Bill Lawvere’s 1973 milestone paper “Metric spaces, generalized logic, and closed categories” helped us to detect categorical structures in previously unexpected surroundings. His revolutionary idea was not only to regard individual metric spaces as categories (enriched over the monoidal-closed category given by the non-negative extended real half-line, with arrows provided by ≥ and tensor by +...

The most basic concept is that of an infinite sequence (of real or complex numbers in these notes). For p ∈ Z, let Np = {k ∈ Z : k ≥ p}. An infinite sequence of (complex) numbers is a function a : Np → C. Usually, for n ∈ Np we write a(n) = an, and denote the sequence by a = {an}n=p. The sequence {an}n=p is said to converge to the limit A ∈ C provided that for each > 0 there is an N ∈ Z such th...

‎let $g$ be a locally compact abelian group‎. ‎the concept of a generalized multiresolution structure (gms) in $l^2(g)$ is discussed which is a generalization of gms in $l^2(mathbb{r})$‎. ‎basically a gms in $l^2(g)$ consists of an increasing sequence of closed subspaces of $l^2(g)$ and a pseudoframe of translation type at each level‎. ‎also‎, ‎the construction of affine frames for $l^2(g)$ bas...

Proof. To see L2-convergence, we will prove that (Sn)n>1 is Cauchy in L 2. It is enough to show that ||Sn − Sm||2 < ε for all n,m > N(ε). Suppose n > m, then we obtain ||Sn − Sm||2 = E(Sn − Sm) = E(Xm+1 +Xm+2 + · · ·+Xn) = Var(Xm+1 +Xm+2 + · · ·+Xn) = Var(Xm+1) + Var(Xm+2) + · · ·+ Var(Xn). For any ε > 0, there exists N = N(ε) with ∑∞ i=N Var(Xi) < ε 2, thus we have ||Sn−Sm||2 < ε for all n,m >...

We study the Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion and with monotonically increasing initial data using the Riemann-Hilbert (RH) approach. The solution of the Cauchy problem, in the zero dispersion limit, is obtained using the steepest descent method for oscillatory Riemann-Hilbert problems. The asymptotic solution is completely described by a scalar func...

In this paper, we establish a new version of Ekeland’s variational principle in a new setting of cone pseudo-quasimetric spaces. In constrast to metric spaces, we do not require that each forward Cauchy sequence is forward convergent and that each forward convergent sequence has the unique forward limit. The motivation of this paper comes from applications in Behavioral Sciences since a cone ps...

1. The Set Q 1 2. Addition and multiplication of rational numbers 4 2.1. Definitions and properties. 4 2.2. Comments 7 2.3. Connections with Z. 9 2.4. Better notation. 10 2.5. Solving the equations Ea,b and Ma,b. 12 3. Ordering the rational numbers 13 4. Sequences and limits in Q 18 5. Non-convergent Cauchy sequences of rationals 20 5.1. An irrational sequence of rationals 20 5.2. An irrational...

A family of generalized Cauchy distributions, T-Cauchy{Y} family, is proposed using the T-R{Y} framework. The family of distributions is generated using the quantile functions of uniform, exponential, log-logistic, logistic, extreme value, and Fréchet distributions. Several general properties of the T-Cauchy{Y} family are studied in detail including moments, mean deviations and Shannon’s entrop...

Observations of the multi-TeV spectra of the nearby BL objects Mkn 421 and Mkn 501 exhibit the high energy cutoffs predicted to be the result of intergalactic annihilation interactions, primarily with infrared photons having a flux level as determined by various astronomical observations. After correction for this absorption effect, the derived intrinsic spectra of these multi-TeV sources can b...

We derive an algorithm of optimal complexity which determines whether a given matrix is a Cauchy matrix, and which exactly recovers the Cauchy points defining a Cauchy matrix from the matrix entries. Moreover, we study how to approximate a given matrix by a Cauchy matrix with a particular focus on the recovery of Cauchy points from noisy data. We derive an approximation algorithm of optimal com...