All posts under tag "Mathematics"
Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane induced by the action of $PSL(2,mathbb{Z})$. Chapter 3 gives bac
In this paper we go into the study of 2-limits and 2-colimits in the 2-category CAT the category of small categories. More precisely we show the commutation of filtered 2-colimits and finite 2-limits. It is a generalization of a classical result in category theory.
We give a specific method to solve with quadratic complexity the linear systems arising in known algorithms to deal with the sign determination problem. In particular, this enable us to improve the complexity bound for sign determination in the univariate case.
We find a necessary and sufficient condition for a compact 7-manifold to admit a $tilde G_2$-structure. As a result we find a sufficient condition for an open 7-manifold to admit a closed 3-form of $tilde G_2$-type.
Consider a sequence of Markov processes $X^1, X^2,...$ with state space $E$, where $X^N$ has a strong drift to $D subseteq E$, such that $Φ(X^N)$ is slow for some appropriate $Φ: Eto D$. Using the method of martingale problems, we give a limit result, such that $Φ(X^N) xRightarrow{Ntoinfty} Z$ in th
Here, we present a family of time series with a simple growth constraint. This family can be the basis of a model to apply to emerging computation in business and micro-economy where global functions can be expressed from local rules. We explicit a double statistics on these series which allows to e
The paper presents new clustering algorithm. The proposed algorithm gives less distortion as compared to well known Linde Buzo Gray (LBG) algorithm and Kekre's Proportionate Error (KPE) Algorithm. Constant error is added every time to split the clusters in LBG, resulting in formation of cluster in o
We state a general formula for the number of binomial coefficients $n$ choose $k$ that are divided by a fixed power of a prime $p$, i.e., the number of binomial coefficients divided by $p^j$ and not divided by $p^{j+1}$.
The close relation between properness and closedness of maps is well-known. For instance, for Fredholm mappings of infinite dimensional Banach manifolds, these properties are equivalent. On the other hand, properness of maps plays an important role for construction of various topological degree theo
We propose a modified condition of consistency on cubic lattices for some special classes of two-dimensional discrete equations and prove that the discrete nonlinear equations defined by determinants of matrices of orders N > 2 are consistent on cubic lattices in this sense.
Using flag algebras, we prove that the minimum density of $8$-cliques in a large graph without an independent set of size $3$ is $491411/268435456+o(1)$, thus resolving a new case of an old problem of Erdős [Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962) 459-464]. Also, we establish some other re
We consider the relations between well-known Fourier transform algorithms.
Assuming $mathfrak p=mathfrak c$, we show that for every Eberlein compact space $L$ of weight $mathfrak c$ there exists a short exact sequence $0to c_0to Xto C(L)to 0$, where the Banach space $X$ is not isomorphic to a $C(K)$-space.
We prove that the character of an irreducible cuspidal representation of $GL_n(mathbb{F}_{ell}((t)))$ is locally bounded up to a logarithmic factor by the orbital integral of a matrix coefficient of this representation. The characteristic $0$ analog of this result is part of the proof of the celeb
In this paper we consider tandem error control coding and cryptography in the setting of the {em wiretap channel} due to Wyner. In a typical communications system a cryptographic application is run at a layer above the physical layer and assumes the channel is error free. However, in any real applic
Over an arbitrary commutative ring $R$, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space $mathbf{Q}(X)$ of quantum cellular automata (QCA) on a given metric space $X$. In most cases of interest, $π_0 mathbf{Q}(X)$ classifies QCA up to quantum circ
In this paper, we build a procedure that allows to establish regularity and controls in time for probabilistic solutions to PDEs. Probabilistic approaches to global wellposedness problems usually provide ensemble bounds on the solutions. These bounds are the main tools to ensure convergence procedur
The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if $Asubset mathbb{Z}$ is a finite set of integers with $lvert Arvert=N$ then $| widehat{1_A}|_1geq clog N$ for some absolute constant $c > 0$. We explore what structure $A$ must have if $| widehat{1_A}|_
In this paper, we describe the solution for a problem dealing with definite properties of binary sequences. This problem, proposed by Xavier Grandsart in the form of a mathematical contest, has been solved also by Maher Younan, Ph.D. student of Theoretical Physics at the University of Geneva, and Pi
We solve some forms of non homogeneous differential equations in one and two dimensions. By expanding the solution into whell-posed closed form-Eisenstein series the solution itself is quite simple and elementary. Also we consider Fourier series solutions of linear differential operator equations. I
We investigate the stochastic homogenization of a class of turbulent diffusions generated by non-local symmetric Lévy operators with divergence-free drift fields in ergodic random environments, where neither the drift fields nor their associated stream functions are assumed to be bounded. A pivotal
In this article we present evaluations of continued fractions studied by Ramanujan. More precisely we give the complete polynomial equations of Rogers-Ramanujan and other continued fractions, using tools from the elementary theory of the Elliptic functions. We see that all these fractions are roots
Comparison between the various impact factors of a few Russian journals demonstrates the deficiencies of the popular citation indices.
In this article we give, for the fist time the solution of the general difference equation of 2-degree. We also give as application the expansion of a continued fraction into series, which was first proved, found in the past by the author.
In this paper we construct a cofibrantly generated model category structure on the category of all small symmetric multicategories enriched in simplicial sets.
We solve some decision problems for timed automata which were recently raised by S. Tripakis in [ Folk Theorems on the Determinization and Minimization of Timed Automata, in the Proceedings of the International Workshop FORMATS'2003, LNCS, Volume 2791, p. 182-188, 2004 ] and by E. Asarin in [ Challe
In a previous paper, we derived a recursive formula determining the weight distributions of the [n=(q^m-1)/(q-1)] Hamming code H(m,q), when (m,q-1)=1. Here q is a prime power. We note here that the formula actually holds for any positive integer m and any prime power q, without the restriction (m, q
We describe a sufficient condition for the process of left Kan extension to be a conservative functor. This is useful in the study of graphic Fourier transforms and quantum categories and groupoids.
The presence of infinitesimals is traced back to some of the most general algebraic structures, namely, semigroups, and in fact, magmas, [1], in which none of the structures of linear order, field, or the Archimedean property need to be present. Such a clarification of the basic structures from wher
To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same length. We say that an infinite word w over A is 'fine' if there
We characterize simplicial localization functors among relative functors from relative categories to simplicial categories as any choice of homotopy inverse to the delocalization functor of Dwyer and the second author.
The Jacobi identities play an important role in constructing the explicit exact solutions of a broad class of integrable systems in soliton theory. In the paper, a direct and simple proof of the Jacobi identities for determinants is presented by employing the Pl$ddot{u}$cker relations.
We give a (consistent) example of a first-countable continuum that is not a remainder of the real line.
The necessary and sufficient condition of the channel capacity is rigorously formulated for the N-user discrete memoryless multiple-access channel (MAC). The essence of the formulation is to invoke an {em elementary} MAC where sizes of input alphabets are not greater than the size of output alphabet
As we known, the {it Seifert-Van Kampen theorem} handles fundamental groups of those topological spaces $X=Ucup V$ for open subsets $U, Vsubset X$ such that $Ucap V$ is arcwise connected. In this paper, this theorem is generalized to such a case of maybe not arcwise-connected, i.e., there are $C_1$,
We outline an alternative approach to the geometric notion of a saddle point for real-valued functions of two variables. It is argued that this is more natural compared to the usual treatment of this topic in standard texts on Calculus.
Adaptive systems must strike a balance between prediction and surprise to thrive in uncertain environments. We propose an information-theoretic balance function, $ f(p) = -(1 - p)ln(1 - p) + ln p $, which quantifies the net informational gain from contrasting explained variance $p$ with unexplained
Starting from an abelian rigid braided monoidal category C we define an abelian rigid monoidal category C_F which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT co
It is well known that sin(api/b), cos(api/b), etc., are only rational numbers for a few select integers a and b. We show that this is equivalent to the fact that only for d = 1,2,3,4, and 6 is the primitive dth root of unity of degree 2 over Q.
Suppose that the Riemann hypothesis for the Riemann $zeta$-function is false. There have yet been no workable results that the falsity of the Riemann hypothesis implies. The result of this article may be considered as more workable for the sake of deducing other new results on the Riemann hypothesis
This note studies an issue relating to essential smoothness that can arise when the theory of large deviations is applied to a certain option pricing formula in the Heston model. The note identifies a gap, based on this issue, in the proof of Corollary 2.4 in cite{FordeJacquier10} and describes how
Proving the quantum Hamming bound for degenerate nonbinary stabilizer codes has been an open problem for a decade. In this note, I prove this bound for double error-correcting degenerate stabilizer codes. Also, I compute the maximum length of single and double error-correcting MDS stabilizer codes o
This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; fo
We introduce the notion of an ACP process algebra and the notion of a meadow enriched ACP process algebra. The former notion originates from the models of the axiom system ACP. The latter notion is a simple generalization of the former notion to processes in which data are involved, the mathematical
In this paper, we discuss the Cram'er-Lundberg model with investments, where the price of the invested risk asset follows a geometric Brownian motion with drift $a$ and volatility $sigma> 0.$ By assuming there is a cap on the claim sizes, we prove that the probability of ruin has at least an algebra
A set S is independent if no two vertices from S are adjacent. In this paper we prove that if F is a collection of maximum independent sets of a graph, then there is a matching from S-{intersection of all members of F} into {union of all members of F}-S, for every independent set S. Based on this fi
A simple algorithm for decoding both errors and erasures of Reed-Solomon codes is described.
The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots $z_i$ of a complex polynomial $P$ lie on the unit circle $U$, the points of $U$ different from the $z_i$, and such that $Arg(P(z))=theta$, form a regular $n$-gon, where $n$ is the deg
In this paper, we propose a general approach for improving the efficiency of computing distribution functions. The idea is to truncate the domain of summation or integration.
We extend Stein's maximal theorem to the bilinear setting. Let $M$ be a homogeneous space with a transitive action of a compact abelian group, and let $1 le p,q le 2$ and $1/2 le r le 1$ satisfy $1/p + 1/q = 1/r$. For a family of translation-invariant bilinear operators [ T_m : L^p(M) times L^q(M) t
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