Search results for " Algebra"
showing 10 items of 2082 documents
Primitive sets of words
2020
Given a (finite or infinite) subset $X$ of the free monoid $A^*$ over a finite alphabet $A$, the rank of $X$ is the minimal cardinality of a set $F$ such that $X \subseteq F^*$. We say that a submonoid $M$ generated by $k$ elements of $A^*$ is {\em $k$-maximal} if there does not exist another submonoid generated by at most $k$ words containing $M$. We call a set $X \subseteq A^*$ {\em primitive} if it is the basis of a $|X|$-maximal submonoid. This definition encompasses the notion of primitive word -- in fact, $\{w\}$ is a primitive set if and only if $w$ is a primitive word. By definition, for any set $X$, there exists a primitive set $Y$ such that $X \subseteq Y^*$. We therefore call $Y$…
A Quantum Lovasz Local Lemma
2012
The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions. Our result immediately applies to the k-QSAT problem: For instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most $2^k/(e \cdot k)$ of them, has a joint satisfiable state. We then …
Generalized Logical Operations among Conditional Events
2018
We generalize, by a progressive procedure, the notions of conjunction and disjunction of two conditional events to the case of n conditional events. In our coherence-based approach, conjunctions and disjunctions are suitable conditional random quantities. We define the notion of negation, by verifying De Morgan’s Laws. We also show that conjunction and disjunction satisfy the associative and commutative properties, and a monotonicity property. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals; in particular we examine the Frechet-Hoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction $\mathcal…
Fractal surfaces from simple arithmetic operations
2015
Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is shown that these models give rise to a roughness exponent $H$ that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.
Thresholding projection estimators in functional linear models
2008
We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove these estimators are minimax and rates of convergence are given for some particular cases.
Isotonic regression for metallic microstructure data: estimation and testing under order restrictions
2021
Investigating the main determinants of the mechanical performance of metals is not a simple task. Already known physical inspired qualitative relations between 2D microstructure characteristics and 3D mechanical properties can act as the starting point of the investigation. Isotonic regression allows to take into account ordering relations and leads to more efficient and accurate results when the underlying assumptions actually hold. The main goal in this paper is to test order relations in a model inspired by a materials science application. The statistical estimation procedure is described considering three different scenarios according to the knowledge of the variances: known variance ra…
Binary jumbled string matching for highly run-length compressible texts
2012
The Binary Jumbled String Matching problem is defined as: Given a string $s$ over $\{a,b\}$ of length $n$ and a query $(x,y)$, with $x,y$ non-negative integers, decide whether $s$ has a substring $t$ with exactly $x$ $a$'s and $y$ $b$'s. Previous solutions created an index of size O(n) in a pre-processing step, which was then used to answer queries in constant time. The fastest algorithms for construction of this index have running time $O(n^2/\log n)$ [Burcsi et al., FUN 2010; Moosa and Rahman, IPL 2010], or $O(n^2/\log^2 n)$ in the word-RAM model [Moosa and Rahman, JDA 2012]. We propose an index constructed directly from the run-length encoding of $s$. The construction time of our index i…
Random Walk in a N-cube Without Hamiltonian Cycle to Chaotic Pseudorandom Number Generation: Theoretical and Practical Considerations
2017
Designing a pseudorandom number generator (PRNG) is a difficult and complex task. Many recent works have considered chaotic functions as the basis of built PRNGs: the quality of the output would indeed be an obvious consequence of some chaos properties. However, there is no direct reasoning that goes from chaotic functions to uniform distribution of the output. Moreover, embedding such kind of functions into a PRNG does not necessarily allow to get a chaotic output, which could be required for simulating some chaotic behaviors. In a previous work, some of the authors have proposed the idea of walking into a $\mathsf{N}$-cube where a balanced Hamiltonian cycle has been removed as the basis o…
Determinantal sets, singularities and application to optimal control in medical imagery
2016
International audience; Control theory has recently been involved in the field of nuclear magnetic resonance imagery. The goal is to control the magnetic field optimally in order to improve the contrast between two biological matters on the pictures. Geometric optimal control leads us here to analyze mero-morphic vector fields depending upon physical parameters , and having their singularities defined by a deter-minantal variety. The involved matrix has polynomial entries with respect to both the state variables and the parameters. Taking into account the physical constraints of the problem, one needs to classify, with respect to the parameters, the number of real singularities lying in som…
A Unified SVM Framework for Signal Estimation
2013
This paper presents a unified framework to tackle estimation problems in Digital Signal Processing (DSP) using Support Vector Machines (SVMs). The use of SVMs in estimation problems has been traditionally limited to its mere use as a black-box model. Noting such limitations in the literature, we take advantage of several properties of Mercer's kernels and functional analysis to develop a family of SVM methods for estimation in DSP. Three types of signal model equations are analyzed. First, when a specific time-signal structure is assumed to model the underlying system that generated the data, the linear signal model (so called Primal Signal Model formulation) is first stated and analyzed. T…