The class of strong stochastic Runge{Kutta (SRK) methods for stochas-tic differential equations with a Commutative noise proposed by R o ler (2010) is considered. Motivated by Komori and Burrage (2013), we design a class of explicit stochastic orthogonal Runge{Kutta Chebyshev (SROCKC2) meth-ods of strong order one for the approximation of the solution of It^o SDEs with an m-dimensional Commutative noise. The mean-square and asymptotic stability analysis of the newly proposed methods applied to a scalar linear test equation with a multiplicative noise is presented. Finally, some numer-ical experiments for stochastic models arising in applications are given that con rm the theoretical discussion.

We prove that the non-Commutative perspective of an operator convex function is the unique extension of the corresponding Commutative perspective that preserves homogeneity and convexity

Introduction: Cochlear dead zones are defined as areas where the inner hair cells have been destroyed.Thresholds on the audiograms show the integrity of those parts of the ear that are tested. Care must be taken in interpretating audiograms. Thanks to the advances in understanding of cochlear functions, it is now possible to spot false responses that come from dead zones of the cochlea. Recently, cochlear dead regions have been detected via TEN (Threshold Equalizing noise) test in which ipsilateral broadband noise and threshold shifting are used.Materials and Methods: A review of the literature on the subject of dead zones published from 1993 to 2003 was performed using Pubmed, Ebsco, Science Direct, Google Scholar Thieme ProQuest databases and library sources. key word: were "cochlear dead zone", "traveling wave", "ten (threshold equalizing noise) test", "ipsilateral noise" and "real-ear measurement for hearing aids prescription".Conclusion: Hearing aids fitting process for patients with severe and sloping sensory neural hearing loss must be noted specially by amplifying active zone and avoiding amplification for dead region i.e., offering amplification to the transition frequencies that have better hearing than others, those among the fine regions and the dead zones. Dead zone detection may help in hearing aids fitting and fine tuning.

In this paper, a new method for extending and relaxing the noise-coupling (NC) technique is proposed to enhance the noise-shaping order without adding the number of integrators. The noise-shaping order of the introduced ∑∆ modulator whit applying a second-order noise-coupling technique is enhanced and its performance with optimizing the noise transfer function (NTF) zeros is improved. Also, by removing the analog adder at feedforward path and transferring it to a new feedback branch before the last integrator and adding second-order NC path can be decreased the input voltage swing to the quantizer. Thus, by improving the modulator resolution, power consumption can be reduced. Mathematical analyses and behavioural simulation results confirm the effectiveness of the new NC method. To examine its performance, a 2nd-order single loop ΣΔ modulator was designed. The new noise-coupling method is used to achieve the three-order noise shaping to increase the resolution with low complexity and low-power. The results show an outstanding improvement in signal-to-noise and distortion ratio (SNDR) compared to the conventional structure.

The aim of this paper is to introduce the notion of Commutative pseudo BE-algebras and investigate their properties. We generalize some results proved by A. Walendziak for the case of Commutative BE-algebras. We prove that the class of Commutative pseudo BE-algebras is equivalent to the class of Commutative pseudo BCK-algebras. Based on this result, all results holding for Commutative pseudo BCK-algebras also hold for Commutative pseudo BE-algebras. For example, any finite Commutative pseudo BE-algebra is a BE-algebra, and any Commutative pseudo BE-algebra is a join-semilattice. Moreover, if a Commutative pseudo BE-algebra is a meet-semilattice, then it is a distributive lattice. We define the pointed pseudo-BE algebras, and introduce and study the relative negations on pointed pseudo BE-algebras. Based on the relative negations we construct two closure operators on a pseudo BE-algebra. We also define relative involutive pseudo BE-algebras, we investigate their properties and prove equivalent conditions for a relative involutive pseudo BE-algebra. We define the relative Glivenko property for a relative good pseudo BE-algebra and show that any relative involutive pseudo BE-algebra has the relative Glivenko property.

In this paper, it is proven that a semigroup is regular and RGC_n-Commutative if, and only if, it is a spined product of a Commutative Clifford semigroup and a right regular band

In this paper, we introduce the notion of a block commu-tativity in several groupoids, and show that the class of block Commutative groupoids and the class of d/BCK-algebras are Smarandache disjoint. The block commuta-tivity in linear/quadratic groupoids is investigated, and we prove that every group is a normal groupoid. More-over, we discuss block n-Commutative groupoids and block ranks.

By studying and using the quasi-pure part concept, we im-prove some statements and show that some assumptions in some articles are super uous. We give some characterizations of Gelfand rings. For example: we prove that R is Gelfand if and only if m (Σ,, 2A I,) Σ,= , 2A m(I, ), for each family fI, g, 2A of ideals of R, in addition if R is semiprimitive and Max(R) ,Y ,Spec(R), we show that R is a Gelfand ring if and only if Y is normal. We prove that if R is reduced ring, then R is a von Neumann regular ring if and only if Spec(R) is regular. It has been shown that if R is a Gelfand ring, then Max(R) is a quotient of Spec(R), and sometimes hM(a)'s behave like the zerosets of the space of maximal ideal. Finally, it has been proven that Z ( Max(C(X)) ) = fhM(f): f 2 C(X)g if and only if fhM(f): f 2 C(X)g is closed under countable intersection if and only if X is pseudocompact.

We consider pseudoquotient extensions of positive linear functionals on a Commutative Banach algebra A and give conditions under which the constructed space of pseudoquotients can be identified with all Radon measures on the structure space Â.

It is shown that a Commutative reduced ring R is a Baer ring if and only if it is a CS-ring; if and only if every dense subset of Spec (R) containing Max (R) is an extremally disconnected space; if and only if every non-zero ideal of R is essential in a principal ideal generated by an idempotent.