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1
Lee, Ang Wee, Nayef Mohamed Ghasem, and Mohamed Azlan Hussain. "Utilization of Mathematical Software Packages in Chemical Engineering Research." ASEAN Journal of Chemical Engineering 5, no. 2 (December 1, 2005): 125. http://dx.doi.org/10.22146/ajche.50180.
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Using Fortran taken as the starting point, we are now on the sixth decade of high-level programming applications. Among the programming languages available, computer algebra systems (CAS) appear to be a good choice in chemical engineering can be applied easily. Until the emergence of CAS, the assistance from a specialized group for large-scale programming is justified. Nowadays, it is more effective for the modern chemical engineer to rely on his/her own programming ability for problem solving. In the present paper, the abilities of Polymath, Maple, Matlab, Mathcad, and Mathematica in handling differential equations are illustrated for differential-algebraic equations, large system of nonlinear differential equations, and partial differential equations. The programming of solutions with these CAS are presented, contrasted, and discussed in relation to chemical engineering problems. Keywords: Computer algebra systems (CAS),computer simulation,Mathcad, Mathematica,Mathlab and numerical methods.
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Mezhennaya, Natalia M., and Oleg V. Pugachev. "ON PERCEPTION OF COMPUTER ALGEBRA SYSTEMS AND MICROSOFT EXCEL BY ENGINEERING STUDENTS." Problems of Education in the 21st Century 77, no. 3 (June 18, 2019): 379–95. http://dx.doi.org/10.33225/pec/19.77.379.
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This research presents the analysis of image of the popular program packages: CAS (Computer Algebra Systems) Matlab, Mathcad, Mathematica, and the spreadsheet ‘Microsoft Excel’ among students of engineering specializations studying in Bauman Moscow State Technical University (BMSTU). The research has explored perception of program packages by all students (N=164), and by male students (N=110) and female students (N=54) separately. Image of the program packages, convenience of their interfaces, usage scenarios, and applicability for solution of tasks of certain types have been explored via questioning the students. Moreover, the students assessed likelihood of using the programs in the future. It was established that all the program packages under consideration can be successfully used in the process of education, but some of them (Matlab and Mathematica) require additional lessons to learn their interface and peculiarities. Gender differences in perception of the program packages do not determine the education outcomes. The differences of perception of the programs are mainly caused by their popularity and the students’ skills of using certain programs. The following gender differences have been detected: in the female students’ sample, the programs Mathcad and Excel are leading, Matlab is performed a little worse; Matlab is associated with the statement “difficult / inconvenient interface”. In the male students’ sample, the three programs Matlab, Mathcad, and Microsoft Excel got nearly equal grades. Female students estimate future usage of Microsoft Excel higher than male students do. The CAS Mathematica takes the lowest places for both samples; as a possible cause of such result, it has been found that Mathematica is not widespread as compared to the other programs under consideration. Keywords: computer algebra system, education statistics, engineering education, mathematical education, Mathcad, Mathematica, Matlab, Microsoft Excel.
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Velychko, Vladyslav Ye, Andrii V. Stopkin, and Olena H. Fedorenko. "USE OF COMPUTER ALGEBRA SYSTEM MAXIMA IN THE PROCESS OF TEACHING FUTURE MATHEMATICS TEACHERS." Information Technologies and Learning Tools 69, no. 1 (February 25, 2019): 112. http://dx.doi.org/10.33407/itlt.v69i1.2284.
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The article is devoted to the research of a problem of the application of computer algebra systems in the course of skill training of future mathematics teachers. The possibility of study organization for students of pedagogical institutions of higher education receiving "014 secondary education (mathematics)" speciality with the use of computer algebra systems is considered. The term "computer algebra system" is understood as the software which is intended for the solving of various mathematical tasks by means of a computer with a possibility of visualization of all stages of calculations. Proceeding from the fact that the computer algebra systems are actively used in educational process all over the world and informational educational environments in the course of teaching mathematical disciplines are created on their basis, it is possible to make a conclusion that the computer algebra systems are very important components of the process of training future experts. The current state of the problem of introduction and application of the computer algebra systems in the training of future mathematics teachers is investigated. The role of the computer algebra systems in teaching mathematics subjects and general natural sciences in pedagogical establishments of higher education is defined. Such computer algebra systems as MathCAD, MatLab, Maple, Mathematica, Maxima, SAGE, and Axiom are marked out among the most modern and successfully used systems aimed at solving mathematical tasks. Arguments in favour of the use of the free computer algebra system Maxima in educational and scientific activity of students are adduced in the article. Its main advantages and disadvantages are presented. The main functional of this system is considered which allows to optimize the time of the solution of various mathematical tasks, to adjust a conclusion of the received result, to perform staged verification of correctness of calculations.
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JARA, P., and D. ŞTEFAN. "HOPF–CYCLIC HOMOLOGY AND RELATIVE CYCLIC HOMOLOGY OF HOPF–GALOIS EXTENSIONS." Proceedings of the London Mathematical Society 93, no. 1 (June 9, 2006): 138–74. http://dx.doi.org/10.1017/s0024611506015772.
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Let $H$ be a Hopf algebra and let $\mathcal{M}_s (H)$ be the category of all left $H$-modules and right $H$-comodules satisfying appropriate compatibility relations. An object in $\mathcal{M}_s (H)$ will be called a stable anti-Yetter–Drinfeld module (over $H$) or a SAYD module, for short. To each $M \in \mathcal{M}_s (H)$ we associate, in a functorial way, a cyclic object $\mathrm{Z}_\ast (H, M)$. We show that our construction can be used to compute the cyclic homology of the underlying algebra structure of $H$ and the relative cyclic homology of $H$-Galois extensions.Let $K$ be a Hopf subalgebra of $H$. For an arbitrary $M \in \mathcal{M}_s (K)$ we define a right $H$-comodule structure on $\mathrm{Ind}_K^H M := H \otimes_K M$ so that $\mathrm{Ind}_K^H M$ becomes an object in $\mathcal{M}_s (H)$. Under some assumptions on $K$ and $M$ we compute the cyclic homology of $\mathrm{Z}_\ast (H, \mathrm{Ind}_K^H M)$. As a direct application of this result, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the cyclic homology of group algebras and quantum tori.Finally, when $H$ is the enveloping algebra of a Lie algebra $\mathfrak{g}$, we construct a spectral sequence that converges to the cyclic homology of $H$ with coefficients in a given SAYD module $M$. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of $H$ with coefficients in a certain SAYD module.
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Radzevich, Stephen P. "A Novel Design of Cylindrical Hob for Machining of Precision Involute Gears." Journal of Mechanical Design 129, no. 3 (March 6, 2006): 334–45. http://dx.doi.org/10.1115/1.2406105.
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This paper aims at development of a novel design of precision gear hob for machining involute gear on a conventional gear-hobbing machine. The reported research is based on use of fundamental results obtained in analytical mechanics of gearing. For solving the problem, both the descriptive-geometry-based (DGB) methods together with pure analytical methods have been employed. The use of DGB methods is insightful for solving most of the principal problems, which consequently were analytically solved. The analytical methods used provide an example of application of the DG∕K-method of surface generation developed earlier by the author. For interpretation of the results of research, several computer codes in the commercial software MathCAD∕Scientific were composed. Ultimately, a method of computation of parameters of design of a hob with straight-line lateral cutting edges for machining of precision involute gears is developed in the paper. Coincidence of the straight-line lateral cutting edges of the hob with the straight-line characteristics of it generating surface eliminates the major source of deviations of the hobbed involute gears. The relationships between major principal design parameters that affect the gear hob performance are investigated with the use of vector algebra, matrix calculus, and elements of differential geometry. Gear hobs of the proposed design yield elimination of the principal and major source of deviation of the desired hob tooth profile from the actual hob tooth profile. The reported results of this research are ready to put in practice.
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Di Nola, Antonio, Revaz Grigolia, Nunu Mitskevich, and Gaetano Vitale. "Dynamic Łukasiewicz logic and its application to immune system." Soft Computing 25, no. 15 (June 23, 2021): 9773–80. http://dx.doi.org/10.1007/s00500-021-05955-3.
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AbstractIt is introduced an immune dynamic n-valued Łukasiewicz logic $$ID{\L }_n$$ I D Ł n on the base of n-valued Łukasiewicz logic $${\L }_n$$ Ł n and corresponding to it immune dynamic $$MV_n$$ M V n -algebra ($$IDL_n$$ I D L n -algebra), $$1< n < \omega $$ 1 < n < ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras $$(\mathcal {M}, \mathcal {R}, \Diamond )$$ ( M , R , ◊ ) that combine the varieties of $$MV_n$$ M V n -algebras $$\mathcal {M} = (M, \oplus , \odot , \sim , 0,1)$$ M = ( M , ⊕ , ⊙ , ∼ , 0 , 1 ) and regular algebras $$\mathcal {R} = (R,\cup , ;, ^*)$$ R = ( R , ∪ , ; , ∗ ) into a single finitely axiomatized variety resembling R-module with “scalar” multiplication $$\Diamond $$ ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic $$ID{\L }_n$$ I D Ł n with application in immune system.
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Mironov, Mikhail, and Roman Mudrik. "On assessment of dynamic strength in waves for detailed support-free models of hull structures." Transactions of the Krylov State Research Centre, SPECIAL ISSUE 1 (April 16, 2019): 82–90. http://dx.doi.org/10.24937/2542-2324-2019-1-s-i-82-90.
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Cross-disciplinary CAE-based calculations of ship movement and strains in fluid, at launching device, during ice interaction or in case of a navigation accident require very performant computers, and their results can neither differentiate specific movement shapes nor be checked by means of a physical experiment, so their verification is a very important task. Automated algorithms based on common assessment procedures for movement components and external forces made it possible to not only verify numerical calculations of apparent dynamics, but also to obtain justified and rational layouts for arrangement of external load monitoring sensors, as well as to refine these common procedures as a separate field of studies and use more accurate estimates of external forces in design algorithms. The task is to correctly transfer the obtained information on movement and pressure fields to the analytical model under straining. The model is support-free, so kinematic conditions must be imposed on it so as not to distort realistic stressed state. In the unsteady formulation, it is necessary to correctly consider inertial and damping properties of fluid and structure itself, ensuring correct transition to the quasi-static formulation. Computer-based algebra of PTC MathCAD software made it possible for the authors to develop software modules for three-dimensional motion calculations of ships with arbitrary hull shape and loading, as well as APDL ANSYS applications for applying the data on external loads in stress-strain state calculations of support-free FE model of ship in frequency and time domain. The authors managed to develop a viable open-source solution for a cross-software system meant for dynamic strength analysis of ships in waves, which can be upgraded by non-programmers.
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Verbovyi, A., C. Neamtu, M. Sieryk, B. Vashyst, V. Pavlenko, V. Simonovskiy, and I. Pavlenko. "Ensuring the Vibration Reliability of Rotors Connected by Spline Joints." Journal of Engineering Sciences 6, no. 2 (2019): d14—d19. http://dx.doi.org/10.21272/jes.2019.6(2).d3.
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This article is devoted to the development of refined numerical mathematical models of rotor dynamics of high-performance turbomachines having a spline connection. These models consider the dependence of the critical frequencies of the shaft on the angular stiffness of the spline connection, as well as the procedure of virtual balancing. As a result of the complex application of this approach, the methods of calculation of vibration characteristics taking into ac-count variable values of angular rigidity of splined connection are offered. In addition, the method of evaluating the system of initial imbalances with the corresponding displacements of the rotor axis in the correction and calculation sections has also been improved. The proposed approaches, based on the integrated application of CAE software and computational intelligent systems, allow for modal and harmonic analysis and implement virtual balancing with a significant reduction in preparation and machine time without loss of relative accuracy. In addition, the developed mathematical model of free and forced vibrations of rotor systems have been implemented in the program code operational files “Critical Frequencies of the Rotor” and “Forced Oscillations of the Rotor” of the computer algebra system MathCAD that allows improving the dynamic balancing procedure for evaluating primary imbalances. The high accuracy of the proposed approach is confirmed by checking the dynamic deviations of the rotor axis by the system of residual imbalances in accordance with the standards of vibration stability. Keywords: turbomachine, spline connection, angular stiffness, virtual balancing, modal analysis, harmonic analysis.
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Roskladka, Andrii, Nataliia Roskladka, Oleksandr Dluhopolskyi, Ganna Kharlamova, and Mehmet Kiziloglu. "Data analysis and forecasting of tourism development in Ukraine." Innovative Marketing 14, no. 4 (December 13, 2018): 19–33. http://dx.doi.org/10.21511/im.14(4).2018.02.
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The article contains a thorough study of tourist flows in Ukraine during the last 18 years. The tendencies of the development of international tourism during the last 20 years and their influence on the development of tourism in Ukraine have been explored. Particular attention is paid to the analysis of inbound tourist flows. The negative dynamics of tourist arrivals in Ukraine led to the construction of a forecast model for the development of this component of tourism activity with the aim of an objective assessment of future tourist arrivals and the adoption of effective management decisions on improving the situation of tourism in Ukraine. Tourist destinations today are rigorously competing for consumer interest in tourism products. In view of this, tourist facilities operating in the tourist market place particular emphasis on the elements of tourism marketing (product, price, distribution, advertising, human capital, actual data, processes).Modern forecasting systems and methods have been used to build a forecast of tourism development in Ukraine and, in particular, arrivals of foreign tourists to Ukraine. An analytical forecasting model was built by the system of CurveExpert in the form of a polynomial function, analytical forecasting models were built by the system of computer algebra Maple in the form of piecewise linear and piecewise polynomial functions. Numerical prediction models in the MathCAD system using different types of spline-interpolation and predictive functions have been constructed. A comparative analysis of the results of forecasting in different systems was carried out. The results of the comparative analysis give confidence in the development of inbound tourism in Ukraine.
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Conti, Roberto, Jason Kimberley, and Wojciech Szymański. "More localized automorphisms of the Cuntz algebras." Proceedings of the Edinburgh Mathematical Society 53, no. 3 (August 5, 2010): 619–31. http://dx.doi.org/10.1017/s0013091508000886.
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AbstractWe completely determine the localized automorphisms of the Cuntz algebras $\mathcal{O}_n$ corresponding to permutation matrices in Mn ⊗ Mn for n = 3 and n = 4. This result is obtained through a combination of general combinatorial techniques and large scale computer calculations. Our analysis proceeds according to the general scheme proposed in a previous paper, where we analysed in detail the case of $\mathcal{O}_2$ using labelled rooted trees. We also discuss those proper endomorphisms of these Cuntz algebras which restrict to automorphisms of their respective diagonals. In the case of $\mathcal{O}_3$ we compute the number of automorphisms of the diagonal induced by permutation matrices in M3 ⊗ M3 ⊗ M3.
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Быкова, Валентина Прохоровна, and Valentina Prokhorovna Bykova. "Application of training and methodological complex Electrical Engineering." Vestnik of Astrakhan State Technical University. Series: Marine engineering and technologies 2019, no. 4 (November 15, 2019): 130–40. http://dx.doi.org/10.24143/2073-1574-2019-4-130-140.
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The paper presents the laboratory practicum on electrical engineering and physics worked out on the basis of the teaching guide on electrical engineering including 5 test benches.
It includes 24 laboratory works with elements of independent research. The detailed and illustrated test benches of the training complex are the following: Electrical Engineering and Electronics, Electric Machines and Electric Drive, Electric Machine Assembly, Electric Energy, its Transmission and Quality, Principles of Electrical Safety. Some stands are connected to a software and hardware complex designed for measuring the studied values, reflection and processing of analog signals. The compactness and versatility of the training complex Electrical Engineering are noted; one can do 6–9 laboratory works on electrical engineering and physics at one stand. Mathematical calculations of the readings are recommended to perform in Mathcad computer algebra system.
A manual based on the laboratory works, which contains general safety rules, guidelines for per-forming tasks and requirements for their processing, is now under consideration. The advantages
of the elaborated laboratory workshop have been listed; the educational, developmental, pedagogical tasks for solving at the lesson have been analyzed; a laboratory work “Three-phase circuit connection in star scheme” is given as an example. The diagram of a star connection is presented;
a vector diagram of linear and phase voltages relations in a star connection is built. The oscillo-grams of currents under symmetrical load are illustrated. It has been noted that the individual tasks have an exploratory nature, they develop cognitive, independent and creative skills of students. The proposed methodological guidelines are recommended for specialists and bachelors in such areas as “Ship and Power Plants”, “Shipbuilding”, “Heat and Power Engineering and Heat Engineering”, etc., and can be used by teachers of general engineering disciplines
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Bichon, Julien. "Hochschild homology of Hopf algebras and free Yetter–Drinfeld resolutions of the counit." Compositio Mathematica 149, no. 4 (December 17, 2012): 658–78. http://dx.doi.org/10.1112/s0010437x12000656.
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AbstractWe show that if$A$and$H$are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter–Drinfeld resolution of the counit of$A$to the same kind of resolution for the counit of$H$, exhibiting in this way strong links between the Hochschild homologies of$A$and$H$. This enables us to obtain a finite free resolution of the counit of$\mathcal {B}(E)$, the Hopf algebra of the bilinear form associated with an invertible matrix$E$, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case$E=I_n$. It follows that$\mathcal {B}(E)$is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when$E$is an antisymmetric matrix, the$L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of$\mathcal {B}(E)$in the cosemisimple case.
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Sato, Ryo. "Modular Invariant Representations of the Superconformal Algebra." International Mathematics Research Notices 2019, no. 24 (February 9, 2018): 7659–90. http://dx.doi.org/10.1093/imrn/rny007.
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Abstract We compute the modular transformation formula of the characters for a certain family of (finitely or uncountably many) simple modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge $c_{p,p^{\prime }}=3\left (1-\frac{2p^{\prime }}{p}\right ),$ where (p, p′) is a pair of coprime positive integers such that p ≥ 2. When p′ = 1, the formula coincides with that of the $\mathcal{N}=2$ unitary minimal series found by F. Ravanini and S.-K. Yang. In addition, we study the properties of the corresponding “modular S-matrix”, which is no longer a matrix if p′≥ 2.
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Kukharska, N. "PROGRAM IMPLEMENTATION OF ALGORITHMS OF HIDING OF INFORMATION BY METHODS OF A RANDOM INTERVAL." Bulletin of Lviv State University of Life Safety, no. 18 (December 31, 2018): 41–48. http://dx.doi.org/10.32447/20784643.18.2018.03.
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At the current stage of the development of information systems and technologies, global computer systems and multimedia tools, as never before, there is an urgent need to ensure the reliability, security of storage of digital data and their transmission through open channels of information communications.
One of the most promising and popular approaches to solving this problem is based on the applying of computer steganography methods.
The purpose of the article is systematizing information about methods of textual steganography, namely methods of arbitrary intervals, carrying out a comparative bandwidth analysis.
Methods of research - methods of textual steganography: the method of double spaces between words, the method of changing the space code, the method of changing the number of spaces at the end of text strings, the method of changing the number of spaces between words aligned to the width of the text.
Methods of arbitrary interval are used to hide the data in the free space in the text. They use intervals between sentences, spaces at the end of text strings, intervals between words in the text, including those, which are aligned in width, manipulate symbols of spaces, which have different ASCII codes. They are used to organize secret transmission of confidential information through open communication channels.
In the article, on the basis of the software complexes developed in the environment of the computer algebra MathCAD, the steganographic transformations, which consistent with the algorithms of the methods, are sequentially tracked.
The question of the bandwidth of the constructed steganosystems was considered.
Bandwidth is the maximum amount of additional information that can be embedded in one element (symbol) of the text container.
So, the bandwidth of the double-space method between words and the method of changing the space code in the case of the Ukrainian-language text container is 1.75%. The method of changing the number of spaces between aligned by the width of the text of the words has a lower bandwidth of 0.4%. The bandwidth of the method of changing the number of spaces at the end of the text strings depends on the difference between the number of symbols in the longest line and all other lines.
The article also outlines the advantages and disadvantages of each method.
Conclusions. The arbitrary interval methods are effective provided that the text is presented in ASCII format. In general, text files are "inconvenient" containers. They lack redundancy in comparison, for example, with graphic or audio files.
For such arbitrary interval methods as a method of changing the space code, the method of changing the number of spaces at the end of the text strings is characterized by the fact that the hidden data can not be obtained from a hard copy of the text file.
Despite the drawbacks, arbitrary interval methods have reason to be applied because of the prevalence of text file files. Users of computer networks are constantly exchanging text messages. This is a routine everyday action, so text files, even those, which contain hidden confidential information, should not cause unnecessary interest among outsiders.
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LU, BIN, and JIANYUAN K. ZHONG. "THE KAUFFMAN POLYNOMIALS OF PRETZEL KNOTS." Journal of Knot Theory and Its Ramifications 17, no. 02 (February 2008): 157–69. http://dx.doi.org/10.1142/s0218216508006026.
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Let ℚ(α, s) be the field of rational functions in α, s. We compute the Kauffman polynomials of pretzel knots [1,6] using the Kauffman skein theory and linear algebra tools. We give a formula for the Kauffman polynomial of a pretzel knot such that after inputting the sequence notation of the pretzel knot, the output is its Kauffman polynomial. Our calculation can be implemented in Mathematica, Maple, Mathcad, etc.
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Ettahri, S., O. Ramaré, and L. Surel. "Fast multi-precision computation of some Euler products." Mathematics of Computation 90, no. 331 (April 5, 2021): 2247–65. http://dx.doi.org/10.1090/mcom/3630.
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For every modulus q ≥ 3 q\ge 3 , we define a family of subsets A \mathcal {A} of the multiplicative group ( Z / q Z ) × (\mathbb {Z}/q\mathbb {Z})^\times for which the Euler product ∏ p + q Z ∈ A ( 1 − p − s ) \prod _{p+q\mathbb {Z}\in \mathcal {A}}(1-p^{-s}) can be computed with high numerical precision, where s > 1 s>1 is some given real number. We provide a Sage script to do so, and extend our result to compute Euler products ∏ p + q Z ∈ A F ( 1 / p s ) / H ( 1 / p s ) \prod _{p+q\mathbb {Z}\in \mathcal {A}}F(1/p^s)/H(1/p^s) where F F and H H are polynomials with real coefficients, when this product converges absolutely. This enables us to give precise values of several Euler products occurring in number theory.
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CACCIATORI, SERGIO LUIGI, BIANCA LETIZIA CERCHIAI, and ALESSIO MARRANI. "ADAMS-IWASAWA $\mathcal{N} = 8$ BLACK HOLES." International Journal of Modern Physics: Conference Series 13 (January 2012): 44–53. http://dx.doi.org/10.1142/s2010194512006721.
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We study some of the properties of the geometry of the exceptional Lie group E7(7), which describes the U-duality of the [Formula: see text], d = 4 supergravity. In particular, based on a symplectic construction of the Lie algebra 𝔢7(7) due to Adams, we compute the Iwasawa decomposition of the symmetric space [Formula: see text], which gives the vector multiplets' scalar manifold of the corresponding supergravity theory. The explicit expression of the Lie algebra is then used to analyze the origin of [Formula: see text] as scalar configuration of the "large" ⅛-BPS extremal black hole attractors. In this framework it turns out that the U(1) symmetry spanning such attractors is broken down to a discrete subgroup ℤ4, spoiling their dyonic nature near the origin of the scalar manifold. This is a consequence of the fact that the maximal manifest off-shell symmetry of the Iwasawa parametrization is determined by a completely non-compact Cartan subalgebra of the maximal subgroup SL(8, ℝ) of E7(7), which breaks down the maximal possible covariance SL(8, ℝ) to a smaller SL(7, ℝ) subgroup. These results are compared with the ones obtained in other known bases, such as the Sezgin-van Nieuwenhuizen and the Cremmer-Julia /de Wit-Nicolai frames.
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Kohel, David, Kristin Lauter, Christophe Petit, and Jean-Pierre Tignol. "On the quaternion -isogeny path problem." LMS Journal of Computation and Mathematics 17, A (2014): 418–32. http://dx.doi.org/10.1112/s1461157014000151.
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AbstractLet $\mathcal{O}$ be a maximal order in a definite quaternion algebra over $\mathbb{Q}$ of prime discriminant $p$, and $\ell $ a small prime. We describe a probabilistic algorithm which, for a given left $\mathcal{O}$-ideal, computes a representative in its left ideal class of $\ell $-power norm. In practice the algorithm is efficient and, subject to heuristics on expected distributions of primes, runs in expected polynomial time. This solves the underlying problem for a quaternion analog of the Charles–Goren–Lauter hash function, and has security implications for the original CGL construction in terms of supersingular elliptic curves.
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Lyuminarskaja, E. S. "Determination of transient resistance low-current elektric contacts in dusty conditions the contact surfaces." Transportation systems and technology 2, no. 3 (September 15, 2016): 84–96. http://dx.doi.org/10.17816/transsyst20162384-96.
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Introduction. Now on transport, in automotive, aviation, space, household and other vehicles the tiny mechanically managed switches are widely used. The small weight and small dimensions belong to their benefits. However bad contact connection can lead to failure of devices on which work of all difficult technique depends. In household appliances it will not lead to serious problems, but for transport failure of contact system can lead to catastrophic consequences. The most important indicator of reliability of microswitches is non-failure operation of work. Failures of microswitches arise because of the thermal, electric, mechanical and chemical phenomena in a transitional zone of electric contacts. Now big distribution was gained by low-current contacts. Refusals and failures in such contacts appear generally because of hit in intercontact space of dust and other pollution, formation of the films which are not collapsing in case of short circuit of contacts, sudden change of force of pressing of contacts because of breakdown of an elastic element or an expansion spring. In article the mathematical model of contact systems in case of failure on the example of modern electronic microswitches of rod type is provided. Purpose. Creation of mathematical model of contact systems at failure on the example of modern electronic microswitches of rod type, research of operation of microswitches in actual practice operation. Methodology. Theoretical methods of research with use of personal electronic computer were applied to achievement of an effective objective. The theory of contact interaction of rough surfaces, the theory of electric contacts, methods of linear algebra, etc. have been used. The program complexes Mathcad, ANSYS were used. In case of pilot studies methods of mathematical statistics were applied. Results. In article the technique of determination of transitional resistance of low-current electric contacts considering possible dust content of contact surfaces, formation of films on surfaces of contacts and change of parameters of a roughness of contact surfaces as a result of plastic deformations is offered. For the studied contacts dependence of probability of failure on concentration of particles of dust is received. Practical importance. The technique described in article allows to determine the transitional resistance of electric contacts both during the normal work, and at failure. Conclusion. Development of mathematical models of contacts at failure gives the chance to estimate reliability of connections in actual practice of work, to choose key parameters of switching elements taking into account possible pollution, and also to develop new and to specify the available techniques of detection of failures.
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Parker, Chris, and Jason Semeraro. "Algorithms for fusion systems with applications to 𝑝-groups of small order." Mathematics of Computation 90, no. 331 (April 23, 2021): 2415–61. http://dx.doi.org/10.1090/mcom/3634.
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For a prime p p , we describe a protocol for handling a specific type of fusion system on a p p -group by computer. These fusion systems contain all saturated fusion systems. This framework allows us to computationally determine whether or not two subgroups are conjugate in the fusion system for example. We describe a generation procedure for automizers of every subgroup of the p p -group. This allows a computational check of saturation. These procedures have been implemented using Magma. We describe a computer program which searches for saturated fusion systems F \mathcal {F} on p p -groups with O p ( F ) = 1 O_p(\mathcal {F})=1 and O p ( F ) = F O^p(\mathcal {F})=\mathcal {F} . Employing these computational methods we determine all such fusion systems on groups of order p n p^n where ( p , n ) ∈ { ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 3 , 7 ) , ( 5 , 4 ) , ( 5 , 5 ) , ( 5 , 6 ) , ( 7 , 4 ) , ( 7 , 5 ) } (p,n) \in \{(3,4),(3,5),(3,6),(3,7),(5,4),(5,5),(5,6),(7,4),(7,5)\} . This gives the first complete picture of which groups can support saturated fusion systems with O p ( F ) = 1 O_p(\mathcal {F})=1 and O p ( F ) = F O^p(\mathcal {F})=\mathcal {F} on small p p -groups of odd order.
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Bitoun, Thomas, and Travis Schedler. "On -modules related to the -function and Hamiltonian flow." Compositio Mathematica 154, no. 11 (October 12, 2018): 2426–40. http://dx.doi.org/10.1112/s0010437x18007492.
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Let $f$ be a quasi-homogeneous polynomial with an isolated singularity in $\mathbf{C}^{n}$. We compute the length of the ${\mathcal{D}}$-modules ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ generated by complex powers of $f$ in terms of the Hodge filtration on the top cohomology of the Milnor fiber. When $\unicode[STIX]{x1D706}=-1$ we obtain one more than the reduced genus of the singularity ($\dim H^{n-2}(Z,{\mathcal{O}}_{Z})$ for $Z$ the exceptional fiber of a resolution of singularities). We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the quotient ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ is nonzero when $\unicode[STIX]{x1D706}$ is a root of the $b$-function of $f$ (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these ${\mathcal{D}}$-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.
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Gainutdinov, Azat M., and Ingo Runkel. "Projective objects and the modified trace in factorisable finite tensor categories." Compositio Mathematica 156, no. 4 (March 26, 2020): 770–821. http://dx.doi.org/10.1112/s0010437x20007034.
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For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:(1)${\mathcal{C}}$ always contains a simple projective object;(2)if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$-action;(3)the action of the Grothendieck ring of ${\mathcal{C}}$ on the span of internal characters of projective objects can be diagonalised;(4)the linearised Grothendieck ring of ${\mathcal{C}}$ is semisimple if and only if ${\mathcal{C}}$ is semisimple.Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in ${\mathcal{C}}$ carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of $S$-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular $S$-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.
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Grushevsky, Samuel, Klaus Hulek, and Orsola Tommasi. "Stable cohomology of the perfect cone toroidal compactification of 𝒜 g." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 741 (August 1, 2018): 211–54. http://dx.doi.org/10.1515/crelle-2015-0067.
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Abstract We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification {{{\mathcal{A}}_{g}^{\operatorname{Perf}}}} of the moduli space of complex principally polarized abelian varieties stabilizes in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification {{{\mathcal{A}}_{g}^{\operatorname{Matr}}}} stabilizes in fixed degree, and forms a polynomial algebra. For degree up to 8, we describe explicitly the generators of the cohomology, and discuss various approaches to computing all of the stable cohomology in general.
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Li, Huishi. "Computation of minimal homogeneous generating sets and minimal standard bases for ideals of free algebras." Georgian Mathematical Journal 25, no. 3 (September 1, 2018): 451–59. http://dx.doi.org/10.1515/gmj-2017-0006.
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AbstractLet {K\langle X\rangle=K\langle X_{1},\ldots,X_{n}\rangle} be the free algebra generated by {X=\{X_{1},\ldots,X_{n}\}} over a field K. It is shown that, with respect to any weighted {\mathbb{N}}-gradation attached to {K\langle X\rangle}, minimal homogeneous generating sets for finitely generated graded two-sided ideals of {K\langle X\rangle} can be algorithmically computed, and that if an ungraded two-sided ideal I of {K\langle X\rangle} has a finite Gröbner basis {{\mathcal{G}}} with respect to a graded monomial ordering on {K\langle X\rangle}, then a minimal standard basis for I can be computed via computing a minimal homogeneous generating set of the associated graded ideal {\langle\mathbf{LH}(I)\rangle}.
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Ciubotaru, Dan, and Marcelo De Martino. "Dirac Induction for Rational Cherednik Algebras." International Mathematics Research Notices 2020, no. 17 (July 5, 2018): 5155–214. http://dx.doi.org/10.1093/imrn/rny153.
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Abstract We introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex reflection group acting on a finite-dimensional vector space $\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the graded $G$-character of the module, the Euler–Poincaré pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for $\mathsf{H}_{t,c}(G,\mathfrak{h})$ constructed from finite-dimensional $G$-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function $c$. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl–Opdam operators.
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Hassan, A. S. M., and P. S. Joag. "Separability criterion for multipartite quantum states based on the Bloch representation of density matrices." Quantum Information and Computation 8, no. 8&9 (September 2008): 773–90. http://dx.doi.org/10.26421/qic8.8-9-7.
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We give a new separability criterion, a necessary condition for separability of N-partite quantum states. The criterion is based on the Bloch representation of a N-partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to arbitrary N-partite quantum states in $\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes \cdots \otimes \mathcal{H}^{d_N}.$ The criterion can test whether a N-partite state is entangled and can be applied to different partitions of the $N$-partite system. We provide examples that show the ability of this criterion to detect entanglement. We show that this criterion can detect bound entangled states. We prove a sufficiency condition for separability of a 3-partite state, straightforwardly generalizable to the case N > 3, under certain condition. We also give a necessary and sufficient condition for separability of a class of N-qubit states which includes N-qubit PPT states.
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NOVYTSKA, Lyudmyla. "THE ROLE OF MATHEMATICAL EDUCATION IN PROFESSIONAL TRAINING OF SPECIALISTS FOR THE TOURISM SECTOR." "EСONOMY. FINANСES. MANAGEMENT: Topical issues of science and practical activity", no. 3 (43) (March 2019): 78–85. http://dx.doi.org/10.37128/2411-4413-2019-3-7.
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At the current stage of social and economic development of Ukraine, the processes of globalization, integration and informatization of society raise qualitatively new challenges in the field of higher education. Ukraine has recently been focusing on European countries, which consider the tourism sector as a priority area for their economic development. This area of investment is very attractive, creates new jobs and is characterized by a high level of efficiency. Therefore, the socio-economic success of the country as a whole depends on the professional training of future tourists. The basis for the development of specialist-tourist skills is mathematical disciplines, including the discipline "Higher and Applied Mathematics". There is an intensive process of mathematization of knowledge, involving mathematical apparatus in the study of natural sciences, humanities, economics and social sciences, which in turn forms a certain level of mathematical culture of students of universities. The purpose of teaching the course "Higher and Applied Mathematics" is to provide students with basic knowledge of mathematics, which enable them to further master the professional disciplines based on mathematical concepts. In this case, considerable attention should be paid to the formation of practical skills, the ability to apply mathematical methods when solving the tasks of professional activity. One of the tools for improving the quality of mathematical education for students-tourists is to update the content of courses that are read in the context of the future profession and modern socio-economic order. When considering a particular material, the idea of its connection with the future profession should be put on the foreground. In this regard, the course "Higher and Applied Mathematics" for students-tourists VNAU contains several critical sections covering all the main areas of application of mathematical methods in the tourism sector. Teaching is carried out in accordance with the typical curriculum for higher education institutions in the discipline "Higher and Applied Mathematics" for the specialty "Tourism", developed by the teachers of the Department of Mathematics, Physics and Computer Technologies, VNU taking into account the principle of professional orientation. Given the fact that for students the most important is the practical aspect of mathematics, the purpose of the course is to familiarize with the basic concepts and methods of studying modern mathematics necessary for further study of professional disciplines, as well as the formation of skills for the correct mathematical formulation of applied problems and the construction of simple mathematical models. The solution of many applications is due to cumbersome mathematical calculations, the need to visually present the results of calculations, check them, so it is advisable to use computer mathematical systems, in particular, Maccats, in practical classes in mathematics. Teachers of the Department of Mathematics of Physics and Computer Technologies are the ones who prefer it. To work with Mathcad, there are basic elementary skills in working with Windows applications, so you need to implement this environment from the first lessons when studying topics such as Linear Algebra, Differential and Integral Computing, Elements of Probability Theory and Mathematical Statistics. The use of information and communication technologies (ICT) in the educational process greatly enhances the opportunities of both the teacher and the student's cognitive abilities. ICTs allow to visualize and dynamize the process of studying the discipline "Higher and Applied Mathematics", to personalize and direct the educational process for the comprehensive development of personality, to substantially increase the volume, sources of receipt and the nature of the information necessary for a person, the means of its processing. They can have a significant impact on the methodology of holding higher and applied mathematics classes, requiring the teacher to use electronic textbooks, lecture notes, practical work, the testing system for assessing student knowledge, and global sources of information. They also facilitate the organization of effective independent work in non-attendance time for students. It should be noted that to date there are quite a few tutorials, collections of tasks that contain applied tasks that can be considered at lectures and practical classes on higher and applied mathematics. Work on the creation of such methodological materials is conducted, in particular, at the Department of Mathematics, Physics and Computer Technologies. Thus, changing requirements for a specialist in the field of tourism, which is conditioned by modernity, requires a permanent rearrangement of the structural elements of its preparation. Fundamental disciplines to a large extent are applied, and therefore a qualitative assimilation of basic and basic concepts of mathematical disciplines will contribute to the competitiveness and qualitative training of the future specialist.
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Lokshtanov, Daniel, Andreas BjÖrklund, Saket Saurabh, and Meirav Zehavi. "Approximate Counting of k -Paths: Simpler, Deterministic, and in Polynomial Space." ACM Transactions on Algorithms 17, no. 3 (August 2021): 1–44. http://dx.doi.org/10.1145/3461477.
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Recently, Brand et al. [STOC 2018] gave a randomized mathcal O(4 k m ε -2 -time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ε based on exterior algebra. Prior to our work, this has been the state-of-the-art. In this article, we revisit the algorithm by Alon and Gutner [IWPEC 2009, TALG 2010], and obtain the following results: • We present a deterministic 4 k + O (√ k (log k +log 2 ε -1 )) m -time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. • Additionally, we present a randomized 4 k +mathcal O(log k (log k +logε -1 )) m -time polynomial-space algorithm. Our algorithm is simple—we only make elementary use of the probabilistic method. Here, n and m are the number of vertices and the number of edges, respectively. Additionally, our approach extends to approximate counting of other patterns of small size (such as q -dimensional p -matchings).
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LÓPEZ PEÑA, J., S. MAJID, and K. RIETSCH. "Lie theory of finite simple groups and the Roth property." Mathematical Proceedings of the Cambridge Philosophical Society 163, no. 2 (January 9, 2017): 301–40. http://dx.doi.org/10.1017/s030500411600102x.
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AbstractIn noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset $\mathcal{C}$ stable under inversion. We study the associated Killing form K. For the universal calculus associated to $\mathcal{C}$ = G \ {e} we show that the magnitude $\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$ of the Killing form is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ ≠ 1/(N − 1) where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N − 1)-dimensional subspace of invariant vectors is invertible iff the finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in any Sn has invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groups D2n with n odd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.
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De Boeck, M., and P. Vandendriessche. "On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$." Advances in Mathematics of Communications 8, no. 3 (2014): 281–96. http://dx.doi.org/10.3934/amc.2014.8.281.
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"Practical use of mathcad®: Solving mathematical problems with a computer algebra system." Computers & Mathematics with Applications 40, no. 2-3 (July 2000): 417. http://dx.doi.org/10.1016/s0898-1221(00)90187-1.
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Ara, Pere, and Joan Claramunt. "$$L^2$$-Betti numbers arising from the lamplighter group." Journal of Algebraic Combinatorics, June 21, 2021. http://dx.doi.org/10.1007/s10801-021-01044-8.
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AbstractWe apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $$\ell ^2$$ ℓ 2 -Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational $$\ell ^2$$ ℓ 2 -Betti numbers arising from the lamplighter group algebra $${\mathbb Q}[{\mathbb Z}_2 \wr {\mathbb Z}]$$ Q [ Z 2 ≀ Z ] . This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $$\ell ^2$$ ℓ 2 -Betti numbers from the algebras $${\mathbb Q}[{\mathbb Z}_n \wr {\mathbb Z}]$$ Q [ Z n ≀ Z ] , where $$n \ge 2$$ n ≥ 2 is a natural number. We also apply the techniques developed to the generalized odometer algebra $${\mathcal {O}}({\overline{n}})$$ O ( n ¯ ) , where $${\overline{n}}$$ n ¯ is a supernatural number. We compute its $$*$$ ∗ -regular closure, and this allows us to fully characterize the set of $${\mathcal {O}}({\overline{n}})$$ O ( n ¯ ) -Betti numbers.
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Tout, Omar. "Structure Coefficients of the Hecke Algebra of $(\mathcal{S}_{2n},\mathcal{B}_n)$." Electronic Journal of Combinatorics 21, no. 4 (November 13, 2014). http://dx.doi.org/10.37236/3592.
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The Hecke algebra of the pair $(\mathcal{S}_{2n},\mathcal{B}_n)$, where $\mathcal{B}_n$ is the hyperoctahedral subgroup of $\mathcal{S}_{2n}$, was introduced by James in 1961. It is a natural analogue of the center of the symmetric group algebra. In this paper, we give a polynomiality property of its structure coefficients. Our main tool is a combinatorial algebra which projects onto the Hecke algebra of $(\mathcal{S}_{2n},\mathcal{B}_n)$ for every $n$. To build it, by using partial bijections we introduce and study a new class of finite dimensional algebras.
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Virmaux, Aladin. "Partial categorification of Hopf algebras and representation theory of towers of \mathcalJ-trivial monoids." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AT,..., Proceedings (January 1, 2014). http://dx.doi.org/10.46298/dmtcs.2438.
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International audience This paper considers the representation theory of towers of algebras of $\mathcal{J} -trivial$ monoids. Using a very general lemma on induction, we derive a combinatorial description of the algebra and coalgebra structure on the Grothendieck rings $G_0$ and $K_0$. We then apply our theory to some examples. We first retrieve the classical Krob-Thibon's categorification of the pair of Hopf algebras QSym$/NCSF$ as representation theory of the tower of 0-Hecke algebras. Considering the towers of semilattices given by the permutohedron, associahedron, and Boolean lattices, we categorify the algebra and the coalgebra structure of the Hopf algebras $FQSym , PBT$ , and $NCSF$ respectively. Lastly we completely describe the representation theory of the tower of the monoids of Non Decreasing Parking Functions.
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Goldbring, Isaac, and Bradd Hart. "Operator algebras with hyperarithmetic theory." Journal of Logic and Computation, September 23, 2020. http://dx.doi.org/10.1093/logcom/exaa059.
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Abstract We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C^*(\varGamma )$ for $\varGamma $ a finitely presented group, $C^*_\lambda (\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C(2^\omega )$ and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $\textrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $\textrm{C}^*$-algebras).
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Berest, Yuri, Ajay C. Ramadoss, and Yining Zhang. "Hodge decomposition of string topology." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.26.
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Abstract Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$ -equivariant homology $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ of the free loop space of X preserves the Hodge decomposition of $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].
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Tout, Omar. "Structure coefficients of the Hecke algebra of $(\mathcal{S}_{2n}, \mathcal{B}_n)$." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AS,..., Proceedings (January 1, 2013). http://dx.doi.org/10.46298/dmtcs.2323.
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International audience The Hecke algebra of the pair $(\mathcal{S}_{2n}, \mathcal{B}_n)$, where $\mathcal{B}_n$ is the hyperoctahedral subgroup of $\mathcal{S}_{2n}$, was introduced by James in 1961. It is a natural analogue of the center of the symmetric group algebra. In this paper, we give a polynomiality property of its structure coefficients. Our main tool is a combinatorial universal algebra which projects on the Hecke algebra of $(\mathcal{S}_{2n}, \mathcal{B}_n)$ for every $n$. To build it, we introduce new objects called partial bijections. L’algèbre de Hecke de la paire $(\mathcal{S}_{2n}, \mathcal{B}_n)$ , où $\mathcal{B}_n$ est le sous-groupe hyperoctaèdral de $\mathcal{S}_{2n}$, aété introduite par James en 1961. C’est un analogue naturel du centre de l’algèbre du groupe symétrique. Dans ce papier, on donne une propriété de polynomialité de ses coefficients de structure. On utilise une algèbre universelle construite d’une façon combinatoire et qui se projette sur toutes les algèbres de Hecke de $(\mathcal{S}_{2n}, \mathcal{B}_n)$. Pour la construire, on introduit de nouveaux objets appelés bijections partielles.
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Belliard, Raphaël, Bertrand Eynard, and Sylvain Ribault. "The geometry of Casimir W-algebras." SciPost Physics 5, no. 5 (November 23, 2018). http://dx.doi.org/10.21468/scipostphys.5.5.051.
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Let \mathfrak{g}𝔤 be a simply laced Lie algebra, \widehat{\mathfrak{g}}_1𝔤̂1 the corresponding affine Lie algebra at level one, and \mathcal{W}(\mathfrak{g})𝒲(𝔤) the corresponding Casimir W-algebra. We consider \mathcal{W}(\mathfrak{g})𝒲(𝔤)-symmetric conformal field theory on the Riemann sphere. To a number of \mathcal{W}(\mathfrak{g})𝒲(𝔤)-primary fields, we associate a Fuchsian differential system. We compute correlation functions of \widehat{\mathfrak{g}}_1𝔤̂1-currents in terms of solutions of that system, and construct the bundle where these objects live. We argue that cycles on that bundle correspond to parameters of the conformal blocks of the W-algebra, equivalently to moduli of the Fuchsian system.
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Watanabe, Yuta. "An Algebra Associated with a Flag in a Subspace Lattice over a Finite Field and the Quantum Affine Algebra." Electronic Journal of Combinatorics 25, no. 4 (November 2, 2018). http://dx.doi.org/10.37236/7008.
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In this paper, we introduce an algebra $\mathcal{H}$ from a subspace lattice with respect to a fixed flag which contains its incidence algebra as a proper subalgebra. We then establish a relation between the algebra $\mathcal{H}$ and the quantum affine algebra $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$, where $q$ denotes the cardinality of the base field. It is an extension of the well-known relation between the incidence algebra of a subspace lattice and the quantum algebra $U_{q^{1/2}}(\mathfrak{sl}_2)$. We show that there exists an algebra homomorphism from $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$ to $\mathcal{H}$ and that any irreducible module for $\mathcal{H}$ is irreducible as an $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$-module.
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Bergeron-Brlek, Anouk. "Words and Noncommutative Invariants of the Hyperoctahedral Group." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AN,..., Proceedings (January 1, 2010). http://dx.doi.org/10.46298/dmtcs.2870.
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International audience Let $\mathcal{B}_n$ be the hyperoctahedral group acting on a complex vector space $\mathcal{V}$. We present a combinatorial method to decompose the tensor algebra $T(\mathcal{V})$ on $\mathcal{V}$ into simple modules via certain words in a particular Cayley graph of $\mathcal{B}_n$. We then give combinatorial interpretations for the graded dimension and the number of free generators of the subalgebra $T(\mathcal{V})^{\mathcal{B}_n}$ of invariants of $\mathcal{B}_n$, in terms of these words, and make explicit the case of the signed permutation module. To this end, we require a morphism from the Mantaci-Reutenauer algebra onto the algebra of characters due to Bonnafé and Hohlweg. Soit $\mathcal{B}_n$ le groupe hyperoctaédral agissant sur un espace vectoriel complexe $\mathcal{V}$. Nous présentons une méthode combinatoire donnant la décomposition de l'algèbre $T(\mathcal{V})$ des tenseurs sur $\mathcal{V}$ en modules simples via certains mots dans un graphe de Cayley donné. Nous donnons ensuite des interprétations combinatoires pour la dimension graduée et le nombre de générateurs libres de la sous-algèbre $T(\mathcal{V})^{\mathcal{B}_n}$ des invariants de $\mathcal{B}_n$, en termes de ces mots, et explicitons le cas du module de permutation signé. À cette fin, nous utilisons un morphisme entre l'algèbre de Mantaci-Reutenauer et l'algèbre des caractères introduit par Bonnafé et Hohlweg.
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Castillejos, Jorge, Samuel Evington, Aaron Tikuisis, Stuart White, and Wilhelm Winter. "Nuclear dimension of simple $$\mathrm {C}^*$$-algebras." Inventiones mathematicae, December 22, 2020. http://dx.doi.org/10.1007/s00222-020-01013-1.
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AbstractWe compute the nuclear dimension of separable, simple, unital, nuclear, $${\mathcal {Z}}$$ Z -stable $$\mathrm {C}^*$$ C ∗ -algebras. This makes classification accessible from $${\mathcal {Z}}$$ Z -stability and in particular brings large classes of $$\mathrm {C}^*$$ C ∗ -algebras associated to free and minimal actions of amenable groups on finite dimensional spaces within the scope of the Elliott classification programme.
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Kiyoshige, Kazuki, and Takahiro Nishinaka. "The chiral algebra of genus two class $$ \mathcal{S} $$ theory." Journal of High Energy Physics 2021, no. 2 (February 2021). http://dx.doi.org/10.1007/jhep02(2021)199.
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Abstract We construct the chiral algebra associated with the A1-type class $$ \mathcal{S} $$ S theory for the genus two Riemann surface without punctures. By solving the BRST cohomology problem corresponding to a marginal gauging in four dimensions, we find a set of chiral algebra generators that form closed OPEs. Given the fact that they reproduce the spectrum of chiral algebra operators up to large dimensions, we conjecture that they are the complete set of generators. Remarkably, their OPEs are invariant under an action of SU(2) which is not associated with any conserved one-form current in four dimensions. We find that this novel SU(2) strongly constrains the OPEs of non-scalar Schur operators. For completeness, we also check the equivalence of Schur indices computed in two S-dual descriptions with a non-vanishing flavor fugacity turned on.
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Xie, Dan, and Wenbin Yan. "Schur sector of Argyres-Douglas theory and $W$-algebra." SciPost Physics 10, no. 3 (March 30, 2021). http://dx.doi.org/10.21468/scipostphys.10.3.080.
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We study the Schur index, the Zhu’s C_2C2 algebra, and the Macdonald index of a four dimensional \mathcal{N}=2𝒩=2 Argyres-Douglas (AD) theories from the structure of the associated two dimensional WW-algebra. The Schur index is derived from the vacuum character of the corresponding WW-algebra and can be rewritten in a very simple form, which can be easily used to verify properties like level-rank dualities, collapsing levels, and S-duality conjectures. The Zhu’s C_2C2 algebra can be regarded as a ring associated with the Schur sector, and a surprising connection between certain Zhu’s C_2C2 algebra and the Jacobi algebra of a hypersurface singularity is discovered. Finally, the Macdonald index is computed from the Kazhdan filtration of the WW-algebra.
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Kvamme, Sondre. "$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories." Mathematische Zeitschrift, June 5, 2020. http://dx.doi.org/10.1007/s00209-020-02534-4.
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Abstract For an exact category $${{\mathcal {E}}}$$ E with enough projectives and with a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory, we show that the singularity category of $${{\mathcal {E}}}$$ E admits a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of $${{\mathcal {E}}}$$ E and $${\underline{{{\mathcal {E}}}}}$$ E ̲ . We also deduce that the Gorenstein projectives of $${{\mathcal {E}}}$$ E admit a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory under some assumptions. Finally, we compute the $$d\mathbb {Z}$$ d Z -cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga–Gorenstein.
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Etingof, Pavel, and Shlomo Gelaki. "Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof." International Mathematics Research Notices, May 28, 2019. http://dx.doi.org/10.1093/imrn/rnz093.
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Abstract We prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber functor to the category of supervector spaces. This proves Ostrik’s conjecture [25, Conjecture 1.3] in this case. Equivalently, we prove that there exists a unique finite supergroup scheme $\mathcal{G}$ over $k$ and a grouplike element $\epsilon \in k\mathcal{G}$ of order $\le 2$, whose action by conjugation on $\mathcal{G}$ coincides with the parity automorphism of $\mathcal{G}$, such that $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(\mathcal{G},\epsilon )$. In particular, when $\mathcal{C}$ is unipotent, the functor lands in $\textrm{Vec}$, so $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(U)$ for a unique finite unipotent group scheme $U$ over $k$. We apply our result and the results of [17] to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over $k$ (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field $k$ of characteristic $p>0$, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field $k$ of characteristic $p>0$. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper [4], and more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic $\ne 2$ is always a Serre subcategory.
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van Ittersum, Jan-Willem M. "A symmetric Bloch–Okounkov theorem." Research in the Mathematical Sciences 8, no. 2 (March 15, 2021). http://dx.doi.org/10.1007/s40687-021-00253-8.
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AbstractThe algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.
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Kang, Liying, Lele Liu, and Erfang Shan. "Sharp Lower Bounds on the Spectral Radius of Uniform Hypergraphs Concerning Degrees." Electronic Journal of Combinatorics 25, no. 2 (April 13, 2018). http://dx.doi.org/10.37236/6644.
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Let $\mathcal{A}(H)$ and $\mathcal{Q}(H)$ be the adjacency tensor and signless Laplacian tensor of an $r$-uniform hypergraph $H$. Denote by $\rho(H)$ and $\rho(\mathcal{Q}(H))$ the spectral radii of $\mathcal{A}(H)$ and $\mathcal{Q}(H)$, respectively. In this paper we present a lower bound on $\rho(H)$ in terms of vertex degrees and we characterize the extremal hypergraphs attaining the bound, which solves a problem posed by Nikiforov [Analytic methods for uniform hypergraphs, Linear Algebra Appl. 457 (2014) 455–535]. Also, we prove a lower bound on $\rho(\mathcal{Q}(H))$ concerning degrees and give a characterization of the extremal hypergraphs attaining the bound.
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Posur, Sebastian. "A Constructive Approach to Freyd Categories." Applied Categorical Structures, October 13, 2020. http://dx.doi.org/10.1007/s10485-020-09612-y.
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Abstract We discuss Peter Freyd’s universal way of equipping an additive category $$\mathbf {P}$$ P with cokernels from a constructive point of view. The so-called Freyd category $$\mathcal {A}(\mathbf {P})$$ A ( P ) is abelian if and only if $$\mathbf {P}$$ P has weak kernels. Moreover, $$\mathcal {A}(\mathbf {P})$$ A ( P ) has decidable equality for morphisms if and only if we have an algorithm for solving linear systems $$X \cdot \alpha = \beta $$ X · α = β for morphisms $$\alpha $$ α and $$\beta $$ β in $$\mathbf {P}$$ P . We give an example of an additive category with weak kernels and decidable equality for morphisms in which the question whether such a linear system admits a solution is computationally undecidable. Furthermore, we discuss an additional computational structure for $$\mathbf {P}$$ P that helps solving linear systems in $$\mathbf {P}$$ P and even in the iterated Freyd category construction $$\mathcal {A}( \mathcal {A}(\mathbf {P})^{\mathrm {op}} )$$ A ( A ( P ) op ) , which can be identified with the category of finitely presented covariant functors on $$\mathcal {A}(\mathbf {P})$$ A ( P ) . The upshot of this paper is a constructive approach to finitely presented functors that subsumes and enhances the standard approach to finitely presented modules in computer algebra.
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Rhoades, Brendon, Tianyi Yu, and Zehong Zhao. "Harmonic Bases for Generalized Coinvariant Algebras." Electronic Journal of Combinatorics 27, no. 4 (October 30, 2020). http://dx.doi.org/10.37236/9610.
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Let $k \leq n$ be nonnegative integers and let $\lambda$ be a partition of $k$. S. Griffin recently introduced a quotient $R_{n,\lambda}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which simultaneously generalizes the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology rings of Springer fibers studied by Tanisaki and Garsia-Procesi. We describe the space $V_{n,\lambda}$ of harmonics attached to $R_{n,\lambda}$ and produce a harmonic basis of $R_{n,\lambda}$ indexed by certain ordered set partitions $\mathcal{OP}_{n,\lambda}$. The combinatorics of this basis is governed by a new extension of the Lehmer code of a permutation to $\mathcal{OP}_{n, \lambda}$.
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Rezapour, Shahram, Sina Etemad, and Hakimeh Mohammadi. "A mathematical analysis of a system of Caputo–Fabrizio fractional differential equations for the anthrax disease model in animals." Advances in Difference Equations 2020, no. 1 (September 9, 2020). http://dx.doi.org/10.1186/s13662-020-02937-x.
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Abstract We study a fractional-order model for the anthrax disease between animals based on the Caputo–Fabrizio derivative. First, we derive an existence criterion of solutions for the proposed fractional $\mathcal {CF}$ CF -system of the anthrax disease model by utilizing the Picard–Lindelof technique. By obtaining the basic reproduction number $\mathcal{R}_{0}$ R 0 of the fractional $\mathcal{CF}$ CF -system we compute two disease-free and endemic equilibrium points and check the asymptotic stability property. Moreover, by applying an iterative approach based on the Sumudu transform we investigate the stability of the fractional $\mathcal{CF}$ CF -system. We obtain approximate series solutions of this system by means of the homotopy analysis transform method, in which we invoke the linear Laplace transform. Finally, after the convergence analysis of the numerical method HATM, we present a numerical simulation of the $\mathcal{CF}$ CF -fractional anthrax disease model and review the dynamical behavior of the solutions of this $\mathcal {CF}$ CF -system during a time interval.