Related papers: The p-norm of some matrices

The arithmetic partial derivative (with respect to a prime $p$) is a function from the set of integers that sends $p$ to 1 and satisfies the Leibniz rule. In this paper, we prove that the $p$-adic valuation of the sequence of higher order…

Given an infinite matrix $M=(m_{nk})$ we study a family of sequence spaces $\ell_M^p$ associated with it. When equipped with a suitable norm $\|\cdot\|_{M,p}$ we prove some basic properties of the Banach spaces of sequences…

We analyze the bit complexity of efficient algorithms for fundamental optimization problems, such as linear regression, $p$-norm regression, and linear programming (LP). State-of-the-art algorithms are iterative, and in terms of the number…

We design an algorithm for computing the $p$-curvature of a differential system in positive characteristic $p$. For a system of dimension $r$ with coefficients of degree at most $d$, its complexity is $\softO (p d r^\omega)$ operations in…

For every $p \geq 2$, we obtained an explicit construction of a family of $\mathcal{W}(2,2p-1)$-modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual…

Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for $n$-point subsets of $\ell_p$, for $p >…

We exhibit a probabilistic algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence. Its bit complexity is roughly quadratic in the B\'ezout number of the system and linear in its bit size. Our…

In this article, we characterize the bounded and the compact multiplication operators between distinct iterated logarithmic Lipschitz spaces, and between the Lipschitz space and an iterated logarithmic Lipschitz space of an infinite tree.…

We study algorithms for the fast computation of modular inverses. Newton-Raphson iteration over $p$-adic numbers gives a recurrence relation computing modular inverse modulo $p^m$, that is logarithmic in $m$. We solve the recurrence to…

The notion of $p$-summing Bloch mapping from the complex unit open disc $\mathbb{D}$ into a complex Banach space $X$ is introduced for any $1\leq p\leq\infty$. It is shown that the linear space of such mappings, equipped with a natural…

For a prime number $p,$ let $\mathbb{Q}_p$ be the field of $p$-adic numbers. In this paper, we established the boundedness of a class of $p$-adic singular integral operators on the $p$-adic generalized Morrey spaces. The corresponding…

We show that the $p$-adic valuation of the sequence of Fibonacci numbers is a $p$-regular sequence for every prime $p$. For $p \neq 2, 5$, we determine that the rank of this sequence is $\alpha(p) + 1$, where $\alpha(m)$ is the restricted…

We construct complex a-priori bounds for certain infinitely renormalizable Lorenz maps. As a corollary, we show that renormalization is a real-analytic operator on the corresponding space of Lorenz maps.

For an $n\times n$ matrix $A_n$, the $r\to p$ operator norm is defined as $$\|A_n\|_{r\to p}:= \sup_{\mathbf{x}\in\mathbb{R}^n:\|\mathbf{x} \|_r\leq 1 } \|A_n\mathbf{x} \|_p\quad\text{for}\quad r,p\geq 1.$$ For different choices of $r$ and…

In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients ${p^a n \choose n}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending…

Let $A$ be a complex Banach algebra. If the spectrum of an invertible element $a\in A$ does not separate the plane, then $a$ admits a logarithm. We present two elementary proofs of this classical result which are independent of the…

Related to the concept of $p$-compact operator with $p\in [1,\infty]$ introduced by Sinha and Karn, this paper deals with the space $\mathcal{H}^\infty_{\mathcal{K}_p}(U,F)$ of all Banach-valued holomorphic mappings on an open subset $U$ of…

We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$. In particular, in the integer case, we improve a recent bound…

We prove that if an $n\times n$ matrix defined over ${\mathbb Q}_p$ (or more generally an arbitrary complete, discretely-valued, non-Archimedean field) satisfies a certain congruence property, then it has a strictly maximal eigenvalue in…

In this article, the open problem of finding the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces $A^p_\alpha$ is adressed. The norm was conjectured to be $\frac{\pi}{\sin \frac{(2+\alpha)\pi}{p}}$ by…