Recently, a construction of group divisible designs (GDDs) derived from the decoding of quadratic residue (QR) codes was given. In this paper, we extend the idea to obtain a new family of GDDs, which is also involved with a well-known balanced incomplete block design (BIBD).

The circular descent of a permutation $\sigma$ is a set $\{ \sigma(i) \mid \sigma(i) \gt \sigma(i+1) \}$. In this paper, we focus on the enumerations of permutations by the circular descent set. Let $\operatorname{cdes}_n(S)$ be the number of permutations of length $n$ which have the circular descent set $S$. We derive the explicit formula for $\operatorname{cdes}_n(S)$. We describe a class of generating binary trees $T_k$ with weights. We find that the number of permutations in the set $\operatorname{CDES}_n(S)$ corresponds to the weights of $T_k$. As a application of the main results in this paper, we also give the enumeration of permutation tableaux according to their shape.

In this paper we prove a conjecture of Farkas and Kra, which is a modular equation involving a half sum of certain modular form of weight $1$ for congruence subgroup $\Gamma_1(k)$ with any prime $k$. We prove that their conjecture holds for all odd integers $k \geq 3$. A new modular equation of Farkas and Kra type is also established.

In this paper, we find the non-symplectic indexes of all supersingular K3 surfaces in terms of their period spaces when the base field is of odd characteristic. And we show that the maximal non-symplectic index of a K3 surface defined over a field of odd characteristic $p$ is $p^{10}+1$.

We investigate the commutators $[b,I_{\rho}]$ of generalized fractional integral operators $I_{\rho}$ with functions $b$ in generalized Campanato spaces and give a necessary and sufficient condition for the boundedness of the commutators on Orlicz spaces. To do this we define Orlicz spaces with generalized Young functions and prove the boundedness of generalized fractional maximal operators on the Orlicz spaces.

Recently, Lenzmann and the author observed how to obtain a large class of sharp commutator estimates by a combination of an integration by parts, an harmonic extension, and trace space estimates. In this survey we review this approach in three concrete examples: the Jacobian estimate by Coifman-Lions-Meyer-Semmes, the Coifman-Rochberg-Weiss commutator estimate for Riesz transforms, and a Kato-Ponce-Vega-type inequality.

In this paper, we are interested in the multiple solutions of the following fractional critical problem \[ \begin{cases} (-\Delta)^s u = |u|^{2_s^*-2} u + \lambda u &\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\mathbb{R}^N \setminus \Omega$}, \end{cases} \] where $s \in (0,1)$, $N \gt 4s$, $2^*_s = 2N/(N-2s)$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ and $(-\Delta)^s$ is the fractional Laplace operator. Because the nonlocal property of fractional Laplacian makes the variational functional of the fractional critical problem different from the one of local operator $-\Delta$. To the best of our knowledge, it is still unknown whether multiple solutions of the fractional critical problem exist for all $\lambda \gt 0$. In this paper, we give a partial answer. Precisely, by introducing some new ideas and careful estimates, we prove that for any $s \in (0,1)$, the fractional critical problem has at least $[(N+1)/2]$ pairs of nontrivial solutions if $0 \lt \lambda \neq \lambda_n$, and has $[(N+1-l)/2]$ pairs if $\lambda = \lambda_n$ with multiplicity number $0 \lt l \lt \min \{n,N+2\}$, via constraint method and Krasnoselskii genus. Here $\lambda_n$ denotes the $n$-th eigenvalue of $(-\Delta)^s$ with zero Dirichlet boundary data in $\Omega$ and $[a]$ denotes the least positive integer $k$ such that $k \geq a$.

In this paper we explore the structure of certain surjective generalized isometries (which are transformations that leave any given member of a large class of generalized distance measures invariant) of the set of positive invertible elements in a finite von Neumann factor with unit Fuglede-Kadison determinant. We conclude that any such map originates from either an algebra $^*$-isomorphism or an algebra $^*$-antiisomorphism of the underlying operator algebra.

The aim of this paper is to study the traveling waves in a spatial SIRI epidemic model arising from herpes viral infection. We obtain the complete information about the existence and non-existence of traveling waves in the model. Namely, we prove that when the basic reproduction number $\mathcal{R}_0 \gt 1$, there exists a critical wave speed $c^* \gt 0$ such that for each $c \gt c^*$, the model admits positive traveling waves; and for $c \lt c^*$, the model has no non-negative and bounded traveling wave. We also give some numerical simulations to illustrate our analytic results.

This paper deals with lower and upper bounds for the lifespan of solutions to a fourth-order nonlinear hyperbolic equation with strong damping: \[ u_{tt} + \Delta^{2} u - \Delta u - \omega \Delta u_t + \alpha(t) u_t = |u|^{p-2} u. \] First of all, the authors construct a new control function and apply the Sobolev embedding inequality to establish some qualitative relationships between initial energy value and the norm of the gradient of the solution for supercritical case ($2(N-2)/(N-4) \lt p \lt 2N/(N-4)$, $N \geq 5$). And then, the concavity argument is used to prove that the solution blows up in finite time for initial data at low energy level, at the same time, an estimate of the upper bound of blow-up time is also obtained.

Subsequently, for initial data at high energy level, the authors prove the monotonicity of the $L^{2}$ norm of the solution under suitable assumption of initial data, furthermore, we utilize the concavity argument and energy methods to prove that the solution also blows up in finite time for initial data at high energy level.

At last, for the supercritical case, a new control functional with a small dissipative term and an inverse Hölder inequality with correction constants are employed to overcome the difficulties caused by the failure of the embedding inequality ($H^{2}(\Omega) \cap H^{1}_{0}(\Omega) \hookrightarrow L^{2p-2}$ for $2(N-2)/(N-4) \lt p \lt 2N/(N-4)$) and then an explicit lower bound for blow-up time is obtained. Such results extend and improve those of [S. T. Wu, J. Dyn. Control Syst. 24 (2018), no. 2, 287--295].

In this paper, a system of fractional elliptic equation is investigated, which involving fractional critical Sobolev-Hardy exponent and concave-convex terms. By means of variational methods and analytic techniques, the existence and multiplicity of positive solutions to the system is established.

The palindromic inverse eigenvalue problem (PIEP) of constructing matrices $A$ and $Q$ of size $n \times n$ for the quadratic palindromic polynomial $P(\lambda) = \lambda^2 A^{\star} + \lambda Q + A$ so that $P(\lambda)$ has $p$ prescribed eigenpairs is considered. This paper provides two different methods to solve PIEP, and it is shown via construction that PIEP is always solvable for any $p$ ($1 \leq p \leq (3n+1)/2$) prescribed eigenpairs. The eigenstructure of the resulting $P(\lambda)$ is completely analyzed.

In this paper, we focus on the study of finding efficient solutions for a multicriteria optimization problem (MP), where both the objective and constraint functions are SOS-convex polynomials. By using the well-known $\epsilon$-constraint method (a scalarization technique), we substitute the problem (MP) to a class of scalar ones. Then, a zero duality gap result for each scalar problem, its sum of squares polynomial relaxation dual problem, the semidefinite representation of this dual problem, and the dual problem of the semidefinite programming problem, is established, under a suitable regularity condition. Moreover, we prove that an optimal solution of each scalar problem can be found by solving its associated semidefinite programming problem. As a consequence, we show that finding efficient solutions for the problem (MP) is tractable by employing the $\epsilon$-constraint method. A numerical example is also given to illustrate our results.