The present paper is devoted to the study of Jensen type inequalities for fractional integration on finite subintervals of the real axis. The complete form of Jensen's inequality and the generalized Jensen's inequality are investigated by using the Chebyshev inequality. As applications, some new integral inequalities, including Holder's and Minkowski's inequalities, are obtained by using Jensen's inequality via Riemann-Liouville fractional integrals.

This paper deals with the finite time blow-up of solutions to the initial boundary value problem of a non-Newton filtration equation with special medium void. A new criterion for the solutions to blow up in finite time is established by using the Hardy inequality. Moreover, the upper and lower bounds for the blow-up time are also estimated. The results solve an open problem proposed by Liu in 2016.

This paper primarily studies monomial ideals by their associated lcm-lattices. It first introduces notions of weak coordinatizations of finite atomic lattices which have weaker hypotheses than coordinatizations and shows the characterizations of all such weak coordinatizations. It then defines a finite super-atomic lattice in $\mathcal {L}(n)$, investigates the structures of $\mathcal {L}(n)$ by their super-atomic lattices and proposes an algorithm to calculate all of the super-atomic lattices in $\mathcal {L}(n)$. It finally presents a specific labeling of finite atomic lattice and obtains the conditions that the specific labelings of finite atomic lattices are the weak coordinatizations or the coordinatizations by using the terminology of super-atomic lattices.

In this paper, we study the following $p(x)$-biharmonic problem in Sobolev spaces with variable exponents \begin{equation} \begin{cases} \triangle ^{2}_{p(x)}u=\lambda ({\partial F } (x,u)/{\partial u}) & x\in \Omega , \\ {\partial u}/{\partial n}=0 & x\in \partial \Omega ,\\ {\partial }(|\triangle u|^{p(x)-2}\triangle u)/{\partial n} =a(x)|u|^{p(x)-2}u & x\in \partial \Omega . \end{cases} \end{equation} By means of the variational approach and Ekeland's principle, we establish that the above problem admits a nontrivial weak solution under appropriate conditions.

A quadrature mirror filter (QMF) function can be considered as the transition function for a Markov process on the unit interval. The QMF functions that generate scaling functions for multiresolution analyses are then distinguished by properties of their invariant sets. By characterizing these sets, we answer in the affirmative a question raised by Gundy.

We establish conditions ensuring either existence or blow up of nonnegative solutions for the following parabolic problem: \begin{equation} \begin{cases} Hu -Vu + ({\partial u}/{\partial t}) =0 & \mbox {in } X\times (0,T), \\ u(x,0)=u_{0}(x) & \mbox {in } X, \end{cases} \end{equation} where $T>0$, $X$ is a locally compact separable metric space, $H$ is a selfadjoint operator associated with a regular Dirichlet form $\mathcal E$; the initial value $u_{0}\in L^{2}(X,m)$, where $m$ is a positive Radon measure on Borel subset $U$ of $X$ such that $m(U)>0$ and $V$ is a Borel locally integrable function on $X$.

In this paper, we study the degenerate convective Cahn-Hilliard equation, which is a special case of the general convective Cahn-Hilliard equation with $M(u,\nabla u)=diag(0,1,\ldots ,1)$. We obtain the uniform a priori decay estimates of a solution by use of the long-short wave method and the frequency decomposition method. We prove the existence of the unique global classical solution with small initial data by establishing the uniform estimates of the solution. Decay estimates are also discussed.

We characterize the Cohen-Macaulay property for generalized Petersen graphs and $3$-regular graphs. In particular, we prove that these graphs are vertex decomposable. Also, we characterize pure vertex decomposability for $4$-transitive graphs without $5$-holes. Finally, we study the small cycles of well-covered and Cohen-Macaulay regular graphs.

We give simple proofs of the fact that, for certain parameters, the roots of the generalized Rogers-Ramanujan function are irrational numbers and, for example, that at least one of the following two numbers is irrational: $ \{\sum _{n=1}^\infty {F_n }/({m^n \prod _{i=0}^{n-1}\phi (k+i)}), \sum _{n=1}^\infty {F_n }/(m^n \prod _{i=0}^{n-1}$ $\phi (k+i+1)) \}$, where $F_{n+2}=F_{n+1}+F_n$, $F_0=0,F_1=1$ (the Fibonacci sequence), $m$ is a natural number $> ({1+\sqrt 5})/{2}$ and $\phi (k)$ is any function taking positive integer values such that $\limsup _{k\rightarrow \infty } \phi (k)= \infty $.

In this paper, we develop the theory of weighted persistent homology. In 1990, Dawson was the first to study in depth the homology of weighted simplicial complexes. We generalize the definitions of weighted simplicial complex and the homology of weighted simplicial complex to allow weights in an integral domain $R$. Then, we study the resulting weighted persistent homology. We show that weighted persistent homology can distinguish between filtrations that ordinary persistent homology does not distinguish. For example, if there is a point considered as special, weighted persistent homology can tell when a cycle containing the point is formed or has disappeared.

Filip Najman examined the possibilities for the group of torsion points on elliptic curves over the number fields $\mathbb {Q}(\sqrt {-1})$ and $\mathbb {Q}(\sqrt {-3})$ in Najman (2011, 2010). In this article, we study the possible torsion structures of elliptic curves over the remaining imaginary quadratic fields of class numbers $1$, i.e., over the fields $\mathbb {Q}(\sqrt {-2})$, $\mathbb {Q}(\sqrt {-7})$, $\mathbb {Q}(\sqrt {-11})$, $\mathbb {Q}(\sqrt {-19})$, $\mathbb {Q}(\sqrt {-43})$, $\mathbb {Q}(\sqrt {-67})$ and $\mathbb {Q}(\sqrt {-163})$.

Let $\mathcal {A}$ be a prime $\ast $-algebra and $\Phi $ a $\lambda $-Jordan triple derivation on $A$, that is, for every $A,B,C \in \mathcal {A}$, $$\Phi (A\diamond _{\lambda } B \diamond _{\lambda }C)=\Phi (A)\diamond _{\lambda }B\diamond _{\lambda } C+A\diamond _{\lambda }\Phi (B)\diamond _{\lambda }C+A\diamond _{\lambda } B\diamond _{\lambda }\Phi (C),$$ where $A\diamond _{\lambda } B = AB + \lambda BA^{\ast }$ such that a complex scalar $|\lambda |\neq 0,1$, and $\Phi $ is additive. Moreover, if $\Phi (I)$ is self-adjoint, then $\Phi $ is a $\ast $-derivation.

We investigate the radial positive solutions of the Henon equation. It is known that this equation has three different types of radial solutions: the M-solutions (singular at $r=0$), the E-solutions (regular at $r=0$) and the F-solutions (whose existence begins away from $r=0$). For the M-solutions and E-solutions, by virtue of some prior estimates, we adopt a circulating iterative method, step-by-step, to derive their precise asymptotic expansions. In particular, the M-solution has an extremely plentiful structure, and its asymptotic expansions are more complicated. In contrast to previous research Bratt and Pfaffelmoser, and Gidas and Spruck, our results are more accurate.

The objective of this paper is to explore the existence and roughness of the nonuniform $(h,k,\mu ,\nu )$-trichotomy for nonautonomous differential equations. We first propose a more general notion of trichotomies called the nonuniform $(h,k,\mu ,\nu )$-trichotomy for linear nonautonomous differential equations. Then, we give a complete characterization of the notion of nonuniform $(h,k,\mu ,\nu )$-trichotomy for linear nonautonomous differential equations and prove that any linear nonautonomous differential equation admits a nonuniform $(h,k,\mu ,\nu )$-trichotomy if it has an $(H,K,L)$ Lyapunov exponent with different signs in a finite-dimensional space. Finally, we establish the roughness of nonuniform $(h,k,\mu ,\nu )$-trichotomies in a very concise manner, which implies that the nonuniform $(h,k,\mu ,\nu )$-trichotomy persists under sufficiently small linear perturbations. This study exhibits some new interesting findings in trichotomy that extend the corresponding results for uniform and nonuniform trichotomies.

This paper concerns regularity criteria for the $2$D generalized MHD system and shows that, if we can control the Besov norm of the vorticity and/or the current density, then the solution is, in fact, smooth. This improves the recent result of Jiang, Wang, and Zhou (2016).