Let $s$ and $t$ be two coprime integers. Bessenrodt and Olsson obtained the number of $(\overline{s},\overline{t})$-cores for odd $s$ and odd $t$ by establishing a bijection between the lattice paths in $(s,t)$ Yin-Yang diagram and $(\overline{s},\overline{t})$-cores. In this paper, motivated by their results, we extend the definition of Yin-Yang diagram and the bijection to all possible coprime pairs $(s,t)$, then obtain that the number of $(\overline{s},\overline{t})$-cores is $\binom{\lfloor s/2 \rfloor + \lfloor t/2 \rfloor}{\lfloor s/2 \rfloor}$. Furthermore, based on the identities of Chen-Huang-Wang, we determine the average size of an $(\overline{s},\overline{t})$-core depending on the parity of $s$, which is $(s-1) (t-1) (s+t-2)/48$ if $s$ and $t$ are both odd, or $(t-1) (s^2+st-3s+2t+2)/48$ if $s$ is even and $t$ is odd.

Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_{n-1} \geq \mu_n = 0$ be the Laplacian eigenvalues of graph $G$ and let $\sigma = \sigma(G)$ ($1 \leq \sigma \leq n$) be the largest positive integer such that $\mu_{\sigma} \geq 2m/n$. In this paper, we prove that $\mu_2(G) \geq 2m/n$ for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in [8], that is, the characterization of all graphs with $\sigma = 1$.

Assume that $\psi$ is a function of positive variable $t$, monotonically increasing to infinity and $0 \lt \psi(t) \ll \log t/(\log \log t)$. Let $\mathcal{R}_{3}(n)$ denote the number of representations of the integer $n$ as sums of two squares and three biquadrates of primes and we write $\mathcal{E}_{3}(N)$ for the number of integers $n$ satisfying $n \leq N$, $n \equiv 5, 53, 101 \pmod{120}$ and \[ \left| \mathcal{R}_{3}(n) - \frac{\Gamma^{2}(1/2) \Gamma^{3}(1/4)}{\Gamma(7/4)} \frac{\mathfrak{S}_{3}(n) n^{3/4}}{\log^{5}n} \right| \geq \frac{n^{3/4}}{\psi(n) \log^{5}n}, \] where $0 \lt \mathfrak{S}_{3}(n) \ll 1$ is the singular series. In this paper, we prove \[ \mathcal{E}_{3}(N) \ll N^{23/48+\varepsilon} \psi^{2}(N) \] for any $\varepsilon \gt 0$. This result constitutes a refinement upon that of Friedlander and Wooley [2].

Let $k$ be an integer with $k \geq 3$. Let $\lambda_1$, $\lambda_2$, $\lambda_3$ be non-zero real numbers, not all negative. Assume that $\lambda_1/\lambda_2$ is irrational and algebraic. Let $\mathcal{V}$ be a well-spaced sequence, and $\delta \gt 0$. In this paper, we prove that, for any $\varepsilon \gt 0$, the number of $\upsilon \in \mathcal{V}$ with $\upsilon \leq X$ such that the inequality \[ |\lambda_1 p_1^2 + \lambda_2 p_2^2 + \lambda_3 p_3^k - \upsilon| \lt \upsilon^{-\delta} \] has no solution in primes $p_1$, $p_2$, $p_3$ does not exceed $O(X^{1-2/(7m_2(k))+2\delta+\varepsilon})$, where $m_2(k)$ relies on $k$. This refines a recent result. Furthermore, we briefly describe how a similar method can refine a previous result on a Diophantine problem with two squares of primes, one cube of primes and one $k$-th power of primes.

In this paper, character formulas are explicitly characterized for all simple restricted modules of Hamiltonian Lie superalgebras of odd type over an algebraically closed field of characteristic $p \gt 3$. In the process we use the lengths and highest weights of simple quotients of restricted Kac modules of atypical weights with respect to a series of Borel subalgebras to determine the composition factors, composition series and the character formulas for the restricted Kac modules of atypical weights for the Lie superalgebras under consideration.

Let $A$ be a noetherian local ring of dimension $d \geq 1$ and $\operatorname{depth}(A) \geq d-1$. In this paper, we study the non-positivity for the Hilbert coefficients of parameter ideals in the ring $A$. Moreover, we establish a bound for the Castelnuovo-Mumford regularity of associated graded ring of $A$ with respect to parameter ideal in terms of the first Hilbert coefficient and the dimension.

Let $\mathcal{L}$ be a Schrödinger operator of the form $\mathcal{L} = -\Delta + V$ acting on $L^2(\mathbb{R}^n)$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q \geq n$. Let $L^{p,\lambda}(\mathbb{R}^{n})$, $0 \leq \lambda \lt n$ denote the Morrey space on $\mathbb{R}^{n}$. In this paper, we will show that a function $f \in L^{2,\lambda}(\mathbb{R}^{n})$ is the trace of the solution of $\mathbb{L}u := u_{t} + \mathcal{L}u = 0$, $u(x,0) = f(x)$, where $u$ satisfies a Carleson-type condition \[ \sup_{x_B,r_B} r_B^{-\lambda} \int_0^{r_B^2} \!\! \int_{B(x_B,r_B)} |\nabla u(x,t)|^2 \, dx dt \leq C \lt \infty. \] Conversely, this Carleson-type condition characterizes all the $\mathbb{L}$-carolic functions whose traces belong to the Morrey space $L^{2,\lambda}(\mathbb{R}^{n})$ for all $0 \leq \lambda \lt n$. This result extends the analogous characterization found by Fabes and Neri in [8] for the classical BMO space of John and Nirenberg.

In this paper, we consider a class of eigenvalue problems for fractional $p(x,y)$-Laplacian equations with indefinite weight in fractional Sobolev space with variable exponent. Under some suitable conditions on the growth rates involved in the problem, we establish some results on the existence of a continuous family of eigenvalues using variational techniques and Ekeland's variational principle.

In this paper we characterize Toeplitz matrices with entries in the space of bounded operators on Hilbert spaces $\mathcal{B}(H)$ which define bounded operators acting on $\ell^2(H)$ and use it to get the description of the right Schur multipliers acting on $\ell^2(H)$ in terms of certain operator-valued measures.

In this paper, we consider a transmission problem in the presence of a boundary control condition of nonlocal type. We prove well-posedness by using the semigroup theory. Also we establish an optimal decay result by frequency domain method and Borichev-Tomilov theorem.

In previous work [23], Mustafa considered a viscoelastic laminated beam system with structural damping in the case of equal-speed wave propagations, and established explicit energy decay formula which gives the best decay rates. In this paper, we continue to consider the similar problems and establish the general decay result for the energy, to system with structural damping in the case of non-equal wave speeds and to system without structural damping in the case of equal wave speeds, respectively. For the first case, we use the second-order energy method to overcome the difficulty of estimating the non-equal speeds term. For the second case, we construct an appropriated perturbed functional to estimate $\|w_{t}\|^{2}_{2}$ so as to overcome the absence of structural damping.

The early exercise property of American option changes the original Black-Scholes equation to an inequality that cannot be solved via traditional finite difference method. Therefore, finding the early exercise boundary prior to spatial discretization is a must in each time step. This overhead slows down the computation and the accuracy of solution relies on if the early exercise boundary can be accurately located. A simple numerical method based on finite difference and method of lines is proposed here to overcome this difficulty in American option valuation. Our method averts the otherwise necessary procedure of locating the optimal exercise boundary before applying finite difference discretization. The method is efficient and flexible to all kinds of pay-off. Computations of American put, American call with dividend, American strangle, two-factor American basket put option, and two-factor convertible bond with embedded call and put options are demonstrated to show the efficiency of the current method.

In this paper, we consider an inverse problem of identifying an unknown time dependent source function in a time-fractional diffusion-wave equation. First, some basic properties of the shifted Jacobi polynomials (SJPs) are presented. Then, the analytical solution of the direct problem is given and used to obtain an approximation of the unknown source function in a series of SJPs. Due to ill-posedness of this inverse problem, the Tikhonov regularization method with Morozov's discrepancy principle criterion is applied to find a stable solution. After that, an error bound is obtained for the approximation of the unknown source function. Finally, some numerical examples are provided to show effectiveness and robustness of the proposed algorithm.