In this paper, we investigate the existence of solutions for a nonlinear fractional-order coupled system involving both Caputo and Riemann–Liouville generalized fractional derivatives of different orders equipped with coupled integral boundary conditions. We transform the given system into an equivalent fixed point problem and solve it by applying the standard fixed point theorems. Examples are constructed for the illustration of the obtained results.

We obtain oscillation and nonoscillation criteria for solutions to four-dimensional advanced and delay systems of first-order dynamic equations on time scales. To establish oscillation criteria, we eliminate nonoscillatory solutions of the systems based on the sign of components of the solutions. Furthermore, some of our results are new in the discrete case.

We study the existence of vector-valued solutions to quasilinear elliptic systems with nonpolynomial growth, namely N-functions. Existence of solutions is shown by means of the theory of Young measures, where the right-hand side is in the divergence form.

Let $X,Y$ be asymmetric normed spaces and $L_c\left(X,Y\right)$ the convex cone of all linear continuous operators from $X$ to $Y$. It is known that in general, $L_c\left(X,Y\right)$ is not a vector space. The aim of this note is to give, using the Baire category theorem, a complete characterization on $X$ and a finite dimensional $Y$ so that $L_c\left(X,Y\right)$ is a vector space. For this, we introduce an index of symmetry of the space $X$ denoted $c\left(X\right)\in \left[0,1\right]$ and we give the link between the index $c\left(X\right)$ and the fact that $L_c\left(X,Y\right)$ is in turn an asymmetric normed space for every asymmetric normed space $Y$. Our study leads to a topological classification of asymmetric normed spaces.

Let $T$ be a complete local (Noetherian) ring, and let $A$ be a local subring of $T$ such that the completion of $A$ with respect to its maximal ideal is $T$. We investigate the possible structures of the partially ordered set $Spec\left(A\right)$. Specifically, we explore the minimal prime ideals of $A$ and their formal fibers, the maximal chains of prime ideals in $A$, and the number of prime ideals in $A$ containing combinations of minimal prime ideals of $A$.

In this expository paper, we show how the pressing down lemma and Ulam matrices can be used to study the topology of subsets of ${\omega }_1$. We prove, for example, that if $S$ and $T$ are stationary subsets of ${\omega }_1$ with $S\mathrm{\Delta }T=\left(S-T\right)\cup \left(T-S\right)$ stationary, then $S$ and $T$ cannot be homeomorphic. Because Ulam matrices provide ${\omega }_1$-many pairwise disjoint stationary subsets of any given stationary set, it follows that there are $2^{{\omega }_1}$-many stationary subsets of any stationary subset of ${\omega }_1$ with the property that no two of them are homeomorphic to each other. We also show that if $S$ and $T$ are stationary sets, then the product space $S\times T$ is normal if and only if $S\cap T$ is stationary. In addition, we prove that for any $X\subseteq {\omega }_1$, $X\times X$ is normal, and that if $X\times X$ is hereditarily normal, then $X\times X$ is metrizable.

We have established some new weakly singular integral inequalities, and our obtained inequalities generalize some recently obtained inequalities in the literature. Also, by involving a Caputo-type fractional derivative with respect to another function, we have obtained some applications to fractional differential equations.

We give a new Hermite–Hadamard inequality for a function $f:\left[a,b\right]\times \left[c,d\right]\subset {ℝ}^2\to ℝ$ which is semiconvex of rate $\left(k_1,k_2\right)$ on the coordinates. This generalizes some existing results on Hermite–Hadamard inequalities of S. S. Dragomir. In addition, we explain the Hermite–Hadamard inequality from the point of view of optimal mass transportation with cost function $c\left(x,y\right):=f\left(y-x\right)+\frac{k_1}{2}|x_1-y_1{|}^2+\frac{k_2}{2}|x_2-y_2{|}^2$, where $f\left(\cdot \right):\left[a,b\right]\times \left[c,d\right]\to \left[0,\infty \right)$ is semiconvex of rate $\left(k_1,k_2\right)$ on the coordinates and $x=\left(x_1,x_2\right)$, $y=\left(y_1,y_2\right)\in \left[a,b\right]\times \left[c,d\right]$.

For the field $\mathbb{𝔽}$ and $\mathbb{𝔽}$-vector space $A$, let $G$ be a subgroup of $GL\left(\mathbb{𝔽},A\right)$, the group of invertible $\mathbb{𝔽}$-linear transformations of $A$. A subspace $B$ of $A$ is $G$-contrainvariant if $GB=A$. We discuss groups $G$ and vector spaces $A$ such that every subspace of $A$ is either $G$-invariant or $G$-contrainvariant.

Let $\alpha$ be a nonzero algebraic integer of degree $d$ whose all conjugates ${\alpha }_1=\alpha ,{\alpha }_2,\dots ,{\alpha }_d$ lie in a sector $|argz|\le 𝜃$, $0\le 𝜃\le 90^{\circ }$. We define the N-measure of $\alpha$ by $N\left(\alpha \right)={\prod }_{i=1}^d\left(\left|{\alpha }_i\right|+1∕\left|{\alpha }_i\right|\right)$ and the absolute N-measure of $\alpha$ by $\nu \left(\alpha \right)=N{\left(\alpha \right)}^{1∕deg\left(\alpha \right)}$. Firstly, we consider the case $𝜃=0$. We prove that $N\left(\alpha \right)\in \mathbb{N}$ and that, if $\alpha$ is a reciprocal algebraic integer, $N\left(\alpha \right)$ is a square. Then, we study the set $\mathcal{𝒩}$ of the quantities $\nu \left(\alpha \right)$. We prove that there exists a number $l$ such that $\mathcal{𝒩}$ is dense in $\left(l,\infty \right)$. Finally, using the method of auxiliary functions, we find the seven smallest points of $\mathcal{𝒩}$ in $\left(2,l\right)$. In case of $0<𝜃\le 90^{\circ }$, we compute the greatest lower bound $c\left(𝜃\right)$ of the absolute N-measure of $\alpha$, for $\alpha$ belonging to eight subintervals of $$]

The Riccati equation method is used to establish some new oscillatory criteria for Hamiltonian systems in a new direction, which is to break the positive definiteness restriction imposed on one of coefficients of the Hamiltonian system. The obtained results are compared with some well known ones.

We establish sufficient conditions for the existence of solutions of boundary value problems for a fractional differential inclusion with nonlocal multipoint boundary conditions involving the Caputo type derivative with order $\alpha \in \left(1,2\right]$. Both convex- and nonconvex-valued right-hand sides are considered.

Let $G$ be a finite group. The bipartite divisor graph $B\left(G\right)$ for the set of irreducible complex character degrees $cd\left(G\right)$ is the undirected graph with vertex set consisting of the prime numbers dividing some element of $cd\left(G\right)$ and of the nonidentity character degrees in $cd\left(G\right)$, where a prime number $p$ is declared to be adjacent to a character degree $m$ if and only if $p$ divides $m$. The graph $B\left(G\right)$ is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: the prime graph and the divisor graph on the set of irreducible character degrees.

The scope of this paper is two-fold. We draw some attention to $B\left(G\right)$ by outlining the main results that have been proved so far, see for instance [10; 11; 25; 26; 27]. In this process we improve some of these results.

We imitate a classical construction of a counterexample to the local-global principle of cubic forms of 4 variables which was discovered first by Swinnerton-Dyer (Mathematika (1962)). Our construction gives new explicit families of counterexamples in homogeneous forms of $4,5,6,\dots ,2n+2$ variables of degree $2n+1$ for infinitely many integers $n$. It is contrastive to Swinnerton-Dyer’s original construction that we do not need any concrete calculation in the proof of local solubility.

In 1970, Chou showed there are $\left|{\mathbb{N}}^{\ast }\right|=2^{2^{\mathbb{N}}}$ topologically invariant means on $L_{\infty }\left(G\right)$ for any noncompact, $\sigma$-compact amenable group. Over the following 25 years, the sizes of the sets of topologically invariant means on $L_{\infty }\left(G\right)$ and $VN\left(G\right)$ were determined for any locally compact group. Each paper on a new case reached the same conclusion — “the cardinality is as large as possible” — but a unified proof never emerged. In this paper, I show $L_1\left(G\right)$ and $A\left(G\right)$ always contain orthogonal nets converging to invariance. An orthogonal net indexed by $\mathrm{\Gamma }$ has $\left|{\mathrm{\Gamma }}^{\ast }\right|$ accumulation points, where $\left|{\mathrm{\Gamma }}^{\ast }\right|$ is determined by ultrafilter theory.

Among a smattering of other results, I prove Paterson’s conjecture that left and right topologically invariant means on $L_{\infty }\left(G\right)$ coincide if and only if $G$ has precompact conjugacy classes.

Finally, I discuss some open problems arising from the study of the sizes of sets of invariant means on groups and semigroups.

Let ${\mathbb{Z}}_K$ denote the ring of algebraic integers of an algebraic number field $K=\mathbb{Q}\left(𝜃\right)$ where the algebraic integer $𝜃$ is a root of an irreducible quadrinomial $f\left(x\right)=x^n+a{x}^{n-1}+b{x}^{n-2}+c$ belonging to $\mathbb{Z}\left[x\right]$ with $a^2=4b$. We give necessary and sufficient conditions involving only $a,b,c,n$ for a prime $p$ to divide the index of the subgroup $\mathbb{Z}\left[𝜃\right]$ in ${\mathbb{Z}}_K$. As a consequence, we obtain necessary and sufficient conditions for ${\mathbb{Z}}_K$ to be equal to $\mathbb{Z}\left[𝜃\right]$. Moreover, when ${\mathbb{Z}}_K\ne \mathbb{Z}\left[𝜃\right]$, we provide an explicit formula for the index $\left[{\mathbb{Z}}_K:\mathbb{Z}\left[𝜃\right]\right]$ in some cases.

We prove global well-posedness of solution to the 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) equations, where the initial data is in critical Besov space. Since the initial vertical velocity may be large enough, our work improves some recent results in WanZ2,WZ19.

By Dhage fixed point theory of three operators, and Banach contraction principle, we study the existence and uniqueness of solution for fractional neutral hybrid nonlinear differential equations with finite delay. An example is introduced to explain the abstract results.

Let $H=\mathbb{Q}\left({\zeta }_n+{{\zeta }_n}^{-1}\right)$ and $\ell$ be an odd prime such that $q\equiv 1\left(mod\ell \right)$ for a prime factor $q$ of $n$. We get a bound on the $\ell$-rank of the class group of $H$ in terms of the $\ell$-rank of the class group of a real quadratic subfield contained in $H$. At the end we look into few numerical examples.

Let $R=K\left[x_1,\dots ,x_n\right]$ be the standard graded polynomial ring over a field $K$. We’ll study the Lyubeznik numbers of two homogeneous ideals of $R$ whose initial ideals are linked for some monomial order over $R$. We’ll also investigate some relations between Lyubeznik and Betti numbers of homogeneous ideals.

We find all rational solutions to the equation $y^2=x^6+k$, where $k=-39$ or $k=-47$. These are the two unsolved cases for $k$ in the range $\left|k\right|\le 50$. Our method is elementary and can be applied to other situations to find all rational points on certain curves of genus $2$.

By extending the method of the author’s earlier paper, a purely recursive approach is devised for counting the total number of unfolded self-avoiding walks terminating along the line $x=n$, within the finite lattice strip $\left\{0,1,\dots ,n\right\}\times \left\{0,1,2,3\right\}$ of width three. This approach yields a generating function for the sequence of numbers just described, without the need for any exact enumeration, usually entailed with an application of the transfer matrix algorithm. As a result of determining this generating function, the total number of unfolded walks within the lattice strip can then be easily computed.

This paper aims to employ the implicit radial point interpolation method, which is intrinsically meshless, for the numerical simulation of nonlinear space fractional advection-diffusion equations. The space fractional derivative is defined in the Caputo sense and calculated by the Gauss’s Jacobi quadrature formula. The accuracy and convergency of the proposed meshless method are demonstrated by several numerical examples with different regions and different nodal distributions. It is proved that the presented method is computational efficiency for modeling and simulation of nonlinear SFADEs.

The Hopf bifurcation problem of the fractional-order Hindmarsh–Rose neuron model with time-delay is investigated. The effects of time-delay to the bifurcation behavior of system is considered by linearizing the system at the equilibria method. With the bifurcation theory, the condition for the existence of Hopf bifurcation is obtained when considering time-delay $\tau$ as variable parameter. As illustrated by the numerical example, the proposed theoretical results work effectively and accurately.

Let $\mathrm{\Lambda }\left(n\right)$ be the von Mangoldt function and $\tau \left(n\right)$ the divisor function. We focus on the summation

$$T\left(x\right)=\sum _{1<n\le x}\mathrm{\Lambda }\left(n\right)\tau \left(n-1\right).$$

Under the Riemann hypothesis related to Dirichlet $L$-functions, it is proved that

$$T\left(x\right)=x{P}_1\left(logx\right)+O\left(x^{1-𝜃}\right)$$

holds for $𝜃<\frac{1}{6}\times 10^{-4}$. Here $P_1\left(t\right)$ is a polynomial of degree $1$.

We investigate the existence of three positive solutions of fourth order problems with clamped beam boundary conditions

$$\left\{\begin{array}{c}{u}^{\prime \prime \prime \prime }\left(x\right)=\lambda f\left(u\left(x\right)\right),x\in \left(-1,1\right),\hfill \\ u\left(1\right)=u\left(-1\right)=u^{\prime }\left(1\right)=u^{\prime }\left(-1\right)=0,\hfill \end{array}\right.$$

where $\lambda >0$ is a parameter, $f\in C\left[0,\infty \right)$ is nondecreasing, $f\left(0\right)=0$, $f\left(s\right)>0$ for all $s>0$. Especially, we obtain an S-shaped unbounded connected component in the symmetric positive solutions set. The proof is based on the directions of a bifurcation.

Combined with radial Minkowski addition, radial Blaschke addition and harmonic Blaschke addition, we establish cyclic Brunn–Minkowski inequalities for dual mixed volumes of star bodies.

Let $\bar{p}\left(n\right)$ denote the number of overpartitions of $n$. We show that $\bar{p}\left(3n\right)\equiv \bar{p}\left(9\cdot 3n\right)\left(mod3\right)$ for $n\ge 0$ and $\bar{p}\left(3n\right)\equiv {\left(-1\right)}^n\bar{p}\left(16\cdot 3n\right)\left(mod3\right)$ for $n\ge 0$ by using a relation of the generating function of $\bar{p}\left(3n\right)$ modulo $3$ and elementary dissection manipulations. Furthermore, by studying a 4-dissection formula of the generating function of $\bar{p}\left(3n\right)$ modulo 3 and iterating the above two congruence relations, we derive that $\bar{p}\left(16^{\alpha }{9}^{\beta }\cdot \left(72n+51\right)\right)\equiv 0\left(mod3\right)$ for $\alpha ,\beta ,n\ge 0$. Moreover, applying the fact that $\varphi {\left(q\right)}^5$ is a Hecke eigenform in $M_{5∕2}\left({\tilde{\mathrm{\Gamma }}}_0\left(4\right)\right)$, we obtain an infinite family of congruences $\bar{p}\left(16^{\alpha }{9}^{\beta }\cdot 3{\ell }^{2}n\right)\equiv 0\left(mod3\right)$, where $\alpha ,\beta \ge 0$ and $\ell$ is a prime such that $\ell \equiv 1\left(mod3\right)$ and $\left(n∕\ell \right)=-1$. In this way, we find various Ramanujan-type congruences for $\bar{p}\left(n\right)$ modulo $3$ such as $\bar{p}\left(1029n+441\right)\equiv 0\left(mod3\right)$, $\bar{p}\left(1029n+735\right)\equiv 0\left(mod3\right)$ and $\bar{p}\left(1029n+882\right)\equiv 0\left(mod3\right)$ for $n\ge 0$.

Let $\mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)$ and ${\mathfrak{𝔅}}^s\left(\mathcal{\mathscr{H}}\right)$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathcal{\mathscr{H}}$ and the real Jordan algebra of all self-adjoint operators on $\mathcal{\mathscr{H}}$, respectively. Suppose $\mathcal{𝒲}$ and $\mathcal{𝒱}$ are subsets of $\mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)$ or ${\mathfrak{𝔅}}^s\left(\mathcal{\mathscr{H}}\right)$ and $F\left(\cdot \right)$ is the $c$-numerical radius or the diameter of an operator. Under an assumption on $\mathcal{𝒲}$ and $\mathcal{𝒱}$, characterizations are obtained for surjective maps $\mathrm{\Phi }:\mathcal{𝒲}\to \mathcal{𝒱}$ satisfying

$$F\left(AB\right)=F\left(\mathrm{\Phi }\left(A\right)\mathrm{\Phi }\left(B\right)\right)\left(A,B\in \mathcal{𝒲}\right).$$

When $dim\mathcal{\mathscr{H}}\ge 3$, to establish the proofs, some general results are obtained for functions $F:\mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)\to \left[0,+\infty \right)$ satisfying $\left(P_1\right)$ $F\left(UA{U}^{\ast }\right)=F\left(A\right)$ for all $A\in \mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)$ and unitary $U$ on $\mathcal{\mathscr{H}}$; $\left(P_2\right)$ For $A\in \mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)$, $F\left(A\right)=0$ if and only if $A$ is a multiple of the identity; $\left(P_3\right)$ There are nonnegative real numbers $\alpha ,\beta$ with ${\alpha }^2+{\beta }^2\ne 0$ such that $F\left(T\right)=\alpha \parallel T\parallel +\beta |tr\left(T\right)|$ for each rank-one $T\in \mathfrak{𝔅}\left(\mathcal{\mathscr{H}}\right)$.