We generalize the $2$-tensor paraproduct decomposition result of arXiv:2503.12629 to $d$-tensors and expound on its caricatures. In particular, we show that for $A \in C^{d}, f \in \Lambda^{\alpha}([0,1]^d)$, $A(f)$ has an approximation $\tilde{A}(f) = (\sum_{\alpha=1}^d A^{\alpha}(P^{j_1,j_2, \ldots, j_d}(f)) \tilde{\mathbf{v}}^{\alpha}(f) ) $ for a fixed sequence of scales, $\mathbf{j} = (j_1, j_2, \ldots, j_d)$, and the series $\sum_{\alpha=1}^d A^{\alpha}(P^{j_1,j_2, \ldots, j_d}(f)) \tilde{\mathbf{v}}^{\alpha}(f) $ is a taylor expansion of $A$. Additionally, we show the sequence of operators $(\tilde{\mathbf{v}}^1(f),\tilde{\mathbf{v}}^2(f), \ldots, \tilde{\mathbf{v}}^d(f))$ form a basis for the subspace comprised of linear functionals of the form $A(f)$, and the residual, $\Delta(A,f) = \tilde{A}(f) - A(f) \in \Lambda^{2\alpha}([0,1]^d)$. Consequentially, we show one can obtain a Calderon-Zygmund type decomposition, $\Delta(A,f)_\lambda + \tilde{A}(f)_{\lambda}$ with the obtained paraproduct decomposition $A(f) = \Delta(A,f) + \tilde{A}(f)$. Our theoretical findings are supported by computational examples for d=2,3.

Oluwadamilola Fasina

Let $f$ be a transcendental entire function. The fast escaping set, $A(f)$, plays a key role in transcendental dynamics. The quite fast escaping set, $Q(f)$, defined by an apparently weaker condition is equal to $A(f)$ under certain conditions. Here we introduce $Q_2(f)$ defined by what appears to be an even weaker condition. Using a new regularity condition we show that functions of finite order and positive lower order satisfy $Q_2(f)=A(f)$. We also show that the finite composition of such functions satisfies $Q_2(f)=A(f)$. Finally, we construct a function for which $Q_2(f) \neq Q(f)= A(f)$.

Vasiliki Evdoridou

Suppose that $f(z)$ is a transcendental entire function and that the Fatou set $F(f)\neq\emptyset$. Set $$B_1(f):=\sup_{U}\frac{\sup_{z\in U}\log(|z|+3)}{\inf_{w\in U}\log(|w|+3)}$$ and $$B_2(f):=\sup_{U}\frac{\sup_{z\in U}\log\log(|z|+30)}{\inf_{w\in U}\log(|w|+3)},$$ where the supremum $\sup_{U}$ is taken over all components of $F(f)$. If $B_1(f)<\infty$ or $B_2(f)<\infty$, then we say $F(f)$ is strongly uniformly bounded or uniformly bounded respectively. In this article, we will show that, under some conditions, $F(f)$ is (strongly) uniformly bounded.

Xiaoling Wang Wang Zhou

In this note we give a new separation between sensitivity and block sensitivity of Boolean functions: $bs(f)=(2/3)s(f)^2-(1/3)s(f)$.

Andris Ambainis Xiaoming Sun

These notes deal with some recent assertions about truncations $f \mapsto |f|$ and compositions $f \mapsto g\circ f$ in the spaces $A^s_{p,q}(\mathbb{R}^n)$, $A \in \{B,F \}$.

Hans Triebel

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=f"(0)=0, P. T. Mocanu (Mathematica (Cluj), 42(2000)) has considered some sufficient arguments of f'(z)+zf"(z) for |\arg(zf'(z)/f(z))|<\pi\mu/2. The object of the present paper is to discuss those probrems for f(z) with f"(0)=f"'(0)=...=f^{(n)}(0)=0 and f^{(n+1)}(0) \ne 0.

Hitoshi Shiraishi Shigeyoshi Owa

P. Hrube\v s, S. Natarajan Ramamoorthy, A. Rao and A. Yehudayoff proved the following result: Let $p$ be a prime and let $f\in \mathbb F _p[x_1,\ldots,x_{2p}]$ be a polynomial. Suppose that $f(\mathbf{v_F})=0$ for each $F\subseteq [2p]$, where $|F|=p$ and that $f(\mathbf{0})\neq 0$. Then $\mbox{deg}(f)\geq p$. We prove here the following generalization of their result. Let $p$ be a prime and $q=p^\alpha>1$, $\alpha\geq 1$. Let $n>0$ be a positive integer and $q-1\leq d\leq n-q+1$ be an integer. Let $\mathbb F$ be a field of characteristic $p$. Suppose that $f(\mathbf{v_F})=0$ for each $F\subseteq [n]$, where $|F|=d$ and $\mbox{deg}(f)\leq q-1$. Then $f(\mathbf{v_F})=0$ for each $F\subseteq [n]$, where $|F|\equiv d \mbox{ (mod }q)$. Let $t=2d$ be an even number and $L\subseteq [d-1]$ be a given subset. We say that $\mbox{$\cal F$}\subseteq 2^{[t]}$ is an {\em $L$-balancing family} if for each $F\subseteq [t]$, where $|F|=d$ there exists a $G\subseteq [n]$ such that $|F\cap G|\in L$. We give a general upper bound for the size of an $L$-balancing family.

Gábor Hegedüs

Block sensitivity ($bs(f)$), certificate complexity ($C(f)$) and fractional certificate complexity ($C^*(f)$) are three fundamental combinatorial measures of complexity of a boolean function $f$. It has long been known that $bs(f) \leq C^{\ast}(f) \leq C(f) =O(bs(f)^2)$. We provide an infinite family of examples for which $C(f)$ grows quadratically in $C^{\ast}(f)$ (and also $bs(f)$) giving optimal separations between these measures. Previously the biggest separation known was $C(f)=C^{\ast}(f)^{\log_{4.5}5}$. We also give a family of examples for which $C^{\ast}(f)=\Omega(bs(f)^{3/2})$. These examples are obtained by composing boolean functions in various ways. Here the composition $f \circ g$ of $f$ with $g$ is obtained by substituting for each variable of $f$ a copy of $g$ on disjoint sets of variables. To construct and analyse these examples we systematically investigate the behaviour under function composition of these measures and also the sensitivity measure $s(f)$. The measures $s(f)$, $C(f)$ and $C^{\ast}(f)$ behave nicely under composition: they are submultiplicative (where measure $m$ is submultiplicative if $m(f \circ g) \leq m(f)m(g)$) with equality holding under some fairly general conditions. The measure $bs(f)$ is qualitatively different: it is not submultiplicative. This qualitative difference was not noticed in the previous literature and we correct some errors that appeared in previous papers. We define the composition limit of a measure $m$ at function $f$, $m^{\lim}(f)$ to be the limit as $k$ grows of $m(f^{(k)})^{1/k}$, where $f^{(k)}$ is the iterated composition of $f$ with itself $k$-times. For any function $f$ we show that $bs^{\lim}(f) = (C^*)^{\lim}(f)$ and characterize $s^{\lim}(f), (C^*)^{\lim}(f)$, and $C^{\lim}(f)$ in terms of the largest eigenvalue of a certain set of $2\times 2$ matrices associated with $f$.

Justin Gilmer Michael Saks Srikanth Srinivasan

Sufficient conditions are given for a function $F(p)$, analytic in Re$p>0$, to be a Laplace transform of a function $f(t)$, such that $max_{t\ge 0}|f(t)|<\infty$, $f(0)=0$.

Alexander G. Ramm

The famous Jacobian problem asks: Is a morphism $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ having an invertible Jacobian, invertible? If we add the assumption that $\mathbb{C}(f(x),f(y))=\mathbb{C}(x,y)$, then $f$ is invertible; this result is due to O. H. Keller (1939). We suggest the following slight generalization of Keller's theorem: If $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ is a morphism having an invertible Jacobian, and if there exist $n \geq 1$, $a \in \mathbb{C}(f(x),f(y))^*$ and $b \in \mathbb{C}(f(x),f(y))$ such that $(ax +b)^n \in \mathbb{C}(f(x),f(y))$, then $f$ is invertible. A similar result holds for $\mathbb{C}[x_1,\ldots,x_m]$.

Vered Moskowicz