IA Scholar Query: Combining Algebraic Computing and Term-Rewriting for Geometry Theorem Proving.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 04 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440LIPIcs, Volume 238, DNA 28, Complete Volume
https://scholar.archive.org/work/627o3xn4vbbgpdox5dwolgcuny
LIPIcs, Volume 238, DNA 28, Complete VolumeThomas E. Ouldridge, Shelley F. J. Wickhamwork_627o3xn4vbbgpdox5dwolgcunyThu, 04 Aug 2022 00:00:00 GMTComputing Real Numbers with Large-Population Protocols Having a Continuum of Equilibria
https://scholar.archive.org/work/fdjsn7ibbff3ziqp2fqu65g54i
Bournez, Fraigniaud, and Koegler [Bournez et al., 2012] defined a number in [0,1] as computable by their Large-Population Protocol (LPP) model, if the proportion of agents in a set of marked states converges to said number over time as the population grows to infinity. The notion, however, restricts the ordinary differential equations (ODEs) associated with an LPP to have only finitely many equilibria. This restriction places an intrinsic limitation on the model. As a result, a number is computable by an LPP if and only if it is algebraic, namely, not a single transcendental number can be computed under this notion. In this paper, we lift the finitary requirement on equilibria. That is, we consider systems with a continuum of equilibria. We show that essentially all numbers in [0,1] that are computable by bounded general-purpose analog computers (GPACs) or chemical reaction networks (CRNs) can also be computed by LPPs under this new definition. This implies a rich series of numbers (e.g., the reciprocal of Euler's constant, π/4, Euler's γ, Catalan's constant, and Dottie number) are all computable by LPPs. Our proof is constructive: We develop an algorithm that transfers bounded GPACs/CRNs into LPPs. Our algorithm also fixes a gap in Bournez et al.'s construction of LPPs designed to compute any arbitrary algebraic number in [0,1].Xiang Huang, Rachel N. Huls, Thomas E. Ouldridge, Shelley F. J. Wickhamwork_fdjsn7ibbff3ziqp2fqu65g54iThu, 04 Aug 2022 00:00:00 GMTOn the Identity Problem and the Group Problem for subsemigroups of unipotent matrix groups
https://scholar.archive.org/work/uws4p6lo55g7hjqf6a44oeig7e
Let 𝒢 be a finite set of matrices in a unipotent matrix group G over ℚ, where G has nilpotency class at most ten. We exhibit a polynomial time algorithm that computes the subset of 𝒢 which generates the group of units of the semigroup ⟨𝒢⟩ generated by 𝒢. In particular, this result shows that the Identity Problem and the Group Problem are decidable in polynomial time for finitely generated subsemigroups of the groups 𝖴𝖳(11, ℚ)^n. Another important implication of our result is the decidability of the Identity Problem and the Group Problem within finitely generated nilpotent groups of class at most ten. Our main idea is to analyze the logarithm of the matrices appearing in ⟨𝒢⟩. This allows us to employ Lie algebra methods to study this semigroup. In particular, we prove several new properties of the Baker-Campbell-Hausdorff formula, which help us characterize the convex cone spanned by the elements in log⟨𝒢⟩. Furthermore, we formulate a sufficient condition in order for our results to generalize to unipotent matrix groups of class d > 10. For every such d, we exhibit an effective procedure that verifies this sufficient condition in case it is true.Ruiwen Dongwork_uws4p6lo55g7hjqf6a44oeig7eThu, 04 Aug 2022 00:00:00 GMTSparse Continuous Distributions and Fenchel-Young Losses
https://scholar.archive.org/work/2yqsquvnjzhirjdhvhwim4eawe
Exponential families are widely used in machine learning, including many distributions in continuous and discrete domains (e.g., Gaussian, Dirichlet, Poisson, and categorical distributions via the softmax transformation). Distributions in each of these families have fixed support. In contrast, for finite domains, recent work on sparse alternatives to softmax (e.g., sparsemax, α-entmax, and fusedmax), has led to distributions with varying support. This paper develops sparse alternatives to continuous distributions, based on several technical contributions: First, we define Ω-regularized prediction maps and Fenchel-Young losses for arbitrary domains (possibly countably infinite or continuous). For linearly parametrized families, we show that minimization of Fenchel-Young losses is equivalent to moment matching of the statistics, generalizing a fundamental property of exponential families. When Ω is a Tsallis negentropy with parameter α, we obtain "deformed exponential families," which include α-entmax and sparsemax (α=2) as particular cases. For quadratic energy functions, the resulting densities are β-Gaussians, an instance of elliptical distributions that contain as particular cases the Gaussian, biweight, triweight, and Epanechnikov densities, and for which we derive closed-form expressions for the variance, Tsallis entropy, and Fenchel-Young loss. When Ω is a total variation or Sobolev regularizer, we obtain a continuous version of the fusedmax. Finally, we introduce continuous-domain attention mechanisms, deriving efficient gradient backpropagation algorithms for α∈{1, 4/3, 3/2, 2}. Using these algorithms, we demonstrate our sparse continuous distributions for attention-based audio classification and visual question answering, showing that they allow attending to time intervals and compact regions.André F. T. Martins, Marcos Treviso, António Farinhas, Pedro M. Q. Aguiar, Mário A. T. Figueiredo, Mathieu Blondel, Vlad Niculaework_2yqsquvnjzhirjdhvhwim4eaweThu, 04 Aug 2022 00:00:00 GMTMathematical Structures of Cohomological Field Theories
https://scholar.archive.org/work/clnhfznkiram7ehcxpsg6bq6cu
A mathematical definition of Cohomological Lagrangian Field Theories (CohLFTs) is formulated in the language of graded/bi-graded manifolds. Algebraic properties of observables in CohLFTs are studied. Methods of constructing CohLFTs, with or without gauge symmetries, are discussed. In particular, a generalization of the Mathai-Quillen formalism is given. Examples such as topological quantum mechanics, topological sigma model, topological M-theory, and topological Yang-Mills theory can be obtained uniformly using this new formalism.Shuhan Jiangwork_clnhfznkiram7ehcxpsg6bq6cuThu, 04 Aug 2022 00:00:00 GMTLIPIcs, Volume 239, TYPES 2021, Complete Volume
https://scholar.archive.org/work/uxb5tcwg6bflpdk4rslta5pn6a
LIPIcs, Volume 239, TYPES 2021, Complete VolumeHenning Basold, Jesper Cockx, Silvia Ghilezanwork_uxb5tcwg6bflpdk4rslta5pn6aThu, 04 Aug 2022 00:00:00 GMTPeriodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms
https://scholar.archive.org/work/6fmk3xmrtzexvm7c567aq2tona
We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the grading and the degree, and recovering the Calabi invariant of Hamiltonians in the limit. We also prove a strong non-vanishing result, showing that under a monotonicity assumption which holds for a dense set of maps, the Periodic Floer Homology has infinite rank. An application of these results yields that a C^∞-generic area-preserving diffeomorphism of a closed surface has a dense set of periodic points. This settles Smale's tenth problem in the special case of area-preserving diffeomorphisms of closed surfaces.Dan Cristofaro-Gardiner, Rohil Prasad, Boyu Zhangwork_6fmk3xmrtzexvm7c567aq2tonaWed, 03 Aug 2022 00:00:00 GMTOn ergodic invariant measures for the stochastic Landau-Lifschitz-Gilbert equation in 1D
https://scholar.archive.org/work/wskthnr725hf7foefbcqp3asrq
We establish existence of an ergodic invariant measure on H^1(D,ℝ^3)∩ L^2(D,𝕊^2) for the stochastic Landau-Lifschitz-Gilbert equation on a bounded one dimensional interval D. The conclusion is achieved by employing the classical Krylov-Bogoliubov theorem. In contrast to other equations, verifying the hypothesis of the Krylov-Bogoliubov theorem is not a standard procedure. We employ rough paths theory to show that the semigroup associated to the equation has the Feller property in H^1(D,ℝ^3)∩ L^2(D,𝕊^2). It does not seem possible to achieve the same conclusion by the classical Stratonovich calculus. On the other hand, we employ the classical Stratonovich calculus to prove the tightness hypothesis. The Krein-Milman theorem implies existence of an ergodic invariant measure. In case of spatially constant noise, we show that there exists a unique Gibbs invariant measure and we establish the qualitative behaviour of the unique stationary solution. In absence of the anisotropic energy and for a spatially constant noise, we are able to provide a path-wise long time behaviour result: in particular, every solution is recurrent for large times.Emanuela Gussettiwork_wskthnr725hf7foefbcqp3asrqWed, 03 Aug 2022 00:00:00 GMTSuper Catalan Numbers and Fourier Summation over Finite Fields
https://scholar.archive.org/work/waql5bdudfdfxc3v4ikyn7apze
We study polynomial summation over unit circles over finite fields of odd characteristic, obtaining a purely algebraic integration theory without recourse to infinite procedures. There are nonetheless strong parallels to classical integration theory over a circle, and we show that the super Catalan numbers and closely related rational numbers lie at the heart of both theories. This gives a uniform analytic meaning to these up to now somewhat mysterious numbers. Our derivation utilises the three-fold symmetry of chromogeometry between Euclidean and relativistic geometries, and we find that the Fourier summation formulas we derive in these two different settings are closely connected.Kevin Limanta, Norman J. Wildbergerwork_waql5bdudfdfxc3v4ikyn7apzeWed, 03 Aug 2022 00:00:00 GMTMemory as a Topological Structure on a Surface Network
https://scholar.archive.org/work/gcx6efgzfvgdvfw7b4ortgfodi
A special charged surface network with surface spin half particles on it, that can be arranged in topologically inequivalent ways, is introduced. It is shown that action potential-like signals can be generated in the network in response to local surface deformations of a particular kind. Signals generated in this way carry details of the deformation that create them as a form of plasticity that influences the pathways they traverse leaving a topologically stable helical array of spins: a potential memory substrate. The structure is a non transient alignment of surface spins in response to the transient magnetic field generated by the moving charges present in the action potential-like voltage signals generated since particles with spin have magnetic properties. The structure has a natural excitation frequency that may play a role in memory retrieval. Signal generation and memory storage are proposed to depend on the existence of a surface spin structure. We show that such a surface network can capture the intricate topological features of any connectome in the brain. In addition biophysical properties of such a network are examined in order to constrain predictions of how it may function.Siddhartha Senwork_gcx6efgzfvgdvfw7b4ortgfodiWed, 03 Aug 2022 00:00:00 GMTOn Iwasawa invariants of modular forms with reducible and non-p-distinguished residual Galois representations
https://scholar.archive.org/work/3xml2i6gkrgfnfkwvnveas5urm
In the present paper, we study the p-adic L-functions and the (strict) Selmer groups over ℚ_∞, the cyclotomic ℤ_p-extension of ℚ, of the p-adic weight one cusp forms f, obtained via the p-stabilization of weight one Eisenstein series, under the assumption that a certain Eisenstein component of the p-ordinary universal cuspidal Hecke algebra is Gorenstein. As an application, we compute the Iwasawa invariants of ordinary modular forms of weight k≥ 2 with the same residual Galois representations as the one of f, which in our setting, is reducible and non-p-distinguished. Combining this with a result of Kato , we prove the Iwasawa main conjecture for these forms. Also, we give numerical examples that satisfy the Gorenstein hypothesis. The crucial point on the analytic counter part is that under the Gorenstein hypothesis, we are able to define, following Greenberg–Vatsal, the p-adic L-functions of p-adic weight one forms f as an element in the one-dimensional Iwasawa algebra by using Mazur–Kitagawa two-variable p-adic L-function and then, to compute them explicitly via local explicit reciprocity law. On the algebraic counter part, we compute the (strict) Selmer groups of f over ℚ_∞ via the knowledge of the Galois representations of f studied in .Sheng-Chi Shih, Jun Wangwork_3xml2i6gkrgfnfkwvnveas5urmWed, 03 Aug 2022 00:00:00 GMTA Dynamical Analogue of Sen's Theorem
https://scholar.archive.org/work/gcroxuv7vbg23gq6g7l7jqjtge
We study the higher ramification structure of dynamical branch extensions, and propose a connection between the natural dynamical filtration and the filtration arising from the higher ramification groups: each member of the former should, after a linear change of index, coincide with a member of the latter. This is an analogue of Sen's theorem on ramification in p-adic Lie extensions. By explicitly calculating the Hasse-Herbrand functions of such branch extensions, we are able to show that this description is accurate for some families of polynomials, in particular post-critically bounded polynomials of p-power degree. We apply our results to give a partial answer to a question of Berger (in arXiv:1411.7064) and a partial answer to a question about wild ramification in arboreal extensions of number fields (raised in both arXiv:math/0408170 and arXiv:1511.00194).Mark O.-S. Singwork_gcroxuv7vbg23gq6g7l7jqjtgeWed, 03 Aug 2022 00:00:00 GMTGeodesic surfaces in the complement of knots with small crossing number
https://scholar.archive.org/work/ojakxk6zkjfqnefi3mzaqijuue
In this article, we investigate the problem of counting totally geodesic surfaces in the complement of hyperbolic knots with at most 9 crossings. Adapting previous counting techniques of boundary slope and intersection, we establish uniqueness of a totally geodesic surface for the knots 7_4 and 9_35. Extending an obstruction to the existence of totally geodesic surfaces due to Calegari, we show that there is no totally geodesic surface in the complement of 47 knots.Khanh Le, Rebekah Palmerwork_ojakxk6zkjfqnefi3mzaqijuueWed, 03 Aug 2022 00:00:00 GMTFormalizing the Ring of Adèles of a Global Field
https://scholar.archive.org/work/zauu5tivj5b2xk3n7ga464y4tm
The ring of adèles of a global field and its group of units, the group of idèles, are fundamental objects in modern number theory. We discuss a formalization of their definitions in the Lean 3 theorem prover. As a prerequisite, we formalize adic valuations on Dedekind domains. We present some applications, including the statement of the main theorem of global class field theory and a proof that the ideal class group of a number field is isomorphic to an explicit quotient of its idèle class group.María Inés de Frutos-Fernández, June Andronick, Leonardo de Mourawork_zauu5tivj5b2xk3n7ga464y4tmWed, 03 Aug 2022 00:00:00 GMTClassifying decomposition and wavelet coorbit spaces using coarse geometry
https://scholar.archive.org/work/nkwgptkryvak3n37c5xi2viftq
This paper is concerned with the study of Besov-type decomposition spaces, which are scales of spaces associated to suitably defined coverings of the euclidean space ℝ^d, or suitable open subsets thereof. A fundamental problem in this domain, that is currently not well understood, is deciding when two different coverings give rise to the same scale of decomposition spaces. In this paper, we establish a coarse geometric approach to this problem, and show how it specializes for the case of wavelet coorbit spaces associated to a particular class of matrix groups H < GL(ℝ^d) acting via dilations. This class can be understood as a special case of decomposition spaces, and it turns out that the question whether two different dilation groups H_1,H_2 have the same coorbit spaces can be decided by investigating whether a suitably defined map ϕ: H_1 → H_2 is a quasi-isometry with respect to suitably defined word metrics. We then proceed to apply this criterion to a large class of dilation groups called shearlet dilation groups, where this quasi-isometry condition can be characterized algebraically. We close with the discussion of selected examples.Hartmut Führ, René Kochwork_nkwgptkryvak3n37c5xi2viftqWed, 03 Aug 2022 00:00:00 GMTSurface Reconstruction from Point Clouds without Normals by Parametrizing the Gauss Formula
https://scholar.archive.org/work/f7gvwz4mfbhv3ktpbpvo6gj5qi
We propose Parametric Gauss Reconstruction (PGR) for surface reconstruction from point clouds without normals. Our insight builds on the Gauss formula in potential theory, which represents the indicator function of a region as an integral over its boundary. By viewing surface normals and surface element areas as unknown parameters, the Gauss formula interprets the indicator as a member of some parametric function spaces. We can solve for the unknown parameters using the Gauss formula and simultaneously obtain the indicator function. Our method bypasses the need for accurate input normals as required by most existing non-data-driven methods, while also exhibiting superiority over data-driven methods since no training is needed. Moreover, by modifying the Gauss formula and employing regularization, PGR also adapts to difficult cases such as noisy inputs, thin structures, sparse or nonuniform points, for which accurate normal estimation becomes quite difficult. Our code is publicly available at https://github.com/jsnln/ParametricGaussRecon.Siyou Lin, Dong Xiao, Zuoqiang Shi, Bin Wangwork_f7gvwz4mfbhv3ktpbpvo6gj5qiWed, 03 Aug 2022 00:00:00 GMTClusters and semistable models of hyperelliptic curves in the wild case
https://scholar.archive.org/work/shraia6znrgzhmbepmhlpgdbp4
Given a Galois cover Y → X of smooth projective geometrically connected curves over a complete discrete valuation field K with algebraically closed residue field, we define a semistable model of Y over the ring of integers of a finite extension of K, which we call the relatively stable model 𝒴^rst of Y, and we discuss its properties. We focus on the case when Y : y^2 = f(x) is a hyperelliptic curve, viewed as a degree-2 cover of the projective line X := ℙ_K^1, and demonstrate a practical way to compute the relatively stable model. In the case of residue characteristic p ≠ 2, the components of the special fiber (𝒴^rst)_s correspond precisely to the non-singleton clusters of roots of the defining polynomial f, i.e. the subsets of roots of f which are closer to each other than to the other roots of f with respect to the induced discrete valuation on the splitting field; this relationship, however, is far less straightforward in the p=2 case, which is our main focus (the techniques we introduce nevertheless also allow us to recover the simpler, already-known results in the p≠ 2 case). We show that, when p = 2, for each cluster containing an even number of roots of f, there are 0, 1, or 2 components of (𝒴^rst)_s corresponding to it, and we determine a direct method of finding and describing them. We also define a polynomial F(T) ∈ K[T] whose roots allow us to find the components of (𝒴^rst)_s which are not connected to even-cardinality clusters.Leonardo Fiore, Jeffrey Yeltonwork_shraia6znrgzhmbepmhlpgdbp4Wed, 03 Aug 2022 00:00:00 GMTRunning coupling and non-perturbative corrections for O(N) free energy and for disk capacitor
https://scholar.archive.org/work/a7tymapltnar7gj7lbp7qkosji
We reconsider the complete solution of the linear TBA equation describing the energy density of finite density states in the O(N) nonlinear sigma models by the Wiener-Hopf method. We keep all perturbative and non-perturbative contributions and introduce a running coupling in terms of which all asymptotic series appearing in the problem can be represented as pure power series without logs. We work out the first non-perturbative contribution in the O(3) case and show that (presumably because of the instanton corrections) resurgence theory fails in this example. Using the relation of the O(3) problem to the coaxial disks capacitor problem we work out the leading non-perturbative terms for the latter and show that (at least to this order) resurgence theory, in particular the median resummation prescription, gives the correct answer. We demonstrate this by comparing the Wiener-Hopf results to the high precision numerical solution of the original integral equation.Zoltan Bajnok, Janos Balog, Arpad Hegedus, Istvan Vonawork_a7tymapltnar7gj7lbp7qkosjiWed, 03 Aug 2022 00:00:00 GMTThe Picard group of the universal moduli stack of principal bundles on pointed smooth curves
https://scholar.archive.org/work/i7fn4rqcqfd2legdbj5ok7vgqe
For any smooth connected linear algebraic group G over an algebraically closed field k, we describe the Picard group of the universal moduli stack of principal G-bundles over pointed smooth k-projective curvesRoberto Fringuelli, Filippo Vivianiwork_i7fn4rqcqfd2legdbj5ok7vgqeWed, 03 Aug 2022 00:00:00 GMTHomological stability for Iwahori-Hecke algebras
https://scholar.archive.org/work/ltbs33atbbaxtnrbxt644mdyxq
We show that the Iwahori-Hecke algebras H_n of type A_n-1 satisfy homological stability, where homology is interpreted as an appropriate Tor group. Our result precisely recovers Nakaoka's homological stability result for the symmetric groups in the case that the defining parameter is equal to 1. We believe that this paper, and our joint work with Boyd on Temperley-Lieb algebras, are the first time that the techniques of homological stability have been applied to algebras that are not group algebras.Richard Hepworthwork_ltbs33atbbaxtnrbxt644mdyxqWed, 03 Aug 2022 00:00:00 GMT