Bipolar neutrosophic set theory and rough neutrosophic set theory are emerging as powerful tool for dealing with uncertainty, and indeterminate, incomlete, and inprecise information. In the present study we develop a hybrid structure called “rough bipoar neutrsophic set”. In the study, we define rough bipoar neutrsophic set and define union, complement, intersection and containment of rough bip...

Appendices A. Derivation of the CTC-EM Equations In the maximization step, the objective is to obtain the derivative of Q τ (w|x, z, w (n)) with respect to the input of the softmax layer, a t k , at time t. We first differentiate Q τ with respect to y t k at w = w (n) : ∂Q τ (w|x, z, w (n)) ∂y t k w=w (n) = |z| m=0 p(z 1:m |Z, x 1:τ , w (n)) ∂ ln p(z 1:m |x 1:τ , w (n)) ∂y t k. (1) With Bayes' ...

We prove: Let P(z) = P n k=0 a k z k be a complex polynomial with n 1 and a 0 an 6 = 0. If z is a zero of P, then we have for all real numbers t > 0: (*) jzj > ja 0 jt ja 0 j + Kn(t) with Kn(t) = 1 1 ? n(t) n min 1mn h (n(t) m ? n(t) n) max mpn Ap(t) + (1 ? n(t) m) max 1pn Ap(t) i ; n(t) = ja 0 j ja 0 j + max 1pn Ap(t) ; Ap(t) = 1 p p X k=1 ja k jt k : Inequality (*) sharpens a result of E. Lan...

We compute the leading-twist T-odd quasidistributions of proton in a spectator model with scalar and axial-vector diquarks: quasi-Sivers function ${\stackrel{\texttildelow{}}{f}}_{1T}^{\ensuremath{\perp}}(x,{\mathbit{k}}_{T}^{2};{P}_{z})$ quasi-Boer-Mulders ${\stackrel{\texttildelow{}}{h}}_{1}^{\ensuremath{\perp}}(x,{\mathbit{k}}_{T}^{2};{P}_{z})$. obtain quark-quark correlators four-dimensiona...

form, Problem P . Find fu; ; g such that 0 +K1 +K2 + C1v + S(v; ; ) = Q in V 0; Bv + Au + C2 + 4j(v; ; ; v) 3 f in E 0: Here f 2 E 0 and Q 2 V 0 are given by hf; wi = Z T 0 Z f(t)wi(t) dxdt + Z T 0 Z ij (t)wi;j(t) dxdt + Z T 0 Z N fN(t)wi(t)d dt; hQ; i = Z T 0 hq(t); (t)i dt Z T 0 Z 0(t) (t) dxdt Z T 0 Z C hR( (t) R(t)) (t)d dt Z T 0 Z kij ;i(t) ;j(t) dxdt; and 4j(v; ; ; w) denotes the partial ...

Introduction 0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Z p) of its Z p-rational points. For every n in N, there is a natural map π n : X(Z p) → X(Z/p n+1) assigning to a Z p-rational point its class modulo p n+1. The image Y n,p of X(Z p) by π n is exactly the set of Z/p n+1-rational points which can be lifted to Z p-r...

Abstract We study the homogeneous Dirichlet problem for parabolic equations u t ? div ( class="MJX-tex-caligraphic" mathvariant="script">A z , ? ?</m:m...