81 research outputs found

    UNOmaha Problem of the Week (2021-2022 Edition)

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    The University of Omaha math department\u27s Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester\u27s end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes. Now there are three difficulty tiers to POW problems, roughly corresponding to easy/medium/hard difficulties, with each tier getting twelve problems per semester, and three problems (one of each tier) per week posted online and around campus. The tiers are named after the EPH classification of conic sections (which is connected to many other classifications in math), and in the present compilation they abide by the following color-coding: Cyan, Green, and Magenta. In practice, when creating the problem sets, we begin with a large enough pool of problem drafts and separate out the ones which are most obviously elliptic or hyperbolic, and then the remaining ones fall into parabolic. The tiers don\u27t necessarily reflect workload, though, only prerequisite mathematical background. Ideally, the solutions to elliptic problems, and any parts of solutions to parabolic and hyperbolic problems not covered in standard undergraduate courses, are meant to test participants\u27 creativity. Beware, though, many solutions also include additional commentary which varies wildly in the reader\u27s assumed mathematical maturity

    Critical site percolation on the triangular lattice: From integrability to conformal partition functions

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    Critical site percolation on the triangular lattice is described by the Yang-Baxter solvable dilute A2(2)A_2^{(2)} loop model with crossing parameter specialized to λ=π3\lambda=\frac\pi3, corresponding to the contractible loop fugacity β=2cos4λ=1\beta=-2\cos4\lambda=1. We study the functional relations satisfied by the commuting transfer matrices of this model and the associated Bethe ansatz equations. The single and double row transfer matrices are respectively endowed with strip and periodic boundary conditions, and are elements of the ordinary and periodic dilute Temperley-Lieb algebras. The standard modules for these algebras are labeled by the number of defects dd and, in the latter case, also by the twist eiγe^{i\gamma}. Nonlinear integral equation techniques are used to analytically solve the Bethe ansatz functional equations in the scaling limit for the central charge c=0c=0 and conformal weights Δ,Δˉ\Delta,\bar\Delta. For the groundstates, we find Δ=Δ1,d+1\Delta=\Delta_{1,d+1} for strip boundary conditions and (Δ,Δˉ)=(Δγ/π,d/2,Δγ/π,d/2)(\Delta,\bar\Delta)=(\Delta_{\gamma/\pi,d/2},\Delta_{\gamma/\pi,-d/2}) for periodic boundary conditions, where Δr,s=124((3r2s)21)\Delta_{r,s}=\frac1{24}((3r-2s)^2-1). We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For d8d\le8, these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading 2020 or more conformal eigenenergies. With these conjectures, we apply the Markov traces to obtain the conformal partition functions on the cylinder and torus. These precisely coincide with our previous results for critical bond percolation on the square lattice described by the dense A1(1)A_1^{(1)} loop model with λ=π3\lambda=\frac\pi3. The concurrence of all this conformal data provides compelling evidence supporting a strong form of universality between these two stochastic models as logarithmic CFTs.Comment: 81 page

    Gauge theories on quantum spaces

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    We review the present status of gauge theories built on various quantum space-times described by noncommutative space-times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with particular emphasis on some explored quantum properties. Recent attempts aiming to include gravity dynamics within a noncommutative framework are also considered.Comment: 141 pages. Review article. This is a preliminary versio

    Images of multilinear polynomials on n×nn\times n upper triangular matrices over infinite fields

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    In this paper we prove that the image of multilinear polynomials evaluated on the algebra UTn(K)UT_n(K) of n×nn\times n upper triangular matrices over an infinite field KK equals JrJ^r, a power of its Jacobson ideal J=J(UTn(K))J=J(UT_n(K)). In particular, this shows that the analogue of the Lvov-Kaplansky conjecture for UTn(K)UT_n(K) is true, solving a conjecture of Fagundes and de Mello. To prove that fact, we introduce the notion of commutator-degree of a polynomial and characterize the multilinear polynomials of commutator-degree rr in terms of its coefficients. It turns out that the image of a multilinear polynomial ff on UTn(K)UT_n(K) is JrJ^r if and only if ff has commutator degree rr.Comment: To appear in Israel Journal of Mathematic

    Indutseeritud 3-Lie superalgebrad ja nende rakendused superruumis

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    Väitekirja elektrooniline versioon ei sisalda publikatsiooneKäesoleva doktoritöö eesmärk on uurida selliste n-Lie superalgerbrate omadusi, mis on konstrueeritud kasutades (n-1)-Lie superalgebra aluseks olevat (n-1)-aarset tehet, seda eriti juhul n=3. Tavalise Lie algebra mõistet on võimalik super- (või Z_2-gradueeritud) struktuuridele üle kanda kui toome sisse Lie superalgebra mõiste. Sarnaselt on võimalik n-Lie algebra, kus binaarne tehe on asendatud n-aarse tehtega, üldistada superstruktuuridele, kui kasutame Filippov-Jacobi samasuse gradueeritud analoogi, saades n-Lie superalgebra. Väitekirjas on esitatud madaladimensionaalsete 3-Lie superalgebrate klassifikatsioon. Lisaks näitame, et n-Lie superalgebra abil, mille tehtele leidub superjälg, saab konstrueerida (n+1)-Lie superalgebra, mida me nimetame indutseeritud (n+1)-Lie superalgebraks. Enamgi veel, on tõestatud, et kommutatiivse superalgebra korral on võimalik indutseerida erinevad 3-Lie superalgebra struktuurid kasutades involutsiooni, derivatsiooni või neid mõlemad korraga. Dissertatsioonis on toodud Nambu-Hamiltoni võrrandi üldistus superruumis jaoks, ja on näidatud, et selle abil on võimalik indutseerida ternaarsete Nambu-Poissoni sulgude pere superruumi paarisfunktsioonide jaoks. Järgnevalt on konstrueeritud indutseeritud 3-Lie superalgebrate indutseeritud esitused, kasutades selleks vastavalt kas esialgset binaarset Lie algebrat koos jäljega või Lie superalgebrat koos superjäljega. Töös on näidatud, et 3-Lie algebra indutseeritud esitus on sisestatav jäljeta maatriksite Lie algebrasse sl(V), kus sümboliga V on tähistatud esituse ruum. Kahedimensionaalse indutseeritud esituse korral on esitatud tingimused, mida vastav esitus peab rahuldama, et ta oleks taandumatu.The aim of the present thesis is to study the properties and characteristics of n-Lie superalgebras that are constructed using an operation from (n-1)-Lie superalgebras, especially in the case n=3. A regular Lie algebra can be extended to super- (or Z_2-graded) structures by introducing the notion of Lie superalgebra. Similarly n-Lie algebra, where binary operation is replcaed with n-ary multiplication law, can be extended to superstructures by making use of a graded Filippov-Jacobi identity, giving a notion of n-Lie superalgebra. In the dissertation a classification of low dimensional 3-Lie superalgebras is presented. We show that an n-Lie superalgebra equipped with a supertrace can be used to construct a (n+1)-Lie superalgebra, which is referred to as the induced (n+1)-Lie superalgebra. It is proved that one can construct induced 3-Lie superalgebras from commutative superalgebras by using involution, even degree derivation, or combination of both of them together. In the thesis a generalization of Nambu-Hamilton equation to a superspace is proposed, and it is shown that it induces a family of ternary Nambu-Poisson brackets of even degree functions on a superspace. Finally a representations of induced 3-Lie algebras and Lie superalgebras are constructed by means of a representation of the initial binary Lie algebra and trace or Lie superalgebra and supertrace, respectively. It is shown that the constructed induced representation of 3-Lie algebra is a representation by traceless matrices, that is, lies in the Lie algebra sl(V), where V is a representation space. For 2-dimensional representations the irreduciblility condition of the induced representation of induced 3-Lie algebra is found.https://www.ester.ee/record=b536058
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