We study a polynomial interpolation of finite multiple zeta and zeta-star values with variable $t$, which is an analogue of interpolated multiple zeta values introduced by Yamamoto. We introduce several relations among them and, in particular, prove the cyclic sum formula, the Bowman--Bradley type formula, and the weighted sum formula. The harmonic relation, the shuffle relation, the duality relation, and the derivation relation are also presented.

In this paper, we investigate a quantum-classical correspondence property of a system of a massless Klein-Gordon field coupled to a static source. Under momentum regularity conditions, the total Hamiltonian is a self-adjoint operator on a boson Fock space. We introduce annihilation operators in the Heisenberg picture and define a sesquilinear form. Under additional momentum regularity conditions, it is proven that the sesquilinear form is a solution of the classical field equation.

A nonlinear superposition operator $T_g$ related to a Borel measurable function $g:\ {\mathbb C}\to {\mathbb C}$ is defined via $T_g(f):=g\circ f$ for any complex-valued function $f$ on ${\mathbb R^n}$. This article is devoted to investigating the mapping properties of $T_g$ on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083--2101], as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion and the continuity properties of $T_g$ on these spaces are obtained.

Let $f$ be an operator convex function on $I$ and $A,$ $B\in \mathcal{SA}_{I}\left( H\right) ,$ the convex set of selfadjoint operators with spectra in $I.$ If $A\neq B$ and $f,$ as an operator function, is Gâteaux differentiable on \begin{equation*} [ A,B] :=\left\{ ( 1-t) A+tB \mid t\in [ 0,1] \right\} \,, \end{equation*} while $p:[ 0,1] \rightarrow \lbrack 0,\infty )$ is Lebesgue integrable and symmetric, namely $p\left( 1-t\right) $ $=p\left( t\right) $ for all $t\in [ 0,1] ,$ then \begin{align*} 0& \leq \int_{0}^{1}p\left( t\right) f\left( \left( 1-t\right) A+tB\right) dt-\left( \int_{0}^{1}p\left( t\right) dt\right) f\left( \frac{A+B}{2}\right) \\ & \leq \frac{1}{2}\left( \int_{0}^{1}\left\vert t-\frac{1}{2}\right\vert p\left( t\right) dt\right) \left[ \nabla f_{B}\left( B-A\right) -\nabla f_{A}\left( B-A\right) \right] \end{align*} and \begin{align*} 0& \leq \left( \int_{0}^{1}p\left( t\right) dt\right) \frac{f\left( A\right) +f\left( B\right) }{2}-\int_{0}^{1}p\left( t\right) f\left( \left( 1-t\right) A+tB\right) dt \\ & \leq \frac{1}{2}\int_{0}^{1}\left( \frac{1}{2}-\left\vert t-\frac{1}{2} \right\vert \right) p\left( t\right) dt\left[ \nabla f_{B}\left( B-A\right) -\nabla f_{A}\left( B-A\right) \right] \,. \end{align*} Two particular examples of interest are also given.

For locally compact Hausdorff spaces $X$ and $Y$, and function algebras $A$ and $B$ on $X$ and $Y$, respectively, surjections $T:A \longrightarrow B$ satisfying norm multiplicative condition $\|Tf\, Tg\|_Y =\|fg\|_X$, $f,g\in A$, with respect to the supremum norms, and those satisfying $\||Tf|+|Tg|\|_Y=\||f|+|g|\|_X$ have been extensively studied. Motivated by this, we consider certain (multiplicative or additive) subsemigroups $A$ and $B$ of $C_0(X)$ and $C_0(Y)$, respectively, and study surjections $T$ from $A$ onto $B$ satisfying the norm condition $\rho(Tf, Tg)=\rho(f,g)$, $f,g \in A$, for some classes of two variable positive functions $\rho$. It is shown that such a map $T$ is also a composition in modulus map.

By using the Fourier transform method combining with the weighted energy method and some new skills, the global stability of oscillatory traveling waves for a monostable delayed equation is established. Particularly, the nonlinearity is nonlocal in the equation.

We consider a differential system $x\frac{d}{dx} Y=GY$ where $G(0)$ is a nilpotent matrix. Then there exists a solution matrix of the form $Y=F \exp(G(0)\log x)$. If a solution matrix at a generic point is of log-growth $\delta$, then we prove that $F$ is of log-growth $\delta$.

This is a sequel to the paper “Large time asymptotics for a cubic nonlinear Schrödinger system in one space dimension” by the same authors. We continue to study the Cauchy problem for the two-component system of cubic nonlinear Schrödinger equations in one space dimension. We provide criteria for large time decay or non-decay in $L^2$ of the small amplitude solutions in terms of the Fourier transforms of the initial data.

Let $X$ be a smooth complex projective variety of dimension $n\geq 2$ polarized by an ample line bundle $H$. For $n=2$, the structure of pairs $(X,H)$ as above is described under the assumption that $m-3d\leq 4$, where $m$ and $d$ stand for the class and the degree of $(X,H)$, respectively. In particular, special attention is payed to the case $m-2d \leq 1$, in which $X$ turns out to be a ruled surface. In higher dimensions, adding to the above setting an ample vector bundle $\mathcal{E}$ of rank $r\leq n-2$ on $X$ such that $\mathcal{F}:=\mathcal{E}\oplus H^{\oplus (n-r-2)}$ admits a regular section vanishing on a smooth surface, by relying on the analysis made in the surface case, the structure of the generalized polarized manifold $(X,\mathcal F)$ is described for small values of $m_2-3d$, where $m_2=m_2(X,\mathcal{E},H)$ is the generalized second class defined in a previous paper and $d:=c_{n-2}(\mathcal{F})H^2$.

The main goal of this article is to present a new approach, made up of integrals, to refine some numerical radius inequalities. Let $A$ be a bounded linear operator on a complex Hilbert space. If $1\le r\le 2$, it is shown that \[{{\omega }^{2r}}\left( A \right)\le \left\| \int_{0}^{1}{{{\left( \left( 1-t \right)\left( \frac{{{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}}}{2} \right)+t\omega \left( {{A}^{2}} \right)I \right)}^{r}}dt} \right\|\,.\] Here $\omega \left( \cdot \right)$, $\left\| \cdot \right\|$ are the numerical radius and the usual operator norm, $\left| A \right|={{\left( {{A}^{*}}A \right)}^{{1}/{2}\;}}$, and $I$ is the identity operator.

We show the ergodicity of Tanaka–Ito type $\alpha$-continued fraction maps and construct their natural extensions. We also discuss the relation between entropy and the size of the natural extension domain.

We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.

Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not necessarily containing the cyclotomic $\mathbb{Z}p$-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the $p$-adic Lie extension is one-dimensional, we show that the Pontryagin dual of the classical Selmer group has no nontrivial finite submodules. Finally, we show that the aforementioned conclusions carry over to the Selmer group of a non-ordinary cuspidal modular form.

For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces), we show that the set of points for which the Birkhoff average of a continuous function does not exist (called the irregular set) is either empty or has full topological entropy. This generalizes Thompson's theorem for irregular sets of $\beta$-transformations, and reduces a complete description of irregular sets of transitive piecewise monotonic maps to the Hofbauer-Raith problem on the density of periodic measures.

Let $k$ be a real quadratic field. A criterion for the Diophantine equation $X^3=u+v$ to have no solutions in a nonzero integer $X$ of $k$ and units $u,v$ of $k$ is given.

In this paper we generalize Bauer's Identical Congruence appearing in Hardy and Wright's book [6], Theorems 126 and 127. Bauer's Identical Congruence asserts that the polynomial $\prod_t(x-t)$, where the product runs over a reduced residue system modulo a prime power $p^a$, is congruent (mod $p^a$) to the “simple” polynomial $(x^{p-1}-1)^{p^{a-1}}$ if $p>2$ and $(x^2-1)^{2^{a-2}}$ if $p=2$ and $a\geqslant2$. Our article generalizes these results to a broader context, in which we find a “simple” form of the polynomial $\prod_t(x-t)$, where the product runs over the solutions of the congruence $t^n\equiv 1\pmod{\mathrm{P}^a}$ in the framework of the ring of algebraic integers of a given number field $\mathbb{K}$, and where $\mathrm{P}$ is a prime ideal.