IA Scholar Query: Finding Irreducible Polynomials over Finite Fields
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 04 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Sampled Characteristic Modeling and Forgetting Gradient Learning Algorithm for Robot Servo Systems
https://scholar.archive.org/work/m2wvj2ywijgbxpvjf72gzy43v4
Servo systems of robotic exhibit nonlinear coupling with multidimensional characteristics, which poses a challenge to existing modeling and identification techniques. According to a kind of robot servo system which runs repetitively operations over a prespecified finite time interval, a low-order sampling characteristic modeling method is derived in this work. Characteristic parameters are allowed to vary from both time axis and iteration one; the forgetting gradient learning algorithm is utilized to estimate characteristic parameters. Furthermore, the effectiveness of the proposed algorithms is proved via theoretical analyses and numerical simulations.Hongbo Bi, Dong Chen, Yanjuan Li, Ting You, Sai Zouwork_m2wvj2ywijgbxpvjf72gzy43v4Tue, 04 Oct 2022 00:00:00 GMTModuli of Lie p-algebras
https://scholar.archive.org/work/3pnhhiyuerg6pamfjyildv6z5m
In this paper, we study moduli spaces of finite-dimensional Lie algebras with flat center, proving that the forgetful map from Lie p-algebras to Lie algebras is an affine fibration, and we point out a new case of existence of a p-mapping. Then we illustrate these results for the special case of Lie algebras of rank 3, whose moduli space we build and study over Z. We extend the classical equivalence of categories between locally free Lie p-algebras of finite rank with finite locally free group schemes of height 1, showing that the centers of these objects correspond to each other. We finish by analysing the smoothness of the moduli of p-Lie algebras of rank 3, in particular identifying some smooth components.Alice Bouilletwork_3pnhhiyuerg6pamfjyildv6z5mMon, 03 Oct 2022 00:00:00 GMTSpectrahedral Shadows and Completely Positive Maps on Real Closed Fields
https://scholar.archive.org/work/zgfmnn6lqndknms2efdbtmgpze
In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a given support is a spectrahedral shadow in terms of sums of squares. As an application of this result we prove that the cone of copositive matrices of size n≥5 is not a spectrahedral shadow, answering a question of Scheiderer. Our arguments are based on the model theoretic observation that any formula defining a spectrahedral shadow must be preserved by every unital ℝ-linear completely positive map R→ R on a real closed field extension R of ℝ.Manuel Bodirsky, Mario Kummer, Andreas Thomwork_zgfmnn6lqndknms2efdbtmgpzeMon, 03 Oct 2022 00:00:00 GMTComplete Positivity of Comultiplication and Primary Criteria for Unitary Categorification
https://scholar.archive.org/work/s7vueqs3vzdj7pomfw2lezusny
In this paper, we investigate quantum Fourier analysis on subfactors and unitary fusion categories. We prove the complete positivity of the comultiplication for subfactors and derive a primary n-criterion of unitary categorifcation of multifusion rings. It is stronger than the Schur product criterion when n≥3. The primary criterion could be transformed into various criteria which are easier to check in practice even for noncommutative, high-rank, high-multiplicity, multifusion rings. More importantly, the primary criterion could be localized on a sparse set, so that it works for multifusion rings with sparse known data. We give numerous examples to illustrate the efficiency and the power of these criteria.Linzhe Huang and Zhengwei Liu and Sebastien Palcoux and Jinsong Wuwork_s7vueqs3vzdj7pomfw2lezusnyMon, 03 Oct 2022 00:00:00 GMTPoles of finite-dimensional representations of Yangians
https://scholar.archive.org/work/csfsvnsdyra3hbjjztaeueoehu
Let 𝔤 be a finite-dimensional simple Lie algebra over ℂ, and let Y_ħ(𝔤) be the Yangian of 𝔤. In this paper, we study the sets of poles of the rational currents defining the action of Y_ħ(𝔤) on an arbitrary finite-dimensional vector space V. Using a weak, rational version of Frenkel and Hernandez' Baxter polynomiality, we obtain a uniform description of these sets in terms of the Drinfeld polynomials encoding the composition factors of V and the inverse of the q-Cartan matrix of 𝔤. We then apply this description to obtain a concrete set of sufficient conditions for the cyclicity and simplicity of the tensor product of any two irreducible representations, and to classify the finite-dimensional irreducible representations of the Yangian double.Sachin Gautam, Curtis Wendlandtwork_csfsvnsdyra3hbjjztaeueoehuMon, 03 Oct 2022 00:00:00 GMTStructure vs. Randomness for Bilinear Maps
https://scholar.archive.org/work/qj5k5m7aezhn5kue2njwha3efq
We prove that the slice rank of a 3-tensor (a combinatorial notion introduced by Tao in the context of the cap-set problem), the analytic rank (a Fourier-theoretic notion introduced by Gowers and Wolf), and the geometric rank (an algebro-geometric notion introduced by Kopparty, Moshkovitz, and Zuiddam) are all equal up to an absolute constant. As a corollary, we obtain strong trade-offs on the arithmetic complexity of a biased bilinear map, and on the separation between computing a bilinear map exactly and on average. Our result settles open questions of Haramaty and Shpilka [STOC 2010], and of Lovett [Discrete Anal. 2019] for 3-tensors.Alex Cohen, Guy Moshkovitzwork_qj5k5m7aezhn5kue2njwha3efqMon, 03 Oct 2022 00:00:00 GMTOrnstein-Zernike behavior for Ising models with infinite-range interactions
https://scholar.archive.org/work/rgdzdnbj5venlfuafm4yeswm3e
We prove Ornstein-Zernike behavior for the large-distance asymptotics of the two-point function of the Ising model above the critical temperature under essentially optimal assumptions on the interaction. The main contribution of this work is that the interactions are not assumed to be of finite range. To the best of our knowledge, this is the first proof of OZ asymptotics for a nontrivial model with infinite-range interactions. Our results actually apply to the Green function of a large class of "self-repulsive in average" models, including a natural family of self-repulsive polymer models that contains, in particular, the self-avoiding walk, the Domb-Joyce model and the killed random walk. We aimed at a pedagogical and self-contained presentation.Yacine Aoun, Sébastien Ott, Yvan Velenikwork_rgdzdnbj5venlfuafm4yeswm3eMon, 03 Oct 2022 00:00:00 GMTRank 2 Amalgams and Fusion Systems
https://scholar.archive.org/work/fgd6aognqjgp5b6zqc4sapng6m
We classify fusion systems ℱ in which O_p(ℱ)={1}, and there are two Aut_ℱ(S)-invariant essential subgroups whose normalizer systems generate ℱ. We employ the amalgam method and, as a bonus, obtain p-local characterizations of certain rank 2 group amalgams whose parabolic subgroups involve strongly p-embedded subgroups.Martin van Beekwork_fgd6aognqjgp5b6zqc4sapng6mMon, 03 Oct 2022 00:00:00 GMTA note on morphisms to wreath products
https://scholar.archive.org/work/ladckudjwvckxiwhoq37yfexfy
Given a morphism φ : G → A ≀ B from a finitely presented group G to a wreath product A ≀ B, we show that, if the image of φ is a sufficiently large subgroup, then ker(φ) contains a non-abelian free subgroup and φ factors through an acylindrically hyperbolic quotient of G. As direct applications, we classify the finitely presented subgroups in A ≀ B up to isomorphism and we deduce that a group having a wreath product (non-trivial) ≀ (infinite) as a quotient must be SQ-universal (extending theorems of Baumslag and Cornulier-Kar). Finally, we exploit our theorem in order to describe the structure of the automorphism groups of several families of wreath products, highlighting an interesting connection with the Kaplansky conjecture on units in group rings.Anthony Genevois, Romain Tesserawork_ladckudjwvckxiwhoq37yfexfyMon, 03 Oct 2022 00:00:00 GMTThe Triangulated Auslander–Iyama Correspondence
https://scholar.archive.org/work/v2wbug5wknfwboaa7oteyqi4mm
We work over a perfect field. Recent work of the third-named author established a Triangulated Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted 3-periodic to algebraic triangulated categories of finite type. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. In this article we prove a higher-dimensional version of this result that, given an integer d≥1, relates twisted (d+2)-periodic algebras to algebraic triangulated categories with a dℤ-cluster tilting object. We also show that the latter triangulated categories admit a unique differential graded enhancement. Our result yields recognition theorems for interesting algebraic triangulated categories, such as the Amiot cluster category of a self-injective quiver with potential in the sense of Herschend and Iyama and, more generally, the Amiot-Guo-Keller cluster category associated with a d-representation finite algebra in the sense of Iyama and Oppermann. As an application of our result, we obtain infinitely many triangulated categories with a unique differential graded enhancement that is not strongly unique. In the appendix, B. Keller explains how - combined with crucial results of August and Hua-Keller - our main result yields the last key ingredient to prove the Donovan-Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds.Gustavo Jasso, Bernhard Keller, Fernando Murowork_v2wbug5wknfwboaa7oteyqi4mmMon, 03 Oct 2022 00:00:00 GMTHolographic BCFT Spectra from Brane Mergers
https://scholar.archive.org/work/amb54iox2bdjtnz6nkxq52h7zi
We use holography to study the spectra of boundary conformal field theories (BCFTs). To do so, we consider a 2-dimensional Euclidean BCFT with two circular boundaries that correspond to dynamical end-of-the-world branes in 3-dimensional gravity. Interactions between these branes inform the operator content and the energy spectrum of the dual BCFT. As a proof of concept, we first consider two highly separated branes whose only interaction is taken to be mediated by a scalar field. The holographic computation of the scalar-mediated exchange reproduces a light scalar primary and its global descendants in the closed-string channel of the dual BCFT. We then consider a gravity model with point particles. Here, the interaction of two separated branes corresponds to a heavy closed-string operator which lies below the black hole threshold. However, we may also consider branes at finite separation that "merge" non-smoothly. Such brane mergers can be used to describe unitary sub-threshold boundary-condition-changing operators in the open-string spectrum of the BCFT. We also find a new class of sub-threshold Euclidean bra-ket wormhole saddles with a factorization puzzle for closed-string amplitudes.Shovon Biswas, Jani Kastikainen, Sanjit Shashi, James Sullywork_amb54iox2bdjtnz6nkxq52h7ziMon, 03 Oct 2022 00:00:00 GMTRational curves on del Pezzo surfaces in positive characteristic
https://scholar.archive.org/work/t5miooq4o5eaxmiaw2yf2dqssm
We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic 0. We also investigate the principles of Geometric Manin's Conjecture for weak del Pezzo surfaces. In the course of this investigation, we give examples of weak del Pezzo surfaces defined over 𝔽_2(t) or 𝔽_3(t) such that the exceptional sets in Manin's Conjecture are Zariski dense.Roya Beheshti, Brian Lehmann, Eric Riedl, Sho Tanimotowork_t5miooq4o5eaxmiaw2yf2dqssmMon, 03 Oct 2022 00:00:00 GMTPeriodic de Rham bundles over curves
https://scholar.archive.org/work/pi42xde7cndnhd5hzvm7ha2lua
In this article, we introduce the notion of periodic de Rham bundles over smooth complex curves. We prove that motivic de Rham bundles over smooth complex curves are periodic. We conjecture that irreducible periodic de Rham bundles over smooth complex curves are motivic. We show that the conjecture holds for rank one objects and rigid objects.Raju Krishnamoorthy, Mao Shengwork_pi42xde7cndnhd5hzvm7ha2luaMon, 03 Oct 2022 00:00:00 GMTGalois actions of finitely generated groups rarely have model companions
https://scholar.archive.org/work/2zhrzmlf5jhdfmni4prgwb6uwa
We show that if G is a finitely generated group such that its profinite completion G is "far from being projective" (that is the kernel of the universal Frattini cover of G is not a small profinite group), then the class of existentially closed G-actions on fields is not elementary. Since any infinite, finitely generated, virtually free, and not free group is "far from being projective", the main result of this paper corrects an error in our paper "Model theory of fields with virtually free group actions", Proc. London Math. Soc., (2) 118 (2019), 221–256 by showing the negation of Theorem 3.26 in that paper.Özlem Beyarslan, Piotr Kowalskiwork_2zhrzmlf5jhdfmni4prgwb6uwaMon, 03 Oct 2022 00:00:00 GMTFans and polytopes in tilting theory I: Foundations
https://scholar.archive.org/work/sxslnmb73jhtvislnmjdjiunoy
For a finite dimensional algebra A over a field k, the 2-term silting complexes of A gives a simplicial complex Δ(A) called the g-simplicial complex. We give tilting theoretic interpretations of the h-vectors and Dehn-Sommerville equations of Δ(A). Using g-vectors of 2-term silting complexes, Δ(A) gives a nonsingular fan Σ(A) in the real Grothendieck group K_0( proj A)_ℝ called the g-fan. We give several basic properties of Σ(A) including sign-coherence, sign decomposition, idempotent reductions, Jasso reductions, pairwise positivity and a connection with Newton polytopes of A-modules. Moreover, Σ(A) gives a (possibly infinite and non-convex) polytope P(A) in K_0( proj A)_ℝ called the g-polytope of A. We call A g-convex if P(A) is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of A. There are precisely 7 convex g-polyogons up to isomorphism. We give a classification of algebras whose g-polytopes are smooth Fano. We study g-fans and g-polytopes of 3 classes of algebras. First, the g-fan of a classical or generalized preprojective algebra is given by the Coxeter fan. It is g-convex if and only if it is of type A or B, and in this case, its g-polytope is the dual polytope of the short root polytope. Secondly, the g-fan of a Jacobian algebra of a non-degenerate quiver with potential is given by the g-fan of the corresponding cluster algebra. It is g-convex if and only if it is mutation equivalent to a quiver of type A. Thirdly, we classify Brauer graph algebras which are g-convex, and describe their g-polytopes as the root polytopes of type A or C.Toshitaka Aoki, Akihiro Higashitani, Osamu Iyama, Ryoichi Kase, Yuya Mizunowork_sxslnmb73jhtvislnmjdjiunoySun, 02 Oct 2022 00:00:00 GMTPolynomial representations of the Witt Lie algebra
https://scholar.archive.org/work/4334y74y7bgexaqzytvsrasxgy
The Witt algebra W_n is the Lie algebra of all derivations of the n-variable polynomial ring V_n=C[x_1, ..., x_n] (or of algebraic vector fields on A^n). A representation of W_n is polynomial if it arises as a subquotient of a sum of tensor powers of V_n. Our main theorems assert that finitely generated polynomial representations of W_n are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of Fin^op, where Fin is the category of finite sets. We also show that polynomial representations of W_n are equivalent to polynomial representations of the endomorphism monoid of A^n. These equivalences are a special case of an operadic version of Schur--Weyl duality, which we establish.Steven V Sam, Andrew Snowden, Philip Tostesonwork_4334y74y7bgexaqzytvsrasxgySun, 02 Oct 2022 00:00:00 GMTThe F-signature Function on the Ample Cone
https://scholar.archive.org/work/rlgnpgah65ccnpfk2unw7itgze
For any fixed globally F-regular projective variety X over an algebraically closed field of positive characteristic, we study the F-signature of section rings of X with respect to the ample Cartier divisors on X. In particular, we define an F-signature function on the ample cone of X and show that it is locally Lipschitz continuous. We further prove that the F-signature function extends to the boundary of the ample cone. We also establish an effective comparison between the F-signature function and the volume function on the ample cone. As a consequence, we show that for divisors that are nef but not big, the extension of the F-signature is zero.Seungsu Lee, Swaraj Pandework_rlgnpgah65ccnpfk2unw7itgzeSun, 02 Oct 2022 00:00:00 GMTBounded Cohomology of Groups acting on Cantor sets
https://scholar.archive.org/work/alrysfnl6ffjlgpk55p2yo7u5u
We study the bounded cohomology of certain groups acting on the Cantor set. More specifically, we consider the full group of homeomorphisms of the Cantor set as well as Thompson's group V. We prove that both of these groups are boundedly acyclic, that is the bounded cohomology with trivial real coefficients vanishes in positive degrees. Combining this result with the already established ℤ-acyclicity of Thompson's group V, will make V the first example of a finitely generated group, in fact the first example of a group of type F_∞, which is universally boundedly acyclic. Before proving bounded acyclicity, we gather various properties of the groups under consideration and certain subgroups thereof. As a consequence the proofs of bounded acyclicity will be relatively short. It will turn out that the approaches to handle these groups are very similar. This suggests that there could be a unifying approach which would imply the bounded acyclicity of a larger class of groups acting on the Cantor set, including the discussed ones.Konstantin Andritschwork_alrysfnl6ffjlgpk55p2yo7u5uSun, 02 Oct 2022 00:00:00 GMTSphere Packing Densities of Sublattices of the Mordell-Weil Lattices of two Families of Elliptic Curves
https://scholar.archive.org/work/us7qh6koincr3dgdx2jzhu5bo4
In this paper, we examine certain maximal rank sublattices of the Mordell-Weil lattices of two families of elliptic curves over fields of characteristic p > 0. We compute explicit lower bounds on the densest sphere packings of these sublattices by finding lower bounds on the minimal norms of the sublattices and explicitly computing the volumes of their fundamental domains.Arjun Nigamwork_us7qh6koincr3dgdx2jzhu5bo4Sun, 02 Oct 2022 00:00:00 GMTCo-axial metrics on the sphere and algebraic numbers
https://scholar.archive.org/work/3anwv3ajdvfdlmukojokzyklo4
In this paper, we study properties of the set of tuples (t_1,...t_n) such that a conical metric of constant curvature 1 with conic singularities at t_1,...,t_n with prescribed angles exists on the Riemann sphere and the monodromy group of its developing map is isomorphic to a subgroup of the unit circle.Zhijie Chen, Chang-Shou Lin, Yifan Yangwork_3anwv3ajdvfdlmukojokzyklo4Sun, 02 Oct 2022 00:00:00 GMT