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In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

When u is a von Neumann-Morgenstern utility function on X ⊗ Y. Y is ‘utility independent’ of X if u can be written as u(x, y)=f(x)g(y) + a(x) with f positive. This paper introduces a fundamental extension of utility independence that is base on induced indifference relations over gambles on one factor when the level of the other factor is fixed. It is proved that Y is ‘degree-n utility independent’ of X if and only if u can be written as u(x, y)= f1(x)g1(y)+ … + fn(x)gn(y)+ a(x) and cannot be written in a similar way with fewer than n products of single-factor functions. A similar theorem holds when the roles of Y and X are interchanged. It follows that if Y is degree-n utility independent of X, then X is degree-m utility independent of Y for some m ∈ ¦ n - l . n . n + l ¦; it is then shown that u can be represented in terms of n conditional utility functions on Y . m conditional utility functions on X, and at most (n + l)(m + 1) scaling constants. [ABSTRACT FROM AUTHOR]

In this article, we investigate the chaotic behaviours in the fractional-order Lorenz system. By utilizing the fractional calculus techniques, we found that chaos exists in the fractional-order Lorenz system of order less than 3. The lowest order we found to have chaos in this system is 2.97. [ABSTRACT FROM AUTHOR]

A finite set of generators for a free product of two groups of type F3 with a subgroup amalgamated, and an estimation for the upper bound of the second order Dehn functions of the amalgamated free product are carried out. [ABSTRACT FROM AUTHOR]

The article reports on the sharpening of Wielandt's characterization of the gamma function. It presents the two conditions that is satisfied with Euler's gamma function. According to the author, the two conditions by themselves do not characterize the gamma function and presents characterization of the gamma function through theorems that Wielandt discovered. He adds that a characterization in the real domain, found in 1922 by other scholars, is well known, while Wielandt's theorem is function-theoretic.

The article presents a discussion about a summation formula that is used in certain transformations on generating algebraic and differential functions and the extraction and creation of coefficients. The goal of the article is to offer the method of coefficients. Mathematical functions are manipulated in order to arrive at a single and simple mathematical expression. It mentions that generating functions have emerged as one of the most popular approaches to combanitorial problems, which includes the analysis of algorithms.

Let R be a prime ring with characteristic different from 2, let U be a nonzero Lie ideal of R, and let f be a generalized derivation associated with d. We prove the following results: (i) If a ∊ R and [a, f(U)] = 0 then a ∊ Z or d(a) = 0 or U ⊂ Z; (ii) If f²(U) = 0 then U ⊂ Z; (iii) If u² ∊ U for all u ∊ U and f acts as a homomorphism or antihomomorphism on U then either d = 0 or U ⊂ Z. [ABSTRACT FROM AUTHOR]

A group G is called strongly real if its every nonidentity element is strongly real, i.e. conjugate with its inverse by an involution of G. We address the classical Lie-type groups of rank l, with l ≤ 4 and l ≥ 13, over an arbitrary field, and the exceptional Lie-type groups over a field K with an element η such that the polynomial X² + X + η is irreducible either in K[X] or K0[X] (in particular, if K is a finite field). The following question is answered for the groups under study: What unipotent subgroups of the Lie-type groups over a field of characteristic 2 are strongly real? [ABSTRACT FROM AUTHOR]

Let $H$ be a Hopf algebra and let $\mathcal{M}_s (H)$ be the category of all left $H$-modules and right $H$-comodules satisfying appropriate compatibility relations. An object in $\mathcal{M}_s (H)$ will be called a stable anti-Yetter–Drinfeld module (over $H$) or a SAYD module, for short. To each $M \in \mathcal{M}_s (H)$ we associate, in a functorial way, a cyclic object $\mathrm{Z}_\ast (H, M)$. We show that our construction can be used to compute the cyclic homology of the underlying algebra structure of $H$ and the relative cyclic homology of $H$-Galois extensions.Let $K$ be a Hopf subalgebra of $H$. For an arbitrary $M \in \mathcal{M}_s (K)$ we define a right $H$-comodule structure on $\mathrm{Ind}_K^H M := H \otimes_K M$ so that $\mathrm{Ind}_K^H M$ becomes an object in $\mathcal{M}_s (H)$. Under some assumptions on $K$ and $M$ we compute the cyclic homology of $\mathrm{Z}_\ast (H, \mathrm{Ind}_K^H M)$. As a direct application of this result, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the cyclic homology of group algebras and quantum tori.Finally, when $H$ is the enveloping algebra of a Lie algebra $\mathfrak{g}$, we construct a spectral sequence that converges to the cyclic homology of $H$ with coefficients in a given SAYD module $M$. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of $H$ with coefficients in a certain SAYD module. [ABSTRACT FROM AUTHOR]

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions f( x) ∈ L( r ∈ ℤ+) and expressions containing moduli of continuity of the kth order ωk( f(r), t). Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from L2. For the classes $$\mathcal{F}$$ ( k, r, Ψ*) defined by ω k( f( r), t) and the majorant $$\Psi _ (t) = t^{4k/\pi ^2 }$$ , we determine the exact values of different widths in the space L2. [ABSTRACT FROM AUTHOR]