Related papers: Non-symmetrically $t$-affine functions revisited

An iterative optimization method applied to a function $f$ on $\mathbb{R}^n$ will produce a sequence of arguments $\{\mathbf{x}_k\}_{k \in \mathbb{N}}$; this sequence is often constrained such that $\{f(\mathbf{x}_k)\}_{k \in \mathbb{N}}$…

Let $s: [1, \infty) \to \C$ be a locally integrable function in Lebesgue's sense on the infinite interval $[1, \infty)$. We say that $s$ is summable $(L, 1)$ if there exists some $A\in \C$ such that $$\lim_{t\to \infty} \tau(t) = A, \quad…

In this paper, we study lower bounds of a general family of $L$-functions on the $1$-line. More precisely, we show that for any $F(s)$ in this family, there exists arbitrary large $t$ such that $F(1+it)\geq e^{\gamma_F} (\log_2 t + \log_3…

We suggest the necessary/sufficient criteria for the existence of a (order-by-order) solution y(x) of a functional equation F(x,y)=0 over a ring. In full generality, the criteria hold in the category of filtered groups, this includes the…

In this work we extend the theory of the classical Hardy space $H^1$ to the rational Dunkl setting. Specifically, let $\Delta$ be the Dunkl Laplacian on a Euclidean space $\mathbb{R}^N$. On the half-space $\mathbb{R}_+\times\mathbb{R}^N$,…

In this note we give a simple sufficient condition for an affine iterated function system to admit an invariant affine subspace persistently with respect to changes in the translation parameters. This yields further examples of tuples of…

Ext-int.\ one affine functions are functions affine in the direction of one-divisible exterior forms, with respect to exterior product in one variable and with respect to interior product in the other. The purpose of this article is to…

In what follows we improve an inequality related to matrix theory. T. Laffey proved (2013) a weaker form of this inequality [2].

Sharp affine fractional $L^p$ Sobolev inequalities for functions on $\mathbb R^n$ are established. The new inequalities are stronger than (and directly imply) the sharp fractional $L^p$ Sobolev inequalities. They are fractional versions of…

We present a general approach to the parametrix construction. We apply it to prove the uniqueness and existence of a weak fundamental solution for the equation $\partial_t =\mathcal{L}$ with non-symmetric non-local operators $$…

It is well known that if a function $f$ satisfies $$\|f(x) e^{\pi \alpha |x|^2}\|_p + \| \widehat{f}(\xi) e^{\pi \alpha |\xi|^2} \|_q<\infty \qquad\qquad\qquad(*)$$ with $\alpha=1$ and $1\le p,q<\infty$, then $f\equiv 0.$ We prove that if…

We prove that if $f:I\subset \Bbb R\to \Bbb R$ is of bounded variation, then the noncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\|DMf\|_1\le |Df|(I)$. This allows us obtain,…

This note deals with the operator $T^*T$, where $T$ is a densely defined operator on a complex Hilbert space. We reprove a recent result of Z. Sebesty\'en and Zs. Tarcsay [13]: If $T^*T$ and $TT^*$ are self-adjoint, then $T$ is closed. In…

We introduce a new family of symmetric functions, which are $q$-analogues of products of Schur functions defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation of the quantum affine…

For entire functions $f(z)=\sum_{n=0}^{+\infty}a_nz^n, z\in {\Bbb C},$ P. L${\rm \acute{e}}$vy (1929) established that in the classical Wiman's inequality $M_f(r)\leq\mu_f(r)\times $ $\times(\ln\mu_f(r))^{1/2+\varepsilon},\ \varepsilon>0,$…

We establish a Sobolev-type inequality in Lorentz spaces for $\mathcal{L}$-superharmonic functions \[ \|u\|_{L^{\frac{nq}{n-\alpha q},t}(\mathbb{R}^n)} \leq c \left\| \frac{u(x) - u(y)}{|x-y|^{\frac{n}{q}+\alpha}}…

In this article, we study algebraic decompositions and secondary constructions of almost perfect nonlinear (APN) functions. In many cases, we establish precise criteria which characterize when certain modifications of a given APN function…

We consider the function $x^{-1}$ that inverses a finite field element $x \in \mathbb{F}_{p^n}$ ($p$ is prime, $0^{-1} = 0$) and affine $\mathbb{F}_{p}$-subspaces of $\mathbb{F}_{p^n}$ such that their images are affine subspaces as well. It…

If $f$ is a Henstock--Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $\|f\|=\sup_I|\int_I f|$ where the supremum is taken over all intervals $I\subset\R$. Define the translation $\tau_x$ by $\tau_xf(y)=f(y-x)$.…

Recently it has been proved that, assuming that there is an almost disjoint family of cardinality (2^{\mathfrak c}) in (\mathfrak c) (which is assured, for instance, by either Martin's Axiom, or CH, or even $2^{<\mathfrak c=\mathfrak c$})…