Search Results (23)

We investigate the onset of hyperbolicity in Jensen polynomials $J_{d,n}$ associated with the Riemann $\Xi$-function and identify a robust parity-driven bifurcation with a natural geometric interpretation. Numerical analysis for degrees $5\le d\le 16$ reveals two distinct regimes. For even d, the roots form a compact complex cluster whose imaginary extent decreases smoothly, and the transition to hyperbolicity is governed by a single complex-conjugate pair, consistent with a low-dimensional (tame) geometric structure. For odd d, a hierarchy of more intricate onset mechanisms emerges, including single-event transitions ($d=11$) and intermittent regimes ($d\ge 13$) with decoupled geometric invariants, suggestive of dynamics on decorated (wild) character varieties. We interpret this dichotomy through a connection with the $P_{\mathrm{III}}\left(D_6\right)$ tau-function arising in the Painlevé confluence diagram. Defining $\tau \left(t\right)=\Xi ({\displaystyle \frac{1}{2}}+\sqrt{-t})/\Xi \left({\displaystyle \frac{1}{2}}\right)$, we construct a generating function $B\left(w\right)={\sum }_{j\ge 0}{b}_j{w}^j$ from the cumulants of $log\Xi ({\displaystyle \frac{1}{2}}+z)$ using high-precision Cauchy/DFT methods (280–400-digit arithmetic), without explicit use of the zero expansion. Two independent numerical diagnostics indicate that B exhibits Stieltjes-type behavior: (i) positivity of Hankel determinants up to order $N=30$ and (ii) Padé approximants whose poles converge to ${\gamma }_k^2$ (squares of Riemann-zero ordinates) with stabilizing residues. These results provide strong evidence that the parity bifurcation observed in Jensen polynomial onset reflects a finite-dimensional manifestation of an underlying moment-based positivity structure. Motivated by this correspondence, we formulate a conjecture relating the Stieltjes nature of $B\left(w\right)$ to the eventual hyperbolicity of Jensen polynomials. This conjecture suggests a bridge between finite-dimensional root geometry and an infinite-dimensional kernel-based positivity framework, while leaving open the problem of establishing such positivity independently of the zero expansion. Full article

In his seminars, Deleuze claims that Spinoza is ‘an absolute geometrist’. This article contextualizes, explains and substantiates this aspect of Deleuze’s interpretation of Spinoza. I position Deleuze’s reading within both the long-running scholarly debate on Spinoza’s relationship to mathematics and within the evolution of Deleuze’s own relation to Spinoza. Deleuze’s idea that Spinoza is a geometrist is shown to consist of three elements. First, according to Spinoza, geometry is more fundamental than arithmetic. Second, Spinoza frees geometry from the realm of fiction and abstract and develops, as Deleuze says, a ‘mathematics of the real’. Third, Spinoza finds in geometry a language of univocity, by which he can avoid the equivocity and hierarchy of the Aristotelian worldview. Full article

The imminent threat of large-scale quantum computers to modern public-key cryptographic devices has led to extensive research into post-quantum cryptography (PQC). Lattice-based schemes have proven to be the top candidate among existing PQC schemes due to their strong security guarantees, versatility, and relatively efficient operations. However, the computational cost of lattice-based algorithms—including various arithmetic operations such as Number Theoretic Transform (NTT), polynomial multiplication, and sampling—poses considerable performance challenges in practice. This survey offers a comprehensive review of hardware acceleration for lattice-based cryptographic schemes—specifically both the architectural and implementation details of the standardized algorithms in the category CRYSTALS-Kyber, CRYSTALS-Dilithium, and FALCON (Fast Fourier Lattice-Based Compact Signatures over NTRU). It examines optimization measures at various levels, such as algorithmic optimization, arithmetic unit design, memory hierarchy management, and system integration. The paper compares the various performance measures (throughput, latency, area, and power) of Field-Programmable Gate Array (FPGA) and Application-Specific Integrated Circuit (ASIC) implementations. We also address major issues related to implementation, side-channel resistance, resource constraints within IoT (Internet of Things) devices, and the trade-offs between performance and security. Finally, we point out new research opportunities and existing challenges, with implications for hardware accelerator design in the post-quantum cryptographic environment. Full article

Decision-making theory has developed over many decades at the intersection of economics, mathematics, psychology, and engineering. Its classical foundations include Bernoulli’s expected utility theory, von Neumann and Morgenstern’s rational choice theory, and the criteria proposed by Savage, Wald, Hurwicz, and others. However, in real-world contexts, decisions are made under uncertainty, incompleteness, and unreliability of information, which classical approaches do not adequately address. To overcome these limitations, modern multi-criteria decision-making methods such as Analytic Hierarchy Process (AHP), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), VlseKriterijumska Optimizacija I Kompromisno Resenje (Compromise solution approach) (VIKOR), and ELimination Et Choix Traduisant la REalité (Elimination and Choice Expressing Reality) (ELECTRE), as well as their fuzzy and Z-number extensions, are widely applied to the modeling and evaluation of complex systems. These Z-number extensions are based on the concept of Z-numbers introduced by Lotfi Zadeh in 2011 to formalize higher-order uncertainty. This study introduces the Z-Regret principle, which extends Savage’s regret criterion through the use of Z-numbers. Supported by Rafik Aliev’s mathematical justifications concerning arithmetic operations on Z-numbers, the model evaluates regret not only as a loss relative to the best alternative but also by incorporating the degree of confidence and reliability of this evaluation. Calculations for the selection of digital advertising platforms in terms of performance assessment under various scenarios demonstrate that the Z-Regret principle enables more stable and well-founded decision-making under uncertainty. Full article

By imposing finite order constraints on Fibonacci anyon braid relations, we construct the finite quotient $G={\mathbb{Z}}_5⋊2I$, where $2I$ is the binary icosahedral group. The Gröbner basis decomposition of its $SL(2,\mathbb{C})$ character variety yields elliptic curves whose L-function derivatives $L^{\prime }(E,1)$ remarkably match fundamental biological structural ratios. Specifically, we demonstrate that the Birch–Swinnerton-Dyer conjecture’s central quantity: the derivative $L^{\prime }(E,1)$ of the L-function at 1 encodes critical cellular geometries: the crystalline B-DNA pitch-to-diameter ratio ($L^{\prime }(E,1)=1.730$ matching $34\AA /20\AA =1.70$), the B-DNA pitch to major groove width ($L^{\prime }=1.58$) and, additionally, the fundamental cytoskeletal scaling relationship where $L^{\prime }(E,1)=3.570\approx 25/7$, precisely matching the microtubule-to-actin diameter ratio. This pattern extends across the hierarchy ${\mathbb{Z}}_5⋊2P$ with $2P\in \{2O,2T,2I\}$ (binary octahedral, tetrahedral, icosahedral groups), where character tables of $2O$ explain genetic code degeneracies while $2T$ yields microtubule ratios. The convergence of multiple independent mathematical pathways on identical biological values suggests that evolutionary optimization operates under deep arithmetic-geometric constraints encoded in elliptic curve L-functions. Our results position the BSD conjecture not merely as abstract number theory, but as encoding fundamental organizational principles governing cellular architecture. The correspondence reveals arithmetic geometry as the mathematical blueprint underlying major biological structural systems, with Gross–Zagier theory providing the theoretical framework connecting quantum topology to the helical geometries that are essential for life. Full article

A hierarchical structure of isomorphic arithmetics is defined by a bijection $g_{\mathbb{R}}:\mathbb{R}\to \mathbb{R}$. It entails a hierarchy of probabilistic models, with probabilities $p_k=g^k\left(p\right)$, where g is the restriction of $g_{\mathbb{R}}$ to the interval $[0,1]$, $g^k$ is the kth iterate of g, and k is an arbitrary integer (positive, negative, or zero; $g^0\left(x\right)=x$). The relation between p and $g^k\left(p\right)$, $k>0$, is analogous to the one between probability and neural activation function. For $k\ll -1$, $g^k\left(p\right)$ is essentially white noise (all processes are equally probable). The choice of $k=0$ is physically as arbitrary as the choice of origin of a line in space, hence what we regard as experimental binary probabilities, $p_{\exp }$, can be given by any k, $p_{\exp }=g^k\left(p\right)$. Quantum binary probabilities are defined by $g\left(p\right)={sin}^2{\displaystyle \frac{\pi }{2}}p$. With this concrete form of g, one finds that any two neighboring levels of the hierarchy are related to each other in a quantum–subquantum relation. In this sense, any model in the hierarchy is probabilistically quantum in appropriate arithmetic and calculus. And the other way around: any model is subquantum in appropriate arithmetic and calculus. Probabilities involving more than two events are constructed by means of trees of binary conditional probabilities. We discuss from this perspective singlet-state probabilities and Bell inequalities. We find that singlet state probabilities involve simultaneously three levels of the hierarchy: quantum, hidden, and macroscopic. As a by-product of the analysis, we discover a new (arithmetic) interpretation of the Fubini–Study geodesic distance. Full article

The theory of semisimple rings plays a fundamental role in noncommutative algebra. We study the complexity of the problem of semisimple rings using the tools of computability theory. Following the general idea of computably enumerable (c.e. for short) universal algebras, we define a c.e. ring as the quotient ring of a computable ring modulo a c.e. congruence relation and view such rings as structures in the language of rings, together with a binary relation. We formalize the problem of being semisimple for a c.e. ring by the corresponding index set and prove that the index set of c.e. semisimple rings is ${\mathsf{\Sigma }}_3^0$-complete. This reveals that the complexity of the definability of c.e. semisimple rings lies exactly in the ${\mathsf{\Sigma }}_3^0$ of the arithmetic hierarchy. As applications of the complexity results on semisimple rings, we also obtain the optimal complexity results on other closely connected classes of rings, such as the small class of finite direct products of fields and the more general class of semiperfect rings. Full article

Developing universities to become green and sustainable universities is important. This is in line with the world’s sustainable development guidelines. In developing a university towards being green and sustainable, different sustainability assessment criteria have been used, including the UI GreenMetric, GRI, STARS, AUN, THE’S Impact Ranking, and AISHE. Each criterion is designed in a broad-spectrum manner that is not specific. Therefore, this research aims to develop sustainable and green university assessment indicators that suit the national context in Thailand, both in terms of the size and location of universities. Based on the criteria from the UI GreenMetric, there are six categories, totaling 51 indicators. Thus, this study aims to analyze the weaknesses and strengths of each indicator using a questionnaire distributed by the Sustainable University Network of Thailand to all 36 universities and to organize a meeting with sustainability experts (focus group) in each area. To obtain appropriate indicators and to prioritize each indicator, the analytic hierarchy process (AHP) and weighted arithmetic mean (WAM) method were used to develop an index of sustainability assessment criteria (GU SI) suitable for the Thai context. The results of this study show that the newly developed GU SI comprises seven aspects, including a total of 27 indicators. These are criteria that can be used to assess the sustainability of small, medium, and large universities. They are not complex, are straightforward to use, and more importantly reflect the Thai context. By applying these new sustainability assessment criteria, it was found that the universities sampled in this study yielded significantly high scores. This study is limited by its application of sustainability assessments to only universities in Thailand. Though there are different contexts across different Thai universities, the proposed indicators can still be used to evaluate the sustainability of universities in Thailand. Full article

A model of set theory ZFC is defined in our recent research, in which, for a given $n\ge 3$, $\left({\mathsf{A}}_n\right)$ there exists a good lightface ${\mathsf{\Delta }}_n^1$ well-ordering of the reals, but $\left({\mathsf{B}}_n\right)$ no well-orderings of the reals (not necessarily good) exist in the previous class ${\mathsf{\Delta }}_{n-1}^1$. Therefore, the conjunction $\left({\mathsf{A}}_n\right)\wedge \left({\mathsf{B}}_n\right)$ is consistent, modulo the consistency of ZFC itself. In this paper, we significantly clarify and strengthen this result. We prove the consistency of the conjunction $\left({\mathsf{A}}_n\right)\wedge \left({\mathsf{B}}_n\right)$ for any given $n\ge 3$ on the basis of the consistency of ${\mathbf{PA}}_2$, second-order Peano arithmetic, which is a much weaker assumption than the consistency of ZFC used in the earlier result. This is a new result that may lead to further progress in studies of the projective hierarchy. Full article

We consider the problem of the existence of well-orderings of the reals, definable at a certain level of the projective hierarchy. This research is motivated by the modern development of descriptive set theory. Given $n\ge 3$, a finite support product of forcing notions similar to Jensen’s minimal-${\mathbf{\Delta }}_3^1$-real forcing is applied to define a model of set theory in which there exists a good ${\mathbf{\Delta }}_n^1$ well-ordering of the reals, but there are no ${\mathbf{\Delta }}_{n-1}^1$ well-orderings of the reals (not necessarily good). We conclude that the existence of a good well-ordering of the reals at a certain level $n\ge 3$ of the projective hierarchy is strictly weaker than the existence of a such well-ordering at the previous level $n-1$. This is our first main result. We also demonstrate that this independence theorem can be obtained on the basis of the consistency of ${\mathbf{ZFC}}^{-}$ (that is, a version of $\mathbf{ZFC}$ without the Power Set axiom) plus ‘there exists the power set of $\omega$’, which is a much weaker assumption than the consistency of $\mathbf{ZFC}$ usually assumed in such independence results obtained by the forcing method. This is our second main result. Further reduction to the consistency of second-order Peano arithmetic ${\mathbf{PA}}_2$ is discussed. These are new results in such a generality (with $n\ge 3$ arbitrary), and valuable improvements upon earlier results. We expect that these results will lead to further advances in descriptive set theory of projective classes. Full article

Aggregation methods in group decision-making refer to techniques used to combine the individual preferences, opinions, or judgments of group members into a collective decision. Each aggregation method has its advantages and disadvantages, and the best method to use depends on the specific situation and the goals of the decision-making process. In certain cases, final rankings of alternatives in the decision-making process may depend on the way of combining different attitudes. The focus of this paper is the application and comparative analysis of the aggregation operators, specifically, arithmetic mean (AM), geometric mean (GM), and Dombi Bonferroni mean (DBM), to the process of criteria weights determination in a fuzzy environment. The criteria weights are determined using Fuzzy Multi-Criteria Decision-Making (F-MCDM) methods, such as Fuzzy Analytic Hierarchy Process (F-AHP), Fuzzy Pivot Pairwise Relative Criteria Importance Assessment (F-PIPRECIA), and Fuzzy Full Consistency Method (F-FUCOM), while the final alternative ranking is obtained by Fuzzy Weighted Aggregated Sum Product Assessment (F-WASPAS). A comparison of aggregation operators is done for the real case of location selection problem for a used motor oil transfer station in the regional center of Southern and Eastern Serbia, the city of Niš. The results obtained in this study showed that the views of different experts and application of a certain aggregation approach may have a significant impact on the values of criteria weight coefficients and further on the final ranking of alternatives. This paper is expected to stimulate future research into the impact of aggregation methods on final rankings in the decision-making process, especially in the field of waste management. Full article

In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series. Full article

At present, diagnostics of the current technical condition of high-voltage power equipment in power systems have become more important. This allows the estimation of the real technical condition of power equipment more accurately with its removal into repair based on the results of the diagnostics. This paper presents the comparative analysis of expert evaluations with the use of the arithmetical mean and median values of expert evaluations. In this case, individual expert opinions, influenced by a level of competence, correspond to each other in a different manner, depending on the applied approach. As the comparison of the consistency of expert opinions is the basis for decision-making, it is recommended to make a decision on the technical condition using median estimations because these estimations are less subjected to distortions from single outliers of judgments. This provides more reliable information for making key decisions. Three approaches are considered in this paper: the method of arithmetical mean estimations, the method of median estimations based on the Kemeny median method, and the analytic hierarchy process of Saaty. The considered methods allow decisions on power equipment operation to be made very quickly; namely, if the power equipment is in an operable state and may remain operated, or it has considerable defects and should be removed from operation for routine maintenance, or it has reached the final technical state and needs to be removed from operation. Full article

This article deals with the Analytic Hierarchy Process (AHP) method, which can be calculated in several ways. The aim of the paper is to analyze and describe the AHP method as essential for strategic managerial decision-making to determine which method is efficient for the calculation and to set the proper order of criteria. In the contribution, we show how the AHP method can be used through different techniques. In the article, there are included methods that can be used in order to calculate the matrix in the AHP process for setting criteria. This study also focused on the accuracy of various methods used to compute AHP. The paper contains the procedure of using the Saaty method through the Excel program. The results of the research show that the most accurate method is the Saaty method. In comparison with the Saaty method is the geometric mean method with the slightest deviation (CI = 0.00010), followed by the Row sum of the adjusted Saaty matrix with deviation (CI = 0.00256), reverse sums of the Saaty matrix columns (CI = 0.00852), Arithmetic mean and Row sums of the Saaty matrix (CI = 0.01261). All of these methods are easy to calculate and can be performed without major mathematical calculations. The AHP method is often used with other methods such as SWOT, FUZZY, etc. The survey was carried out through an inquiry with managers who graduated from universities in Slovakia and showed that the respondents considered the Saaty method as the most complex and the most difficult. The geometric mean and average mean methods were regarded as the simplest methods. Respondents (44%) stated that they were able to use a program to calculate the AHP. Respondents (46%) had experience with some method related to the strategic managerial decision-making process. Managers (72%) regarded this skill as important for decision-making in their managerial position. The contribution of this paper is to show the advantages of the AHP method in its wide use in various fields. Full article

Novel spatial models for appraising arable land resources using data processing techniques can increase insight into agroecosystem services. Hence, the principal component analysis (PCA), hierarchal cluster analysis (HCA), analytical hierarchy process (AHP), fuzzy logic, and geographic information system (GIS) were integrated to zone and map agricultural land quality in an arid desert area (Matrouh Governorate, Egypt). Satellite imageries, field surveys, and soil analyses were employed to define eighteen indicators for terrain, soil, and vegetation qualities, which were then reduced through PCA to a minimum data set (MDS). The original and MDS were weighted by AHP through experts’ opinions. Within GIS, the raster layers were generated, standardized using fuzzy membership functions (linear and non-linear), and assembled using arithmetic mean and weighted sum algorithms to produce eight land quality index maps. The soil properties (pH, salinity, organic matter, and sand), slope, surface roughness, and vegetation could adequately express the land quality. Accordingly, the HCA could classify the area into eight spatial zones with significant heterogeneity. Selecting salt-tolerant crops, applying leaching fraction, adopting sulfur and organic applications, performing land leveling, and using micro-irrigation are the most recommended practices. Highly significant (p < 0.01) positive correlations occurred among all the developed indices. Nevertheless, the coefficient of variation (CV) and sensitivity index (SI) confirmed the better performance of the index developed from the non-linearly scored MDS and weighted sum model. It could achieve the highest discrimination in land qualities (CV > 35%) and was the most sensitive (SI = 3.88) to potential changes. The MDS within this index could sufficiently represent TDS (R² = 0.88 and Kappa statistics = 0.62), reducing time, effort, and cost for estimating the land performance. The proposed approach would provide guidelines for sustainable land-use planning in the studied area and similar regions. Full article