The generalized Beurling–Ahlfors operator $S$ on ${{L}^{p}}({{\mathbb{R}}^{n}};\,\Lambda )$, where $\Lambda \,:=\,\Lambda ({{\mathbb{R}}^{n}})$ is the exterior algebra with its natural Hilbert space norm, satisfies the estimate

$$||S||L\left( {{L}^{p}}\left( {{\mathbb{R}}^{n}};\Lambda \right) \right)\le \left( n/2+1 \right)\left( {{p}^{*}}-1 \right),\,\,\,{{p}^{*}}:=\,\max \{p,\,{{p}^{'}}\}.$$

This improves on earlier results in all dimensions $n\,\ge \,3$. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.

We prove, under some conditions on the domains, that the adjoint of the sum of two unbounded operators is the sum of their adjoints in both Hilbert and Banach space settings. A similar result about the closure of operators is also proved. Some interesting consequences and examples “spice up” the paper.

Let $(X,\,d)$ be a metric space, and let $\text{Lip(}X\text{)}$ denote the Banach space of all scalar-valued bounded Lipschitz functions $f$ on $X$ endowed with one of the natural norms

$$\left\| f \right\|\,=\,\max \{{{\left\| f \right\|}_{\infty }},\,L(f)\}\,\,\text{or}\,\,\left\| f \right\|\,=\,{{\left\| f \right\|}_{\infty }}\,+\,L(f),$$

where $L(f)$ is the Lipschitz constant of $f$. It is said that the isometry group of $\text{Lip(}X\text{)}$ is canonical if every surjective linear isometry of $\text{Lip(}X\text{)}$ is induced by a surjective isometry of $X$. In this paper we prove that if $X$ is bounded separable and the isometry group of $\text{Lip(}X\text{)}$ is canonical, then every 2-local isometry of $\text{Lip(}X\text{)}$ is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of $\text{Lip(}X\text{)}$ when $X$ is bounded.

We show that closed $\widetilde{\mathbb{S}\mathbb{L}}\,\times \,{{\mathbb{E}}^{n}}$-manifolds are topologically rigid if $n\,\ge \,2$, and are rigid up to $s$-cobordism, if $n\,=\,1$.

We show that if the family $\mathcal{O}$ of orbits of all vector fields on a subcartesian space $P$ is locally finite and each orbit in $\mathcal{O}$ is locally closed, then $\mathcal{O}$ defines a smooth Whitney A stratification of $P$. We also show that the stratification by orbit type of the space of orbits $M/G$ of a proper action of a Lie group $G$ on a smooth manifold $M$ is given by orbits of the family of all vector fields on $M/G$.

Let ${{T}_{n}}$ denote the $n$-th Chebyshev polynomial of the first kind, and let ${{U}_{n}}$ denote the $n$-th Chebyshev polynomial of the second kind. We give an explicit formula for the resultant res$({{T}_{m}},\,{{T}_{n}})$. Similarly, we give a formula for res$({{U}_{m}},\,{{U}_{n}})$.

In this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm–Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm–Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm–Liouville equation with turning point to the equation with singularity, then we study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem.

We prove that the fundamental solutions of Kohn sub-Laplacians$\Delta +i\alpha {{\partial }_{t}}$ on the anisotropic Heisenberg groups are tempered distributions and have meromorphic continuation in α with simple poles. We compute the residues and find the partial fundamental solutions at the poles. We also find formulas for the fundamental solutions for some matrix-valued Kohn type sub-Laplacians on $\text{H}$-type groups.

We provide a simple example showing that the tangential derivative of a continuous function $\phi $ can vanish everywhere along a curve while the variation of $\phi $ along this curve is nonzero. We give additional regularity conditions on the curve and/or the function that prevent this from happening.

We give a new factorisation of the classical Dynkin operator, an element of the integral group ring of the symmetric group that facilitates projections of tensor powers onto Lie powers. As an application we show that the iterated Lie power ${{L}_{2}}({{L}_{n}})$ is a module direct summand of the Lie power ${{L}_{2n}}$ whenever the characteristic of the ground field does not divide $n$. An explicit projection of the latter onto the former is exhibited in this case.

For ${{C}^{*}}$-algebras $\mathcal{A}$ of real rank zero, we describe linear maps $\phi $ on $\mathcal{A}$ that are surjective up to ideals $\mathcal{J}$, and $\text{ }\pi \text{ (}A\text{)}$ is invertible in $\mathcal{A}/\mathcal{J}$ if and only if $\text{ }\pi \text{ (}\phi (A))$ is invertible in $\mathcal{A}/\mathcal{J}$, where $A\,\in \,\mathcal{A}$ and $\text{ }\pi \text{ }\text{:}\mathcal{A}\to \mathcal{A}\text{/}\mathcal{J}$ is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra.

We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants that further generalize the quandle counting invariant.

We give a characterization of contact manifolds in terms of symplectic Lie–Rinehart–Jacobi algebras. We also give a sufficient condition for a Jacobi manifold to be a contact manifold.

Let $X$ be a separable non-reflexive Banach space. We show that there is no Borel class which contains the set of norm-attaining functionals for every strictly convex renorming of $X$.

Building on the work of Nogin, we prove that the braid group ${{B}_{4}}$ acts transitively on full exceptional collections of vector bundles on Fano threefolds with ${{b}_{2}}\,=\,1$ and ${{b}_{3}}\,=\,0$. Equivalently, this group acts transitively on the set of simple helices (considered up to a shift in the derived category) on such a Fano threefold. We also prove that on threefolds with ${{b}_{2}}\,=\,1$ and very ample anticanonical class, every exceptional coherent sheaf is locally free.

We prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line.

We establish a discrete-time criteria guaranteeing the existence of an exponential dichotomy in the continuous-time behavior of an abstract evolution family. We prove that an evolution family $\mathcal{U}\,=\,{{\{U(t,\,s)\}}_{t\ge s\ge 0}}$ acting on a Banach space $X$ is uniformly exponentially dichotomic (with respect to its continuous-time behavior) if and only if the corresponding difference equation with the inhomogeneous term from a vector-valued Orlicz sequence space ${{l}^{\Phi }}(\mathbb{N},\,X)$ admits a solution in the same ${{l}^{\Phi }}(\mathbb{N},\,X)$. The technique of proof effectively eliminates the continuity hypothesis on the evolution family (i.e., we do not assume that $U(\,\cdot \,,\,s)x$ or $U(t,\,\cdot \,)x$ is continuous on $[s,\,\infty )$, and respectively $[0,\,t])$. Thus, some known results given by Coffman and Schaffer, Perron, and Ta Li are extended.

We give some extension to theorems of Jiménez López and Soler López concerning the topological characterization for limit sets of continuous flows on closed orientable surfaces.

Let ${{B}_{Y}}$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a sufficient enlargement for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:\,Y\,\to \,X$ such that $P({{B}_{Y}})\,\subset \,A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.