Mathematics and Computer Science: Contemporary Developments Vol. 1

In recent eras, storing and processing data on third-party remote cloud servers has been widely used, showing explosive growth. The growing popularity of data outsourcing to third-party cloud servers has a downside, related to the serious concerns of data owners about their security due to possible leakage. The desire to reduce the risk of loss of data confidentiality has become a motivating start to developing mechanisms that provide the ability to effectively use encryption to protect data. However, the use of traditional encryption methods faces a problem. Namely, traditional encryption, by making it impossible for insiders and outsiders to access data without knowing the keys, excludes the possibility of searching. This paper presents a solution that provides a strong level of confidentiality when searching, inserting, modifying, and deleting the required sensitive data in a remote database whose data are encrypted.

 The proposed SQL query processing technique allows the DBMS server to perform search functions over encrypted data in the same way as in an unencrypted database. This study also offers a basis for implementing the solution on the server side of the Oracle DBMS using unmodified DBMS software using our own developed persistent stored modules (PSM). This is achieved through the organization of automatic decryption by specially developed secure software of the corresponding data required for search, without the possibility of viewing these data itself. We tested our prototype on a single machine that simulates both a proxy and a database server. At that, we guarantee the integrity of the stored procedures used and special tables that store encrypted modules of special software and decryption keys, the relevance and completeness of the results returned to the application. The results of the analysis of the feasibility and effectiveness of the proposed solution show that the proper privacy of the stored data can be achieved at a reasonable overhead. Future studies may focus on the conduct of an in-depth performance evaluation of this proposed solution, including a comparison with existing implementations, to show its practicality in various real-life situations.

In this work, we consider the problem of solving strongly nonlinear oscillators making use of a generalized averaging method. We illustrate the efficacy of this method for the generalized Van der Pol-Duffing oscillator. We show that the amplitude of the limit cycle may be obtained by solving a quintic.

We say the k-Fibonacci numbers F_k,i and F_k,j are equidistant if j = n - i and then we study some properties of these pairs of numbers. As a main result, we look for the formula to find the generating function of the product of the equidistant numbers, their sums and their binomial transforms. Next, we apply this formula to some simple cases but more common than the general cases. In particular, we define the half-self-convolution of the k-Fibonacci and k-Lucas sequences. Finally, we study the sum of these new sequences, their recurrence relations, and their generating functions.

Finite projective geometry is used to construct many series of symmetric orthogonal arrays of strength two and more. This chapter proposes a method for constructing nearly orthogonal arrays mappable into fully symmetric orthogonal arrays of strength two using finite projective geometry. Mappable nearly orthogonal arrays are designs used in optimization, particularly in situations where factors interact with each other. They provide an efficient way to study stratification and understand complex systems by minimizing experimental runs while maximizing the number of factors.

Let \(p(z)\) be a polynomial of degree \(n\) having all its zeros in \(|z| \leq 1\), then famous inequality due to Turán [Compos. Math. 7 (1939), 89-95] is

                                  \[\max_{ |z|=1 }\left|p^{\prime}(z)\right| \geq \frac{n}{2} \max _{|z|=1}|p(z)| \text {. }\]

This Turán's inequality was generalized for the first time by Malik [J. London Math. Soc., 1(1969),57-60] that if \(p(z)\) is a polynomial of degree \(n\) having all its zeros in \(|z| \leq k, k \leq 1\), then

                                \[\max_{ |z|=1 }\left|p^{\prime}(z)\right| \geq \frac{n}{1+k} \max _{|z|=1}|p(z)| \text {. }\]

While for the case \(k \geq 1\), Govil [Proc. Amer. Math. Soc. 41(1973), 543-546] prove that

                                \[\max_{ |z|=1 }\left|p^{\prime}(z)\right| \geq \frac{n}{1+k^n} \max _{|z|=1}|p(z)| \text {. }\]

The above inequalities play a vital role in approximation theory. Frequently, further progress in this theory has depended on first obtaining a corresponding generalization or analogue of these inequalities. In this article, we discuss in brief some of the recent improvements of the above inequalities particularly the later type and make a comparative study of them using an example with detail graphical illustrations.

A non-parametric test procedure is proposed to address the random-interval observation with imbalanced count data in a medical follow study. Proposed test statistics are constructed based on the integrated weighted differences between the mean cumulative function of the recurrences event with conditions on different treatment groups. The proposed non-parametric test procedure can detect the departure from the null hypothesis based on the weighted difference between conditional mean cumulative functions. The proposed test procedure is concerned with non-parametric comparisons with time-independent covariates and non-informative censoring processes. The mixed Poisson process is constructed, and a multivariate non-homogeneous Poisson process serves as a benchmark for the multivariate mixed Poisson process. The performance of the proposed non-parametric test procedure is investigated through a simulation study with an illustration of a numerical example in a medical follow-up study obtained from the skin cancer chemoprevention trial conducted by the University of Wisconsin Comprehensive Cancer Center.

The aim of this paper is to give some new charaterizations of additive functions in the tensor product of two hermitian Banach *-algebras. Our results extend Vukman’s work on Cauchy functional equation in hermitian Banach *-algebras to the projective tensor product. An example is provided in support of our result.

This study is an extended version of the work published in Kumar and Vishwakarma (Proceedings of the National Academy of Sciences, India, Section A: Physical Sciences, 90(5): 933-939, 2020). The present study provides theoretical, as well as empirical, investigations of various sample allocation strategies for the estimation of population mean in stratified random sampling. The mathematical expressions for mean square errors (MSEs) of several well-known estimators of population mean are derived under the various allocation schemes considered under investigation. The pre-existing estimators are compared with that of the proposed classes of estimators using the MSE criterion, and the necessary and sufficient conditions (NASCs) for the dominance of proposed classes of estimators are revealed. Furthermore, the empirical results of the study exhibit the superiority of Neyman allocation scheme over the Equal and Proportional allocation schemes for the considered estimators.

In this chapter, the homotopy perturbation and Adomain's decomposition methods are applied to obtain the approximate solutions of the equation of motion and heat equation for the harmonic waves propagation in a nonlinear generalized thermoelasticity with magnetic field. The nonlinear coupled system of partial differential equations often appear in the study of circled fuel reactor, high-temperature hydrodynamics and thermo-elasticity problems. The problem is solved in one-dimensional elastic half-space model subjected initially to a prescribed harmonic displacement and the temperature of the medium. The displacement and temperature are calculated for the two methods with the variations of the magnetic field and the relaxation times considering Green Lindsay theory (GL). The results obtained are displayed graphically to show the in uences of the new parameters and the differs between the methods technique. It is obvious that the homotopy perturbation method and adomain decomposition method give the same results that indicates to the origin of the approximate solutions and the methods powerful. The homotopy perturbation method and adomain decomposition method give the same results that indicates to the origin of the approximate solutions and the methods powerful.

We deal with the stabilization problem of discrete nonlinear systems. We construct a control Lyapunov function on discrete nonlinear systems. Then, we present a new method to construct a continuous state feedback law.