In this note we give three short results concerning the elliptic Harnack inequality (EHI), in the context of random walks on graphs. The first is that the EHI implies polynomial growth of the number of points in balls, and the second that the EHI is equivalent to an annulus type Harnack inequality for Green’s functions. The third result uses the lamplighter group to give a counterexample concer...

objectives major differences exist in the anatomy and biomechanics of the growing spine that causes failure patterns different from those in adults. spinal injury in the pediatric patient is a main concern because timely diagnosis and appropriate treatment can prevent further neurologic damage and deformity and potentiate recovery. we conducted a retrospective clinical study of 137 cases (93 bo...

this study attempts to examine the multilingualism in algeria, in other words, the phenomenon of linguistic diversity which characterizes the algerian reality and its relationship. for that matter the article visualizes the linguistic level of learners especially children, highlighting the characteristics of the multiplicity of its causes and manifestations. here, the descriptive methodology is...

some children of borazjan area have minute white flecks, yellow or brown spot areas scattered irregularly over the tooth surface, the causative factor was attributed to excess of fluoride in water. to verify this matter short chemical examination of water samples from endemic area was carried out. the results of water analysis by colorimetric method, using zirconium a1izarin reagent after disti...

The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...

and Applied Analysis 3 The following Hardy-Sobolev inequality is due to Caffarelli et al. 12 , which is called Caffarelli-Kohn-Nirenberg inequality. There exist constants S1, S2 > 0 such that (∫ RN |x|−bp |u|pdx )p/p∗ ≤ S1 ∫ RN |x|−ap|∇u|pdx, ∀u ∈ C∞ 0 ( R N ) , 1.8 ∫ RN |x|− a 1 |u|dx ≤ S2 ∫ RN |x|−ap|∇u|pdx, ∀u ∈ C∞ 0 ( R N ) , 1.9 where p∗ Np/ N − pd is called the Sobolev critical exponent. ...

Let A and B be bounded linear operators on a Hilbert space satisfying A ≥ B ≥ 0. The well-known Furuta inequality is given as follows: Let r ≥ 0 and p > 0; then A r 2 Amin{1,p}A r 2 ≥ (A r 2 BpA r 2 ) min{1,p}+r p+r . In order to give a self-contained proof of it, Furuta (1989) proved that if 1 ≥ r ≥ 0, p > p0 > 0 and 2p0 + r ≥ p > p0, then (A r 2 Bp0A r 2 ) p+r p0+r ≥ (A r 2 BpA r 2 ) p+r p+r ...

This study addresses whether/how monolingual and bilingual Spanish-speaking children differ in their acquisition of grammar by examining direct object clitics in contexts where either proclisis or enclisis is possible (Lo voy a ver vs. Voy a verlo). The current study examines variable clitic placement in sociolinguistic interviews conducted with 21 Spanish-English bilingual children of Mexican ...

Despite the great success of quantum mechanics, questions regarding its application still exist and the boundary between quantum and classical mechanics remains unclear. Based on the philosophical assumptions of macrorealism and noninvasive measurability, Leggett and Garg devised a series of inequalities (LG inequalities) involving a single system with a set of measurements at different times. ...