m-Nearly k-Universal Words -- Investigating Simon Congruence

• URL: http://arxiv.org/abs/2202.07981v1
• Date: Wed, 16 Feb 2022 10:45:10 GMT
• Title: m-Nearly k-Universal Words -- Investigating Simon Congruence
• Abstract summary: Two words $u$ and $v$ are called Simon congruent if they have the same set of scattered factors. We show some results if in addition $w$ is $(k-1)$-universal as well as some further insights for different $m$.
• Score: 1.3229510087215552
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Determining the index of the Simon congruence is a long outstanding open problem. Two words $u$ and $v$ are called Simon congruent if they have the same set of scattered factors, which are parts of the word in the correct order but not necessarily consecutive, e.g., $\mathtt{oath}$ is a scattered factor of $\mathtt{logarithm}$. Following the idea of scattered factor $k$-universality, we investigate $m$-nearly $k$-universality, i.e., words where $m$ scattered factors of length $k$ are absent, w.r.t. Simon congruence. We present a full characterisation as well as the index of the congruence for $m=1$. For $m\neq 1$, we show some results if in addition $w$ is $(k-1)$-universal as well as some further insights for different $m$.

Related papers

• A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results [1.8051706864605614]
  Tang and Zhang conjectured an explicit formula for every finite $p>1$.<n>We prove the conjectured sharp bound for every family of rank-at-most-one summands when $2leq pinfty$.
  arXiv  Detail & Related papers  (2026-08-16T06:10:33Z)
• Rényi exponent landscape of multipartite entanglement in free-fermion systems [51.56484100374058]
  We show that the Rényi tripartite information $I_3() exhibits a qualitatively $exclusion-dependent scaling at small Fermi momentum.<n>A direct consequence is a replica obstruction: $I_m(n)/I_m(1) sim zm-1 to 0$ for all integer $n geq 2$, so the leading von Neumann signal cannot be reconstructed from integer Rényi data.
  arXiv  Detail & Related papers  (2026-03-09T22:27:00Z)
• Approximating the operator norm of local Hamiltonians via few quantum states [53.16156504455106]
  Consider a Hermitian operator $A$ acting on a complex Hilbert space of $2n$.<n>We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian.<n>We show that whenever $A$ is $d$-local, textiti.e., $deg(A)le d$, we have the following discretization-type inequality.
  arXiv  Detail & Related papers  (2025-09-15T14:26:11Z)
• Non-representable quantum measures [55.2480439325792]
  Grade-$d$ measures on a $sigma$-algebra $mathcalAsubseteq 2X$ over a set $X$ are generalizations of measures satisfying one of a hierarchy of weak additivity-type conditions.<n>Every signed polymeasure $lambda$ on $(X,mathcalA)d$ produces a grade-$d$ measure as its diagonal $widetildelambda(A):=lambda(A,cdots,A)$.
  arXiv  Detail & Related papers  (2025-08-20T00:47:24Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Rényi divergence-based uniformity guarantees for $k$-universal hash functions [59.90381090395222]
  Universal hash functions map the output of a source to random strings over a finite alphabet. We show that it is possible to distill random bits that are nearly uniform, as measured by min-entropy.
  arXiv  Detail & Related papers  (2024-10-21T19:37:35Z)
• LevAttention: Time, Space, and Streaming Efficient Algorithm for Heavy Attentions [54.54897832889028]
  We show that for any $K$, there is a universal set" $U subset [n]$ of size independent of $n$, such that for any $Q$ and any row $i$, the large attention scores $A_i,j$ in row $i$ of $A$ all have $jin U$. We empirically show the benefits of our scheme for vision transformers, showing how to train new models that use our universal set while training as well.
  arXiv  Detail & Related papers  (2024-10-07T19:47:13Z)
• Dimension-free Remez Inequalities and norm designs [48.5897526636987]
  A class of domains $X$ and test sets $Y$ -- termed emphnorm -- enjoy dimension-free Remez-type estimates. We show that the supremum of $f$ does not increase by more than $mathcalO(log K)2d$ when $f$ is extended to the polytorus.
  arXiv  Detail & Related papers  (2023-10-11T22:46:09Z)
• $\alpha$-$\beta$-Factorization and the Binary Case of Simon's Congruence [1.5935205681539144]
  In 1991 H'ebrard introduced a factorization of words that turned out to be a powerful tool for the investigation of a word's scattered factors. Based on this, first Karandikar and Schnoebelen introduced the notion of $k$-richness and later on Barker et al. the notion of $k$-universality. In 2022 Fleischmann et al. presented a generalization of the arch factorization by intersecting the arch factorization of a word and its reverse.
  arXiv  Detail & Related papers  (2023-06-25T10:16:49Z)
• Constructions of $k$-uniform states in heterogeneous systems [65.63939256159891]
  We present two general methods to construct $k$-uniform states in the heterogeneous systems for general $k$. We can produce many new $k$-uniform states such that the local dimension of each subsystem can be a prime power.
  arXiv  Detail & Related papers  (2023-05-22T06:58:16Z)
• Realization of an arbitrary structure of perfect distinguishability of states in general probability theory [0.0]
  All subsets with a single element are of course in $mathcal A$, and since smaller collections are easier to distinguish, if $Hin mathcal A$ and $L subset H$ then $Lin mathcal A$; in other words, $mathcal A$ is a so-called $textitindependence system$ on the set of indices $[n]$.
  arXiv  Detail & Related papers  (2023-01-16T18:33:39Z)
• Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
  We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
  arXiv  Detail & Related papers  (2022-02-10T16:10:41Z)
• A rigidity property of complete systems of mutually unbiased bases [0.0]
  We prove that for some unit vectors $b_1,ldots b_n$ in $mathbb Cd$, they can be arranged into $d+1$ orthonormal bases.
  arXiv  Detail & Related papers  (2021-11-30T20:47:16Z)
• Gray Cycles of Maximum Length Related to k-Character Substitutions [0.0]
  Given a word binary relation $tau$ we define a $tau$-Gray cycle over a finite language $X$ to be a permutation $left(w_[i]right)_0le ile |X|-1$ of $X$. We compute the bound $lambda(n)$ for all cases of the alphabet cardinality and the argument $n$.
  arXiv  Detail & Related papers  (2021-08-31T07:49:15Z)
• Scattered Factor Universality -- The Power of the Remainder [1.8065361710947976]
  Scattered factor (circular) was firstly introduced by Barker et al. in 2020. We prove two problems, namely a generalisation of one of their main theorems to arbitrary alphabets and a slight modification of another theorem.
  arXiv  Detail & Related papers  (2021-04-19T05:45:44Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)
• Tight Quantum Lower Bound for Approximate Counting with Quantum States [49.6558487240078]
  We prove tight lower bounds for the following variant of the counting problem considered by Aaronson, Kothari, Kretschmer, and Thaler ( 2020) The task is to distinguish whether an input set $xsubseteq [n]$ has size either $k$ or $k'=(1+varepsilon)k$.
  arXiv  Detail & Related papers  (2020-02-17T10:53:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.