We study the existence of solutions for a class of boundary value problem for fractional $q$-difference equations involving $q$-derivative of the Caputo sense. The main results are proved by applying Mönch’s fixed-point theorem associated with the technique of measure of noncompactness. Further, an example is presented to illustrate the main results.

We investigate the finite-time stability of stochastic linear fractional systems with delay (SLFSD) for $\alpha \in \left(\frac{1}{2},1\right)$. By the Banach fixed-point theorem and classical stochastic techniques, the finite-time stability (FTS) of the solution for SLFSD is studied. One example is given to illustrate our theory.

By applying disease-like dynamics to online social networks (OSNs), we propose a susceptible-infected-recovered-susceptible (SIRS) model with infectious recovery dynamics to examine the user adoption and abandonment dynamics of OSNs, where adoption is analogous to infection and abandonment is analogous to recovery. Unlike many traditional infectious disease models available in the literature, our model requires direct contact between recovered and infected members of the population in order to recover from the infected class. We prove the well-posedness of the model and discuss the existence and stability of its equilibria. More specifically, we find the user-free equilibrium and derive the reproduction number ${\mathcal{ℛ}}_0$ for the model and further prove that if , the user-free equilibrium is globally asymptotically stable. When , we establish sufficient conditions under which the model has a unique user-prevailing equilibrium and prove criteria for the local and global asymptotic stability of the user-prevailing equilibrium. We perform numerical simulations to validate the theoretic results. Finally, we demonstrate the effectiveness of the model by fitting it to the historical Facebook daily active user data, and we utilize the fitted model parameter values to predict the numbers of future Facebook active users.

We investigate the limit behavior of the solutions of the Leray-$\alpha$ equations with Navier slip boundary conditions in a 3D smooth bounded domain to the solution of Euler equations as the parameters $\alpha ,\nu$ tend to zero. Firstly, we focus on the convergence in $L^{\infty }(0,T;L^2(\mathrm{\Omega }))$ by the energy method. Secondly, we prove the convergence in $L^{\infty }(0,T;H^1(\mathrm{\Omega }))$ by choosing the proper expansion ansatz and constructing the corresponding boundary layer profile.

We present a new characterization of real homogeneous functions of a negative degree by a new counterpart of Euler’s homogeneous function theorem using quantum calculus and replacing the classical derivative operator by a $(p,q)$-derivative operator. As an application we study the solution of the Cauchy problem associated to the $(p,q)$-analogue of the Euler operator. Using this solution, a probabilistic interpretation is given in some details; more specifically, we prove that this solution is a stochastically continuous markovian transition operator. Finally, we study it’s associated subordinated stochastically markovian transition operator.

Let $\mathcal{ℛ}$ be a unital prime ring with characteristic not $2$ and containing a nontrivial idempotent. We characterize Lie centralizers and generalized Lie derivations at zero products on $\mathcal{ℛ}$. The Lie centralizers and generalized Lie derivations on $\mathcal{ℛ}$ will be described. The obtained results are applied to Banach space standard operator algebras and factor von Neumann algebras.

We study a new problem of nonlinear integrodifferential equations with nonlocal integral conditions. The considered problem is singular at the origin of the time axis and it involves convergent series combined with Riemann–Liouville integrals. We prove an existence and uniqueness result for our problem. Some examples are given to illustrate the uniqueness result. The Ulam–Hyers stability for the problem is also studied. Then, thanks to some numerical techniques, that allow us to approximate the Caputo derivatives, and by using the Runge–Kutta method, we present a numerical study with some simulations to show more comprehension of the proposed examples.

We derive an identity for twice-differentiable functions whose second derivatives are convex. By using this equality, we establish some perturbed Newton type inequalities for twice-differentiable convex functions, which we investigate by using Riemann–Liouville fractional integrals. Furthermore, we present our results by using special cases of the obtained theorems.

We show that if $𝜃\in (0,1)$ is an irrational number, then for any , there exists satisfying the following: For any unital $C^{\ast }$-algebra $A$ with the cancellation property, strict comparison and nonempty tracial state space, any three unitaries $u,v,w\in A$ such that

there exists a triple of unitaries $\tilde{u},\tilde{v},\tilde{w}\in A$ such that

According to the above conclusion, the rotation relation of two unitaries with ${\mathbb{Z}}_3$-action is stable under the conditions in (2). Likewise, we show that the rotation relation of two unitaries with ${\mathbb{Z}}_4$ and ${\mathbb{Z}}_6$-actions is stable under the above similar conditions.

We describe the irreducible decomposition of the facet ideal $\mathcal{ℱ}({\mathrm{\Delta }}_{m,n})$ of the chessboard complex ${\mathrm{\Delta }}_{m,n}$ with $n\ge m$. We also provide some lower bounds for the depth and regularity of the facet ideal $\mathcal{ℱ}({\mathrm{\Delta }}_{m,n})$. When $m\le 3$, we prove that these lower bounds can be obtained.

Various integral inequalities for s-convex functions in the second sense are obtained by means of the Caputo fractional derivative and the Caputo–Fabrizio integral operator. Some generalizations of the Hermite–Hadamard type inequalities including the Caputo–Fabrizio integral operator are expressed for the functions whose derivatives are $s$-convex. Moreover, some inequalities involving these fractional operators are stated for the product of these functions. Inequalities involving special means and the digamma function are given as applications.

We introduce a logarithmic pressure to discuss the formation of delta-shocks for the Chaplygin gas equations. We solve the Riemann problem for the logarithmic perturbed equations by the analysis method in phase plane. The Riemann solutions with four different kinds of structures are obtained. We prove that as the logarithmic pressure vanishes, some Riemann solutions containing the two-shock tend to the delta-shock solution to the Chaplygin gas equations, the other Riemann solutions of the logarithmic perturbed equations tend to the two-contact discontinuity solution to the Chaplygin gas equations. Finally, we present some numerical results to exhibit the formation process of delta-shocks.

We generalize Hirzebruch surfaces from the point of view of toric geometry. We show that the Mukai conjecture holds for this class of generalized surfaces.

We consider diagonal quadratic forms

$$a_1{x}_1^2+a_2{x}_2^2+\cdots +a_{\ell }{x}_{\ell }^2,$$

where $\ell \ge 5$ is an odd integer and $a_i\ge 1$ are integers. By using the extended Shimura correspondence, we obtain explicit formulas for the number of representations of $\left|D\right|n^2$ by such quadratic forms, where $D$ is either a squarefree integer or a fundamental discriminant such that . We demonstrate our method with many examples, in particular recovering results of Cooper, Lam and Ye (2013): all their formulas (when $\ell =5$) for $n^2$ for quinary quadratic forms and all the representation formulas for septenary quadratic forms when $n$ is even. (Those formulas were originally derived by combining certain theta function identities with a method of Hurwitz.) Our method works with arbitrary coefficients $a_i$. As a consequence of some of our formulas, we obtain identities among the representation numbers and also congruences involving the Fourier coefficients of certain newforms of weights $6$ and $8$ and divisor functions.

In this paper, a general inversion pair is established which provides deformed versions of the Humbert polynomial, Wilson polynomials and Racah polynomials along with their inverse series. Their several particular cases have been illustrated which include the deformed polynomials of Kinney, Pincherle, Gegenbauer, Legendre etc. The general inverse pair also deforms the Bessel function together with the Neumann series as its inverse series. The $p$-deformed Gauss sum is used together with the inverse series to derive certain summation formulas. Finally, all six classes of inverse series relations involving combinatorial identities due to John Riordan (Combinatorial identities, Wiley, 1968) are extended.

We study the positive solutions for fractional integral boundary value problem with derivative dependence. Some properties of the associated Green function are established. By using the Guo–Krasnoselskii fixed-point theorem on cones, the existence results for positive solutions are obtained. Under certain monotone-type assumption on the nonlinearity, we establish the criteria of existence, multiplicity and nonexistence of positive solutions via the fixed-point index theory.

Recently, Guo (2019) gave a $q$-analogue of a combinatorial congruence by Sun (2010). Later, Wang and Ni (2022) obtained several conclusions similar to Guo’s $q$-congruence. We establish some $q$-supercongruences involving one free parameter, which are common generalizations of these $q$-congruences of Guo and Wang and Ni just mentioned.

We consider the differential and difference problems associated with the discrete approximation of radially symmetric spacelike solutions of the nonlinear Dirichlet problem for the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime

where $\mathcal{ℬ}$ is the unit ball in ${ℝ}^N$, $\text{div}$ denotes the divergence operator of ${ℝ}^N$, $\text{grad}u$ is the gradient of $u$, $|\cdot |$ denotes the Euclidean norm in ${ℝ}^N$, $f\in C^{\infty }(I)$, , $I$ is an open interval in $ℝ$, $f^{\prime }(u)≔f^{\prime }\circ u$ and $H:I\times [0,+\infty )\to ℝ$ is the prescribed mean curvature function. By using lower and upper solutions, we prove the existence of solutions of the corresponding differential and difference problems, and based on the ideas of a prior bound show the solutions of the discrete problem converge to the solutions of the continuous problem.

We revisit the Rankin–Selberg integral of Shimura type for generic representations of ${\text{SL}}_2\times {\text{GL}}_2$ constructed by Ginzburg, Rallis, and Soudry. We give a different and more “intrinsic” proof of the unramified computation. In contrast to their proof we avoid the local functional equation for the general linear groups but use the Casselman–Shalika formulas for unramified Whittaker functions for ${\text{SL}}_2$ and ${\text{GL}}_2$.

For , we prove Wolff’s ideal theorem for the multiplier algebra of the Besov type space $B_p(s)$, denoted $\mathcal{ℳ}(B_p(s))$; the multiplier algebra of the Möbius invariant space $F(p,p-2,s)$, denoted $\mathcal{ℳ}(F(p,p-2,s))$; and $H^{\infty }\cap F(p,p-2,s)$.