In this paper, we introduce the vectorization of shuffle products of two sums of multiple zeta values, which generalizes the weighted sum formula obtained by Ohno and Zudilin. Some interesting weighted sum formulas of vectorized type are obtained, such as\begin{align*} &\quad \sum_{\substack{\boldsymbol{\alpha}+\boldsymbol{\beta}=\boldsymbol{k} \\ |\boldsymbol{\alpha}|: \textrm{even}}} M(\boldsymbol{\alpha}) M(\boldsymbol{\beta}) \sum_{|\boldsymbol{c}|=|\boldsymbol{k}|+r+3} 2^{c_{|\boldsymbol{\alpha}|+1}} \zeta(c_0, c_1, \ldots, c_{|\boldsymbol{\alpha}|}, \ldots, c_{|\boldsymbol{k}|+1}+1) \\ &= \frac{1}{2} (2|\boldsymbol{k}|+r+5) M(\boldsymbol{k}) \zeta(|\boldsymbol{k}|+r+4),\end{align*}where $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ and $\boldsymbol{k}$ are $n$-tuples of nonnegative integers with $|\boldsymbol{k}| = k_1 + k_2 + \cdots + k_n$ even; $M(\boldsymbol{u})$ is the multinomial coefficient defined by $\binom{u_1 + u_2 + \cdots + u_n}{u_1, u_2, \ldots, u_n}$ with the value $\frac{|\boldsymbol{u}|!}{u_1! u_2! \cdots u_n!}$; and $r$ is a nonnegative integer. Moreover, some newly developed combinatorial identities of vectorized types involving multinomial coefficients by extending the shuffle products of two sums of multiple zeta values in their vectorizations are given as well.

Let $H$ be a co-quasitriangular Hopf algebra with bijective antipode. We prove that the Drinfeld center of the category of $H$-comodules is equivalent to the category of modules over some braided group. In particular, the equivalence holds not only for a finite dimensional $H$, but also for an infinite dimensional one.

In this paper, we study a complete intersection surface singularity of Brieskorn type and provide a condition for the coincidence of the fundamental cycle and the minimal cycle on the minimal resolution space.

Petty proved that a convex body in $\mathbb{R}^{n}$ has the minimal surface area among its $\operatorname{SL}(n)$ images if and only if its surface area measure is isotropic. Recently, Zou and Xiong generalized this result to the Orlicz setting by introducing a new notion of minimal Orlicz surface area, and the analog of Ball's reverse isoperimetric inequality is established. In this paper, we give the dual results in Orlicz space by introducing a new notion of minimal dual Orlicz surface area. And the dual form of Ball's isoperimetric inequality is established.

In the present paper we consider a special class of spacelike surfaces in the Minkowski $4$-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. They are called meridian surfaces of elliptic or hyperbolic type, respectively. We study these surfaces with respect to their Gauss map. We find all meridian surfaces of elliptic or hyperbolic type with harmonic Gauss map and give the complete classification of meridian surfaces of elliptic or hyperbolic type with pointwise $1$-type Gauss map.

The explicit expressions of mixed and componentwise condition numbers for the spectral projections, generalized spectral projections and sign functions for matrices and regular matrix pairs are derived. The condition numbers improve some known results of the normwise type and reveal the structured perturbations. Statistical condition estimation is applied to these problems which can be calculated efficiently.

In this paper we investigate the attraction property of an unstable manifold of admissible classes for solutions to the semi-linear evolution equation of the form $u(t) = U(t,s) u(s) + \int_s^t U(t, \xi) f(\xi, u(\xi)) \, d\xi$. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like $L_p$-spaces, the Lorentz spaces $L_{p, q}$ and many other function spaces occurring in interpolation theory. We then apply our abstract results to study Fisher-Kolmogorov model with time-dependent environmental capacity.

An infinite homogeneous $d$-dimensional medium initially is at zero temperature. A heat impulse is applied at the origin, raising the temperature there to a value greater than a constant value $u_0 \gt 0$. The temperature at the origin then decays, and when it reaches $u_0$, another equal-sized heat impulse is applied at time $t_1$. Subsequent equal-sized heat impulses are applied at the origin at times $t_n$, $n \geq 2$, when the temperature there has decayed to $u_0$. The waiting-time sequence $\{ t_n - t_{n-1} \}$ can be defined recursively by a difference equation and its asymptotic behavior was first proposed as a conjecture by Myshkis in $1997$.

In this paper we study the same heating-time problem set on semi-infinite regions $[-L, L] \times \mathbb{R}$ and $\{(x, y) : x^2 + y^2 \leq L \} \times \mathbb{R}$ with insulated boundary condition and all actions taking place at some point $\boldsymbol{p}$ which needs not be the origin.

Let $\operatorname{Diff}(M)$ be the space of diffeomorphisms of a closed $C^{\infty}$ manifold $M$ $(\dim M \geq 2)$ endowed with the $C^1$ topology. In this paper we show that for $C^1$ generic $f \in \operatorname{Diff}(M)$, any shadowable chain recurrence class $C_f$ is hyperbolic if it contains a hyperbolic periodic point.

In this paper, we study the existence and multiplicity of semiclassical solutions of a modified version of the Schrödinger-Poisson system with critical nonlinearity in $\mathbb{R}^{3}$. Under some given conditions which are given in Section 1, we prove that the problem has at least one nontrivial solution provided that $\epsilon \leq \varepsilon$ and that for any $n^{*} \in \mathbb{N}$, it has at least $n^{*}$ pairs of solutions if $\epsilon \leq \varepsilon_{n^{*}}$, where $\varepsilon$ and $\varepsilon_{n^{*}}$ are sufficiently small positive numbers. Moreover, these solutions $u_{\epsilon} \to 0$ in $H^{1}(\mathbb{R}^{3})$ as $\epsilon \to 0$.

In this paper we consider a class of nonhomogeneous elliptic equations involving multi-polar Hardy type potentials and a Sobolev critical nonlinearity in an open domain of $\mathbb{R}^{N}$, $N \geq 3$. By Ekeland's Variational Principle and the Mountain Pass Lemma, we prove the existence of multiple solutions under sufficient conditions on the data and the considered parameters.

This paper is concerned with a class of $p(x)$-Kirchhoff type problem in $\mathbb{R}^N$. By the theories of nonsmooth critical point and variable exponent Sobolev spaces, we establish the existence and multiplicity of solutions to the $p(x)$-Kirchhoff type problem under weaker hypotheses on the nonsmooth potential at zero (at infinity, respectively). Some recent results in the literature are generalized and improved.

In this paper, we are concerned with the nonlinear singular integral equation\[ u(x) = |x|^\sigma \int_{R^n} \frac{u^p(y) \, dy}{|x-y|^{n-\alpha}},\]where $\alpha \in (0, n)$, $\sigma \in (\max \{-\alpha, \frac{\alpha-n}{2}\}, 0]$. Such an integral equation appears in the study of sharp constants of the Hardy-Sobolev inequality and the Hardy-Littlewood-Sobolev inequality. It is often used to describe the shapes of the extremal functions. If $0 \lt p \leq \frac{n}{n-\alpha-\sigma}$, there is not any positive solution to this equation. Under the assumption of $p = \frac{n+\alpha+2\sigma}{n-\alpha}$, we obtain an integrability result for the integrable solution $u$ (i.e., $u \in L^{\frac{2n}{n-\alpha}}(\mathbb{R}^n)$) of the integral equation. Such an integrable solution is radially symmetric and decreasing about $x_0 \in \mathbb{R}^n$. Furthermore, $x_0$ is also the origin if $\sigma \neq 0$. In addition, this integrable solution is blowing up with the rate $-\sigma$ when $|x| \to 0$. Moreover, if $n + p \sigma \gt 0$, then $u$ decays fast with the rate $n - \alpha - \sigma$ when $|x| \to \infty$.