uniqueness of a locally holomorphic at 0 solution of the differential equation y ^(m) − Ay = f is considered in this paper. In particular, it is shown that this problem is solvable under the condition $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} = 0. It is proved also that if the vector-function f is entire, then there exists a unique entire solution of this equation. Moreover, the necessary and sufficient conditions for the Cauchy problem for such an equation to be correctly posed in the class of locally holomorphic functions are presented.     

Boundedness and compactness of the embedding between spaces with multiweighted derivatives when 1 ⩽ <Emphasis Type="Italic">q</Emphasis> < <Emphasis Type="Italic">p</Emphasis> < ∞

We consider a new Sobolev type function space called the space with multiweighted derivatives $ W_{p,\bar \alpha }^n $ W_{p,\bar \alpha }^n , where $ \bar \alpha $ \bar \alpha = (α ₀, α ₁,…, α _n ), α _i ∈ ℝ, i = 0, 1,…, n, and $ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} $ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} ,

On the hierarchies of higher order mKdV and KdV equations

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $ \hat H_s^r \left( \mathbb{R} \right) $ \hat H_s^r \left( \mathbb{R} \right) defined by the norm

Sharp estimates for derivatives of functions in the Sobolev classes $$ \mathop {W_2^r }\limits^ \circ $$(−1,1)

Explicit formulas are obtained for the maximum possible values of the derivatives f ^(k)(x), x ∈ (−1, 1), k ∈ {0, 1, ..., r − 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to ≤ r − 1 at the points ±1 and are such that $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 . As a corollary, it is shown that the first eigenvalue λ _1,r of the operator (−D ²) ^r with these boundary conditions is $ \sqrt 2 $ \sqrt 2 (2r)! (1 + O(1/r)), r → ∞.     

Local limit theorem for the first crossing time of a fixed level by a random walk

Let X,X(1),X(2),... be independent identically distributed random variables with mean zero and a finite variance. Put S(n) = X(1) + ... + X(n), n = 1, 2,..., and define the Markov stopping time η _y = inf {n ≥ 1: S(n) ≥ y} of the first crossing a level y ≥ 0 by the random walk S(n), n = 1, 2,.... In the case $ \mathbb{E} $ \mathbb{E} |X|³ < ∞, the following relation was obtained in [8]: $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) as n → ∞, where the constant R and the bounded sequence ν _n were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0, and there was found a representation for H(y). The present paper was motivated by the following reason. In [8], the authors unfortunately did not cite papers [1, 5] where the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [5] the existence of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0 under the condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ only; In [1], an explicit form of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) was found under the same condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ in the case when the summand X has an arithmetic distribution. In the present paper, we prove that the main assertion in [5] fails and we correct the original proof. It worth noting that this corrected version was formulated in [8] as a conjecture.     

Convergence rates of the LIL for random fields in Hilbert spaces

Considering the positive d-dimensional lattice point Z ₊ ^d (d ≥ 2) with partial ordering ≤, let {X _k: k ∈ Z ₊ ^d } be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $ S_n = \sum\limits_{k \leqslant n} {X_k } $ S_n = \sum\limits_{k \leqslant n} {X_k } , n ∈ Z ₊ ^d . Let σ _i ², i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ ². Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) . We show that when l ≥ 2 and b > −l/2, E[‖X‖²(log ‖X‖) ^d−2(log log ‖X‖) ^b+4] < ∞ implies $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} , where Γ(·) is the Gamma function and $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } .     

Hamilton-type gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let (M,g) be a complete noncompact Riemannian manifold. In this note, we derive a local Hamilton-type gradient estimate for positive solution to a simple nonlinear parabolic equation

Transient phenomena for random walks in the absence of the expected value of jumps

Let ξ, ξ₁, ξ₂, ... be independent identically distributed random variables, and S _n :=Σ _j=1 ⁿ ,ξ _j , $ \bar S $ \bar S := sup _n≥0 S _n . If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution $ \bar S $ \bar S as a → 0 and $ \bar S $ \bar S tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models:

Real moments of the restrictive factor

Let λ be a real number such that 0 < λ < 1. We establish asymptotic formulas for the weighted real moments Σ _n≤x R ^λ (n)(1 − n/x), where R(n) =$ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu - 1} } $ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu - 1} } is the Atanassov strong restrictive factor function and n =$ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } $ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } is the prime factorization of n.     

A note on nil power serieswise Armendariz rings

A ring R is called nil power serieswise Armendariz if $ \forall f = \sum\limits_{i = 0}^\infty {a_i X^i } $ \forall f = \sum\limits_{i = 0}^\infty {a_i X^i } and $ g = \sum\limits_{i = 0}^\infty {b_i X^i } $ g = \sum\limits_{i = 0}^\infty {b_i X^i } in R[[X]] such that f g ∈ Nil(R)[[X]], then a _i b _j ∈ Nil(R) for all i and j. In this note we characterize completely nil power serieswise Armendariz rings with their nilradical Nil(R) (where the nilradical is the set of nilpotent elements). We prove that a ring is nil power serieswise Armendariz if and only if Nil(R) is an ideal of R. We prove that each power serieswise Armendariz ring is nil power serieswise Armendariz and we give examples of nil power serieswise Armendariz rings.     

Multidimensional analog of the Hardy condition for power series

In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f ⁻¹(x) = $ \sum\limits_{n = 0}^\infty {b_n x^n } $ \sum\limits_{n = 0}^\infty {b_n x^n } b _n x ⁿ be such that b ₀ > 0 and b _n ≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x ₁, x ₂, …, x _m ) with positive coefficients in order that the series f ⁻¹(x ₁, x ₂, …, x _m ) = $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } satisfies the property b _{0, …, 0} > 0, $ bi_1 ,i_2 , \ldots ,i_m $ bi_1 ,i_2 , \ldots ,i_m ≤ 0, i ₁² + i ₂² + … + i _m ² > 0, which is similar to the one-dimensional case.     

On the geometry of metric measure spaces

We introduce and analyze lower (Ricci) curvature bounds  ⩾ K for metric measure spaces . Our definition is based on convexity properties of the relative entropy regarded as a function on the L ₂-Wasserstein space of probability measures on the metric space . Among others, we show that  ⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds,  ⩾ K if and only if  ⩾ K for all . The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation. We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence.     

Strict topology over the space of measures with continuous translations on a locally compact foundation semigroup

Let $ \mathfrak{S} $ \mathfrak{S} be a locally compact semigroup, ω be a weight function on $ \mathfrak{S} $ \mathfrak{S} , and M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) be the weighted semigroup algebra of $ \mathfrak{S} $ \mathfrak{S} . Let L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) be the C*-algebra of all M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)-measurable functions g on $ \mathfrak{S} $ \mathfrak{S} such that g/ω vanishes at infinity. We introduce and study a strict topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) and show that the Banach space L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) can be identified with the dual of M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) endowed with β ¹($ \mathfrak{S} $ \mathfrak{S} , ω). We finally investigate some properties of the locally convex topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω).     

The best extension of algebraic polynomials from the unit circle

We study the class $ \mathfrak{P}_n $ \mathfrak{P}_n of algebraic polynomials P _n (x, y) in two variables of total degree n whose uniform norm on the unit circle Γ₁ centered at the origin is at most 1: $ \left\| {P_n } \right\|_{C(\Gamma _1 )} $ \left\| {P_n } \right\|_{C(\Gamma _1 )} ≤ 1. The extension of polynomials from the class $ \mathfrak{P}_n $ \mathfrak{P}_n to the plane with the least uniform norm on the concentric circle Γ _r of radius r is investigated. It is proved that the values θ _n (r) of the best extension of the class $ \mathfrak{P}_n $ \mathfrak{P}_n satisfy the equalities θ _n (r) = r ⁿ for r > 1 and θ _n (r) = r ⁿ⁻¹ for 0 < r < 1.     

Dual Riemannian spaces of constant curvature on a normalized hypersurface

In this paper, the following results are obtained: 1) It is proved that, in the fourth order differential neighborhood, a regular hypersurface V _n−1 embedded into a projective-metric space K _n , n ≥ 3, intrinsically induces a dual projective-metric space $ \bar K_n $ \bar K_n . 2) An invariant analytical condition is established under which a normalization of a hypersurface V _n−1 ⊂ K _n (a tangential hypersurface $ \bar V_{n - 1} $ \bar V_{n - 1} ⊂ $ \bar K_n $ \bar K_n ) by quasitensor fields H _n ⁱ , H _i ($ \bar H_n^i $ \bar H_n^i , $ \bar H_i $ \bar H_i ) induces a Riemannian space of constant curvature. If the two conditions are fulfilled simultaneously, the spaces R _n−1 and $ \bar R_{n - 1} $ \bar R_{n - 1} are spaces of the same constant curvature $ K = - \tfrac{1} {c} $ K = - \tfrac{1} {c} . 3) Geometric interpretations of the obtained analytical conditions are given.     

Boundedness of parabolic singular integrals and Marcinkiewicz integrals on Triebel-Lizorkin spaces

In this paper,we obtain the boundedness of the parabolic singular integral operator T with kernel in L(logL)1/γ(Sn-1) on Triebel-Lizorkin spaces.Moreover,we prove the boundedness of a class of Marcinkiewicz integrals μΩ,q(f) from ∥f∥ F˙p0,q(Rn) into Lp(Rn).     

New improvements on connectivity of cages

A (δ, g)-cage is a δ-regular graph with girth g and with the least possible number of vertices. In this paper, we show that all (δ, g)-cages with odd girth g ≥ 9 are r-connected, where (r − 1)² ≤ δ + $ \sqrt \delta $ \sqrt \delta − 2 < r ² and all (δ, g)-cages with even girth g ≥ 10 are r-connected, where r is the largest integer satisfying $ \frac{{r\left( {r - 1} \right)^2 }} {4} + 1 + 2r\left( {r - 1} \right) \leqslant \delta $ \frac{{r\left( {r - 1} \right)^2 }} {4} + 1 + 2r\left( {r - 1} \right) \leqslant \delta . These results support a conjecture of Fu, Huang and Rodger that all (δ, g)-cages are δ-connected.     

On $$ \mathfrak{F}_h $$-normal subgroups of finite groups

Let G be a finite group, and let $ \mathfrak{F} $ \mathfrak{F} be a formation of finite groups. We say that a subgroup H of G is $ \mathfrak{F}_h $ \mathfrak{F}_h -normal in G if there exists a normal subgroup T of G such that HT is a normal Hall subgroup of G and (H ∩ T)H _G /H _G is contained in the $ \mathfrak{F} $ \mathfrak{F} -hypercenter $ Z_\infty ^\mathfrak{F} $ Z_\infty ^\mathfrak{F} (G/H _G ) of G/H _G . In this paper, we obtain some results about the $ \mathfrak{F}_h $ \mathfrak{F}_h -normal subgroups and then use them to study the structure of finite groups.     

A sharp form of an embedding into multiple exponential spaces

Let Θ be a bounded open set in ℝ ⁿ , n ⩾ 2. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that