842 research outputs found

    On the irreducible spin representations of symmetric and alternating groups which remain irreducible in characteristic 3 3

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    For any finite group and any prime one can ask which ordinary irreducible representations remain irreducible in characteristic , or more generally, which representations remain homogeneous in characteristic . In this paper we address this question when is a proper double cover of the symmetric or alternating group. We obtain a classification when except in the case of a certain family of partitions relating to spin RoCK blocks. Our techniques involve induction and restriction, degree calculations, decomposing projective characters and recent results of Kleshchev and Livesey on spin RoCK blocks

    Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity

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    In each of the 10 cases with propagators of unit or zero mass, the finite part of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letter words in the 7-letter alphabet of the 1-forms Ω:=dz/z\Omega:=dz/z and ωp:=dz/(λ−p−z)\omega_p:=dz/ (\lambda^{-p}-z), where λ\lambda is the sixth root of unity. Three diagrams yield only ζ(Ω3ω0)=1/90π4\zeta(\Omega^3\omega_0)=1/90\pi^4. In two cases π4\pi^4 combines with the Euler-Zagier sum ζ(Ω2ω3ω0)=∑m>n>0(−1)m+n/m3n\zeta(\Omega^2\omega_3\omega_0)=\sum_{m> n>0}(-1)^{m+n}/m^3n; in three cases it combines with the square of Clausen's Cl2(π/3)=ℑζ(Ωω1)=∑n>0sin⁥(πn/3)/n2Cl_2(\pi/3)=\Im \zeta(\Omega\omega_1)=\sum_{n>0}\sin(\pi n/3)/n^2. The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: ℜζ(Ω2ω3ω1)=∑m>n>0(−1)mcos⁥(2πn/3)/m3n\Re \zeta(\Omega^2\omega_3\omega_1)= \sum_{m>n>0}(-1)^m\cos(2\pi n/3)/m^3n. The previously unidentified term in the 3-loop rho-parameter of the standard model is merely D3=6ζ(3)−6Cl22(π/3)−1/24π4D_3=6\zeta(3)-6 Cl_2^2(\pi/3)-{1/24}\pi^4. The remarkable simplicity of these results stems from two shuffle algebras: one for nested sums; the other for iterated integrals. Each diagram evaluates to 10 000 digits in seconds, because the primitive words are transformable to exponentially convergent single sums, as recently shown for ζ(3)\zeta(3) and ζ(5)\zeta(5), familiar in QCD. Those are SC∗(2)^*(2) constants, whose base of super-fast computation is 2. Mass involves the novel base-3 set SC∗(3)^*(3). All 10 diagrams reduce to SC∗(3)âˆȘ^*(3)\cupSC∗(2)^* (2) constants and their products. Only the 6-mass case entails both bases.Comment: 41 pages, LaTe

    On the Basis Number of the Strong Product of Theta Graphs with Cycles

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    In graph theory, there are many numbers that give rise to a better understanding and interpretation of the geometric properties of a given graph such as the crossing number, the thickness, the genus, the basis number, etc.

    On minimum cycle bases of the wreath product of wheels with stars

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    The length of a cycle basis of a graph is the sum of the lengths of its elements. A minimum cycle basis is a cycle basis with minimum length. In this work, a construction of a minimum cycle basis for the wreath product of wheels with stars is presented. Moreover, the length of minimum cycle basis and the length of its longest cycle are calculated.Scopu

    International Journal of Mathematical Combinatorics, Vol.2

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    The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

    Mechanized Advent Calendar

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    Within this document, the “Naughty and Nice” team will describe the details associated with the completion of this project, which involves creating an animated advent calendar for the Cambria Christmas Market. The problem, the customers and stakeholders, and the general goals of the project are introduced. The team conducted background research; how this research helps in finding a solution or supporting the needs of the sponsor is illustrated. Initial research includes interviews with the sponsor and potential users, patent and existing product searches, and some technical research pertaining to the subject. The team found that there are not many existing products that are similar to this project, but there are some which will lend the team inspiration and ideas for implementation in the design process. The scope of the problem is assessed from the customer’s needs and wants, along with a defined problem statement and a Quality FunctionDevelopment (QFD) table. The team illustrates the process of selecting a chosen Concept Design. The Final Design direction for the façade and each box theme is described in detail, including functionality, evidence that the design will meet specifications, a discussion of safety, maintenance, and repair considerations, and a detailed cost analysis. A Manufacturing Plan is delineated, describing procurement, manufacturing, and assembly of each box, especially with respect to moving components. The team introduces a Design Verification Plan, which describes how each design specification was satisfied via testing. The overall design process of the project is investigated including the key deliverables and their due dates. Finally, the conclusion summarizes the document

    The bubble algebra: structure of a two-colour Temperley–Lieb Algebra

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    We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley–Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang–Baxter equations
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