Given an elliptic curve E over a finite field Fq of q elements, we say that an odd prime l ∤ q is an Elkies prime for E if tE − 4q is a quadratic residue modulo l, where tE = q+1−#E(Fq) and #E(Fq) is the number of Fq-rational points on E. These primes are used in the presently most efficient algorithm to compute #E(Fq). In particular, the bound Lq(E) such that the product of all Elkies primes f...

The aim of this article is to prove the following theorem. Theorem: Let p be in (1,∞), Hn,m a group of Heisenberg type, R the vector of the Riesz transforms on Hn,m. There exists a constant Cp independent of n and m such that for every f ∈ L (Hn,m) C p e ‖f‖Lp(Hn,m) ≤ ‖|Rf |‖Lp(Hn,m) ≤ Cpe‖f‖Lp(Hn,m). It has been proved by F. Lust-Piquard that the operator norm of R does not depend on n (see [L...

In this paper new div-curl results are derived. For any open set Ω of RN , N 2, we study the limit of the product vn · wn where the sequences vn and wn are respectively bounded in Lp(Ω)N and Lq(Ω)N , while divvn and curlwn are compact in some Sobolev spaces, under the condition 1 1 p + 1 q 1 + 1 N . Our approach is based on a suitable decomposition of the functions vn and wn, combined with the ...

In this paper, a vector version of the intermediate value theorem is established. The main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{On fixed point theorems for monotone increasing vector valued mappings via scalarizing}, Positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

boujari [5] proved a fixed point theorem with an old version of the integraltype contraction , his proof is incorrect. in this paper, a new generalizationof integral type contraction is introduced. moreover, a fixed point theorem isobtained.

In this work, we prove the well-posedness and propagation of chaos for a stochastic particle system in mean-field interaction under assumption that interacting kernel belongs to suitable Ltq−Lxp space. Contrary large deviation principle approach recently proposed [2], main ingredient proof here are Partial Girsanov transformations introduced [3] developed general setting work.

For d≥3, we prove that time-inhomogeneous stochastic differential equations driven by additive noises with drifts in critical Lebesgue space Lq([0,T];Lp(Rd)), where (p,q)∈(d,∞]×[2,∞) and d∕p+2∕q=1, or (p,q)=(d,∞) divb∈L∞([0,T];Ld∕2+ε(Rd)), are well-posed. The weak uniqueness is obtained solving corresponding Kolmogorov backward some second-order Sobolev spaces, which analytically interesting it...