Related papers: {\L}ojasiewicz inequalities with explicit exponent…

Let F be a real-valued convex function on the complex plane, and let Ps(b) be a pseudodifferential operator with symbol b. Under certain natural assumptions about properties of pseudodifferential operators, we prove that the sum of F(z)…

Let $\Re_n$ be the set of all rational functions of the type $r(z) = p(z)/w(z),$ where $p(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-a_j)$, $|a_j|>1$ for $1\leq j\leq n$. In this paper, we set up some results…

We show that if f is a nonzero, noninvertible function on a smooth complex variety X and J_f is the Jacobian ideal of f, then lct(f, J_f^2)>1 if and only if the hypersurface defined by f has rational singularities. Moreover, if this is not…

We prove an f-version of Mirsky's singular value inequalities for differences of matrices. This f-version consists in applying a positive concave function f, with f(0)=0, to every singular value in the original Mirsky inequalities.

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable…

The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. We introduce a generic method for increasing the…

It is shown that $\sum_{j=-m}^m (-1)^j \frac{f(x-j)(f(x+j)}{(m-j)! (m+j)!} \ge 0,$ $m=0,1,...,$ where $f(x)$ is either a real polynomial with only real zeros or an allied entire function of a special type, provided the distance between two…

This short note gathers known results to state that the squared distance function to a (nonconvex) closed set of an Euclidean space is Polyak-{\L}ojasiewicz. As a fuzzy reciprocate, we also recall that every Polyak-{\L}ojasiewicz function…

Let $\Lambda_s$ denote the Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$, which consists of all $f\in\mathfrak{C}\cap L^\infty$ such that, for some constant $L\in(0,\infty)$ and some integer $r\in(s,\infty)$, \begin{equation*}…

Let $\mathcal{W}$ be a closed dilation and translation invariant subspace of the space of $\mathbb{R}^\ell$-valued Schwartz distributions in $d$ variables. We show that if the space $\mathcal{W}$ does not contain distributions of the type…

We propose a manifold AdaGrad-Norm method (\textsc{MAdaGrad}), which extends the norm version of AdaGrad (AdaGrad-Norm) to Riemannian optimization. In contrast to line-search schemes, which may require several exponential map computations…

In this paper, we study the D-gap function associated with a nonsmooth and nonmonotone variational inequality problem. We present some exact formulas for the subderivative, the regular subdifferential set, and the limiting subdifferential…

Let $0<\alpha<d$ and $1\leq p<d/\alpha$. We present a proof that for all $f\in W^{1,p}(\mathbb{R}^d)$ both the centered and the uncentered Hardy-Littlewood fractional maximal operator $\mathcal M_\alpha f$ are weakly differentiable and $…

We study pluricomplex Green functions on algebraic sets. Let $f$ be a proper holomorphic mapping between two algebraic sets. Given a compact set $K$ in the range of $f$, we show how to estimate the pluricomplex Green functions of $K$ and of…

Cilleruelo conjectured that for an irreducible polynomial $f \in \mathbb{Z}[X]$ of degree $d \geq 2$ one has $$\log\left[\mathrm{lcm}(f(1),f(2),\ldots f(N))\right]\sim(d-1)N\log N$$ as $N \to \infty$. He proved it in the case $d=2$ but it…

The {\L}ojasiewicz inequality characterizes objective-value convergence along gradient flows and, in special cases, yields exponential decay of the cost. However, such results do not directly give rates of convergence in the state. In this…

Let $f,g:\Bbb{R}^{N}\rightarrow (-\infty ,\infty ]$ be Borel measurable, bounded below and such that $\inf f+\inf g\geq 0.$ We prove that with $ m_{f,g}:=(\inf f-\inf g)/2,$ the inequality $||(f-m_{f,g})^{-1}||_{\phi…

By using the method developed in the paper [G.Pantsulaia, G.Giorgadze, On some applications of infinite-dimensional cellular matrices, {\it Georg. Inter. J. Sci. Tech., Nova Science Publishers,} Volume 3, Issue 1 (2011), 107-129], it is…

The Sz\'asz inequality is a classical result that provides a bound for polynomials with zeros in the upper half of the complex plane, expressed in terms of their low-order coefficients. Generalizations of this result to polynomials in…

Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such…