We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on unit disk. This is equivalent to solving following P.D.E. $$\begin{aligned} {\left\{ \begin{array}{ll}-\Delta u=2K(z)e^u&{}\hbox {in}\;\mathbb {D}^2,\\ \partial _\nu u+2=2h(z)e^\frac{u}{2}&{}\hbox {on}\;\partial \mathbb {D}^2,\end{array}\right. } \end{aligned}$$ where K, h are prescribed curvatures...

A well-known application of the Ramsey Theorem is proof Brunel–Sucheston Theorem. Based on this application, as an intermediate step, we consider concept $$(k,\varepsilon )$$ -oscillation stable sequence, which generalized with notion $$((\mathcal {B}_i)_{i=1}^k,\varepsilon -block oscillation sequence where $$(\mathcal {B}_i)_{i=1}^k$$ a finite barriers $$\mathbb {N}$$ . We prove that equivalen...

The aim of this paper is to extend the truth value table oflattice-valued convergence spaces to a more general case andthen to use it to introduce and study the quantale-valued fuzzy Scotttopology in fuzzy domain theory. Let $(L,*,varepsilon)$ be acommutative unital quantale and let $otimes$ be a binary operationon $L$ which is distributive over nonempty subsets. The quadruple$(L,*,otimes,varep...

The clique removal lemma says that for every r ≥ 3 $r\ge 3$ and ε > 0 $\varepsilon \gt 0$ , there exists some δ $\delta so n $n$ -vertex graph G $G$ with fewer than {n}^{r}$ copies of K ${K}_{r}$ can be made -free by removing at most 2 {n}^{2}$ edges. dependence $ on in this result is notoriously difficult to determine: it known − 1 ${\delta }^{-1}$ must least super-polynomial ${\varepsilon tow...

We study the entropy rate of a hidden Markov process, defined by observing the output of a symmetric channel whose input is a first order Markov process. Although this definition is very simple, obtaining the exact amount of entropy rate in calculation is an open problem. We introduce some probability matrices based on Markov chain's and channel's parameters. Then, we try to obtain an estimate ...

Abstract We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories second order arithmetic. The main result is that any arithmetic theory $T_0$ extending $\mathsf {RCA}_0$ and axiomatizable by a $\Pi ^1_{k+2}$ sentence, $n\geq k+1$ , $$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}}(T) \ = T_0 + \mathrm{TI}_{\varPi^1_n}(\varepsilon_0), \end{align*}$$ \mat...

[PDF] [Full Text] [Abstract] , August 1, 2004; 287 (2): F289-F298. Am J Physiol Renal Physiol S. Redling, I. L. Pfaff, M. Leitges and V. Vallon {varepsilon} in mouse kidney Immunolocalization of protein kinase C isoenzymes {alpha}, {beta}I, {beta}II, {delta}, and [PDF] [Full Text] [Abstract] , April 1, 2005; 313 (1): 403-409. J. Pharmacol. Exp. Ther. T. Rieg, H. Steigele, J. Schnermann, K. ...

Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability hyperfiniteness coincide. In this paper we define uniform version graphed bounded vertex degrees prove these two coincide as well. Roughly speaking, a measured graph $\mathcal {G}$ is uniformly hyperfinite if any ${\varepsilon }>0$ there exists $K\geq 1$ such not only {G...