The RSA cryptosystem has long been a fundamental component of contemporary public key in- frastructure. However, recent developments in quantum computing and mathematical theory have introduced signicant challenges to its security. A fully operational quantum computer would allow for the application of Shor's algorithm, enabling the ecient factoring of large integers, thereby com- promising the integrity of RSA and other cryptographic methods dependent on discrete logarithms. Although Grover's algorithm poses a comparatively smaller threat to symmetric encryption systems, it still presents a vulnerability by accelerating key search processes. The looming threat from quan- tum technologies has driven a surge in research aimed at developing quantum-resistant cryptography. These eorts focus on cryptographic techniques grounded in error-correcting codes, lattice structures, and multivariate public key systems, all of which leverage the complexity of NP-hard problemssuch as solving multivariate quadratic equationsto preserve security in a post-quantum world. This pa- per reviews the current progress in quantum-resistant encryption methods, particularly emphasizing the role of robust trapdoor functions. Additionally, it provides a comprehensive analysis of critical multivariate cryptosystem frameworks, such as Matsumoto-Imai, Oil and Vinegar, and Polly Cracker schemes. It also examines advances in lattice-based systems like Kyber and Crystals-Dilithium, which are undergoing assessment by NIST for potential standardization. As quantum computing continues to evolve, the demand for cutting-edge cryptographic solutions to safeguard digital communications grows increasingly pressing.

This work presents a historical-epistemological analysis of the foundations of quantum mechanics. From a philosophical point of view, the relationships between quantum mechanics and scientific realism are highlighted, as well as the influence of the neo-positivist philosophy of the first decades of the 20th century. The great significance of measurement in the context of quantum mechanics is underlined, as an essential element in providing a coherent description of reality moving from the microcosm to the macrocosm, as well as the role that the measurement apparatus and the consciousness of the observer assume during the act of measurement, the ontological problem, the problem of the objectification of the wave function and the problem of the definition of discernibility. This is followed by an overview of the most important alternative proposals to the standard interpretation (Copenhagen interpretation), with particular reference to hidden variable theories, and other alternatives to standard quantum mechanics that profoundly modify the vision of reality, such as the Everett’s many-worlds theory, the QBism, the unified holistic approach.

This article is some review of results that were obtained at 2007-2021 years development of “The Information as Absolute” concept and the informational physical model, which is based on the concept; including a number of fundamental physical problems are briefly considered in framework of the conception and the model. Recently in physics there are several publications, that present lists of the problems. However, those lists are essentially incomplete, for at least two reasons. Firsts of all, a number of phenomena are studied traditionally by philosophy, and so corresponding problems are usually considered to be “metaphysical”. However, they relate also to some concrete physical phenomena. For example, physics evidently studies Matter, and so the metaphysical problems “what is ontology of Matter”, “what is “Space”, “Time” and a few other physical phenomena and notions as well, are really a Meta-physical problems “what does physics study?” There are other fundamental physical problems, which are not considered as such in physics, and are absent in the “fundamental problems lists”. Those include the problems, which really exist, yet are incorporated into standard physical theories, and so are fundamental “implicitly”, which in physics are “solved by default” ‒ and mostly erroneously. Note, though, that a number of “Meta-physical”, and concrete fundamental, problems more in detail are considered in the paper “The Informational Conception and Basic Physics”, https://arxiv.org/abs/0707.4657, v5 (2021), so this paper is, in certain sense, an expanded conclusion of this paper, which includes, correspondingly, more in detail consideration of some more general physical problems; and, besides, in this article, the problem “what are Gravity and Electric Forces” is essentially clarified comparing with the arXiv 2021 paper version above, and additionally initial model of Nuclear Force is presented. Besides, the concrete problem “What is Life”, and the rational cosmological model, where a few vague points in standard cosmology rather probably are rationally clarified, while the fundamental problem “matter – antimatter asymmetry” in Matter is solved with rather large probability, are considered, and one of recently published rather complete “lists of fundamental problems” is commented in Appendix.

Abstract: The frequency and intensity of air pollution are two of most critical issues the world faces in dealing with global environmental problems. They are both important areas that need to be improved. Our previous research [20] examines the assessment of fine dust pollution in Baekryeong island by statistical reasoning through one specific example of inland city (Yanggu) in Republic of Korea; however, given that Baekryeong Island is a remote island, there has been little focus on in-depth examination of its effect on the high density of fine dust in Baekryeong Island. The purpose of this article is twofold: (i) to examine the effect of a remote island on the high density of fine dust on Baekryeong Island using statistical reasoning and to provide a more specific statistical basis for the claim that one of the non-negligible factors influencing the high density of fine dust on Baekryeong Island is the geographic proximity to China; and (ii) to present brief discussion of fundamental cause of fine dust and global air pollution issues, which are not discussed in [20].

The article proposes various new approximate analytical solutions of the boundary value prob-lem for the non-stationary system of Nernst-Planck-Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the general case consists of a space charge region and a region of local electroneutrality. The proposed analytical solutions of the boundary value problems for the non-stationary system of Nernst-Planck-Poisson equations are based on the derivation of a new singularly perturbed nonlinear partial differential equation for the potential in the space charge region (SCR). This equation can be reduced to a singularly perturbed inhomogeneous Burgers equation, which, by the Hopf-Cole transformation, is reduced to an inhomogeneous singularly perturbed linear equation of parabolic type. Inside the extended SCR, there is a suffi-ciently accurate analytical approximation to the solution of the original boundary value prob-lem. The electroneutrality region has a curvilinear boundary with the SCR, and with an un-known boundary condition on it. The article proposes a solution to this problem. The new ana-lytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transfer in membrane systems.

This text deals with exploring random solutions for generalized nonlinear variational inequality problems. Using the Fan-KKM theorem and Aumann's measurable selection theorem, we are able to prove the existence and uniqueness of random solution sets under the conditions of monotonicity and convexity. Additionally, we use Minty's lemma to demonstrate the compactness and convexity of the random solution sets.

Fractional differential systems have received much attention in recent decades likely due to its powerful ability in modeling memory processes, which are mostly observed in real world. Fractional models have been widely investigated and applied in many fields such as physics, biochemistry, electrical engineering, continuum and statistical mechanics. It is shown by many researchers that fractional derivatives can provide more accurate models than integer order derivatives do. Obviously, bang-bang property is of a significant importance in optimal control theory. In this paper a Bang-Bang property and time-optimal control problem for time-Fractional Differential System (FDS) is under consideration. First we formulate our problem and we prove the existence theorem. We state and prove the bang-bang theorem. Finally we give the optimality conditions that characterized the optimal control. Some illustrate applications are given to clarify our results.

We present a MATLAB-based, open source reconstruction software for Hybrid Imaging problem arising from Current Density Impedance Imaging (CDII) and Acoustic-Electrical Impedance Tomography (AEIT), utilizing mixed finite element method solver of Darcy toolbox {\tt DarcyLite}~. For a formulation of hybrid inverse problems in impedance tomography the successive substitution iterative schemes are adapted and an reconstruction algorithm is developed. The problem formulation is a class of hybrid imaging modality which is current density impedance imaging or Acoustic-Electrical Impedance Tomography. The proposed algorithm is implemented numerically in two dimensions.

This paper examines the correlation between numbers of computer cores in parallel genetic algorithms. The objective to determine the linear polynomial complementary equation in order represent the relation between number of parallel processing and optimum solutions. Model this relation as optimization function (f(x)) which able to produce many simulation results. F(x) performance is outperform genetic algorithms. Compression results between genetic algorithm and optimization function is done. Also the optimization function give model to speed up genetic algorithm. Optimization function is a complementary transformation which maps a TSP given to linear without changing the roots of the polynomials.

The article discusses the problem of digital transformation of school mathematics education. Two traditions of mathematical education in Kazakhstan are considered in historical perspective: “factory” and creative, research. The key features of the current state of the school education system in Kazakhstan in the framework of the mass school system and alternative systems are considered. The system of state final and current student assessment, elements of the digital economy of education are characterized. The socio-economic and political prerequisites for the possibility of significant changes in the education system are formulated. The necessity is substantiated for the effective development of the country in the 21st century of personality qualities brought up in creative mathematical education in the context of tasks that are “not known how to solve”. It is shown that due to the use of digital technologies, in particular AI technologies, the process of transformation of the education system can be started in modern Kazakhstan. This transformation will be aimed at bringing the education system in line with the needs of the modern economy and modern society. The main components and dimensions of this process are considered. In particular, the necessary changes in the system of training and professional development of mathematics teachers and the necessary elements of continuity of this system with a comprehensive school are indicated.

Access to medicines is one of the essential problems in Public Health of low- and middle-income countries (LMICs). The World Health Organization (WHO) defines access to medicines as the possibility of "having continuously accessible and affordable medicines in public or private health facilities that are within a kilometer of the place of residence." Access to medicines, as defined by the WHO, is not fully guaranteed in many LMICs and even in many regions of high-resource countries. The WHO identifies several factors as determinants of limitations in the access to medicines: rational selection, affordable prices, sustainable financing, and reliable health services. The action on these factors makes it possible to improve universal access to medicines with consequent improvement in Public Health. Adequate access to antibiotics and vaccination will avoid a large part of the deaths caused by infectious diseases in the LMICs. However, the emergence of resistance and the difficulties in vaccination campaigns due to socio-political or cultural problems make it challenging to fight many easily treatable infectious diseases. The use and abuse of antibiotics are inevitably associated with the appearance of resistances that make them ineffective. Thus, whereas limited access to antibiotics raises mortality rates from infectious diseases, generalized open access to them ends up eliminating their clinical value. Moreover, the contraction of research in this field for many years has reduced the success in discovering new drugs. Additionally, local market regulations, inadequate selection, inaccessible prices, especially for those of second and third-generation, inefficient health systems, and difficulties of administration and control of prescription compliance, especially in the case of combined therapies, are additional obstacles to universal access to antibiotics. In order to simultaneously improve access to antibiotics and keep resistances under control, it is necessary to develop training and education activities at different social levels (from patients to various Health Care Providers) to complement the national or supranational strategic plans.

In real-life scenario, there are many mathematical tools to handle the incomplete and imprecise data. One of them is the fuzzy approach. This article aims to contribute to the literature of fuzzy multi-objective dynamic programming issues involving the fuzzy objective functions. The piecewise quadratic fuzzy numbers characterize these fuzzy parameters. Some basic notions in the problem under the α-pareto optimal solution concept is redefined and analyzed to study the stability of the problem. Furthermore, a technique, named as enhancing decomposition approach, is presented for achieving a subset for the parametric space that contains the sameα-pareto optimal solution. For a better understanding and comprehension of the suggested concept, a numerical example is provided.

The introduction of any new disruptive technology in a traditionally well-established industry, such as the road construction industry, is usually associated with considerable resistance. This is especially relevant when the new technology is based on the use of granular materials traditionally considered to be of an unacceptable quality in combination with relatively new concepts such as New-age (Nano) Modified Emulsions (NME). In such cases, the fact that the material design methods are based on fundamental scientific principles and have been proven in laboratories and through Accelerated Pavement Testing (APT), may be of little influence. However, the general acceptance of new disruptive technologies, e.g. telecommunications and Information Technologies (IT), have been based on the considerable advantages it presented. The same principles are applicable to the general acceptance and use of NME stabilisation/enhancement of materials in the road construction industry. This article is aimed at the practical demonstration of the advantages of the use of nanotechnologies in the construction of the highest order roads (i.e. inter-city multi-lane highways) to lower order access roads (i.e. Low-Volume-Roads (LVR) and even local accesses to farms and in villages/townships). The implementation of NME technologies is directly associated with ease of use, time and cost savings and the addressing and reduction of risks.

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.