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We study the transformation of "predicate introduction" in non-monotonic logics. By this, we mean the act of replacing a complex formula by a newly defined predicate. From a knowledge representation perspective, such transformations can be used to eliminate redundancy or to simplify a theory. From a more practical point of view, they can also be used to transform a theory into a normal form imposed by certain inference programs or theorems. In a companion paper, we developed an algebraic theory that considers predicate introduction within the framework of "approximation theory", a fixpoint theory for non-monotone operators that generalizes all main semantics of various non-monotonic logics, including logic programming, default logic and autoepistemic logic. We then used these results to show that certain logic programming transformations are equivalence preserving under, among others, both the stable and well-founded semantics. In this paper, we now apply the same algebraic results to autoepistemic logic and prove that a transformation to reduce the nesting depth of modal operators is equivalence preserving under a family of semantics for this logic. This not only provides useful theorems for autoepistemic logic, but also demonstrates that our algebraic theory does indeed capture the essence of predicate introduction in a generally applicable way. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we completely solve a family of Thue inequalities defined over the field of functions $\mathbb{C}(T)$, namely deg (X4?4cX3Y+(6c+2)X2Y2+4cXY3+Y4) ? deg c, where the solutions x,ycome from the ring $\mathbb{C}\left[ T\right] $and the parameter $c\in \mathbb{C}\left[ T\right] $is some non-constant polynomial. [ABSTRACT FROM AUTHOR]

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]

Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato–Fefferman–Phong’s class in Lipschitz domains. An elementary proof of the doubling property for u 2 over balls is presented, if the balls are contained in the domain or centered at some points near an open subset of the boundary on which the solution u vanishes continuously. Moreover, we prove the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations, and we derive the ℬ p weight properties for the solution u near the boundary. [ABSTRACT FROM AUTHOR]

To enrich any model and its dynamics introduction of delay is useful, that models a precise description of real-life phenomena. Differential equations in which current time derivatives count on the solution and its derivatives at a prior time are known as delay differential equations (DDEs). In this study, we are introducing new techniques for finding the numerical solution of fractional delay differential equations (FDDEs) based on the application of neural minimization (NM) by utilizing Chebyshev simulated annealing neural network (ChSANN) and Legendre simulated annealing neural network (LSANN). The main purpose of using Chebyshev and Legendre polynomials, along with simulated annealing (SA), is to reduce mean square error (MSE) that leads to more accurate numerical approximations. This study provides the application of ChSANN and LSANN for solving DDEs and FDDEs. Proposed schemes can be effortlessly executed by using Mathematica or MATLAB software to get explicit solutions. Computational outcomes are depicted, for various numerical experiments, numerically and graphically with error analysis to demonstrate the accuracy and efficiency of the methods. [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.

Based on the Orlov and Shulman’s M operator, the additional symmetries and the string equation of the CKP hierarchy are established, and then the higher order constraints on L l are obtained. In addition, the generating function and some properties are also given. In particular, the additional symmetry flows form a new infinite dimensional algebra $$W^C_{1+\infty}$$ , which is a subalgebra of W 1+∞. [ABSTRACT FROM AUTHOR]

Let R be an exchange ring with primitive factors artinian. We prove that there exists a u∈ U( R) such that 1 R ± u ∈ U( R), if and only if for any a∈ R, there exists a u ∈ U( R) such that a ± u ∈ U( R). Furthermore, we prove that, for any A ∈ M n ( R)( n ≥ 2), there exists a U ∈ GL n ( R) such that A ± U ∈ GL n ( R). [ABSTRACT FROM AUTHOR]

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.

The article presents a study on the mathematical equation of two-dimensional linear differential systems with deviating arguments using Kneser type solutions. It presents an equation with positive coefficient and two linearly independent solutions. The article also evaluates the existence and uniqueness of these types of solutions. It proves that Kneser type solution is equivalent to nonvanishing of the Wronskian fundamental system and positivity of Green's function of the problem.

The article presents a research on the iterative methods for generalized von Foerster equations with functional dependence. It examines the meeting of a natural iterative method to the exact solution of a differential-functional von Foerster-type equation that notes a single population depending on its total size, past time and states densities. It also assumes either Perron comparison conditions and some monotonicity. It also investigates on the natural conditions which ensures convergence of iterative methods.

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 develop a method to prove that some critical levels for functionals invariant by symmetry obtained by minimax methods without any symmetry constraint are attained by symmetric critical points. It is used to investigate the symmetry properties of solutions of elliptic partial differential equations with Dirichlet or Neumann boundary conditions. It is also an alternative to concentration-compactness for some symmetric elliptic problems. [ABSTRACT FROM AUTHOR]

We provide new estimates for the gap function of the Delta function and for the number of nonzero values of the Ramanujan tau function in short intervals. [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]