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Journal articles on the topic 'Probabilistic number theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 June 2025

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1

Stakėnas, Vilius. "The probabilistic number theory and continuum." Lietuvos matematikos rinkinys, no. III (December 17, 1999): 93–99. http://dx.doi.org/10.15388/lmr.1999.35485.

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Let bn be a sequence of real numbers increasing unboundedly, α > 0 and B(n, α) = [bn α]. The conditions on bn are considered, which imply the regularity of distribution of B(n, α) in arithmetic progressions for almost all α > 0. This allows to develop a piece of probabilistic number theory on sequences B(n, α) for almost all α > 0.

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2

Indlekofer, K. H. "Number theory—probabilistic, heuristic, and computational approaches." Computers & Mathematics with Applications 43, no. 8-9 (April 2002): 1035–61. http://dx.doi.org/10.1016/s0898-1221(02)80012-8.

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3

Stakėnas, V. "On some inequalities of probabilistic number theory." Lithuanian Mathematical Journal 46, no. 2 (April 2006): 208–16. http://dx.doi.org/10.1007/s10986-006-0022-2.

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4

Erdõs, P. "Recent problems in probabilistic number theory and combinatorics." Advances in Applied Probability 24, no. 4 (December 1992): 766–67. http://dx.doi.org/10.1017/s0001867800024654.

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5

Stakėnas, Vilius. "Jonas Kubilius and genesis of probabilistic number theory." Lithuanian Mathematical Journal 55, no. 1 (January 2015): 25–47. http://dx.doi.org/10.1007/s10986-015-9263-2.

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6

Zhang, Wen-Bin. "Probabilistic number theory in additive arithmetic semigroups II." Mathematische Zeitschrift 235, no. 4 (December 1, 2000): 747–816. http://dx.doi.org/10.1007/s002090000165.

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7

Kátai, Imre, Bui Minh Phong, and Le Manh Thanh. "Some results and problems in probabilistic number theory." Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio computatorica, no. 43 (2014): 253–65. https://doi.org/10.71352/ac.43.253.

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8

Elliott, P. D. T. A. "Jonas Kubilius and Probabilistic Number Theory Some Personal Reflections." Lithuanian Mathematical Journal 55, no. 1 (January 2015): 2–24. http://dx.doi.org/10.1007/s10986-015-9262-3.

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9

Daili, Noureddine. "DENSITIES AND NATURAL INTEGRABILITY. APPLICATIONS IN PROBABILISTIC NUMBER THEORY." JP Journal of Algebra, Number Theory and Applications 47, no. 1 (July 1, 2020): 51–65. http://dx.doi.org/10.17654/nt047010051.

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10

Lokutsievskiy, Lev V. "Optimal probabilistic search." Sbornik: Mathematics 202, no. 5 (May 31, 2011): 697–719. http://dx.doi.org/10.1070/sm2011v202n05abeh004162.

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11

Aistleitner, Christoph, and Christian Elsholtz. "The Central Limit Theorem for Subsequences in Probabilistic Number Theory." Canadian Journal of Mathematics 64, no. 6 (December 1, 2012): 1201–21. http://dx.doi.org/10.4153/cjm-2011-074-1.

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Abstract Let (nk)k≥1 be an increasing sequence of positive integers, and let f (x) be a real function satisfyingIf the distribution ofconverges to a Gaussian distribution. In the casethere is a complex interplay between the analytic properties of the function f , the number-theoretic properties of (nk)k≥1, and the limit distribution of (2).In this paper we prove that any sequence (nk)k≥1 satisfying lim contains a nontrivial subsequence (mk)k≥1 such that for any function satisfying (1) the distribution ofconverges to a Gaussian distribution. This result is best possible: for any ε > 0 there exists a sequence (nk)k≥1 satisfying lim such that for every nontrivial subsequence (mk)k≥1 of (nk)k≥1 the distribution of (2) does not converge to a Gaussian distribution for some f.Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

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12

Haik, Sumayra. "AFFINE LINES AND ADVANCED PROBABILISTIC MODEL THEORY." Mathematical Statistician and Engineering Applications 70, no. 2 (February 26, 2021): 01–14. http://dx.doi.org/10.17762/msea.v70i2.8.

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Letjbe an almost surely algebraic, meromorphic domain.Recent developments in singular logic [36, 22] have raised the question ofwhether every pseudo-Dedekind number is ultra-Gaussian and singular.We show tha

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13

Barát, Anna, and Erdener Kaya. "On the models of Indlekofer and Kubilius in probabilistic number theory." Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio computatorica, no. 39 (2013): 17–34. https://doi.org/10.71352/ac.39.017.

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14

Keller, Frank, and Ash Asudeh. "Probabilistic Learning Algorithms and Optimality Theory." Linguistic Inquiry 33, no. 2 (April 2002): 225–44. http://dx.doi.org/10.1162/002438902317406704.

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This article provides a critical assessment of the Gradual Learning Algorithm (GLA) for probabilistic optimality-theoretic (OT) grammars proposed by Boersma and Hayes (2001). We discuss the limitations of a standard algorithm for OT learning and outline how the GLA attempts to overcome these limitations. We point out a number of serious shortcomings with the GLA: (a) A methodological problem is that the GLA has not been tested on unseen data, which is standard practice in computational language learning. (b) We provide counterexamples, that is, attested data sets that the GLA is not able to learn. (c) Essential algorithmic properties of the GLA (correctness and convergence) have not been proven formally. (d) By modeling frequency distributions in the grammar, the GLA conflates the notions of competence and performance. This leads to serious conceptual problems, as OT crucially relies on the competence/performance distinction.

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15

Lazorec, Mihai-Silviu. "Probabilistic aspects of ZM-groups." Communications in Algebra 47, no. 2 (October 16, 2018): 541–52. http://dx.doi.org/10.1080/00927872.2018.1482310.

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16

Heijungs, Reinout. "On the number of Monte Carlo runs in comparative probabilistic LCA." International Journal of Life Cycle Assessment 25, no. 2 (October 22, 2019): 394–402. http://dx.doi.org/10.1007/s11367-019-01698-4.

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Abstract Introduction The Monte Carlo technique is widely used and recommended for including uncertainties LCA. Typically, 1000 or 10,000 runs are done, but a clear argument for that number is not available, and with the growing size of LCA databases, an excessively high number of runs may be a time-consuming thing. We therefore investigate if a large number of runs are useful, or if it might be unnecessary or even harmful. Probability theory We review the standard theory or probability distributions for describing stochastic variables, including the combination of different stochastic variables into a calculation. We also review the standard theory of inferential statistics for estimating a probability distribution, given a sample of values. For estimating the distribution of a function of probability distributions, two major techniques are available, analytical, applying probability theory and numerical, using Monte Carlo simulation. Because the analytical technique is often unavailable, the obvious way-out is Monte Carlo. However, we demonstrate and illustrate that it leads to overly precise conclusions on the values of estimated parameters, and to incorrect hypothesis tests. Numerical illustration We demonstrate the effect for two simple cases: one system in a stand-alone analysis and a comparative analysis of two alternative systems. Both cases illustrate that statistical hypotheses that should not be rejected in fact are rejected in a highly convincing way, thus pointing out a fundamental flaw. Discussion and conclusions Apart form the obvious recommendation to use larger samples for estimating input distributions, we suggest to restrict the number of Monte Carlo runs to a number not greater than the sample sizes used for the input parameters. As a final note, when the input parameters are not estimated using samples, but through a procedure, such as the popular pedigree approach, the Monte Carlo approach should not be used at all.

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Corob Cook, Ged, and Matteo Vannacci. "Probabilistic finiteness properties for profinite groups." Journal of Algebra 574 (May 2021): 584–616. http://dx.doi.org/10.1016/j.jalgebra.2021.01.026.

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18

Ragot, Jean-François. "Probabilistic absolute irreducibility test for polynomials." Journal of Pure and Applied Algebra 172, no. 1 (July 2002): 87–107. http://dx.doi.org/10.1016/s0022-4049(01)00165-7.

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19

Dvurecenskij, Anatolij, та Jiri Rachunek. "Probabilistic Averaging in Bounded Rℓ-Monoids". Semigroup Forum 72, № 2 (25 березня 2006): 191–206. http://dx.doi.org/10.1007/s00233-005-0545-6.

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20

Guralnick, Robert M., and William M. Kantor. "Probabilistic Generation of Finite Simple Groups." Journal of Algebra 234, no. 2 (December 2000): 743–92. http://dx.doi.org/10.1006/jabr.2000.8357.

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21

KUBOTA, Hisayoshi, and Hiroshi SUGITA. "PROBABILISTIC PROOF OF LIMIT THEOREMS IN NUMBER THEORY BY MEANS OF ADELES." Kyushu Journal of Mathematics 56, no. 2 (2002): 391–404. http://dx.doi.org/10.2206/kyushujm.56.391.

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22

Thiagarajan, P. S., and Shaofa Yang. "A Theory of Distributed Markov Chains." Fundamenta Informaticae 175, no. 1-4 (September 28, 2020): 301–25. http://dx.doi.org/10.3233/fi-2020-1958.

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We present the theory of distributed Markov chains (DMCs). A DMC consists of a collection of communicating probabilistic agents in which the synchronizations determine the probability distribution for the next moves of the participating agents. The key feature of a DMC is that the synchronizations are deterministic, in the sense that any two simultaneously enabled synchronizations involve disjoint sets of agents. Using our theory of DMCs we show how one can analyze the behavior using the interleaved semantics of the model. A key point is, the transition system which defines the interleaved semantics is—except in degenerate cases—not a Markov chain. Hence one must develop new techniques to analyze these behaviors exhibiting both concurrency and stochasticity. After establishing the core theory we develop a statistical model checking procedure which verifies the dynamical properties of the trajectories generated by the the model. The specifications consist of Boolean combinations of component-wise bounded linear time temporal logic formulas. We also provide a probabilistic Petri net representation of DMCs and use it to derive a probabilistic event structure semantics.

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23

Chan, Yu, Emelie Curl, Jesse Geneson, Leslie Hogben, Kevin Liu, Isaac Odegard, and Michael Ross. "Using Markov Chains to Determine Expected Propagation Time for Probabilistic Zero Forcing." Electronic Journal of Linear Algebra 36, no. 36 (June 7, 2020): 318–33. http://dx.doi.org/10.13001/ela.2020.5127.

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Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices as possible. Probabilistic zero forcing yields a discrete dynamical system governed by a Markov chain. Since in a connected graph any one vertex can eventually color the entire graph blue using probabilistic zero forcing, the expected time to do this is studied. Given a Markov transition matrix for a probabilistic zero forcing process, an exact formula is established for expected propagation time. Markov chains are applied to determine bounds on expected propagation time for various families of graphs.

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24

Pfeifer, Dietmar. "A probabilistic variant of Chernoff's product formula." Semigroup Forum 46, no. 1 (December 1993): 279–85. http://dx.doi.org/10.1007/bf02573572.

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25

Morillas, Patricia. "Optimal dual fusion frames for probabilistic erasures." Electronic Journal of Linear Algebra 32 (February 6, 2017): 191–203. http://dx.doi.org/10.13001/1081-3810.3267.

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For any fixed fusion frame, its optimal dual fusion frames for reconstruction is studied in case of erasures of subspaces. It is considered that a probability distribution of erasure of subspaces is given and that a blind reconstruction procedure is used, where the erased data are set to zero. It is proved that there are always optimal duals. Sufficient conditions for the canonical dual fusion frame being either the unique optimal dual, a non-unique optimal dual, or a non optimal dual, are obtained. The reconstruction error is analyzed, using the optimal duals in the probability model considered here and using the optimal duals in a non-probability model.

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26

Damian, E., and A. Lucchini. "Finite Groups withp-Multiplicative Probabilistic Zeta Function." Communications in Algebra 35, no. 11 (October 23, 2007): 3451–72. http://dx.doi.org/10.1080/00927870701509313.

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27

Breuer, Thomas, Robert M. Guralnick, and William M. Kantor. "Probabilistic generation of finite simple groups, II." Journal of Algebra 320, no. 2 (July 2008): 443–94. http://dx.doi.org/10.1016/j.jalgebra.2007.10.028.

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28

HIRTH, ULRICH MARTIN. "Probabilistic Number Theory, the GEM/Poisson-Dirichlet Distribution and the Arc-sine Law." Combinatorics, Probability and Computing 6, no. 1 (March 1997): 57–77. http://dx.doi.org/10.1017/s0963548396002805.

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The prime factorization of a random integer has a GEM/Poisson-Dirichlet distribution as transparently proved by Donnelly and Grimmett [8]. By similarity to the arc-sine law for the mean distribution of the divisors of a random integer, due to Deshouillers, Dress and Tenenbaum [6] (see also Tenenbaum [24, II.6.2, p. 233]), – the ‘DDT theorem’ – we obtain an arc-sine law in the GEM/Poisson-Dirichlet context. In this context we also investigate the distribution of the number of components larger than ε which correspond to the number of prime factors larger than nε.

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29

Yee, Eugene. "Theory for Reconstruction of an Unknown Number of Contaminant Sources using Probabilistic Inference." Boundary-Layer Meteorology 127, no. 3 (March 20, 2008): 359–94. http://dx.doi.org/10.1007/s10546-008-9270-5.

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30

Zhang, Wenyu, and Yuting Zhu. "The Probabilistic Dual Hesitant Fuzzy Multi-Attribute Decision-Making Method Based on Cumulative Prospect Theory and Its Application." Axioms 12, no. 10 (September 28, 2023): 925. http://dx.doi.org/10.3390/axioms12100925.

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Addressing the complex issue of multi-attribute decision-making within a probabilistic dual hesitant fuzzy context, where attribute weights are unknown, a novel decision-making method based on cumulative prospect theory is proposed, named the probabilistic dual hesitant fuzzy multi-attribute decision-making method based on cumulative prospect theory. Firstly, a decision matrix is formulated, representing probabilistic dual hesitant fuzzy information. Secondly, according to the decision maker’s authentic preference and non-membership information sensitivity, a comprehensive score function suitable for probabilistic dual hesitant fuzzy elements is proposed. The attribute weights are then determined using the entropy method. Next, the value function and decision weight function from the cumulative prospect theory are employed to compute the cumulative prospect value attributed to each available scheme. In addition, a cumulative prospect matrix is constructed, enabling the establishment of scheme rankings based on the comprehensive cumulative prospect value. Finally, the analysis of specific cases and a comparative assessment of methods pertaining to the selection of emergency response schemes collectively demonstrate the rationality and efficacy of the decision-making method presented in this study.

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31

DAMIAN, ERIKA, and ANDREA LUCCHINI. "A PROBABILISTIC GENERALIZATION OF SUBNORMALITY." Journal of Algebra and Its Applications 04, no. 03 (June 2005): 313–23. http://dx.doi.org/10.1142/s0219498805001204.

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A subnormal subgroup X of G has the following property: there exists a Dirichlet polynomial Q(s) with integer coefficients such that, for each t ∈ ℕ, Q(t) is the conditional probability that t random elements generate G given that they generate G together with the elements of X In this paper we analyze how far can a subgroup X be with this property from being a subnormal subgroup.

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32

Matera, Guillermo. "Probabilistic Algorithms for Geometric Elimination." Applicable Algebra in Engineering, Communication and Computing 9, no. 6 (July 1, 1999): 463–520. http://dx.doi.org/10.1007/s002000050115.

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33

Shareshian, John. "On the Probabilistic Zeta Function for Finite Groups." Journal of Algebra 210, no. 2 (December 1998): 703–7. http://dx.doi.org/10.1006/jabr.1998.7560.

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34

Damian, Erika, and Andrea Lucchini. "The probabilistic zeta function of finite simple groups." Journal of Algebra 313, no. 2 (July 2007): 957–71. http://dx.doi.org/10.1016/j.jalgebra.2007.02.055.

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35

Mosbah, Mohamed. "PROBABILISTIC GRAPH GRAMMARS." Fundamenta Informaticae 26, no. 3,4 (1996): 341–62. http://dx.doi.org/10.3233/fi-1996-263406.

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36

Alon, Noga. "Choice Numbers of Graphs: a Probabilistic Approach." Combinatorics, Probability and Computing 1, no. 2 (June 1992): 107–14. http://dx.doi.org/10.1017/s0963548300000122.

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The choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). By applying probabilistic methods, it is shown that there are two positive constants c1 and c2 such that for all m ≥ 2 and r ≥ 2 the choice number of the complete r-partite graph with m vertices in each vertex class is between c1r log m and c2r log m. This supplies the solutions of two problems of Erdős, Rubin and Taylor, as it implies that the choice number of almost all the graphs on n vertices is o(n) and that there is an n vertex graph G such that the sum of the choice number of G with that of its complement is at most O(n1/2(log n)1/2).

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37

Sproston, Jeremy. "Probabilistic Timed Automata with Clock-Dependent Probabilities." Fundamenta Informaticae 178, no. 1-2 (January 13, 2021): 101–38. http://dx.doi.org/10.3233/fi-2021-2000.

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Probabilistic timed automata are classical timed automata extended with discrete probability distributions over edges. We introduce clock-dependent probabilistic timed automata, a variant of probabilistic timed automata in which transition probabilities can depend linearly on clock values. Clock-dependent probabilistic timed automata allow the modelling of a continuous relationship between time passage and the likelihood of system events. We show that the problem of deciding whether the maximum probability of reaching a certain location is above a threshold is undecidable for clock-dependent probabilistic timed automata. On the positive side, we show that the maximum and minimum probability of reaching a certain location in clock-dependent probabilistic timed automata can be approximated using a region-graph-based approach.

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38

Brennan, Zachary. "Probabilistic zero forcing with vertex reversion." Electronic Journal of Linear Algebra 41 (January 21, 2025): 49–73. https://doi.org/10.13001/ela.2025.8653.

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Probabilistic zero forcing is a graph coloring process in which blue vertices "infect" (color blue) white vertices with a probability proportional to the number of neighboring blue vertices. This paper introduces reversion probabilistic zero forcing (RPZF), which shares the same infection dynamics but also allows for blue vertices to revert to being white in each round. A threshold number of blue vertices is produced such that the complete graph is entirely blue in the next round of RPZF with high probability. Utilizing Markov chain theory, a tool is formulated which, given a graph's RPZF Markov transition matrix, calculates the probability of whether the graph becomes all white or all blue as well as the time at which this is expected to occur.

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39

FREIVALDS, RŪSIŅŠ. "NON-CONSTRUCTIVE METHODS FOR FINITE PROBABILISTIC AUTOMATA." International Journal of Foundations of Computer Science 19, no. 03 (June 2008): 565–80. http://dx.doi.org/10.1142/s0129054108005826.

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Size (the number of states) of finite probabilistic automata with an isolated cut-point can be exponentially smaller than the size of any equivalent finite deterministic automaton. However, the proof is non-constructive. The result is presented in two versions. The first version depends on Artin's Conjecture (1927) in Number Theory. The second version does not depend on conjectures not proved but the numerical estimates are worse. In both versions the method of the proof does not allow an explicit description of the languages used. Since our finite probabilistic automata are reversible, these results imply a similar result for quantum finite automata.

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40

Liu, Yuqi, Xuehua Li, and Huan Li. "N-Widths of Multivariate Sobolev Spaces with Common Smoothness in Probabilistic and Average Settings in the Sq Norm." Axioms 12, no. 7 (July 17, 2023): 698. http://dx.doi.org/10.3390/axioms12070698.

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In this article, we give the sharp bounds of probabilistic Kolmogorov N,δ-widths and probabilistic linear N,δ-widths of the multivariate Sobolev space W2A with common smoothness on a Sq norm equipped with the Gaussian measure μ, where A⊂Rd is a finite set. And we obtain the sharp bounds of average width from the results of the probabilistic widths. These results develop the theory of approximation of functions and play important roles in the research of related approximation algorithms for Sobolev spaces.

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41

Bukovics, István, and István Kun. "Non-probabilistic hazard evaluation." Opuscula Theologica et Scientifica 1, no. 1 (May 13, 2023): 131–42. http://dx.doi.org/10.59531/ots.2023.1.1.131-142.

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Quantitative analysis of complex risk systems often faces three major problems. The first problem is the size of the system to be examined, since the number of possible system states increases exponentially with system size. The second problem is the topology of the system which may not be tree-like. The third problem is the consideration of randomness, since risk events are in many cases single, non-repetitive, therefore probability theory is inadequate. In order to handle this deficiency we suggest to apply logical risk analysis which traces back the main risk event to elementary, controllable risk events by means of the logical structure describing the operation of the risk system. For illustrations we give some very simple structures and a soil contamination problem as a real-size example.

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42

Abbes, Samy. "Introduction to Probabilistic Concurrent Systems." Fundamenta Informaticae 187, no. 2-4 (October 19, 2022): 71–102. http://dx.doi.org/10.3233/fi-222133.

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The first part of the paper is an introduction to the theory of probabilistic concurrent systems under a partial order semantics. Key definitions and results are given and illustrated on examples. The second part includes contributions. We introduce deterministic concurrent systems as a subclass of concurrent systems. Deterministic concurrent system are “locally commutative” concurrent systems. We prove that irreducible and deterministic concurrent systems have a unique probabilistic dynamics, and we characterize these systems by means of their combinatorial properties. ACM CSS: G.2.1; F.1.1

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43

SHAKERI, S. "A CONTRACTION THEOREM IN MENGER PROBABILISTIC METRIC SPACES." Journal of Nonlinear Sciences and Applications 01, no. 03 (September 20, 2008): 189–93. http://dx.doi.org/10.22436/jnsa.001.03.07.

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44

Shatanawi, Wasfi, and Mihai Postolache. "Mazur-Ulam theorem for probabilistic 2-normed spaces." Journal of Nonlinear Sciences and Applications 08, no. 06 (December 6, 2015): 1228–33. http://dx.doi.org/10.22436/jnsa.008.06.29.

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45

Beyerlein, I. J., and C. N. Tomé. "A probabilistic twin nucleation model for HCP polycrystalline metals." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2121 (March 17, 2010): 2517–44. http://dx.doi.org/10.1098/rspa.2009.0661.

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This article presents a basic probabilistic theory for the nucleation of deformation twins in hexagonal close packed (HCP) metals. Twin nucleation is assumed to rely on the dissociation of grain boundary defects (GBDs) under stress into the required number of twinning partials to create a twin nucleus. The number of successful conversion events is considered to follow a stochastic Poisson process where the rate is assumed to increase with local stress. From this concept, the probability distribution for the critical stress to form a twin nucleus is derived wherein the parameters of the distribution are related to properties of the GBDs. The theory is implemented into a multi-scale constitutive model for HCP metals in order to test its predictive capability against measurements made previously on pure zirconium deformed at 76 and 300 K.

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46

Krejsa, Martin, Petr Janas, Işık Yilmaz, Marian Marschalko, and Tomas Bouchal. "The Use of the Direct Optimized Probabilistic Calculation Method in Design of Bolt Reinforcement for Underground and Mining Workings." Scientific World Journal 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/267593.

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The load-carrying system of each construction should fulfill several conditions which represent reliable criteria in the assessment procedure. It is the theory of structural reliability which determines probability of keeping required properties of constructions. Using this theory, it is possible to apply probabilistic computations based on the probability theory and mathematic statistics. Development of those methods has become more and more popular; it is used, in particular, in designs of load-carrying structures with the required level or reliability when at least some input variables in the design are random. The objective of this paper is to indicate the current scope which might be covered by the new method—Direct Optimized Probabilistic Calculation (DOProC) in assessments of reliability of load-carrying structures.DOProCuses a purely numerical approach without any simulation techniques. This provides more accurate solutions to probabilistic tasks, and, in some cases, such approach results in considerably faster completion of computations.DOProCcan be used to solve efficiently a number of probabilistic computations. A very good sphere of application forDOProCis the assessment of the bolt reinforcement in the underground and mining workings. For the purposes above, a special software application—“Anchor”—has been developed.

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47

Bruce, Juliette, and Daniel Erman. "A probabilistic approach to systems of parameters and Noether normalization." Algebra & Number Theory 13, no. 9 (December 7, 2019): 2081–102. http://dx.doi.org/10.2140/ant.2019.13.2081.

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48

Ben-Ari, Iddo, and Michael Neumann. "Probabilistic approach to Perron root, the group inverse, and applications." Linear and Multilinear Algebra 60, no. 1 (January 2012): 39–63. http://dx.doi.org/10.1080/03081087.2011.559167.

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49

Huynh, Dung T., and Lu Tian. "On Some Equivalence Relations for Probabilistic Processes1." Fundamenta Informaticae 17, no. 3 (September 1, 1992): 211–34. http://dx.doi.org/10.3233/fi-1992-17304.

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In this paper, we investigate several equivalence relations for probabilistic labeled transition systems: bisimulation equivalence, readiness equivalence, failure equivalence, trace equivalence, maximal trace equivalence and finite trace equivalence. We formally prove the inclusions (equalities) among these equivalences. We also show that readiness, failure, trace, maximum trace and finite trace equivalences for finite probabilistic labeled transition systems are decidable in polynomial time. This should be contrasted with the PSPACE completeness of the same equivalences for classical labeled transition systems. Moreover, we derive an efficient polynomial time algorithm for deciding bisimulation equivalence for finite probabilistic labeled transition systems. The special case of initiated probabilistic transition systems will be considered. We show that the isomorphism problem for finite initiated labeled probabilistic transition systems is NC(1) equivalent to graph isomorphism.

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Markov, Boris A., and Yury I. Sukharev. "Probabilistic and dynamic colloid equations." Butlerov Communications 57, no. 3 (March 31, 2019): 33–41. http://dx.doi.org/10.37952/roi-jbc-01/19-57-3-33.

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We obtained two equations that characterize the structure of the colloid: the equation of the Schrodinger type that specifies the redistribution of heat and potential energy in the colloid and material equation – the diffusion equation with the operator of Liesegang associated directly with a substance that allows you to find the discontinuities of the structures caused by the vibrations of electrically charged particles. This procedure based on the assumption of the instability of the colloidal state, caused by the movement of charged particles. The reality is not collected in parts from the particles of matter in the course of evolution from the past to the future, and is all at once from the past to the future for a given pattern, that is, for specific PATTERNS, as defined by quantum theory. Without going deep into the theory of Kulakov, we will accept its fundamental provisions as a certain given. The forms of this structural data were obtained experimentally and mathematically confirmed. Let there be a certain angle of the skeleton, where due to the unevenness and partial randomness of the structure of the core grids forms "defects" – that is, electrical or magnetic moments of a particular order. Then small mobile clusters are attracted to it by electrostatic or electromagnetic forces, which are then adsorbed and somehow arranged on the "defects" in accordance with their dipole moments. This circumstance can be determined by "magic numbers", that is, as the number of clusters "stuck" to the defect of the core structure, with the formation of chemical bonds in the future. We can assume that the spanning structure of Coxeter can form small clusters form regular polyhedrons, and may occur or other structure having more complicated form.

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