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## Über dieses Buch

This volume is devoted to the most recent discoveries in mathematics and statistics. It also serves as a platform for knowledge and information exchange between experts from industrial and academic sectors. The book covers a wide range of topics, including mathematical analyses, probability, statistics, algebra, geometry, mathematical physics, wave propagation, stochastic processes, ordinary and partial differential equations, boundary value problems, linear operators, cybernetics and number and functional theory. It is a valuable resource for pure and applied mathematicians, statisticians, engineers and scientists.

## Inhaltsverzeichnis

### On Seven-Dimensional Filiform Leibniz Algebras

This paper gives complete classification of a subclass of seven-dimensional complex filiform Leibniz algebras denoted by

TLb

n

in fixed dimension

n

. The classification is carried out by choosing a basis which is adapted. Through this basis, an appropriate table of multiplication of the algebra is constructed. This leads to establishment of necessary and sufficient condition for any two algebras to be isomorphic. According to this condition, we break this class into 30 subsets and as a result, 10 of these subsets are single orbits and the remaining are union of parametric family of orbits. The single orbits are outlined from the parametric ones and their respective representatives are given. In parametric orbits case, the invariants that characterize the parameter are given. The filiform Lie algebras in this dimension are pointed out.

Abdulafeez O. Abdulkareem, I. S. Rakhimov, S. K. Said Husain

### On Exponential Stability of Stochastic Control Systems

In this paper, we study the exponential input-to-state stability in probability of a wider class of composite stochastic control system. Our aim is to establish sufficient conditions for exponential input-to-state stability in probability of this composite system. We also give a numerical example illustrating our results.

Fakhreddin Abedi, Wah June Leong

### On the Solution of Singular Ordinary Differential Equations Using a Composite Chebyshev Finite Difference Method

In this paper, a numerical algorithm based upon a hybrid of Chebyshev polynomials and block-pulse functions is proposed for solving both linear and nonlinear singular boundary value problems. Composite Chebyshev finite difference method is indeed an extension of the well-known Chebyshev finite difference method. We take advantage of the useful properties of Chebyshev polynomials and finite difference method to reduce the computation of the problem to a set of algebraic equations simplifying the problem. Several examples are included to illustrate the applicability and accuracy of the introduced method. Convergence analysis is presented.

A. Kazemi Nasab, Z. Pashazadeh Atabakan, A. Kilicman, Zainidin K. Eshkuvatov

### On 0-Controllability and Pursuit Problems for Linear Discrete Systems Under Total Constraints on Controls

We consider linear discrete control and pursuit game problems. Control vectors are subjected to total constraints, which are discrete analogues of the integral constraint. By definition, (i) the control system is 0-controllable on the whole if there is a control such that the state of the system

z

(

t

) = 0 at some step

t

, (ii) pursuit can be completed if there exists a strategy of the pursuer such that for any strategy of the evader the state of the system

y

(

t

) = 0. We obtained sufficient condition for equivalence of 0-controllability and completion of the game from any initial position of the space.

Atamurat Kuchkarov, Gafurjan Ibragimov, Akmal Sotvoldiev

### Construction of Strategies of Pursuers in a Differential Game of Many Players with State and Integral Constraints

The approach of a group of controlled objects, the pursuers, to another one, the evaders, is considered. The motions of all the objects are described by simple differential equations. The control functions of players are subjected to integral constraints. The amount of control resources such as fuel, energy etc. are described by such constraints. Given a non-empty convex subset of $${{\mathbb{R}}^{n}},$$ all objects move in this set. If the position of each evader $${{y}_{j}},\text{ }j\in \{1,\,2,\,\text{ }...\,\text{ }k\},$$ coincides with the position of a pursuer $${{x}_{i}},\text{ }i\in \{1,\text{ }...\ ,m\},$$ at some time $${{t}_{j}}\text{, i}\text{.e}\text{. }{{x}_{i}}({{t}_{j}})={{y}_{j}}({{t}_{j}}),$$ then we say that pursuit can be completed. The total resource of the pursuers is assumed to be greater than that of the evaders. We show that pursuit can be completed in this differential game.

Gafurjan Ibragimov, Asqar Rahmanov, Idham Arif Alias

### A Triangular Stochastic Facility Layout Problem in a Cellular Manufacturing System

Volatility of manufacturing environments decreases the performance of the system by degrading the efficiency and effectiveness of the layout. In this paper, in order to simulate the effects of variability of demand of products on the layout of facilities in a cellular manufacturing system (CMS), a triangular stochastic facility layout model in a CMS has been developed. To validate the model, two solution approaches have been applied; a developed enumeration method, and also Lingo 12.0 optimization software. Solving the model for a case shows that variation in demand of products may lead to a change in the layout of the facilities.

Shahram Ariafar, Zahra Firoozi, Napsiah Ismail

### Constructing a Three-Stage Asymptotic Coverage Probability for the Mean Using Edgeworth Second-Order Approximation

In this paper we consider a three-stage procedure that was presented by Hall (Ann Stat 9(6):1229–1238, 1981) to yield a fixed-width confidence interval for the mean with a precise confidence level using Edgeworth second-order expansion assuming the underlying continuous distribution has finite but unknown six moments. The procedure is based on expanding an asymptotic second order approximation of a differentiable and bounded function of the final stage stopping rule found in Yousef et al. (J Stat Plan Inference 143(9):1606–1618, 2013) by Edgeworth expansion. The performance of the asymptotic coverage was shown to be controlled by the performance of the Edgeworth approximation for the standardized underlying density and thus sensitive to the skewness and kurtosis of the underlying standardized distribution. The impact of several parameters on the asymptotic coverage is explored under continuous classes of distributions; normal, student’s

t

-distribution, uniform, beta and chi-squared. For brevity, simulation results are given for three types of underlying distributions: standard uniform, standard normal and standard exponential.

Ali S. Yousef

### Mathematical and Numerical Modelling of the Thermoplastic Coupled Problem

The coupled thermoplastic dynamic boundary problem is formulated using the deformation theory of plasticity for small deformations. The explicit and implicit schemes of finite difference equations in one-dimension case are constructed. The discreet equations are numerically solved using the explicit and implicit schemes. Comparison shows the coincidence of the numerical results received using two methods.

Abduvali A. Khaldjigitov, Nik Mohd Asri Nik Long, Aziz Qalandarov, Zainidin K. Eshkuvatov

### New Modification of Laplace–Adomian Decomposition Method for the Fifth-Order KdV Equation

In this paper, for obtaining accurate solutions of the fifth-order KdV equation with initial condition, a new modification of Laplace–Adomian decomposition method is proposed. To see the accuracy of the proposed method,

L

2

and

$L_{\infty}$

error norms are calculated in test problems. The numerical results are found to be in a good agreement with exact solutions and with the literature.

H. O. Bakodah, B. S. Kashkari

### Extended Simpson Rule for Solving First-Order Fuzzy Differential Equations Using Generalized Differentiability

In this present paper, extended Milne–Simpson rule is applied to find the fuzzy solutions of first order fuzzy differential equations (FDEs) under generalized Hukuhara differentiability concept. Based on the work done by Chalco-Cano we show how FDEs can be transformed to a system of ordinary differential equations. Then to find the solutions of fuzzy initial value problems (FIVPs), the Milne–Simpson formula is generalized. The errors, which guarantee pointwise convergence, are compared with other established methods which clearly show the advantage of our method for solving FIVPs.

Reza Afsharinafar, Fudziah Ismail, Mohamed Suleiman, Ali Ahmadian Hosseini

### Modified Decomposition Method for Solving Nonlinear Volterra–Fredholm Integral Equations

In this note, an approximate solution of nonlinear Volterra–Fredholm integral equation is obtained by using modified decomposition method. General cases of nonlinear terms in the equations are considered. Finally, some numerical examples are presented to validate the accuracy and efficiency of the method.

F. S. Zulkarnain, Z. K. Eshkuvatov, Z. Muminov, N. M. A. Nik Long

### Solution of Second Order Ordinary Differential Equations by Direct Diagonally Implicit Block Methods

This paper presents a diagonally implicit method for directly solving stiff second-order ordinary differential equations (ODEs) which produce a block of two new values at each step of application. The formulation of the method is described. Numerical results are given to demonstrate the efficiency of the proposed method.

Nooraini Zainuddin, Zarina Bibi Ibrahim, Mohamed Suleiman, Khairil Iskandar Othman, Yong Faezah Rahim

### Group Algebra Codes Define Over Extra-Special p-Group

In this paper, group algebra code defined over any extra-special

p

-group

G

is constructed. If

$char(F) \nmid |G|$

, then

FG

is semisimple and hence

$FG = \bigoplus_{e_i \in M} FGe_j$

, where

e

j

is an idempotent of

FG

and

M

is the set consisting of all idempotents of

FG

. Any idea

I

of

FG

is a direct sum of some

FGe

J

, say

$I = \bigoplus_{k=1}^{t} FGe_{j_k}$

, for some

t

such that

$1 \leq t \leq |G|$

. Let

$\beta = \{e_{j_k}\}_{k=1}^t$

and

$\mu=M \backslash \beta$

, then

I

is generated by β and for technical reason,

I

denotes

$I_{\mu} = \{u \in FG \mid ue_{j_r} = 0, \forall e_{j_r} \in \mu\}$

. The idempotent

e

j

provides useful information to determine the minimum distance for this family of group algebra code. Our primary task is to identify all such idempotents and thus construct a family of MDS group algebra code by choosing a suitable subset of μ in order to maximize the minimum distance.

Denis C. K. Wong

### New Subclasses of Meromorphic Functions Related to Cho-Kwon-Srivastava Operator

Making use of a linear operator, which is defined here by means of the Hadamard product (or convolution), we introduce some new subclasses of the meromorphically univalent function class Σ and investigate their inclusion relationships.

Firas Ghanim, Maslina Darus

### The Description of Orbits Under an Action of GL 9 ON SLb 9

The paper concerns with the classification problem of a subclass of nine-dimensional complex filiform Leibniz algebras. The concepts of adapted basis and adapted transformations are given to express the action (“transport of structure”) of the adapted transformations group on

SLb

9

. We present

SLb

9

as a disjoint union of its subsets and specify the orbits under the action of adapted transformations. The list of isomorphism classes of

SLb

9

with their representatives and the table of multiplications are given.

I. S. Rakhimov, S. K. Said Husain, F. Deraman

### Effects of Magnetohydrodynamic on the Stagnation Point Flow past a Stretching Sheet in the Presence of Thermal Radiation with Newtonian Heating

In this study, the effects of magnetohydrodynamic on the stagnation point flow past a stretching surface in the presence of thermal radiation generated by Newtonian heating are studied, where the heat transfer rate from the bounding surface with a finite heat capacity is proportional to the local surface temperature. The transformed boundary layer equations are solved numerically using the Keller-box method. Numerical solutions are obtained for the local heat transfer coefficient, the surface temperature as well as the velocity and temperature profiles. The features of the flow and heat transfer characteristics for various values of the magnetic parameter and thermal radiation parameter are analyzed and discussed.

Muhammad Khairul Anuar Mohamed, Muhammad Imran Anwar, Sharidan Shafie, Mohd Zuki Salleh, Anuar Ishak

### Description of Three Dimensional Solvable Evaluation Algebras

The concept of evolution algebras lies between algebras and dynamical system. In this paper we are going to describe three dimensional nilpotent, solvable evolution algebras. It is seen that three dimensional nilpotent evolution algebras are classified into three classes, and solvable ones are classified into five classes.

Farrukh Mukhamedov, Bakhrom Omirov, Izzat Qaralleh

### On Quasi Quantum Quadratic Operators of M 2(C)

In the present paper we study quasi quantum quadratic operators (q.q.o) acting on the algebra of 2 × 2 matrices

M

2

(

C

). We describe quasi q.q.o. with Haar state, and prove that if a symmetric quasi q.q.o. with Haar state is q-pure, then it cannot be positive.

Farrukh Mukhamedov, Abduaziz Abduganiev

### On the Structure of the Essential Spectrum of Four-Particle Schrödinger Operators on a Lattice

The four-particle discrete Schrödinger operator

$H(K),$

$K\in ({-}\pi,\pi]^3$

corresponding to the system of the four particles on the lattice

$\mathbb{Z}^3$

with arbitrary “dispersion functions” not necessarily having compact support and interacting via short-range pair potentials, is described in the coordinate representation as bounded self-adjoint operator on the corresponding Hilbert space. We describe the location and structure of the essential spectrum of the four-particle discrete Schrödinger operator

$H(K),$

$K\in ({-}\pi,\pi]^3$

by means of the spectrum of the three-particle discrete Schrödinger operators and establish the resolvent equation.

Z. Muminov, F. Ismail, Z. Eshkuvatov

### Symmetrizers for Runge–Kutta Methods

L

-stable symmetrizers for symmetric Runge–Kutta methods are constructed to preserve the asymptotic error expansion in even powers of the stepsize and to provide necessary damping for the numerical solution of stiff initial value ordinary differential equations. The process is called symmetrization and has the effect of dampening down undesirable oscillations that may arise when the problem is stiff. The symmetry property can be exploited by Richardson extrapolation in increasing the order by two at a time. The effect of the damping is to stabilize the order behaviour of the extrapolation. In this paper, we discuss two types of symmetrizers; the well-known one-step smoothing formula of Gragg, and the new two-step smoothing formula. We construct these symmetrizers for the implicit midpoint and trapezoidal rules and investigate the active mode of application with extrapolation. Finally, we present numerical results in a constant stepsize setting that show two-step symmetrizers having certain advantages over one-step symmetrizers for stiff linear and nonlinear problems.

N. Razali, R. P. K. Chan

### A Method of Estimating the p-adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Complete Cubic Form

Let

x

$=(x_{1}, x_{2}, {\ldots}, x_{n})$

be a vector in the space

Q

n

with

Q

field of rational numbers and

q

be a positive integer,

f

a polynomial in

x

with coefficient in

Q

. The exponential sum associated with

f

is defined as

$S (\textit{f}; q) = \Sigma_{x mod q}e^{((2i\textit{f}(x))/q)}$

, where the sum is taken over a complete set of residues modulo

q

. The value of

$S (\textit{f}; q)$

depends on the estimate of cardinality

$|V|$

, the number of elements contained in the set

$V =\{\textit{x} mod q | \textit{f}_{\textit{x}}\equiv 0 mod q\}$

where

$\textit{f}_{\textit{x}}$

is the partial derivative of

f

with respect to

x

. To determine the cardinality of

V

, the

p

-adic sizes of common zeros of the partial derivative polynomials need to be obtained. In this paper, we estimate the

p

-adic sizes of common zeros of partial derivative polynomials of

$\textit{f}(x,y)$

in

$Q_{\textit{p}}[x, y]$

with a complete cubic form by using Newton polyhedron technique. The polynomial is of the form

$\textit{f}(x,y)= a\textit{x}^{3}+ b\textit{x}^{2}\textit{y} + c\textit{x}\textit{y}^{2}+d\textit{y}^{3}+ \frac{3}{2} a\textit{x}^{2}+ b\textit{x}\textit{y}+\frac{1}{2}c\textit{y}^{2}+s\textit{x}+t\textit{y}+k.$

S. S. Aminudin, S. H. Sapar, K. A. Mohd Atan

### Feynman Graph Representation to Stochastic Differential Equations Driven by Lévy Noise

Stochastic differential equations driven by Lévy noise are intensively studied. But so far there seems to be no recipe to find out what kind of noise it is, given the general structure of the equation. This can be obtained by recalling a graphical representation of the solution of the stochastic differential equations driven by Lévy noise. The graphs introduced are called generalized Feynman graphs and a numerical value will be assigned to each graph. Our graphs’ formalism can be applied to different kinds of stochastic differential equations. As an example, a graphical representation of the generalized Ornstein–Uhlenbeck process will be given in this work.

Boubaker Smii

### Some Properties of the Concurrent Grammars

Petri nets are becoming one of the most important mathematical tools in Computer Science. In this paper, we study some mathematical properties of concurrent grammars which are controlled by Petri nets under parallel firing strategies, where transitions of Petri nets fire simultaneously in different modes. We propose a notion of concurrent context-free grammar which is a similar case of the context-free Petri nets under parallel firing strategy, where parallel firing modes of context-free Petri nets were converted to rule applications in context-free grammars and we investigate their properties.

Gairatzhan Mavlankulov, Mohamed Othman, Mohd Hasan Selamat, Sherzod Turaev

### Statistical Analysis on LBlock Block Cipher

In this research paper, we present the statistical analysis on full round lightweight block cipher; LBlock. This block cipher has a fixed block size of 64-bit, utilizes an 80-bit key and executes in 32 rounds. To determine the randomness of ciphertext produced by this algorithm, NIST Statistical Test Suite is used. Inputs (plaintext and key) for this block cipher algorithm are generated using nine different types of data. From the analysis done, it is concluded that the LBlock block cipher is not random based on 1 % significance level only.

Nik Azura Nik Abdullah, Kamaruzzaman Seman, Norita Md Norwawi

### A New Threshold-Authenticated Encryption Scheme

An authenticated encryption scheme is a message recovery scheme that provides the authenticity property. In an authenticated encryption scheme, the recipients not only verify the message authentication, but they also could recover the message. In this paper, we propose a new authenticated encryption scheme based on two hard number theoretical problems: factoring and discrete logarithm. In our new scheme,

t

out of

n

signers/senders are required to sign and at the same time encrypt a message, while

k

out of

l

recipients cooperate to verify and recover the original message. We also show that our scheme is secure against some cryptographic attacks and requires reasonable number of operations in both signature/encryption and verification/decryption phases.

### On Differential Invariants of Some Classical Groups

The paper deals with the description of the field of differential rational invariants under simultaneous changing of the differential operator and an action of subgroups of affine transformation group on system of finite points of finite-dimensional differential vector space. In particular, the results in terms of differential geometry mean that we describe the field of differential rational invariants of finite system of geometric curves with respect to various group motions.

Ural D. Bekbaev, Isamiddin S. Rakhimov

### Structure of Relative Relation Modules of Finite Groups

Decomposability properties of modules of finite groups are known to be useful in describing the structure of certain groups. Let

E

be a free product of a finite number of cyclic groups, and

S

a normal subgroup of

E

such that

$E/S \cong G$

is finite. For a prime

p

,

$\hat{S} = S/S^{'}S^{p}$

may be regarded as

$F_{p}G$

-module. Whenever

E

is a free group,

$\hat{S}$

is called relation module (modulo

p

) of

G

; in general

$\hat{S}$

is called relative relation module (modulo

p

). Many researchers have studied relation and relative relation modules. In this paper, we describe the structure of relative relation modules of finite groups, and in particular, those of abelian

p

-groups. We also describe the decomposition of relative relation modules of SL(2,

p

) and PSL(2,

p

). Decomposition of

$\hat{S}$

described in this paper may be useful in studying factor group of PSL(2,

Z

), which is an interesting problem in its own right.

### Block Methods with Off-Steps Points for Solving First Order Ordinary Differential Equations

The block methods with off-step points are proposed for the solution of first-order ordinary differential equations. These methods are applied concurrently in block form to provide the approximation for both the main and off-step points. The stability properties are discussed. Some numerical examples are given to illustrate the efficiency of the methods.

Lee Ken Yap, Fudziah Ismail

### Implicit Finite Difference Solutions of One-Dimensional Burgers’ Equation Using Newton–HSSOR Method

In this paper, we present the application of half-sweep successive over-relaxation (HSSOR) iterative methods together with Newton scheme, collectively Newton–HSSOR, in solving the nonlinear systems generated from the half-sweep Crank–Nicolson finite difference discretization scheme for a one-dimensional Burgers’ equation. To linearize nonlinear systems, the Newton scheme is proposed to transform the nonlinear system into the form of linear system. In addition to that, the basic formulation and implementation of Newton–HSSOR iterative method are also shown. For comparison purpose, we also consider combinations between the full-sweep Gauss–Seidel (FSGS) and full-sweep successive over-relaxation (FSSOR) iterative methods with Newton scheme, which are indicated as Newton–FSGS and Newton–FSSOR methods respectively. Consequently, two illustrative examples are included to demonstrate the validity and applicability of tested methods. Finally, it can be concluded that the Newton–HSSOR method shows superiority over other tested methods.

J. Sulaiman, M. K. Hasan, M. Othman, S.A.A. Karim
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