We prove that for a so-called sticky process S there exists an equivalent probability Q and a Q-martingale S̃ that is arbitrarily close to S in L(Q) norm. For continuous S, S̃ can be chosen arbitrarily close to S in supremum norm. In the case where S is a local martingale we may choose Q arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate t...

In this work‎, ‎an iterative method based on a matrix form of LSQR algorithm is constructed for solving the linear operator equation $mathcal{A}(X)=B$‎ ‎and the minimum Frobenius norm residual problem $||mathcal{A}(X)-B||_F$‎ ‎where $Xin mathcal{S}:={Xin textsf{R}^{ntimes n}~|~X=mathcal{G}(X)}$‎, ‎$mathcal{F}$ is the linear operator from $textsf{R}^{ntimes n}$ onto $textsf{R}^{rtimes s}$‎, ‎$ma...

The d-dimensional Slepian Gaussian random field {S(t), t ∈ R+} is a mean zero Gaussian process with covariance function ES(s)S(t) = ∏d i=1 max(0, ai − |si − ti|) for ai > 0 and t = (t1, · · · , td) ∈ R+. Small ball probabilities for S(t) are obtained under the L2-norm on [0, 1]d, and under the sup-norm on [0, 1]2 which implies Talagrand’s result for the Brownian sheet. The method of proof for t...

exists for each x, y ∈ S(E). In this case E is called smooth. The norm of E is said to be Fréchet differentiable if for each x ∈ S(E) the limit is attained uniformly for y ∈ S(E). The normofE is called uniformlyFréchet differentiable if the limit is attained uniformly for x, y ∈ S(E). It is well known that (uniform) Fréchet differentiability of the norm of E implies (uniform) Gâteaux differenti...

In this paper, we utilized triangular conorms (S-norm). The essence of using S-norm is that the similarity order does not change different norms. fact, are investigating for a new conception calculating two Fermatean fuzzy sets. For purpose, utilizing an S-norm, first present formula values, so they truthful in properties. Following that, generalize sets which prove conditions. Finally, various...

Active Learning Method (ALM) is a soft computing method used for modeling and control based on fuzzy logic. All operators defined for fuzzy sets must serve as either fuzzy S-norm or fuzzy T-norm. Despite being a powerful modeling method, ALM does not possess operators which serve as S-norms and T-norms which deprive it of a profound analytical expression/form. This paper introduces two new oper...

We study the optimal approximation of the solution of an operator equation A(u) = f by linear and different types of nonlinear mappings. In our earlier papers we only considered the error with respect to a certain Hs-norm where s was given by the operator since we assumed that A : Hs 0(Ω) → H−s(Ω) is an isomorphism. The most typical case here is s = 1. It is well known that for certain regular ...