Let _(x, !)r |!| p&2 ! be a p-Laplacian type operator and consider the Hodge decomposition _(x, Du)=D.+H, div H=0. A standard elliptic theory asserts that &D.&q (p&1) C &Du& p&1 q for all q>p&1. There has been considerable recent interest in the validity of the reverse estimate &Du& p&1 q C &D.&q (p&1) for q>p&1 in the regularity study of certain geometrical mappings. In this paper, we give a r...

It is suggested that the (p, q)-hypergeometric series studied by Burban and Klimyk (in Integral Transforms and Special Functions 2 (1994) 15-36) can be considered as a special case of a more general (P, Q)-hypergeometric series. 1. Introduction I propose a general (P, Q)-hypergeometric series. In this, I am inspired mainly by the paper of Burban and Klimyk [1] titled " P, Q-differentiation, P, ...

In this paper, we study the multiplicity and concentration of positive solutions for following (p, q)-Laplacian problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u _{q} +V(\varepsilon x) \left( |u|^{p-2}u + |u|^{q-2}u\right) = f(u) &{} \text{ in } {\mathbb {R}}^{N}, \\ u\in W^{1, p}({\mathbb {R}}^{N})\cap q}({\mathbb {R}}^{N}), \quad u>0 \end{array} \right. \end{aligned}$$ where...

Using the renormalization group approach, the Coulomb gas and the coset techniques, the effect of slightly relevant perturbations is studied for the second parafermionic field theory with the symmetry Z5. New fixed points are found and classified. ∗ Unité Mixte de Recherche UMR 7589 While the first series of parafermionic conformal field theories [1] is well studied and applied in various domai...

Choices between gambles show systematic violations of stochastic dominance. For example, most people choose ($6, .05; $91, .03; $99, .92) over ($6, .02; $8, .03; $99, .95), violating dominance. Choices also violate two cumulative independence conditions: (1) If S 5 (z, r; x, p; y, q) s R 5 (z, r; x8, p; y', q) then S9 5 (x8, r; y, p 1 q) s R9 5 (x8, r 1 p; y8, q). (2) If S8 5 (x, p; y, q; z8, r...

Let E be an elliptic curve over Q and p be a prime of good supersingular reduction for E. Although the Iwasawa theory of E over the cyclotomic Zp-extension of Q is well known to be fundamentally different from the case of good ordinary reduction at p, we are able to combine the method of our earlier paper with the theory of Kobayashi [5] and Pollack [8], to give an explicit upper bound for the ...