Probability Theory

If the ladder of educational opportunity rises high at the doors of some youth and scarcely rises at the doors of others, while at the same time formal education is made a prerequisite to occupational and social advance, then education may become the means, not of eliminating race and class distinctions, but of deepening and solidifying them." President Harry S. "If the rungs on the ladder are getting further apart -which they are, with growing income inequality -then it may become a lot harder to climb that ladder.

A probabilistic version of geometry is introduced. The fifth postulate of Euclid (Playfair's axiom) is adopted in the following probabilistic form: consider a line and a point not on the line-there is exactly one line through the point with probability P, where 0 ≤ P ≤ 1. Playfair's axiom is logically independent of the rest of the Hilbert system of axioms of the Euclidian geometry. Thus, the probabilistic version of the Playfair axiom may be combined with other Hilbert axioms. P = 1 corresponds to the standard Euclidean geometry; P = 0 corresponds to the elliptic-and hyperbolic-like geometries. 0 < P < 1 corresponds to the introduced probabilistic geometry. Parallel constructions in this case are Bernoulli trials. Theorems of the probabilistic geometry are discussed. Given a triangle and a line drawn from a vertex parallel to the opposite side, the event that this line is actually parallel occurs with probability P. Otherwise, the line may intersect the side or diverge. Parallelism is not transitive in the probabilistic geometry. Probabilistic geometry occurs on the surface with a stochastically variable Gaussian curvature. Alternative geometries adopting various versions of the probabilistic Playfair axiom are introduced. Probabilistic non-Archimedean geometry is addressed. Applications of the probabilistic geometry are discussed.

We examine the appearance of power-law behavior in rooted tree graphs in the context of river networks. It has long been observed that the tails of statistical distributions of upstream areas in river networks, measured above every link, obey a power-law relationship over a range of scales. We examine this behavior by considering a subset of all links, defined as those links which drain complete Strahler basins, where the Strahler order defines a discrete measure of scale, for self-similar networks with both deterministic and random topologies. We find an excellent power-law structure in the tail probabilities for complete Strahler basin areas, over many ranges of scale. We show analytically that the tail probabilities converge to a power law under the assumptions of (1) simple scaling of the distributions of complete Strahler basin areas and (2) application of Horton&#39;s law of stream numbers. The convergence to a power law does not occur for all underlying distributions, but for...

This book is titled Forecasting in Mathematics – Recent Advances, New Perspectives and Applications. Additionally, each time I work in the field of mathematical probability and statistics, I have the pleasure of tackling the knowledge, the theorems, the proofs, and the applications of the theory. In fact, each problem is like a riddle to be solved, a conquest to be won, and I am relieved and extremely happy when I find the solution. This proves two important facts: firstly, the power of mathematics and its models to deal with such problems and secondly the power of the human mind that is able to understand such problems and to tame such a wild concept that is randomness, probability, stochasticity, uncertainty, chaos, chance, and nondeterminism.

It gives me great pleasure to introduce as well as discuss, learn, solve, teach, and work with operator theory. This book discusses some fundamental aspects of the theory of operators and explores their use to solve a large array of problems. The book addresses many topics, including the paradigm of complex probability, principal parts extension for a Noether operator A, the sets of fractional operators and some of their applications, the total and partial differentials as algebraically manipulable entities, and more.

Probability theory is a branch of statistics that employs mathematical methods of collection, organization, and interpretation of data, with applications in practically all scientific areas. When working with probability theory, we analyze random phenomena and assess the likelihood that an event will occur. This book, Applied Probability Theory - New Perspectives, Recent Advances and Trends, discusses some
fundamental aspects of probability theory and explores its use to solve a large array of problems. Chapters address such topics as complex probability, the stability of algorithms in statistical modeling, the non-homogeneous Hofmann process, and more.

It gives me great pleasure to introduce this work on simulation modeling. In this book, Simulation Modeling – Recent Advances, New Perspectives, and Applications, we discuss some fundamental aspects of the theory of numerical methods of simulation and explore their use to solve a large array of problems. Therefore, we cover many topics on deterministic and random techniques.
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the simulation represents the evolution of the model over time. Often, computers are used to execute the simulation.
Additionally, simulation is used in many contexts, such as simulation of technology for performance tuning or optimizing, safety engineering, testing, training, education, and video games. Simulation is also used with scientific modeling of natural systems or human systems to gain insight into their functioning, as in economics.
Simulation can be used to show the eventual real effects of alternative conditions and courses of action. Simulation is also used when the real system cannot be engaged because it may not be accessible, or it may be dangerous or unacceptable to engage, or it is being designed but not yet built, or it may simply not exist.
Moreover, key issues in modeling and simulation include the acquisition of valid sources of information about the relevant selection of key characteristics and behaviors used to build the model, the use of simplifying approximations and assumptions within the model, and fidelity and validity of the simulation outcomes. Procedures and protocols for model verification and validation are an ongoing field of academic
study, refinement, research, and development in simulations technology or practice, particularly in the work of computer simulation.
Finally, this book discusses some important aspects of simulation modeling and explores its use to solve a large set of problems. As such, the book will be of interest to all scholars, researchers, and undergraduate and graduate students in pure and applied mathematics, physical sciences, engineering and technology, computer science, numerical analysis, scientific computing, and science in general.

The present research aims to examine the different accounts of induction given by Aristotle, Hume, Leibniz, Carnap and De Finetti, trying to support that probability calculus offers a sufficient grounding of inductive logic. The term induction had been contrasted to deduction, by Aristotle. The Neoplatonic philosopher Alcinous suggested that dialectic firstly investigates the substances and then the accidents. There are five kinds of dialectic reasoning: division, definition, analysis, induction and syllogistic. The first three concern with substances, the last two with accidents. Although Leibniz regarded probability theory as a basis of inductive logic, Hume's skepticism was seminal for the reappraisal of the role of induction in modern philosophy. Enhancing Hume's criticism, Popper and Wittgenstein completely denied that scientists use induction. Hans Reichenbach, however, attempted to build a theory of justification for the use of induction, based on a factual basis of other successful predictive methods that make induction feasible (Earman & Salmon 1999). Moreover, Buchdall (1969) stressed that we must distinguish the inductive process of the scientist from the inductive conclusion, which comes after the completion of observation and experimentation.

Constraint-based causal structure learning algorithms rely on the faithfulness pro- perty. For faithfulness, all conditional in- dependencies should come from the sys- tem&#39;s causal structure. The problem is that even for linear Gaussian models the pro- perty is not tenable. In this paper, we identify 4 non-causal properties that gene- rate conditional independencies and inves- tigate whether they can be recognized by patterns in the conditional independencies.

The paper deals with a load-balancing algorithm for a closed stochastic network with two zones with different demands. The algorithm is motivated by an incentive algorithm for redistribution of cars in a large-scale car-sharing system. The service area is divided into two zones. When cars stay too much long in the low-demand zone, users are encouraged to pick up them and return them in the high-demand zone. The zones are divided in cells called stations. The cars are the network customers. The mean-field limit solution of an ODE gives the large scale distribution of the station state in both clusters for this incentive policy in a discrete Markovian framework. An equilibrium point of this ODE is characterized via the invariant measure of a random walk in the quarter-plane. The proportion of empty and saturated stations measures how the system is balanced. Numerical experiments illustrate the impact of the incentive policy. Our study shows that the incentive policy helps when the high-demand zone observes a lack of cars but a saturation must be prevented especially when the high-demand zone is small.

Risk assessment is an important and significant aid in the decision making process. Risk assessment is performed using 'model' and a model is a function of parameters which are usually affected by uncertainty. Some model parameters are affected by aleatory uncertainty and some others are affected by epistemic uncertainty. In this paper we propose a hybrid method to deal with propagation of both kinds of uncertainty within the same computation of risk.

In this paper the problem of testing a multivariate point hypothesis is considered. Of interest is the relationship between the p-value and the posterior probability. A Bayesian test for simple H 0 : θ = θ 0 versus bilateral H 0 : θ = θ 0 , with a mixed prior distribution for the parameter θ, is developed. The methodology consist on fixing a sphere of radius δ around θ 0 and assigning a prior mass, π 0 , to H 0 by integrating the density π(θ) over this sphere and spreading the remainder, 1-π 0 , over H 1 according to π(θ). A theorem that shows when the frequentist and Bayesian procedures can give rise to the same decision is proved. Then, some examples are revisited.

The concept of mathematical probability was established in 1933 by
Andrey Nikolaevich Kolmogorov through five axioms. The Complex Probability Paradigm enhances this foundation by introducing three additional axioms, thereby extending the real probability space R into a complex universe C = R + M, where M denotes an imaginary probability set. This extension incorporates supplementary imaginary dimensions into any stochastic experiment, enabling the computation of probabilities across R, M, and C. Within the complex probability set C, the total probability is invariantly unity, regardless of the input distribution in the real domain. Consequently, phenomena that are inherently random in R become fully predictable in C. Determinism is achieved by subtracting a chaotic factor (Chf) from the degree of our knowledge (DOK), both encoded in the complex probability vector Z = Pr + Pm = Pr +i(1− Pr ) (with Pr in R and Pm in M). CPP has been applied to classical results and methods – most notably the Central Limit Theorem – where convergence is reformulated deterministically in C. Additional applications include Monte Carlo simulations, Brownian motion analyses, and the reformulation of Markov chains, laying the groundwork for a fully deterministic interpretation of stochastic processes. Here, we apply this innovative paradigm to the well-known Abraham De Moivre problem and to its random algorithm and procedure in a novel way.

Expander graphs are highly connected sparse finite graphs. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. For practical applications it is very important to construct expander and Ramanujan graphs with given regularity and order. In general, constructions of the best expander graphs with a given regularity and order is no easy task. In this paper we present algorithms for generation of Ramanujan graphs and other expanders. We describe properties of obtained graphs in comparison to previously known results. We present a method to obtain a new examples of irregular LDPC codes based on described graphs and we briefly describe properties of this codes.

In this paper we give examples which show that the convolution theorem (Boll, (1), Hajek, (2)) cannot be extended to infinite-dimensional shift experi- ments. This answers a question posed by van der Vaart, (9), and which has been considered also by LeCam, (5).

In this paper, the eight truth values of U8 have been embedded in probability values to form a probability logic. The rules of probability are then given form in terms of the primary truth values {T, F, N}. In particular, negative and positive logic can be employed to calculate the primary truth values {T, F, N} for implication and the additive, multiplicative and complementary rule of probability. Bayesian probability is employed to describe implication and conditional probability and the Boolean rule for material implication, A OR B, is found to be irrelevant to U8.

Like many discussions on the pros and cons of epistemic foundationalism, the debate between C.I. Lewis and H. Reichenbach dealt with three concerns: the existence of basic beliefs, their nature, and the way in which beliefs are related. In this paper we concentrate on the third matter, especially on Lewis's assertion that a probability relation must depend on something that is certain, and Reichenbach's claim that certainty is never needed. We note that Lewis's assertion is prima facie ambiguous, but argue that this ambiguity is only apparent if probability theory is viewed within a modal logic. Although there are empirical situations where Reichenbach is right, and others where Lewis's reasoning seems to be more appropriate, it will become clear that Reichenbach's stance is the generic one. This follows simply from the fact that, if P(E|G) > 0 and P(E|¬G) > 0, then P(E) > 0. We conclude that this constitutes a threat to epistemic foundationalism.

According to the suppositional theory of conditionals, people assess their belief in a conditional statement of the form “if p then q” by conducting a mental simulation on the supposition of p in which they assess their degree of belief in q. This leads to them to the judge the probability of a conditional statement to be equal to the conditional probability, P(q|p). Evidence for this conditional probability hypothesis has been adduced in earlier studies for abstract, causal, and counterfactual conditionals. For the realistic conditionals, it is natural to assume that people perform such mental simulations by building causal mental models from prior causes to later effects. However, in the present study we show that the conditional probability hypothesis extends to diagnostic conditionals, which relate effects to causes. This new finding presents a major challenge for theoretical accounts of the mental processing of conditional statements.

In order to estimate causal relations, the time series econometrics has to be aware of spurious correlation, a problem first mentioned by Yide [21], To solve the problem, one can work with differenced series or use midtivariate models like VAR or VEC models. In this case, the analysed series are going to present a long run relation i.e. a cointegration relation. Even though the Bayesian literature about inference on VARA'EC models is qiute advanced, Bauwens et al. [2] highlight that "the topic of selecting the cointegrating rank has not yet given very useful and convincing results". This paper presents the Fidl Bayesian Significance Test applied to cointegration rank selection tests in multivariate (VAR/VEC) time series models and shows how to implement it using available in the literature and simulated data sets. A standard non-informative prior is assumed.

The Fully Bayesian Significance Test (FBST) is a coherent Bayesian significance test for sharp hypotheses. This paper explores the FBST as a model selection tool for general mixture models, and gives some computational experiments for Multinomial-Dirichlet-Normal-Wishart models.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

The main purpose of this article is to introduce the Dempster-Shafer (DS) Theory of Belief Functions. The DS Theory is founded on the mathematical theory of probability and is a broader framework than the probability theory. It reduces to probability theoryunder a special condition. In addition, the article illustrates problems in representing pure positive, pure negative evidence, and ambiguity under probability theory and shows how this problem is resolved under DS Theory. Next, the article describes and illustrates Dempster’s rule of combination to combine two or more items of evidence. Also, the article introduces Evidential Reasoning approach and its applications to various disciplines such as accounting, auditing, information systems, and information quality. Examples are provided where DS Theory is being used for developing AI and Expert systems within the business disciplines and outside of the business disciplines. 

Multilingualism is generally treated as a catalyst for the careers of individuals and as an asset for entire societies competing on an economic level. It may be a marker of social prestige and a requirement for a good job. It is also believed to open the eyes of citizens to diversity and to promote tolerance. But is such an image not a simplification?
Most of the multilingual and multicultural societies around the world are not as affluent as the inhabitants of Luxembourg or Switzerland. It is also true that multilingual Yugoslavia or Rwanda suffered greatly from their multiculturalism. The largest economy in the world (USA) is practically monolingual while the second largest is linguistically very diverse
(China). The languages spoken in metropolises in post-colonial countries are sometimes an asset and a valued skill (English in India) but also undesirable political baggage (Russian in the Baltic States). Actually, multilingualism is a complex phenomenon that can involve different types and levels of competence (e.g. understanding, speaking, writing, communicating) as well as languages that differ in their economic and cultural power. The first part of the article defines the basic concepts of the analysis (language, multilingualism, information economics). It then formulates the two research hypotheses to be tested, together with a description of the data sources.
The first hypothesis concerns the relation between multilingualism (linguistic diversity) and well-being while the second refers to the relation between the level of English spoken in a society and its prosperity. Greenberg’s Language Diversity Index and the number of official languages in a country were used as a measure of linguistic diversity and the GDP per capita value as a measure of prosperity. The analysis showed that, on a global scale, there is no correlation between social multilingualism and the welfare of a country while the correlation between the level of English and the GDP per capita value is positive and significant.

We investigate the statistical parity of a class of chaos-generated noises on the escape of strongly damped particles out of a potential well. We show that statistical asymmetry in the chaotic fluctuations can lead to a skewed Maxwell-Boltzmann distribution in the well. Depending on the direction of skew, the Kramers escape rate is enhanced or suppressed accordingly. Based on the Perron-Frobenious equation, we determine an analytical expression for the escape rate's prefactor that accounts for this effect. Furthermore, our perturbative analysis proves that in the zeroth-order limit, the rate of particle escape converges to the Kramers rate.

This paper presents the complete framework of the "Multidimensional Truth and Falsity" (MTF) theory, a revolutionary ontological system that evolves from traditional binary logic to infinite-dimensional field networks. The MTF theory has undergone theoretical development from multidimensional vector models to Infinite Binary-Quantified Fields (IBQF), and further to cosmic parent-child set hypernetworks. The core innovation lies in redefining "truth" as a dynamic probability field over infinite-dimensional contextual space, forming stable or changing truth value states through inter-field coupling weights. This paper integrates rigorous mathematical formalization, interdisciplinary theoretical foundations, and practical application prospects, aiming to provide a new theoretical paradigm for reasoning, cognitive modeling, and artificial intelligence systems in complex uncertain environments. This framework transcends the limitations of traditional static logic, establishing a dynamic, computable, and adjustable logical architecture with the potential to redefine the essence of truth evaluation.

This work deals with a general problem of testing multiple hypotheses about the distribution of a discrete-time stochastic process. Both the Bayesian and the conditional settings are considered. The structure of optimal sequential tests is characterized.

Conscious states are characterized by their phenomenal feel; it is like something to experience pain, see the color red, or hear a particular sound. The version of the argument from consciousness that will be discussed in the present paper states that only theism, the postulation of a transcendent God, can explain why consciousness exists in the universe. The argument from consciousness thus takes the very existence of consciousness in the universe as a starting point and concludes that there is a transcendent God. First, a detailed description of AC is provided. Next, some problems with the argument from consciousness are pointed out, and the conclusion is that consciousness is most likely a brute fact-an unexplainable phenomenon that does not raise the probability of theism.

We perform numerical optimization of the axisymmetric flows in a sphere to minimize the critical magnetic Reynolds number Rm cr required for dynamo onset. The optimization is done for the class of laminar incompressible flows of von Kármán type satisfying the steady-state Navier-Stokes equation. Such flows are determined by equatorially antisymmetric profiles of driving azimuthal (toroidal) velocity specified at the spherical boundary. The model is relevant to the Madison plasma dynamo experiment (MPDX), whose spherical boundary is capable of differential driving of plasma in the azimuthal direction. We show that the dynamo onset in this system depends strongly on details of the driving velocity profile and the fluid Reynolds number Re. It is found that the overall lowest Rm cr ≈ 200 is achieved at Re ≈ 240 for the flow, which is hydrodynamically marginally stable. We also show that the optimized flows can sustain dynamos only in the range Rm cr < Rm < Rm cr2 , where Rm cr2 is the second critical magnetic Reynolds number, above which the dynamo is quenched. Samples of the optimized flows and the corresponding dynamo fields are presented.

A space of pseudoquotients P(X, S) is defined as equivalence classes of pairs (x, f ), where x is an element of a non-empty set X, f is an element of S, a commutative semigroup of injective maps from X to X, and (x, f ) ∼ (y, g) if gx = f y. In this note we assume that (X, Σ, µ) is a measure space and that S is a commutative semigroup of measurable injections acting on X and investigate under what conditions there is an extension of µ to P(X, S).

A space of pseudoquotients $\mathcal P (X,S)$ is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f) \sim (y,g)$ if $gx=fy$. In this note we assume that $(X,\Sigma,\mu)$ is a measure space and that $S$ is a commutative semigroup of measurable injections acting on $X$ and investigate under what conditions there is an extension of $\mu$ to $\mathcal P (X,S)$.

Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a Schnyder wood that has proven powerful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a "triangle-reversing" chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. It was shown previously that this chain connects the state space and we show that (i) when restricted to planar triangulations of maximum degree six, the Markov chain is rapidly mixing, and (ii) there exists a triangulation with high degree on which this Markov chain mixes slowly. Next, we consider an "edge-flipping" chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. It was also shown previously that this chain connects the state space and we prove that the chain is always rapidly mixing. The triangle-reversing and edge-flipping Markov chains both arise in the context of sampling other combinatorial structures, such as Eulerian orientations and triangulations of planar pointsets, so our results here may shed light on the mixing rate of these related chains as well.

Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a Schnyder wood that has proven powerful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a &quot;triangle-reversing&quot; chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. It was shown previously that this chain connects the state space and we show that (i) when restricted to planar triangulations of maximum degree six, the Markov chain is rapidly mixing, and (ii) there exists a triangulation with high degree on which this Markov chain mixes slowly. Next, we consider an &quot;edge-flipping&quot; chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixe...

This article introduces a method for exploring liminal horror environments using a modified Wave Function Collapse algorithm. This approach employs real-time generated regions to create infinite, non-deterministic environments, removing the feeling of security due to familiarity. An eventdriven system further reinforces unpredictability by dynamically integrating horror experiences. While still in an exploratory stage, this method provides a foundation for future research on how architectural structures, unpredictability, and environmental cues contribute to unsettling experiences in games.

The U logic system is a set of four logics termed: U8, U4, U2 and U0. U8 is a unique many-valued non-classical logic system in eight variables that more closely models human reasoning. The advantage of U8 is that it can model irrelevancy, N, uncertainty, U, contradiction, 0,  Non-truth, /T, Non-falsity, /F, Non-neutrality, /N, as well as true, T, and false, F, variables.

It has been shown that U8 is sound because the modus ponens inference rule evaluates to designated values and all axioms are tautologies. Furthermore, a Canonical Model construction and proof by contradiction have been employed to show that U8 is complete. As U8 is sound and complete it qualifies as a robust formal logic.

Los físicos más eminentes del último siglo han reflexionado sobre el significado de la más nueva de todas las disciplinas, la física cuántica, sin llegar a determinar con claridad sus postulados filosóficos, o sea, los principios universales que expliquen los resultados incuestionables de los experimentos de esa ciencia. En el presente trabajo se aborda una interpretación de la física moderna a la luz de las teorías cosmológicas de Aristóteles y Wolgang Smith. El objetivo del trabajo es analizar los postulados de la física moderna bajo la luz de las teorías cosmológicas de los filósofos antes citados, identificando sus principios cosmológicos generales e interpretando los fenómenos de la física moderna desde esas perspectivas. El método utilizado es la recolección e interpretación bibliográfica y la confrontación de los principios filosóficos. Luego de la lectura de extensa bibliografía se procedió a elaborar una explicación fundada en la filosofía realista de las más importantes paradojas de la física moderna, que contrasta con la visión positivista de la ciencia y la filosofía de la ciencia poniendo en duda sus principales postulados. La ciencia moderna cae entonces en un positivismo cerrado o en su defecto en una concepción cartesiana de la realidad, negando parte de ella. Es por ello que se hace necesaria una nueva perspectiva, que traiga una respuesta desde la cosmología aristotélica planteada ya varios siglos antes.

This paper revisits the CP=PC problem: the identification of the probability of a conditional with the corresponding conditional probability. Working with a fixed classical probability space and a minimal implicational backbone (rules (p1)-(p4) plus Modus Ponens), we prove a collapse theorem: if a global CP=PC identity is imposed for a single implicational connective and the logic validates a natural symmetry (A-B ⊢ B-A), then the probability assignment degenerates on the positive-probability fragment, trivializing the symmetric extension. We further show that moving to Łukasiewicz logic does not repair the identification: under truth-averaging semantics, the probability of A ⇒ B is a nonlinear expectation that generally diverges from Pr(B | A). The upshot is methodological: CP=PC should not be treated as a law of an implicational connective. Instead, we propose a division of roles: keep the implicational connective for inference, and introduce a dedicated conditional (B | A) whose semantics guarantees that its probability equals Pr(B | A). This reframes the Stalnaker program by separating inferential structure from probabilistic conditionalization and locating CP=PC in the semantics of an internal conditional rather than in the implicational fragment.

This paper explores the emerging field of quantum cognition and its relevance to human decision-making processes. Drawing on mathematical frameworks from quantum theory, it provides an alternative to classical models that struggle to account for order effects, preference reversals, and contextual dependencies. By analyzing key theories and empirical findings, the study highlights how quantum probabilistic structures offer more accurate and nuanced models of human reasoning under uncertainty. It concludes with a critical discussion on the interpretive challenges, practical applications, and future directions for integrating quantum cognition within psychological science.

We investigate spectral statistics of Riemann zeta zeros and their relation to prime-modulated Hamiltonians within the Universal Model Framework (UMF). Using authenticated Odlyzko zeros (indices 10,000–10,599), we confirm GUE universality across multiple metrics: spacing ratios ⟨r⟩ = 0.601 ± 0.012, pair correlation consistent with the sine-kernel, and logarithmic growth of spectral rigidity.

Prime-modulated tight-binding operators display intermediate statistics (⟨r⟩ = 0.543 ± 0.019), while Paley graph Laplacians yield improved agreement (⟨r⟩ ≈ 0.60), suggesting arithmetic structure plus correct symmetry class is essential for universality.

We develop a constructive Li-coefficient positivity framework inspired by Li’s criterion and Nyman–Beurling theory, transforming numerical verification into proof obligations through four theorem pathways. Quantitative falsifiability criteria and a 30/60/90-day development roadmap with acceptance gates are provided.

These results refine the UMF–Hilbert–Pólya program, positioning prime-indexed operators as candidate bridges between number theory, spectral chaos, and physics.

Les techniques d’échantillonnage en recherche quantitative font appel a trois critères : la marge d'erreur, le niveau de confiance et le degré de variabilité de la population-mere. On attribue souvent par de faut une marge d'erreur de 5% et un niveau de confiance de 95% lors de l’échantillonnage. Cependant, on peut convoquer d'autres plages de marge d'erreur et de niveau de confiance pour de terminer la taille de l’échantillon. Cette taille est le nombre d’unités qu'il faut atteindre absolument lors de l'administration du questionnaire quantitatif, l'outil ope rationnel de l’enquête. Hormis la détermination de la taille de l’échantillon en se servant des formules classiques, les tables d’échantillonnage proposent des valeurs d’échantillon correspondants a la population de référence. L'approche de la population-me re qui constitue de fait l’échantillon est abordée pour certains cas rencontres dans les programmes de développement.

In this present study, we propose the concept of tricomplex-controlled metric spaces as a generalization of both controlled metric-type spaces and tricomplex metric-type spaces. In this work, we establish fixed point results using Banach, Kannan and Fisher-type contractions supported with nontrivial examples in the setting of the proposed space. We apply the derived result to find the analytical solution of an integral equation using the fixed point technique under the same metric.

We investigate a question posed by Diaconis & Fulman (2023) regarding modular in-shuffling and the distribution of card positions in binary form. Specifically, we consider the sequence 𝑋 𝑘 , which records whether the position of a tracked card lies in the top half of a deck after 𝑘 iterations of the modular in-shuffle map 𝐼(𝑗) = 2𝑗 mod 𝑝, where 𝑝 is a is a prime for which 2 is a primitive root. We prove that the random sequence {𝑋 𝑘 } is equidistributed and asymptotically behaves like a sequence of independent and identically distributed Bernoulli(1 2) random variables. We provide both theoretical justification and empirical evidence with computed values of 𝑋 𝑘 for large 𝑝 supporting this behavior.

This is a review of global sensitivity indices that were introduced in I.M. Sobol ' (1990). These indices allow to analyze numerically the structure of a nonlinear function defined analytically or by a "black box". As an example the Brownian bridge is considered and an example of the application of global sensitivity indices in finance is presented.

Numerical methods for the Dirichlet problem for linear parabolic stochastic partial differential equations are constructed. The methods are based on the averaging-over-characteristic formula and the weak-sense numerical integration of ordinary stochastic differential equations in bounded domains. Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. The Monte Carlo technique is used for practical realization of the methods. Results of some numerical experiments are presented.

Numerical methods for the Dirichlet problem for linear parabolic stochastic partial differential equations are constructed. The methods are based on the averaging-over-characteristic formula and the weak-sense numerical integration of ordinary stochastic differential equations in bounded domains. Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. The Monte Carlo technique is used for practical realization of the methods. Results of some numerical experiments are presented.