There is a well-known upper bound due to Claesson, Jelínek and Steingrímsson for the growth rate of the merge of two permutation classes. Curiously, there is no known merge for which this bound is not achieved. Using linear algebraic techniques and appealing to the theory of Toeplitz matrices, we provide sufficient conditions for the growth rate to equal this upper bound. In particular, our results apply to all merges of principal permutation classes. We end by demonstrating how our techniques relate to the results of Bóna.

We give a formula for the number of convex sets in a product of two finite totally ordered sets with the product order and discuss related counting problems.

We obtain new, explicit exponential stability conditions for the linear scalar neutral equation with two bounded delays $ (x(t)-a(t)x(g(t)))'+b(t)x(h(t))=0, $ where $|a(t)| \leq A_0 \lt 1$, $0\lt b_0\leq b(t)\leq B_0$, assuming that all parameters of the equation are measurable functions. To analyze the exponential stability, we apply the Bohl-Perron theorem and a reduction of a neutral equation to an equation with an infinite number of non-neutral delay terms. This method has never before been used for this neutral equation; its application allows omitting a usual restriction $|a(t)|\lt {1}/{2}$ in known asymptotic stability tests and the consideration of variable delays.

We study the eight elliptic curves whose rational points correspond to Heron triangles with two rational medians. We show that none of these triangles can have three rational medians.

Bocci, Carlini and Kileel have shown that the square-free Hadamard product of a finite set of points $Z$ that all lie on a line $\ell $ in $\mathbb {P}^n$ produces a star configuration of codimension $n$. In this paper, we introduce a construction using the Hadamard product to construct star configurations of codimension $c$. In the case that $c = n= 2$, our construction produces the star configurations of Bocci, Carlini and Kileel. We will call any star configuration that can be constructed using our approach a Hadamard star configuration. Our main result is a classification of Hadamard star configurations.

We give an analogue of the Rabinowitsch criterion with $\mathbb{Z} $ replaced by the polynomial ring $k[t]$ over a field of characteristic different from $2$. In fact, we expose three different proofs of the Rabinowitsch criterion, using Dedekind-Hasse norms, binary quadratic forms and the Minkowski bound on ideal classes, and adapt each to prove our Polynomial Rabinowitsch criterion. Whereas there are precisely seven cases in which the classical Rabinowitsch criterion holds, working over an arbitrary ground field gives us much more latitude, e.g. recent results about genus $1$ curves yield infinitely many instances in which the Rabinowitsch criterion is satisfied over $k = \mathbb{Q} $. Finally, we take a geometric perspective and relate the Rabinowitsch criterion to the Mordell-Weil group of the Jacobian of the associated hyperelliptic curve.

We show that the recent result of Castañeda and Wu regarding the ramification filtration in certain $p$-extensions of function fields of prime characteristic $p$ is equally valid over local fields of mixed characteristic $(0,p)$. Apart from being applicable to both equicharacteristic and mixed characteristic cases, our method is local, conceptual, natural, and shorter.

We give upper and lower bounds for the trace of a Salem number whose Mahler measure is less than 1.3. We use the method of explicit auxiliary functions, combined with a recursive algorithm developed by the author.

Let $\Omega $ be a smooth open domain in $\mathbb{R} ^n$, $n \geq 3$, with $0 \in \partial \Omega $ and $s \in {]}0,2{[}$. Let $2^\star(s) := {2(n-s)}/({n-2})$ be the critical Hardy-Sobolev exponent. Consider the Dirichlet problem that corresponds to the critical and perturbative Hardy-Sobolev equation \begin{aligned}\begin{cases}-\Delta u = {u^{2^\star(s) -1}}/{|x|^s} + h u^{q-1} \quad &\mbox {in } \Omega ,\\u \equiv 0 &\mbox {on } \partial \Omega, \end{cases}\end{aligned} where $q \in {]} 2,2^\star {[}$ with $2^\star := 2^\star (0)$ and $h \in C^0(\overline {\Omega })$, $h \geq 0$, almost everywhere on $\Omega $. In this article, we investigate the influence of perturbation and mean curvature at a boundary singularity on the existence of a positive Mountain pass solution to the above equation.

In this paper, we derive some identities involving special numbers and moments of random variables by using the generating functions of the moments of certain random variables. Here, the related special numbers are Stirling numbers of the first and second kinds, degenerate Stirling numbers of the first and second kinds, derangement numbers, higher-order Bernoulli numbers and Bernoulli numbers of the second kind.

This paper has two purposes. First, we explain and describe the Besicovitch covering lemma, and we provide a new proof. Applications are given, particularly to the ideas of Nagel and Stein about Fatou theorems through approach regions which are not nontangential. Second, we examine the strong maximal function and give a new, simple, geometric proof of its $L^p$ boundedness.

Let $R$ be a ring and $\mathcal {M}$ the left or right $R$-module category. Let $\mathscr {C}$ be a class of $R$-modules, closed under extensions, finite direct sums, direct summands and isomorphisms. Suppose that $\mathscr {C}$ is both precovering and preenveloping. We prove that every $m$th $\mathscr {C}$-cosyzygy of any $R$-module is contained in $\mathscr {C}$ if and only if every $m$th $\mathscr {C}$-syzygy of any $R$-module is contained in $\mathscr {C}$; if $\mathscr {C}$ is closed under cokernels of monomorphisms, then gl right $\mathscr {C}$-${\dim }\, {\mathcal {M}}\leq m$ and every $m$th and $(m+1)$th $\mathscr {C}$-cosyzygy of any $R$-module have a monic $\mathscr {C}$-preenvelope if and only if gl left $\mathscr {C}$-${\dim }\, {\mathcal {M}} \leq m-2$, $m\geq 2$; if $\mathscr {C}$ is closed under kernels of epimorphisms, then gl left $\mathscr {C}$-${\dim }\, {\mathcal {M}}\leq m$ and every $m$th and $(m+1)$th $\mathscr {C}$-syzygy of any $R$-module have an epic $\mathscr {C}$-precover if and only if gl right $\mathscr {C}$-${\dim }\,{\mathcal {M}}\leq m-2$, $m\geq 2$; if every nonzero $R$-module has a nonzero $\mathscr {C}$-preenvelope and a nonzero $\mathscr {C}$-precover, then gl right $\mathscr {C}$-${\dim }\, {\mathcal {M}} =$ gl left $\mathscr {C}$-${\dim }\, {\mathcal {M}}$. Some applications are given. Some known results are extended or improved.

We study the existence of first integrals for the class of reversible and equivariant quadratic polynomial differential systems in the plane. We put special emphasis in the study of the analytic first integrals.

In this paper, we solve the following bi-additive $s$-functional inequalities: $$ \displaylines { \, \| f(x+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\| \hfill \cr \hfill \quad \le \Bigl \|s \Bigl (2f\Bigl (\frac {x+y}{2}, z-w\Bigr ) + 2f\Bigl (\frac {x-y}{2}, z+w\Bigr ) - 2f(x,z )+ 2 f(y, w)\Bigr )\Bigr \| ,\! \cr \, \Bigl \|2f\Bigl (\frac {x+y}{2}, z-w\Bigr ) +2 f\Bigl (\frac {x-y}{2}, z+w\Bigr ) -2 f(x,z )+2 f(y, w)\Bigr \| \hfill \cr \hfill \le \|s ( f(x+y, z-w) + f(x-y, z+w) -2f(x,z) +2 f(y, w) )\|, \cr \, } $$ where $s$ is a fixed nonzero complex number with $|s |\lt 1$. We also prove the Hyers-Ulam stability of quasi-multipliers on Banach algebras and unital $C^*$-algebras associated with the bi-additive $s$-functional inequalities above.

In this paper, we attempt to establish the existence of at least three positive solutions for a coupled system of $p$-Laplacian fractional order boundary value problems \begin{aligned}\begin{cases} D_{a^+}^{\beta _1}(\phi _{p}(D_{a^+}^{\alpha _1}u(t)))=f_1(t, u(t), v(t)) & \quad a\lt t\lt b,\\ D_{a^+}^{\beta _2}(\phi _{p}(D_{a^+}^{\alpha _2}v(t)))=f_2(t, u(t), v(t)) & \quad a\lt t\lt b, \end{cases}\end{aligned} with the boundary conditions \begin{aligned}\begin{cases} u^{(j)}(a)=0,\ j=0,1,2,\ldots , \ u{''}(b)=\delta u{''}(\xi ),\\ \phi _{p}(D_{a^+}^{\alpha _1}u(a))=0, \ \phi _{p}(D_{a^+}^{\alpha _1} u(b)) =\vartheta \phi _{p}( D_{a^+}^{\alpha _1}u(\eta )),\\ v^{(j)}(a)=0,\ j=0,1,2,\ldots , \ v{''}(b)=\delta v{''}(\xi ),\\ \phi _{p}(D_{a^+}^{\alpha _2}v(a))=0, \ \phi _{p}(D_{a^+}^{\alpha _2} v(b))=\vartheta \phi _{p}( D_{a^+}^{\alpha _2}v(\eta )), \end{cases}\end{aligned} by applying the five functional fixed point theorem.

For $1/2\lt p\lt \infty $, $0\lt q\lt \infty $ and a certain two-sided doubling weight $\omega $, we give a condition for the zeros of a Blaschke product $B$ which guarantees that $$ \|B'\|_{A^{p,q}_\omega }^q=\int _0^1 \bigg (\int _0^{2\pi } |B'(re^{i\theta })|^p d\theta \bigg )^{q/p} \omega (r)\,dr\lt \infty . $$ In addition, it is shown that the condition is necessary if the zero-sequence is a finite union of separated sequences.

We introduce the notion of the partial group algebra with projections and relations and show that this $C^*$-algebra is a partial crossed product. Examples of partial group algebras with projections and relations are the Cuntz-Krieger algebras and the unitization of $C^*$-algebras of directed graphs.

This paper focuses on asymptotic results for linear Hawkes processes with large and small baseline intensities. The intensity process is one of the main tools used to work with the dynamical properties of a general point process. It is of essential interest in credit risk study, in particular. First, we establish a large deviation principle and a moderate deviation principle for the Hawkes process with large baseline intensity. In addition, a law of large numbers and a central limit theorem are also obtained. Second, we observe asymptotic behaviors for the Hawkes process with small baseline intensity. The main idea of the proof relies on the immigration-birth representation and the observations of the moment generating function for the linear Hawkes process.

By means of six specific generating functions, we introduce a type of generalized parametric Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials and systematically study their basic properties. As an application of the new polynomials, we use them in computing a new series of Taylor-type.