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Добірка наукової літератури з теми "Bernoulli number"

Автор: Grafiati

Опубліковано: 8 листопада 2023

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Статті в журналах з теми "Bernoulli number"

Chen, Kwang-Wu. "Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion." Mathematics 10, no. 12 (June 12, 2022): 2033. http://dx.doi.org/10.3390/math10122033.

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Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers. In this paper, we rewrite Ramanujan’s harmonic number expansion into a similar form of Euler’s asymptotic expansion as n approaches infinity: Hn∼γ+c0(h)log(q+h)−∑k=1∞ck(h)k·(q+h)k, where q=n(n+1) is the nth pronic number, twice the nth triangular number, γ is the Euler–Mascheroni constant, and ck(x)=∑j=0kkjcjxk−j, with ck is the negative of the median Bernoulli numbers. Then, 2cn=∑k=0nnkBn+k, where Bn is the Bernoulli number. By using the result obtained, we present two general Ramanujan’s asymptotic expansions for the nth harmonic number. For example, Hn∼γ+12log(q+13)−1180(q+13)2∑j=0∞bj(r)(q+13)j1/r as n approaches infinity, where bj(r) can be determined.

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Jakimczuk, Rafael. "Sequences related to the e number and Bernoulli numbers." Gulf Journal of Mathematics 11, no. 1 (August 9, 2021): 38–42. http://dx.doi.org/10.56947/gjom.v11i1.666.

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Rawlings, Don. "Bernoulli Trials and Number Theory." American Mathematical Monthly 101, no. 10 (December 1994): 948. http://dx.doi.org/10.2307/2975160.

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Rawlings, Don. "Bernoulli Trials and Number Theory." American Mathematical Monthly 101, no. 10 (December 1994): 948–52. http://dx.doi.org/10.1080/00029890.1994.12004573.

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Kaneko, Masanobu. "Poly-Bernoulli numbers." Journal de Théorie des Nombres de Bordeaux 9, no. 1 (1997): 221–28. http://dx.doi.org/10.5802/jtnb.197.

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Gradl, Hans, and Sebastian Walcher. "Bernoulli algebras." Communications in Algebra 21, no. 10 (January 1993): 3503–20. http://dx.doi.org/10.1080/00927879308824745.

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Caratelli, Diego, Pierpaolo Natalini, and Paolo Emilio Ricci. "Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory." Symmetry 15, no. 10 (October 10, 2023): 1900. http://dx.doi.org/10.3390/sym15101900.

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Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers and polynomials to introduce their fractional-index-based types. This reveals a symmetry in relation to the classical numbers and polynomials. We demonstrate some examples of these generalized mathematical entities, which we derive using the computer algebra system Mathematica©.

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CRABB, M. C. "THE MIKI-GESSEL BERNOULLI NUMBER IDENTITY." Glasgow Mathematical Journal 47, no. 2 (July 27, 2005): 327–28. http://dx.doi.org/10.1017/s0017089505002545.

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Xu, Aimin. "Ramanujan’s Harmonic Number Expansion and Two Identities for Bernoulli Numbers." Results in Mathematics 72, no. 4 (September 18, 2017): 1857–64. http://dx.doi.org/10.1007/s00025-017-0748-7.

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Kargın, Levent. "p-Bernoulli and geometric polynomials." International Journal of Number Theory 14, no. 02 (February 8, 2018): 595–613. http://dx.doi.org/10.1142/s1793042118500665.

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We relate geometric polynomials and [Formula: see text]-Bernoulli polynomials with an integral representation, then obtain several properties of [Formula: see text]-Bernoulli polynomials. These results yield new identities for Bernoulli numbers. Moreover, we evaluate a Faulhaber-type summation in terms of [Formula: see text]-Bernoulli polynomials. Finally, we introduce poly-[Formula: see text]-Bernoulli polynomials and numbers, then study some arithmetical and number theoretical properties of them.

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Дисертації з теми "Bernoulli number"

Chellali, Mustapha. "Congruences, nombres de Bernoulli et polynômes de Bessel." Université Joseph Fourier (Grenoble ; 1971-2015), 1989. http://www.theses.fr/1989GRE10091.

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En premiere partie, on donne des congruences entre nombres de bernoulli-hcowitz dans le cas supersingulier. En deuxieme partie, on montre que la suite des nombres de bernoulli verifie des formules de recurrence qui servent a tester si un nombre premier est irregulier. En troisieme partie, on etudie les zeros des polynomes de bessel generalises, en particulier on encadre un zero reel, apres developpement asymptotique, et on donne des estimations uniformes des valeurs de ces polynomes

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Whitaker, Linda M. "The Bernoulli salesman." Diss., Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/24935.

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Smith, Michael J. "Ranking and selection : open sequential procedures for Bernoulli populations." Thesis, Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/25103.

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Martin, Bruno. "Contribution à la théorie des entiers friables." Phd thesis, Université de Lorraine, 2005. http://tel.archives-ouvertes.fr/tel-00795666.

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Un entier naturel est dit $y$-friable lorsque son plus grand facteur premier n'excède pas $y$. Ce travail est consacré à l'étude des entiers friables dans le cadre de la théorie analytique et probabiliste des nombres. La première partie est dévolue à un problème posé par Davenport en 1937, qui consiste à déterminer les conditions de validité de diverses généralisations de son développement de la fonction sinus en série de parties fractionnaires. Ces généralisations peuvent être décrites par un couple de fonctions arithmétiques, liées par la relation de convolution $f=g*\1$. Nous traitons le cas où $g$ est la fonction de Piltz d'ordre $z\in\CC$. La deuxième partie est consacrée à l'étude du comportement asymptotique de la constante optimale dans une version friable de l'inégalité de Turán-Kubilius. Précisant des résultats récents de La Bretèche et Tenenbaum, nous généralisons au cas friable une formule asymptotique de la variance d'une fonction arithmétique additive, établie par Hildebrand en 1983.

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Mirkoski, Maikon Luiz. "Números e polinômios de Bernoulli." Universidade Estadual de Ponta Grossa, 2018. http://tede2.uepg.br/jspui/handle/prefix/2699.

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Submitted by Angela Maria de Oliveira (amolivei@uepg.br) on 2018-11-29T18:07:06Z No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Maikon Luiz.pdf: 959643 bytes, checksum: aaf472f5b8a9a29532793d01234788a9 (MD5)
Made available in DSpace on 2018-11-29T18:07:06Z (GMT). No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Maikon Luiz.pdf: 959643 bytes, checksum: aaf472f5b8a9a29532793d01234788a9 (MD5) Previous issue date: 2018-10-19
Neste trabalho,estudamos os números e os polinomios de Bernoulli,bem como algumas de suas aplicações mais importantes em Teoria dos Números. Com base em uma caracterização ao simples, os polinômios de Bernoulli são introduzidos e, posteriormente, os números de Bernoulli. As séries de Fourier dos polinomios de Bernoulli são utilizadas na demonstração da equação funcional da função teta. Esta equação, por sua vez, é utilizada na demonstração da celebre equação funcional da função zeta, que tem importância central na teoria da distribuição dos números primos. Além das conexões com a funções especiais zeta e teta, discutimos também, em detalhe,conexões entre os números e os polinomios de Bernoulli com a função gama. Essas relações são então exploradas para produzir belas fórmulas para certos valores da função zeta, entre outras aplicações.
In this work we study Bernoulli numbers and Bernoulli polynomials, as well as some of its most important applications in Number Theory. Based on a simple characterization, the Bernoulli polynomials are introduced and, later, the Bernoulli numbers. The Fourier series of the Bernoulli polynomials are used to demonstrate the functional equation of the theta function. This equation, in turn, is used in the proof of the famous functional equation of the zeta function, which is central to the theory of prime number distribution. In addition to the connections with the special functions zeta and theta, we also discuss, in detail, connections between the Bernoulli numbers and Bernoulli polynomials with the gamma function. These relations are then explored to produce beautiful formulas for certain values of the zeta function,among other applications.

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Stacey, Andrew W. "An Adaptive Bayesian Approach to Bernoulli-Response Clinical Trials." CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd2065.pdf.

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Kondo, Pedro Kiochi. "CÁLCULO FINITO: DEMONSTRAÇÕES E APLICAÇÕES." UNIVERSIDADE ESTADUAL DE PONTA GROSSA, 2014. http://tede2.uepg.br/jspui/handle/prefix/1528.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
In this work some topics of the Discrete or Finite Calculus are developed. In particular, we study difference operators, factorial powers, Stirling numbers of the first and second type, the Newton’s formula of differences, the fundamental theorem of the Finite Calculus, the summation process, and the Bernoulli numbers and Bernoulli polynomials. Then we show the effectiveness of the theory for the calculation of closed formulas for the value of many finite sums. We also study the classical problem of obtaining the polynomials which express the value of the sums of powers of natural numbers.
Neste trabalho desenvolvemos alguns tópicos do Cálculo Discreto ou Finito. Em particular, estudamos operadores de diferenças, potências fatoriais, números de Stirling do primeiro e do segundo tipo, a fórmula de diferenças de Newton, o teorema fundamental do Cálculo Finito, o processo de somação e os números e polinômios de Bernoulli. Mostramos então a eficácia da teoria no cálculo de fórmulas fechadas para o valor de diversas somas finitas. Também estudamos o problema clássico de obter os polinômios que expressam o valor de somas de potências de números naturais.

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Perkins, Rudolph Bronson. "On Special Values of Pellarin’s L-series." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1383827548.

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Chung, Yi-Shiu, and 鍾逸修. "The Calculation and Application of Bernoulli number." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/84502958840518031848.

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碩士
國立臺中教育大學
數學教育學系
96
Up to the present, it is an important study for calculating Bernoulli number. There are many different methods to claculate Bernoulli number. But for these methods, we must take lots of steps to calaulate Bernoulli number. Based on this, our research applies Riemann--zeta function and the extended function of the sums of powers of consecutive integers to get an easier method. Then, we will calculate Bernoulli number by using Matlab 7.1, and investigate the relationship between Bernoulli nmuber and Stirling number of second kind. Our results are as follows. 1. The formula of Bernoulli number is B_{2k}=\frac{1}{2k+1} \left \{ C_{2k}^{2k+1}S_{1}^{\prime}(-1) + \sum_{i=1}^{k}C_{2i+1}^{2k+1} S_{2k-2i}^{\prime}(-1) \right \}, k\in N . 2. When $k$ is bigger, Bernoulli number will become bigger and be alternated between plus and minus. 3. The relationship between Bernoulli number and Stirling number of second kind is B_{m+1}=\sum_{k=1}^{m+1}\frac{(-1)^k}{k+1}\cdot k!\cdot S_2(m+1,k).

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Liu, Chih Shiuan, and 劉志璿. "The connection between the functions of Riemann zeta and Bernoulli Number." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/17154599310613619902.

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碩士
國立臺中教育大學
數學教育學系
96
This research hung over from the extended functions for the sum of powers of consecutive integers, we colleted the literatures of the related research about the functions of Riemann zeta and Bernoulli Number, both newly interpreted and predigested the properties of the functions of Riemann zeta and Bernoulli Number. Thus we built the connection between the functions of Riemann zeta and Bernoulli Number, according to \zeta(2 k)=(-1)^{k-1} 2^{2k-1} \frac{B_{2k} \pi^{2k}}{(2k)!}, \ k \in \mathbb{N},and S_{2k}^{\prime}(-1)=\frac{(-1)^{k-1} (2k)!}{2^{2k-1} (\pi)^{2k}}\zeta(2k), S_{2k+1}^{\prime}(-1)=0,Take the function of Riemann zeta as bridge, we find that S_{2k}^{\prime}(-1)=B_{2k},B_{2k}=\frac{1}{2k+1} \left \{ C_{2k}^{2k+1} S_{1}^{\prime}(-1)+ \sum_{i=1}^{k} C_{2i+1}^{2k+1} S_{2k-2i}^{\prime}(-1) \right \},where $S_k^{\prime}(x)$ denotes the first derivative of $S_k(x)$ for each positive integer $k$.

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Книги з теми "Bernoulli number"

1954-, Dilcher Karl, Skula Ladislav, and Slavutskiĭ Ilja Sh, eds. Bernoulli numbers: Bibliography (1713-1990). Kingston, Ont: Queen's University, 1991.

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Arakawa, Tsuneo, Tomoyoshi Ibukiyama, and Masanobu Kaneko. Bernoulli Numbers and Zeta Functions. Tokyo: Springer Japan, 2014. http://dx.doi.org/10.1007/978-4-431-54919-2.

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author, Ibukiyama Tomoyoshi, Kaneko Masanobu author, and Zagier, Don, 1951- writer of supplementary textual content, eds. Bernoulli numbers and Zeta functions. Tokyo: Springer, 2014.

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Kanemitsu, Shigeru. Vistas of special functions. Singapore: World Scientific, 2007.

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Invitation to classical analysis. Providence, R.I: American Mathematical Society, 2012.

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Vistas of Special Functions. World Scientific Publishing Company, 2007.

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Vorlesungen über die Bernoullischen zahlen: Ihren zusammenhang mit den secanten-coefficienten und ihre wichtigeren anwendungen. Berlin: J. Springer, 1991.

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Ibukiyama, Tomoyoshi, Masanobu Kaneko, Tsuneo Arakawa, and Don B. Zagier. Bernoulli Numbers and Zeta Functions. Springer, 2016.

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Ibukiyama, Tomoyoshi, Masanobu Kaneko, and Tsuneo Arakawa. Bernoulli Numbers and Zeta Functions. Springer, 2014.

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Franzosa, Marie M. Densities and dependence for point processes. 1988.

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Частини книг з теми "Bernoulli number"

Ireland, Kenneth, and Michael Rosen. "Bernoulli Numbers." In A Classical Introduction to Modern Number Theory, 228–48. New York, NY: Springer New York, 1990. http://dx.doi.org/10.1007/978-1-4757-2103-4_15.

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Simsek, Yilmaz. "Families of Twisted Bernoulli Numbers, Twisted Bernoulli Polynomials, and Their Applications." In Analytic Number Theory, Approximation Theory, and Special Functions, 149–214. New York, NY: Springer New York, 2014. http://dx.doi.org/10.1007/978-1-4939-0258-3_6.

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Sándor, J., and B. Crstici. "Stirling, bell, bernoulli, euler and eulerian numbers." In Handbook of Number Theory II, 459–618. Dordrecht: Springer Netherlands, 2004. http://dx.doi.org/10.1007/1-4020-2547-5_5.

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Ibukiyama, Tomoyoshi, and Masanobu Kaneko. "Class Number Formula and an Easy Zeta Function of the Space of Quadratic Forms." In Bernoulli Numbers and Zeta Functions, 155–82. Tokyo: Springer Japan, 2014. http://dx.doi.org/10.1007/978-4-431-54919-2_10.

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Wagstaff, Samuel S. "Prime Divisors of the Bernoulli and Euler Numbers." In Number Theory for the Millennium III, 357–74. London: A K Peters/CRC Press, 2023. http://dx.doi.org/10.1201/9780138747022-21.

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Isaacson, Brad. "Generalized Bernoulli Numbers, Cotangent Power Sums, and Higher-Order Arctangent Numbers." In Combinatorial and Additive Number Theory V, 253–61. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-10796-2_12.

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Adam, David, and Jean-Luc Chabert. "Bhargava’s Exponential Functions and Bernoulli Numbers Associated to the Set of Prime Numbers." In Algebraic, Number Theoretic, and Topological Aspects of Ring Theory, 9–35. Cham: Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-28847-0_2.

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Chryssaphinou, O., S. Papastavridis, and T. Tsapelas. "On the Number of Overlapping Success Runs in a Sequence of Independent Bernoulli Trials." In Applications of Fibonacci Numbers, 103–12. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-2058-6_10.

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Ibukiyama, Tomoyoshi, and Masanobu Kaneko. "Bernoulli Numbers." In Bernoulli Numbers and Zeta Functions, 1–24. Tokyo: Springer Japan, 2014. http://dx.doi.org/10.1007/978-4-431-54919-2_1.

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Ribenboim, Paulo. "Bernoulli Numbers." In Classical Theory of Algebraic Numbers, 367–97. New York, NY: Springer New York, 2001. http://dx.doi.org/10.1007/978-0-387-21690-4_18.

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Тези доповідей конференцій з теми "Bernoulli number"

Campos, Richard A., Malvin C. Teich, and B. E. A. Saleh. "Homodyne photon-number statistics for nonclassical states of light at a lossless beam splitter." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1989. http://dx.doi.org/10.1364/oam.1989.thii6.

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Using the quantum theory of a lossless beam splitter, we determine the joint and marginal photon-number distributions at its output ports, for two arbitrary (not necessarily independent) input fields. The joint probability at the output of the beam splitter turns out to be a Fourier series in the relative phase between the output beams. Photomixing with the number states exhibits no phase sensitivity. An elementary example of phase sensitive dependence in the photon-number domain is obtained with the Bernoulli states, which are a subclass of the binomial states.1 For a joint input Bernoulli state, the probability of detecting one photon (or zero photons) at either output port varies sinusoidally with the relative phase. For independent Bernoulli states incident on the beam splitter, the results differ and depend on whether the input is in a pure or mixed state. Photomixing of other nonclassical states exhibit higher orders of phase-induced modulation of the output photon-number probabilities.

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Delande, E. D., D. E. Clark, and J. Houssineau. "Regional variance in target number: Analysis and application for multi-Bernoulli point processes." In IET Conference on Data Fusion & Target Tracking 2014: Algorithms and Applications. Institution of Engineering and Technology, 2014. http://dx.doi.org/10.1049/cp.2014.0531.

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Kuo, Y. L., and W. L. Cleghorn. "Curvature-Based Finite Element Method for Euler-Bernoulli Beams." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34213.

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This paper presents a new method called the curvature-based finite element method to solve Euler-Bernoulli beam problems. An approximated curvature distribution is selected first, and then the approximated transverse displacement is determined by double integrations. Four numerical examples demonstrate the validity of the method, and the results show that the errors are smaller than those generated by a conventional method, the displacement-based finite element method, for comparison based on the same number of degrees of freedom.

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Chih-Wei Yi, Peng-Jun Wan, Xiang-Yang Li, and O. Frieder. "Asymptotic distribution of the number of isolated nodes in wireless ad hoc networks with Bernoulli nodes." In WCNC 2003 - IEEE Wireless Communications and Networking Conference. IEEE, 2003. http://dx.doi.org/10.1109/wcnc.2003.1200623.

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Katariya, Sumeet, Branislav Kveton, Csaba Szepesvári, Claire Vernade, and Zheng Wen. "Bernoulli Rank-1 Bandits for Click Feedback." In Twenty-Sixth International Joint Conference on Artificial Intelligence. California: International Joint Conferences on Artificial Intelligence Organization, 2017. http://dx.doi.org/10.24963/ijcai.2017/278.

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The probability that a user will click a search result depends both on its relevance and its position on the results page. The position based model explains this behavior by ascribing to every item an attraction probability, and to every position an examination probability. To be clicked, a result must be both attractive and examined. The probabilities of an item-position pair being clicked thus form the entries of a rank-1 matrix. We propose the learning problem of a Bernoulli rank-1 bandit where at each step, the learning agent chooses a pair of row and column arms, and receives the product of their Bernoulli-distributed values as a reward. This is a special case of the stochastic rank-1 bandit problem considered in recent work that proposed an elimination based algorithm Rank1Elim, and showed that Rank1Elim's regret scales linearly with the number of rows and columns on "benign" instances. These are the instances where the minimum of the average row and column rewards mu is bounded away from zero. The issue with Rank1Elim is that it fails to be competitive with straightforward bandit strategies as mu tends to 0. In this paper we propose Rank1ElimKL, which replaces the crude confidence intervals of Rank1Elim with confidence intervals based on Kullback-Leibler (KL) divergences. With the help of a novel result concerning the scaling of KL divergences we prove that with this change, our algorithm will be competitive no matter the value of mu. Experiments with synthetic data confirm that on benign instances the performance of Rank1ElimKL is significantly better than that of even Rank1Elim. Similarly, experiments with models derived from real-data confirm that the improvements are significant across the board, regardless of whether the data is benign or not.

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Ishihata, Masakazu, and Takanori Maehara. "Exact Bernoulli Scan Statistics using Binary Decision Diagrams." In Twenty-Eighth International Joint Conference on Artificial Intelligence {IJCAI-19}. California: International Joint Conferences on Artificial Intelligence Organization, 2019. http://dx.doi.org/10.24963/ijcai.2019/795.

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In combinatorial statistics, we are interested in a statistical test of combinatorial correlation, i.e., existence a subset from an underlying combinatorial structure such that the observation is large on the subset. The combinatorial scan statistics has been proposed for such a statistical test; however, it is not commonly used in practice because of its high computational cost. In this study, we restrict our attention to the case that the number of data points is moderately small (e.g., 50), the outcome is binary, and the underlying combinatorial structure is represented by a zero-suppressed binary decision diagram (ZDD), and consider the problem of computing the p-value of the combinatorial scan statistics exactly. First, we prove that this problem is a #P-hard problem. Then, we propose a practical algorithm that solves the problem. Here, the algorithm constructs a binary decision diagram (BDD) for a set of realizations of the random variables by a dynamic programming on the ZDD, and computes the p-value by a dynamic programming on the BDD. We conducted experiments to evaluate the performance of the proposed algorithm using real-world datasets.

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Teng, Shen, Wang Jiong, Sun Dong, Liu Yafeng, and Tian Zhouyu. "Modeling and Numerical Simulation of Flow Resistance Characteristics in Slowly-Varying Rectangular Cross-Section Microchannel." In ASME 2016 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/imece2016-65257.

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In this paper, flow characteristics in slowly-varying rectangular microchannels with different aspect ratios and Re numbers ranging from 7 to 200 are investigated. The obtained simulation results are compared with theoretics values based on the Bernoulli equations. The results show that the simulation results just hold an average discrepancy of 10% with the theoretical calculation value whichis presents a good accord. All the research of microchannel, the Poiseuille number of the flow is inconstant within the range of Re number, except when the contract angle is small where Poiseuille number is essentially unchanged.

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Caddemi, Salvatore, and Ivo Calio`. "Closed Form Buckling Solutions of Euler-Bernoulli Columns With Multiple Singularities." In ASME 2009 International Mechanical Engineering Congress and Exposition. ASMEDC, 2009. http://dx.doi.org/10.1115/imece2009-11168.

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In this paper the Euler-Bernoulli column in presence of multiple concentrated open cracks and for general boundary conditions is studied. The request of an integration procedure able to lead to exact closed form buckling solutions for any number, position and intensity of the cracks is satisfied. A model of concentrated crack based on the adoption of the distributions (generalized functions) is presented. In particular, the exact buckling mode solution and the corresponding exact buckling load equation is obtained. A parametric study for different boundary conditions is presented.

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Yong, Yan. "Vibration of Euler-Bernoulli Beams With Arbitrary Boundaries and Intermediate Constraints." In ASME 1991 Design Technical Conferences. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/detc1991-0284.

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Abstract It is an easy task to obtain the analytical solution for a simple Euler-Bernoulli beam. Difficulties, however, do arise when a beam structure has a large number of different-type intermediate constraints, such as exterior simple supports, interior hinges, rollers and other elastic supports. The purpose of this paper is to develop a systematic approach to obtain exact solutions for any kinds of complex beam configurations. The main feature of this approach is that the vibration motion is treated as the result of wave scattering around different media. The discovery of wave scattering phenomena around intermediate constraints sheds the light, for the first time, on the fact that any constraint can serve as a waveguide capable of transmitting and reflecting waves in either directions. This new finding enables us to treat a complex beam structure as a stepped beam which can be handled easily by using the wave propagation approach, previously developed by Yong and Lin (1989) for the analysis of piece-wise periodic structures.

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Naguleswaran, S. "Vibration of an Euler-Bernoulli Uniform Beam Carrying Several Thin Disks." In ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/detc2003/vib-48361.

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The Euler-Bernoulli uniform beam considered in this paper carry (n+1) thin disks, two of which are at the beam ends. For the analytical method used in the paper, n co-ordinate systems were chosen with origins at the disk locations. The mode shape of the portion of the beam between the jth and (j+1)th disk was expressed in the form Yj(Xj) = A Uj(Xj) + B Vj(Xj) in which Uj(Xj) and Vj(Xj) are ‘modified’ mode shape functions applicable to that portion but the constants A and B are common to all the portions. From the compatibility of moments and forces on the (n+1)th disk, the frequency equation was expressed in closed form as a 2nd order determinant equated to zero. Schemes are presented to compute the 4 elements of the determinant (from a recurrence relationship) and to evaluate the roots of the frequency equation. Computational difficulties were not encountered in the implementation of the schemes to beams carrying very large number of disks. The first three natural frequency parameters of 28 combinations of the boundary conditions (which includes classical clamped, pinned, sliding and free) are tabulated for beams carrying 6, 51, 201 or 1001 thin disks. The approaches in previous publications include those based on various approximate methods like finite element, Rayleighritz, Galerkin, transfer matrix etc. The results in the present paper may be used to judge the accuracy of values obtained by approximate methods. The theory developed in the paper will need modification if axial dimension of the disks are taken into account.

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Звіти організацій з теми "Bernoulli number"

Pengelley, David. Figurate Numbers and Sums of Numerical Powers: Fermat, Pascal, Bernoulli. Washington, DC: The MAA Mathematical Sciences Digital Library, June 2013. http://dx.doi.org/10.4169/loci003987.

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Klammler, Harald. Introduction to the Mechanics of Flow and Transport for Groundwater Scientists. The Groundwater Project, 2023. http://dx.doi.org/10.21083/gxat7083.

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Starting from Newton’s laws of motion and viscosity, this book is an introduction to fundamental aspects of fluid dynamics that are most relevant to groundwater scientists. Based on a perspective of driving versus resisting forces that govern the motion of a fluid, the author derives Darcy’s law for flow through porous media by drawing an analogy to Bernoulli’s law for fluid with negligible viscosity. By combining the effects of gravity and pressure, the author identifies hydraulic head as a convenient numerical quantity to represent the force driving groundwater flow. In contrast to the physical derivation of hydraulic head, hydraulic conductivity emerges as a parameter related to the resisting frictional forces between the mobile fluid and the stationary porous medium. These frictional seepage forces also affect the effective stress state of the porous medium, thus establishing a link to soil stability and quicksand formation. Combining Darcy’s law with the law of mass conservation leads the reader to the fundamental equations of saturated groundwater flow. Finally, the effects of capillary forces are included to establish the governing equations for unsaturated and multi-phase flow. Throughout the book, the author focuses on thoroughly illustrating and deriving the equations while applying order of magnitude analyses. This approach makes it possible to extract the most information, for example in terms of the scale of response time, without requiring explicit solutions. A number of boxes and solved exercises contain further details and links to practical applications such as the water table ratio that reflects ‘fullness’ of an aquifer and the performance of slug tests for in situ measurement of hydraulic conductivity. This book makes an important contribution to groundwater science by providing a progressive introductory explanation of the physical mechanics of groundwater flow and the accompanying socioeconomic and ecological problems that may arise.

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