We prove new weighted Hermite–Hadamard type inequalities for differentiable GA-convex and geometrically quasiconvex functions. These results generalize many results proved in earlier works for these classes of functions.

We prove some new inequalities of Mitrinović–Pečarić inequalities for convex functions on an arbitrary time scale using delta integrals. These inequalities extend and improve some known dynamic inequalities in the literature. The main results will be proved by using Hölder and Jensen inequalities and a simple consequence of Keller’s and Poetzsche’s chain rules on time scales.

Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq \{1,2,\dots ,\lfloor \frac{n}{2}\rfloor \}$, and let $I(G)$ denote the edge ideal in the polynomial ring $R=\mathbb{𝕂}[x_0,x_1,\dots ,x_{n-1}]$ over a field $\mathbb{𝕂}$. In this paper we compute the $\mathbb{N}$-graded Betti numbers of the edge ideals of three families of circulant graphs $C_n(1,2,\dots ,\widehat{\mathit{j}},\dots ,\lfloor \frac{n}{2}\rfloor )$, $C_{lm}(1,2,\dots ,\widehat{2l},\dots ,\widehat{3l},\dots ,\lfloor \frac{lm}{2}\rfloor )$ and $C_{lm}(1,2,\dots ,\widehat{l},\dots ,\widehat{2l},\dots ,\widehat{3l},\dots ,\lfloor \frac{lm}{2}\rfloor )$. Other algebraic and combinatorial properties like regularity, projective dimension, induced matching number and when such graphs are well-covered, Cohen–Macaulay, sequentially Cohen–Macaulay, Buchsbaum and $S_2$ are also discussed.

We establish some upper bounds for Berezin number inequalities including inequalities for $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if , then

$${ber}^r(T)\le 2^{r-2}\left(ber(f^{2r}(\left|X\right|)+g^{2r}(\left|Y^{\ast }\right|))+ber(f^{2r}(\left|Y\right|)+g^{2r}(\left|X^{\ast }\right|))\right)-2^{r-2}\underset{\Vert (k_{{\lambda }_1},k_{{\lambda }_2})\Vert =1}{\inf }\eta (k_{{\lambda }_1},k_{{\lambda }_2}),$$

where $X$, $Y$ are bounded linear operators on a Hilbert space $\mathcal{\mathscr{H}}=\mathcal{\mathscr{H}}(\mathrm{\Omega })$, $r\ge 1$, $f$, $g$ are nonnegative continuous functions on $[0,\infty )$ satisfying the relation $f(t)g(t)=t$ $(t\in [0,\infty ))$ and

$$\eta (k_{{\lambda }_1},k_{{\lambda }_2})={\left({⟨(f^{2r}(\left|X\right|)+g^{2r}(\left|Y^{\ast }\right|))k_{{\lambda }_2},k_{{\lambda }_2}⟩}^{\frac{1}{2}}-{⟨(f^{2r}(\left|Y\right|)+g^{2r}(\left|X^{\ast }\right|))k_{{\lambda }_1},k_{{\lambda }_1}⟩}^{\frac{1}{2}}\right)}^2.$$

Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if $A$ is a bounded linear operator on a complex Hilbert space, then

$$\frac{1}{4}‖A^{*}A+A{A}^{*}‖\le \frac{1}{8}\left({‖A+A^{*}‖}^2+{‖A-A^{*}‖}^2+c^2(A+A^{*})+c^2(A-A^{*})\right)\le w^2(A)$$

and

$$\frac{1}{2}‖A^{\ast }A+A{A}^{\ast }‖-\frac{1}{4}{‖{(A+A^{\ast })}^2{(A-A^{\ast })}^2‖}^{\frac{1}{2}}\le w^2(A)\le \frac{1}{2}‖A^{\ast }A+A{A}^{\ast }‖,$$

where $\parallel \cdot \parallel$, $w(\cdot )$ and $c(\cdot )$ are the operator norm, the numerical radius and the Crawford number, respectively. Further, we prove that if $A,D$ are bounded linear operators on a complex Hilbert space, then

$$‖A{D}^{\ast }‖\le {‖{\displaystyle {\int }_0^1{\left((1-t)({\left|A\right|}^2+{\left|D\right|}^2)/2+t‖A{D}^{\ast }‖I\right)}^{2}dt}‖}^{\frac{1}{2}}\le \frac{1}{2}{‖{\left|A\right|}^2+D‖}^2,$$

where $|A{|}^2=A^{\ast }A$ and $|D{|}^2=D^{\ast }D$. This is a refinement of a well-known inequality obtained by Bhatia and Kittaneh.

We are concerned with the question of when a homeomorphism between $C_p(X\times \tau )$ and $C_p(Y\times \tau )$ implies the existence of a homeomorphism between $C_p(X\times {\tau }^{\prime })$ and $C_p(Y\times {\tau }^{\prime })$, where $\tau$ and ${\tau }^{\prime }$ are spaces of ordinals with some additional desired properties.

This paper is concerned with an equation of the Chern–Simons–Higgs type. We deduce the Euler–Lagrange system of the Chern–Simons–Higgs type energy functional and discuss its stability for radial solutions of degree one in disk domains.

We investigate the relationship between the sign of the discrete fractional difference $({\mathrm{\nabla }}_a^{\nu }f)(t)$ and the convexity of the function $f$ in the case where . We focus on the scenario in which $({\mathrm{\nabla }}_a^{\nu }f)(t)$ satisfies a negative lower bound. Our analytical results are complemented with numerical simulations, the latter providing insight beyond what we are able to produce analytically.

We investigate the mixed second moment involving the second derivative at the central point of a family of quadratic Dirichlet $L$-functions over ${\mathbb{𝔽}}_q(t)$, associated to the hyperelliptic curves of genus $g$. We compute the full degree five polynomial in the asymptotic expansion of the mixed second moment when the cardinality $q$ of the finite field is fixed and the genus $g$ tends to infinity. This is a partial analogue of classical Ingham’s result about the Riemann zeta function.

We introduce the notion of operator $(\phi ,h)$-convex functions and prove Jensen–Hansen–Pedersen-type inequalities and a Hermite–Hadamard-type inequality for this class, which generalize a certain classes of known inequalities. We give equivalent conditions for a function to be operator $(\phi ,h)$-convex. We also obtain a Choi–Davis–Jensen-type inequality for operator $(\phi ,h)$-convex functions.

In this paper, the $L_v^p(ℝ)$-boundedness of the Dunkl–Hausdorff operator

$$H_{\alpha ,\varphi }f(x)={\displaystyle {\int }_{ℝ}\frac{\left|\varphi (t)\right|}{|t{|}^{2\alpha +2}}f\left(\frac{x}{t}\right)dt}$$

has been characterized and for a certain type of weight $v$, the operator norm $\parallel H_{\alpha ,\varphi }{\parallel }_{L_v^p(ℝ)\to L_v^p(ℝ)}$ has been obtained. This covers several of the existing results. Analogous results in two dimensions have also been proved.

The power pseudovariety $PCS$, that is, the pseudovariety of finite semigroups generated by all power semigroups of finite completely simple semigroups has recently been characterized as the pseudovariety $AgBG$ of all so-called aggregates of block groups. This characterization can be expressed as the equality of pseudovarieties $PCS=AgBG$. In fact, a longer sequence of equalities of pseudovarieties, namely the sequence of equalities $PCS=J\ast CS=J\mathit{ⓜ}CS=AgBG$ has been verified at the same time. Here, $J$ is the pseudovariety of all $\mathcal{𝒥}$-trivial semigroups, $CS$ is the pseudovariety of all completely simple semigroups, $J\ast CS$ is the pseudovariety generated by the family of all semidirect products of $\mathcal{𝒥}$-trivial semigroups by completely simple semigroups, and $JⓜCS$ is the pseudovariety generated by the Mal’cev product of the pseudovarieties $J$ and $CS$. In this paper, another different proof of these equalities is provided first. More precisely, the equalities $PCS=J\ast CS=JⓜCS$ are given a new proof, while the equality $JⓜCS=AgBG$ is quoted from a foregoing paper. Subsequently in this paper, this new proof of the mentioned equalities is further refined to yield a proof of the following more general result: For any pseudovariety $H$ of groups, let $CS(H)$ stand for the pseudovariety of all completely simple semigroups whose subgroups belong to $H$. Then it turns out that, for every locally extensible pseudovariety $H$ of groups, the equalities of pseudovarieties $P(CS(H))=J\ast CS(H)=JⓜCS(H)$ hold.

Let $\mathcal{𝒜}$ be a factor von Neumann algebra. For any $\mathcal{𝒜},\mathcal{ℬ}\in \mathcal{𝒜}$, a product ${[\mathcal{𝒜},\mathcal{ℬ}]}_{•}=\mathcal{𝒜}{\mathcal{ℬ}}^{\ast }-\mathcal{ℬ}{\mathcal{𝒜}}^{\ast }$ is called a biskew Lie product. We show that every biskew Lie triple derivation $\psi :\mathcal{𝒜}\to \mathcal{𝒜}$ is an additive $\ast$-derivation.

We study the existence of multiple nontrivial solutions for nonlinear fourth order discrete boundary value problems. We first establish criteria for the existence of at least two nontrivial solutions of the problems and obtain conditions to guarantee that the two solutions are sign-changing. Under some appropriate assumptions, we further prove that the problems have at least three nontrivial solutions, which are positive, negative, and sign-changing, respectively. We include two examples to illustrate the applicability of our results. Our theorems are proved by employing variational approaches, combined with the classic mountain pass lemma and a result from the theory of invariant sets of descending flow.

Starting from a Lax pair, we study the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation by means of the $\overline{\partial }$-dressing method, which can be employed to formulate scattering equations and to solve the inverse scattering problem. Based on that, the solutions of the DJKM equation can be further worked out.

It is known that there exists an explicit function $F$ in $L^2(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is a given bounded open subset of ${ℝ}^N$, such that the corresponding weak solution of the Poisson boundary value problem $-\mathrm{\Delta }u=F(x)$, $u\in H_0^1(\mathrm{\Omega })$, is maximally singular; that is, the singular set of $u$ has Hausdorff dimension equal to ${(N-4)}^{+}$. This constant is optimal, i.e., the largest possible. Here, we show that much more is true: when $N\ge 5$, there exists $F\in L^2(\mathrm{\Omega })$ such that the corresponding weak solution has the pointwise concentration of singular set of $u$, in the sense of the Hausdorff dimension, equal to $N-4$ at all points of $\mathrm{\Omega }$. We also consider the problem of generating weak solutions with the property of contrast; that is, we construct solutions $u$ that are regular (more specifically, of class $C_{\mathrm{loc}}^{2,\alpha }$ for arbitrary $\alpha \in (0,1)$) in any prescribed open subset ${\mathrm{\Omega }}_r$ of $\mathrm{\Omega }$, while they are maximally singular in its complement $\mathrm{\Omega }\setminus {\mathrm{\Omega }}_r$. We indicate several open problems.

We establish Hardy–Sobolev inequalities in the half-space for double phase functionals $\mathrm{\Phi }(x,t)$ of the form $t^p+{(b(x)t)}^q$, where , $b(\cdot )$ is nonnegative and Hölder continuous of order $\theta \in (0,1]$.

Let $R$ be a commutative Noetherian ring and $I$ an ideal of $R$. The purpose of this paper is to show that the topologies defined by the integral filtration ${\{\overline{I^m}\}}_{m\ge 1}$ and the symbolic integral filtration ${\{I^{⟨m⟩}\}}_{m\ge 1}$ are equivalent whenever ${\overline{Q}}^{\ast }(I)$ consists all of the minimal prime ideals of $I$. As an application of this result, by using the Jacobian theorem of Lipman and Sathaye we deduce that the symbolic integral topology ${\{I^{⟨m⟩}\}}_{m\ge 1}$ is equivalent to the $I$-adic topology whenever $R$ is a regular ring. Also, applying these results we provide extensions of classical results of Hartshorne and Zariski on the equivalence of symbolic and adic topologies.

Guo and Zudilin [Adv. Math. 346 (2019), 329–358] developed an analytical method, called “creative microscoping”, to prove many supercongruences by establishing their $q$-analogues. We apply this method to give a $q$-Dwork-type generalization of Rodriguez-Villegas’ supercongruences, which was recently conjectured by Guo and Zudilin.

Consider the first-order nabla dynamic equations

$$x^{\mathrm{\nabla }}(t)-\sum _{i=1}^m{p}_i(t)x({\tau }_i(t))=0,t\in [t_0,\infty {)}_{\mathbb{𝕋}},$$

where $p_i(t)$ $(i=1,2,\dots ,m)\in C_{ld}([t_0,\infty {)}_{\mathbb{𝕋}},{ℝ}^{+})$, ${\tau }_i(t)(i=1,2,\dots ,m)\in C_{ld}({[t_0,\infty )}_{𝕋},𝕋)$ and ${\tau }_i(t)\ge t$. Under the assumption that the ${\tau }_i(t)$ are not necessarily monotone $(i=1,2,\dots ,m)$, we present new sufficient conditions for the oscillation of first-order nabla dynamic equations on time scales. We can say that this paper is the first in the literature in obtaining the oscillations of the solutions of the above equation. An example illustrating the results is also given.

We recall the definition of an $S$-hypersystem and the fundamental normal homomorphism theorems on $S$-hypersystems. We introduce the concept of the category of $S$-hypersystems and study the product and coproduct of $S$-hypersystems. We prove that every $S$-pohypersystem is a subdirect product of subdirectly irreducible $S$-pohypersystems.

Suppose $T_1$ (resp. ${\mathrm{\Delta }}_1$) is an isosceles triangle with base length 1 and with height $\frac{1}{2}\sqrt{2}$ (resp. $\frac{1}{3}\sqrt{2}$). Let $S$ be a square with a side parallel to the base of $T_1$ (resp. ${\mathrm{\Delta }}_1$) and let $\{S_n\}$ be a sequence of the homothetic copies of $S$. We first determine the bound of sums of areas of squares from the sequence $\{S_n\}$ that permits a parallel packing of $T_1$ (resp. ${\mathrm{\Delta }}_1$). Then we generalize the results about packing $T_1$ (resp. ${\mathrm{\Delta }}_1$) with squares to some other triangles. Finally, we consider the parallel packing of the right triangle $T_2^{\prime }$ (with leg lengths 1 and $\sqrt{2}$) with squares.

This article is concerned with the description of transcendental entire solutions of several Fermat type functional equations in ${ℂ}^2$ concerning partial differential-difference equations and partial differential equations. By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we establish some results about the existence and the forms of entire solutions for such equations, which are some improvements and generalizations of the previous theorems. Moreover, some examples are given illustrating that our results are precise to some extent.

We study the boundedness and compactness of the differences of two weighted composition operators acting from $\alpha$-Bloch space to $\beta$-Bloch space on the open unit disk. Such study has a relationship to the topological structure of weighted composition from $\alpha$-Bloch space to $\beta$-Bloch space.

We introduce the concepts of a generalized sequential Feynman integral and a generalized sequential Fourier–Feynman transform. Existence theorems of these integral and transform are obtained for functionals in a Banach algebra $\widehat{\mathcal{𝒮}}$ and some related functionals. Also we give a simple proof of a translation theorem for generalized sequential Feynman integral. Previous results on sequential Feynman integral and sequential Fourier–Feynman transform can be obtained as corollaries of our results. In the last section we introduce the concept of translation invariant generalized sequential Feynman integral.

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. In this paper, we establish an infinite family of congruences $\overline{p}(9^{\alpha }{16}^{\beta }\cdot 3(8\ell n+8i+1))\equiv 0(\mathrm{mod}3)$, where $\alpha ,\beta ,n\ge 0$, $\ell$ is an arbitrary odd prime and is a nonnegative integer such that $\left(\frac{8i+1}{\ell }\right)=-1$. In this way, we find various congruences for $\overline{p}(n)$ modulo $3$ such as $\overline{p}(120n+51,99)\equiv 0(\mathrm{mod}3)$ and $\overline{p}(168n+51,99,123)\equiv 0(\mathrm{mod}3)$. Furthermore, by studying the generating function of $\overline{p}(3n)$ modulo 9 and applying the fact that $\varphi {(q)}^5$ is a Hecke eigenform, we obtain another infinite family of congruences $\overline{p}(3{\ell }^{2}n)\equiv 0(\mathrm{mod}9)$, where $\ell$ is a prime with $\ell \equiv 7(\mathrm{mod}9)$, and $n$ is a nonnegative integer such that $\left(\frac{n}{\ell }\right)=-1$. This leads to lots of congruences for $\overline{p}(n)$ modulo $9$ such as $\overline{p}(1029n+441,735,882)\equiv 0(\mathrm{mod}9)$.

Let $G(V,E)$ be a connected finite graph satisfying the $\mathrm{CDE}(m,-K)$ condition for some , and let $u$ be a positive bounded solution to the equation

$$\mathrm{\Delta }u+c{u}^{\sigma }=0,\sigma \ge 0.$$

Then, for all $x\in V$, we can get the gradient estimate and Harnack inequality for positive solutions to this equation.