On Galilean Invariant and Energy Preserving BBM-Type Equations

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

Parameter Estimation in Convection-Diffusion Model

This paper describes a special one-dimensional convection-diffusion equation and analyzes two types of difference schemes. Numerical solutions of the two difference methods for this equation are implemented to estimate the parameters of the velocity component of the fluid and the diffusion coefficient. Good results of parameters estimated are not achieved because of the larger approximation errors by the difference schemes. Then multiple linear regression is applied to estimating the corresponding parameters by using the analytical solution of this special equation. By this means, the better estimated values of the parameters are obtained.

Efficient k-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly

There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of 2k multi-step formulas (although we will see that this number can be reduced to k+1 in case of a special equation) that provides approximate solutions at k grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each k, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half.

Special equation from Binomial transform and nth order finite difference

Using numerical analysis and tables, nth order backward difference of exponential function is obtained. Further analyzing the obtained equation yields a special identity given as \[\sum\limits_{k\, = \,0}^n {{{( - 1)}^k}\,\frac{{{{(x - \,k)}^{n\,\, - m}}}}{{(n\, - \,k - m)!\,k!}}\,} \, = \,\,\sum\limits_{k\, = \,0}^{n\, - m} {{{( - 1)}^k}\,\frac{{{{(x - \,k)}^{n\, - m}}}}{{(n\, - \,k - m)!\,k!}}\,} = \,\,1\,\,\,\,\,\,:\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{\,\,\,\,\,\,\,\,\,\,\,\,\,x \in R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{n \in \,W{\rm{ }}}\\{m\, \in \,W\,\,{\rm{:}}\,\,m \le \,n}\end{array}} \right.\]This equation yields the value of negative integer factorial, zero factorial and zero to the power zero which is currently indeterminate forms along with other unknown values. With this equation, we also conclude the product of zero and infinity is unity within certain parameters.

A new curvature-like tensor in an almost contact Riemannian manifold

In a M. Prvanović’s paper [5], we can find a new curvature-like tensor in an almost Hermitian manifold.In this paper, we define a new curvature-like tensor, named contact holomorphic Riemannian, briefly (CHR), curvature tensor in an almost contactRiemannian manifold. Then, using this tensor, we mainly research (CHR)-curvature tensor in a Kenmotsu and a Sasakian manifold. We introducethe flatness of a (CHR)-curvature tensor and show that a Kenmotsu anda Sasakian manifold with a flat (CHR)-curvature tensor is flat, see Theorems3.1 and 4.1. Next, we introduce the notion of an (CHR)-n-Einstein inan almost contact Riemannian manifold. In particular, in a Sasakian or aKenmotsu manifold, a (CHR)-n-Einstein manifold is n-Einstein, see Theorem5.3. Finally, from this tensor, we introduce a notion of a (CHR)-spaceform in an almost contact Riemannian manifold. In particular, if a Kenmotsuand a Sasakian manifold are (CHR)-space form, then the (CHR)-curvaturetensor satisfies a special equation, see Theorems 6.2 and 7.1.

A family of Eisenstein polynomials generating totally ramified extensions, identification of extensions and construction of class fields

Let K be a local field with finite residue field, we define a normal form for Eisenstein polynomials depending on the choice of a uniformizer πK and of residue representatives. The isomorphism classes of extensions generated by the polynomials in the family exhaust all totally ramified extensions, and the multiplicity with which each isomorphism class L/K appears is always smaller than the number of conjugates of L over K. An algorithm to recover the set of all special polynomials generating the extension determined by a general Eisenstein polynomial is described. We also give a criterion to quickly establish that a polynomial generates a different extension from that generated by a set of special polynomials, such criterion does not only depend on the usual distance on the set of Eisenstein polynomials considered by Krasner and others. We conclude with an algorithm for the construction of the unique special equation determining a totally ramified class field in general degree, given a suitable representation of a group of norms.

Calculating the number of solutions of a difference equation

The hash algorithms of the MDx family involve cyclic shifts, computation of primitive Boolean functions, and addition of constants. So far, very few works have been published in which the authors attempt to explain the impact that the choice of constants, shifts, and Boolean functions has on the cryptographic properties of the algorithms. G. A. Karpunin and H. T. Nguyen suggested a model in which the resistance against differential cryptanalysis may be quantitatively estimated in terms of the number of solutions of a special equation. In this work, in the framework of the aforementioned model, an equation for the MD5 hash function is derived. Examination of one Boolean function and one value of the cyclic shift through exhaustive search requires 2

FINITE COMPLETIONS VIA FACTORIZING CODES

Several results relate finite maximal codes to factorizations of cyclic groups. In the case of factorizing codes C, i.e. finite maximal codes which satisfy the still open Schützenberger's factorization conjecture, special factorizations, discovered by Hajós, intervene. In particular, given a two-letter alphabet {a,b}, it is known that the set C1 = C ∩ a*ba* satisfies a structural property defined by means of the Hajós factorizations. Conversely, it is not true that a set satisfying this structural property can be embedded in a factorizing code and some partial results are known on the problem of finding additional hypotheses that guarantee the existence of such embedding. Let C be a factorizing code. Inspired by the recursive construction of the Hajós factorizations and starting with a special equation associated with C1 = C ∩ a*ba*, we define a family [Formula: see text] of subsets of a*ba*, each of them still satisfying the above-mentioned structural property. We prove that for each set [Formula: see text], there exists a factorizing code C with C1 = C ∩ a*ba* and as a consequence C1 is a code. C is obtained starting with prefix/suffix codes and by using two types of operations on codes — composition and substitution. We extend all these results to alphabets of size greater than two. We conjecture that for each factorizing code C, we have [Formula: see text]. We also give a method of finding solutions to the above-mentioned equation associated with C1 and we conjecture that this method constructs all these solutions.

On Second Order Nonlinear Equations with Rectilinear Characteristics

A class of quasilinear hyperbolic equations of mixed type whose characteristic roots are simultaneously characteristic invariants is found. For a special equation of this class, a general integral is constructed in terms of invariants in the closed form by using the method of characteristics. Based on the structure of families of characteristics, the initial Cauchy problem is investigated. The structure of the solution definition and regularity domains is defined using the properties of initial perturbations.