The Shannon entropy and the logical entropy of fuzzy σ-algebras are well-known instances of entropy. In this paper, we introduce and study the Tsallis entropy of order α of fuzzy σ−algebras on F −probability measure spaces, where α ∈ (0, 1)∪(1, ∞). Moreover, we study the conditional version of this entropy and examine its basic properties.

Suppose that σ is homomorphism on Banach algebra A. Then in this paper we introduce and study the new two notions σ-cyclic derivation and σ-cyclic amenability for A. We investigate the relation between trace extension property and σ-cyclic amenability; indeed we show that the σ-cyclic amenability of  AI implies that I has the trace extension property. Next, prove that the it’s converse can be true under the special conditions. One of the important result is that every σ-cyclic amenable is essential. Furthermore, for every closed two-sided ideal I of A, the relation between of σ-cyclic amenability of A and σ-cyclic amenability of  AI has been studied. Also, we show that the σ-cyclic amenability of  A and AI is equivalent. Finally, we study this notion on θ-Lau algebras and we investigate its relation with the similar concept on algebras A and B.

When a suitable partial ordered relation is attached to a Γsemihyperring, it results into an ordered Γ-semihyperring. Concepts of an ordered Γ-semihyperring, Γ-band, idempotent Γ-semihyperring, totally ordered Γ-semihyperring, positively ordered Γ-semihyperring, negatively ordered Γ-semihyperring are introduced which are useful to study derivation on ordered Γ-semihyperrings. Derivation is nothing but an additive mapping fulfilling the Leibniz rule. In this paper, we introduce the concept of (σ, τ)-derivation which is a generalization of σ-derivation and derivation on Γ-semihyperring and study some properties of (σ, τ)-derivation on an ordered Γ-semihyperring. Some results reflecting different natures of (σ, τ)-derivation depending on natures of the endomorphisms are encountered.

In this paper, we study some geometric properties of Finsler Σ−spaces with square metric. We prove that Finsler Σ−spaces with square (α, β)−metrics are Riemannian.

In this paper we study the geometric properties of Finsler Σ-spaces . we prove that Infinite series Σ-spaces are Riemannian.

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.

In this paper, we discuss matrix theoretic characterization for weighted conditional operators by properties of conditional expectation operator in some operator classes on $L^{2}(\Sigma)$-semi-Hilbertian space such as self-adjoint, isometry and normal classes of these type operators on this space. Also, we consider the matrix representation of the Moore-Penrose inverse for these types of operators. We also gave examples to show our results.

In this paper,  a new version of a higher-order strongly convex function is introduced which is named    $(\omega-\sigma)$-higher-order strongly convex function. Sufficient conditions for the existence of minimum for  $(\omega-\sigma)$-higher order strongly convex function is provided. The vector version of $(\omega-\sigma)$-higher order strongly convex function is given and by using  KKM theory an existence results for a solution of it is proved. Moreover, the compactness of the solution set of the vector version of $(\omega-\sigma)$-higher order strongly convex function is investigated. The results of this article improve and extend the corresponding results presented in this area.

In this paper, by considering the σ as a BL-homomorphism,we introduce the notion of σ-ideals in BL-algebras and obtain some new results about them. Also, while introducing the primary ideals in BL-algebras, we get some relationship between the primary ideals and obstinate ideals. Finally, by introducing σ-prime, σ-primaryand σ-invariant ideals, we derived some new results related to them.