sup t>0 ‖η( · )m(t · )‖W 2,β(Rd) ≤ Cη, where ‖ · ‖W 2,β(Rd) is the standard Sobolev norm on R, then the multiplier operator f 7→ F−1(mFf), initially defined on L(R) ∩ L(R), is bounded on L(R) for 1 < p < ∞, and is of weak-type (1,1). Here F denotes the Fourier transform. Let (Ω, d(x, y)) be a metric space equipped with a positive measure μ. We assume that (Ω, d, μ) is a space of homogeneous typ...

We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type [ u_{tt}-phi (x)||nabla u(t)||^{2}Delta u+delta u_{t}=|u|^{a}u,, x in mathbb{R}^{N} ,,tgeq 0;,]with initial conditions $u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N geq 3, ; delta geq 0$ and $(phi (x))^{-1} =g (x)$  is a positive function lying in $L^{N/2}(mathb...

let $l$ be a completely regular frame and $mathcal{r}l$ be the ringof continuous real-valued functions on $l$. we study the frame$mathfrak{o}(min(mathcal{r}l))$ of minimal prime ideals of$mathcal{r}l$ in relation to $beta l$. for $iinbeta l$, denoteby $textit{textbf{o}}^i$ the ideal${alphainmathcal{r}lmidcozalphain i}$ of $mathcal{r}l$. weshow that sending $i$ to the set of minimal prime ideals...

Let p be any prime and a be a positive integer. For l, n ∈ {0, 1, . . . } and r ∈ Z, the normalized cyclotomic ψ-coefficient {n r } l,pa := p − ⌊ n−pa−1−lpa pa−1(p−1) ⌋ ∑ k≡r (mod pa) (−1) (n k )( k−r pa l ) is known to be an integer. In this paper, we show that this coefficient behaves like binomial coefficients and satisfies some Lucas-type congruences. This implies that a congruence of Wan i...

i n m a n u f a c t u r i n g a c t i v i t i e s, i s s u e s s u c h a s t h e i n t e n s i t y o f c o m p e t i t i o n, r i s i n g a n d c h a n g i n g c u s t o m e r e x p e c t a t i o n s, a n d e v e r-i n c r e a s i n g t e c h n o l o g i c a l d e v e l o p m e n t s, i n t e n s i f y m a n u f a c t u r e r c o m m i t m e n t t o e l i m i n a t e d e f e c t s i n i t s p r...

Mammalian vestibular organs have two types of hair cell, type I and type II, which differ morphologically and electrophysiologically. Type I hair cells alone express an outwardly rectifying current, I(K, L), which activates at relatively negative voltages. We used whole cell and patch configurations to study I(K,L) in hair cells isolated from the sensory epithelia of rat semicircular canals. I(...

Whole cell recordings from acutely dissociated neocortical pyramidal neurons and striatal medium spiny neurons exhibited a calcium-channel current resistant to known blockers of L-, N-, and P/Q-type Ca(2+) channels. These R-type currents were characterized as high-voltage-activated (HVA) by their rapid deactivation kinetics, half-activation and half-inactivation voltages, and sensitivity to dep...

We provide a probabilistic analysis of the upwind scheme for d-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We...