An analogue of the Moutard transformation for the Goursat equation\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y }

We present a new Bäcklund-type transformation for the nonlinear equation $\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y } $ studied by É. Goursat. Goursat found a linearization transformation and some properties of this equation, which make it similar to the Moutard equation u_xy=M(x, y)u. However, this Goursat transformation does not provide proper superposition formulas. We give the necessary extended superposition formulas.     

Certain imbedding theorems of the functional spacesW\frac{l}{p}_{, \overline a , } \left( G \right)

In the paper one investigates imbedding type theorems for the spaces \(W\frac{l}{p}_{, \overline a , x} \left( G \right)\) of real functions, defined in the domain G ? Eⁿ, which represent a further generalization of the generalized spaces of the Morrey type. One considers also the problem of the extremal choice of the parameters characterizing the indicated properties.     

Asymptotic behavior of multiperiodic functionsG(x) = \prod\limits_{n = 1}^\infty {g(x/2^n )}

Let 0≤g be a dyadic Hölder continuous function with period 1 and g(0)=1, and let $G(x) = \prod\nolimits_{n = 0}^\infty {g(x/{\text{2}}^n )} $ . In this article we investigate the asymptotic behavior of $\smallint _0^{\rm T} \left| {G(x)} \right|^q dx$ and $\frac{1}{n}\sum\nolimits_{k = 0}^n {\log g(2^k x)} $ using the dynamical system techniques: the pressure function and the variational principle. An algorithm to calculate the pressure is presented. The results are applied to study the regulatiry of wavelets and Bernoulli convolutions.     

On the functional equation x + f(y + f(x))=y + f(x + f(y))

For an abelian group (G, + ,0) we consider the functional equation $$f : G \to G, x + f(y + f(x)) = y + f(x + f(y)) \quad (\forall x, y \in G), \quad\quad\qquad (1)$$ most times together with the condition $$f(0) = 0.\qquad\qquad\qquad\qquad\qquad (0)$$ Our main question is whether a solution of ${(1) \wedge (0)}$ must be additive, i.e., an endomorphism of G. We shall answer this question in the negative (Example 3.14) Rätz (Aequationes Math 81:300, 2011).     

Some remarks on the moments of \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| in short intervals

Some new results on power moments of the integral $$ J_k (t,G) = \frac{1} {{\sqrt {\pi G} }}\int_{ - \infty }^\infty { \left| {\varsigma \left( {\tfrac{1} {2} + it + iu} \right)} \right|^{2k} } e^{ - (u/G)^2 } du $$ (t ? T, T ^? ≦ G ? T, κ ∈ N) are obtained when κ = 1. These results can be used to derive bounds for moments of $ \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| $ .     

On the Evaluation of the Integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$

The integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$ where m and n are relatively prime positive integers, is evaluated exactly in terms of elementary functions and the Catalan constant G.     

Some new estimates of the Fourier-Bessel transform in the space \mathbb{L}_2 \left( {\mathbb{R}_ + } \right)

The Fourier-Bessel integral transform $$g\left( x \right) = F\left[ f \right]\left( x \right) = \frac{1} {{2^p \Gamma \left( {p + 1} \right)}}\int\limits_0^{ + \infty } {t^{2p + 1} f\left( x \right)j_p \left( {xt} \right)dt}$$ is considered in the space $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ . Here, j _p (u) = ((2 ^p Γ(p+1))/(u ^p ))J _p (u) and J _p (u) is a Bessel function of the first kind. New estimates are proved for the integral $$\delta _N^2 \left( f \right) = \int\limits_N^{ + \infty } {x^{2p + 1} g^2 \left( x \right)dx, N > 0,}$$ in $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ for some classes of functions characterized by a generalized modulus of continuity.     

Twists and Singular Vectors in \widehat{sl}\left( {2\left| 1 \right.} \right) Representations

We propose new formulas for singular vectors in Verma modules over the affine Lie superalgebra . We analyze the coexistence of singular vectors of different types and identify the twisted modules arising as submodules and quotient modules of Verma modules. We show that with the twists (spectral flow transformations) properly taken into account, a resolution of irreducible representations can be constructed consisting of only the modules.     

Congruences involving $$g_n(x)=\sum \limits _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k$$

Define \(g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k\) for \(n=0,1,2,\ldots \). Those numbers \(g_n=g_n(1)\) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving \(g_n(x)\). For example, for any prime \(p>5\) we have

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$

This is similar to Wolstenholme’s classical congruences

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$

for any prime \(p>3\).

     

The Isometric Extension of an Into Mapping from the Unit Sphere S（t（2）^∞） to S（L^1（μ））

This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.     

Existence and Uniqueness of Limiting Cycle of Equations $\[\dot x = \varphi (y) - F(x),\dot y = - g(x)\]$ and its Existence of Two and only Two Limiting Cycle

In this paper, we consider the existence of limiting cycle of the system of equations $\[\dot x = \varphi (y) - F(x),\dot y = - g(x)\]$(E) its existence and uniqueness, and its existence of two and only two limiting cycle. The theorems of existence and of existence and uniquence include following conditions: 1° All orbits of system (E) rotate round origin and not all orbits rotate round origin; 2° All or some of integral $\[\int_0^{ \pm \infty } {g(x)}dx \]$ and $\[\int_0^{ \pm \infty } {F'(x)} dx\]$ diverge or converge; 3° System (E) has one or two (one of them is saddle point) singular points. These theorems include following results: 1° All orbits of system (E), if not zero, tend to the unique cycle as $\[t \to + \infty \]$. 2° The result allow us to decide the place and the number of cycle etc. In the theorem of existence of two and only two limiting cycle, F(x) and g (x) needn't odd functions; number of zero point of F(jx) may be five or over five; F (x) may ascend or descend repeatedly in certain finite interval. Combining § 2 with § 3, in fact, we can give a result of the existence of n and only n limiting cycle of system (E).     

Applications of fractional complex transform and $\left( \frac{G^{\prime }}{G}\right) $-expansion method for time-fractional differential equations

In this paper, the fractional complex transform and the $\left( \frac{G^{\prime }}{G}\right) $-expansion method are employed to solve the time-fractional modfied Korteweg-de Vries equation (fmKdV),Sharma-Tasso-Olver, Fitzhugh-Nagumo equations, where $G$ satisfies a second order linear ordinary differential equation. Exact solutions are expressed in terms of hyperbolic, trigonometric and rational functions. These solutions may be useful and desirable to explain some nonlinear physical phenomena in genuinely nonlinear fractional calculus.     

Cohomology of Algebras of Dihedral Type. IV: the Family D\left( {{2\mathcal{B}}} \right)

For algebras of dihedral Type that form the family (from K. Erdmann's list), the Yoneda algebras are described in terms of quivers with relations. Bibliography: 9 titles.     

Cartan Invariants in the Category of Restricted Representation for gl（m｜n）

Let k be an algebraically closed field of characteristic p 〉 2, and gl（m｜n） be the general linear Lie superalgebra over k. The Cartan invariants for the restricted supermodule category for gl（m｜n） are presented.     

非线性方程组的符号矩阵，正则分裂与单调包含（Ⅰ）

As a continuation of part I of the paper under the same title, we developgeneral monotonic enclosure methods for the couple systems of the splitting equations {x=G([x]a,[x]b,[y]c) y=G([y]a,[y]b,[x]c),which models the system of equations associated with hybrid and aaynchronotts monotonicity as well as convexity. The resulting algorithms and convergence theorems generalize and unify various known methods and monotonic enclosure theorents established by other authors.     

Mathematical Analysis of a Cauchy Problem for the Time-Fractional Diffusion-Wave Equation with $$ \alpha \in \left( 0,2\right) $$

This paper deals with a theoretical mathematical analysis of a Cauchy problem for the time-fractional diffusion-wave equation in the upper half-plane, \(x\in \mathbb {R}\), \(t\in \mathbb {R}^+\), where the Caputo fractional derivative of order \(\alpha \in \left( 0,2\right) \) is considered. An explicit solution to this Cauchy problem is obtained via separation of variables. A first proof of the validity of the obtained results is provided for a certain kind of initial conditions. Throughout this work a new expression of the solution to this problem and its utility for carrying out rigurous proofs are presented. Finally, several new properties of the solution are obtained.     

THE BEST APPROXIMATION CONSTANT FOR THE JACKSON''''S TYPE OPERATOR J_(n,3) (f;x)

In this paper we obtain the best approximation constant of function f(x)(∈C_(2π))by theJackson's type operator J_(π3)(f;x),i.e.‖J_(n,3)(f,x)-f(x)‖_c≤(4-6/π)ω(f,1/n),‖J_(n,3)(f,x)-f(x)‖_c≤(8-17/π)ω_2(f,1/n)     

On Lagrange interpolatory parabolas to $\left | x\right | ^{\alpha }$ at equally spaced nodes

In this paper we present a generalized quantitative version of a result due to D. L. Berman concerning the exact convergence rate at zero of Lagrange interpolation polynomials to | x| ^a\left | x\right | ^{\alpha } based on equally spaced nodes in [-1, 1]. The estimates obtained turn out to be best possible.     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .