IA Scholar Query: Self-consistency and a generalized principal subspace theorem.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 04 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Primitive permutation groups of degree 3p
https://scholar.archive.org/work/medfr4dq4zbrhdutpcnqyhs55e
This paper presents an analysis of primitive permutation groups of degree 3p, where p is a prime number, analogous to H. Wielandt's treatment of groups of degree 2p. It is also intended as an example of the systematic use of combinatorial methods as surveyed in 6 for distilling information about a permutation group from knowledge of the decomposition of its character. The work is organised into three parts. Part I contains the lesser half of the calculation, the determination of the decomposition of the permutation character. Part II contains a survey of the combinatorial methods and, based on these methods, the major part of the calculation. Part III ties up loose ends left earlier in the paper and gives a tabulation of detailed numerical results.Peter M. Neumannwork_medfr4dq4zbrhdutpcnqyhs55eThu, 04 Aug 2022 00:00:00 GMTL^2 extension of holomorphic functions for log canonical pairs
https://scholar.archive.org/work/ianlfjqedfcehozk7qqpz4gxci
In a general L^2 extension theorem of Demailly for log canonical pairs, the L^2 criterion with respect to a measure called the Ohsawa measure determines when a given holomorphic function can be extended. Despite the analytic nature of the Ohsawa measure, we establish a geometric characterization of this analytic criterion using the theory of log canonical centers from algebraic geometry. Along the way, we characterize when the Ohsawa measure fails to have generically smooth positive density, which answers an essential question arising from Demailly's work.Dano Kimwork_ianlfjqedfcehozk7qqpz4gxciThu, 04 Aug 2022 00:00:00 GMTAdvanced General Relativity Notes
https://scholar.archive.org/work/qsisiimakjbx7pt2h7zeohojyy
These lecture notes are intended as a guide to Graduate level readers that are already familiar with basic General Relativity. They present in a concise way some advanced concepts and problems encountered in the study of gravitation. In these notes are covered: Alternates forms of the Schwarzschild Black Hole solution, including the classic Kruskal extension; An account of the building of Conformal, Carter-Penrose, diagrams; A discussion of Birkhoff Theorem; A discussion of tools for Geodesics and congruences, including Energy Conditions; A discussion of Horizons and an approach to some of the singularity theorems; An exploration of the Kerr Black Hole solution properties, including the Penrose Process and Black Hole Thermodynamics; A discussion of the Eckart and Israel-Stewart Relativistic Thermodynamics; A discussion of Tetrads in Relativity, in Einstein-Cartan theory and in Newman-Penrose formalism; An explicitation of calculations on Geodesics approach from Hamilton-Jacobi Formalism; A derivation from Least action of the equation of Motion of a top in Relativity, the M.P.D. equationsM. Le Delliouwork_qsisiimakjbx7pt2h7zeohojyyThu, 04 Aug 2022 00:00:00 GMTHomogeneous 2-nondegenerate CR manifolds of hypersurface type in low dimensions
https://scholar.archive.org/work/gi3vnbtawndnxfrilkhqm7va2a
In a recent paper, the author and I. Zelenko introduce the concept of modified CR symbols for organizing local invariants of 2-nondegenerate CR structures. Among homogeneous structures with given modified CR symbols it often happens that the maximally symmetric structure is unique. We show that this phenomenon occurs for all modified symbols of homogeneous hypersurfaces in ℂ^4 and classify them, obtaining (up to local equivalence) nine maximally symmetric homogeneous 2-nondegenerate real hypersurfaces in ℂ^4. The methods used to obtain this classification are then applied to find homogeneous hypersurfaces in higher dimensional spaces. In total 20 locally non-equivalent maximally symmetric homogeneous 2-nondegenerate hypersurfaces are described in ℂ^5, and 40 such hypersurfaces are described in ℂ^6, of which some have been described in other works while many are new. Two new sequences, indexed by n, of homogeneous 2-nondegenerate hypersurfaces in ℂ^n+1 are described. Notably, all examples from one of these latter sequences can be realized as left-invariant structures on nilpotent Lie groups.David Sykeswork_gi3vnbtawndnxfrilkhqm7va2aThu, 04 Aug 2022 00:00:00 GMTFMM-LU: A fast direct solver for multiscale boundary integral equations in three dimensions
https://scholar.archive.org/work/ky5cabg2pfhtrluyaezbvenbvi
We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced strong recursive skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an LU-like hierarchical factorization of the dense system matrix, permitting application of the inverse in O(N) time, where N is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted adaptive octree data structure, and therefore it is compatible with highly nonuniform discretizations. Furthermore, the scheme is coupled with high-order accurate locally-corrected Nystrom quadrature methods to integrate the singular and weakly-singular Green's functions used in the integral representations. Our method has immediate applications to a variety of problems in computational physics. We concentrate here on studying its performance in acoustic scattering (governed by the Helmholtz equation) at low to moderate frequencies.Daria Sushnikova, Leslie Greengard, Michael O'Neil, Manas Rachhwork_ky5cabg2pfhtrluyaezbvenbviThu, 04 Aug 2022 00:00:00 GMTDerived categories of hyper-Kähler manifolds: extended Mukai vector and integral structure
https://scholar.archive.org/work/yixupm25ibenbdp73u4x7zqvoi
We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-K\"ahler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-K\"ahler manifolds as well as an integral lattice associated to the derived category of hyper-K\"ahler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.Thorsten Beckmannwork_yixupm25ibenbdp73u4x7zqvoiWed, 03 Aug 2022 00:00:00 GMTFrom behaviour to the brain: representation learning and sensorimotor control
https://scholar.archive.org/work/dqnab2gp7vbyvomttpeezkqpbu
Behaviour, the only approach for living creatures to interact with the environment, is the consequence of sensorimotor transformations in the central nervous system. Successful implementation of behaviour tasks requires high accuracy in sensory perception and motor execution, and more importantly, experience-based plasticities. We seek to investigate neuronal accomplishment of these sensorimotor functions in the brain. Due to technical obstacles, neural recording experiments fail to provide a complete understanding. Thus, we intend to bridge the gap by reverse-engineering the brain with machine learning algorithms as functional and quantitative frameworks. For this purpose, we develop a series of computational models and algorithms, covering different aspects of control and representational learning as sensorimotor transformations. We propose a spiking neural circuit model of the cerebellum, learning from movement errors to generate motor correction against external perturbation to ensure accurate execution. This allows us to discover sensorimotor error-based learning in a bottom-up manner, bringing insights to the functional structure and disorders of the cerebellum. Moving from sensorimotor control to representational learning, we emphasise more on the statistical structure of natural behaviour, i.e. the prior p(x), rather than individual executions. We assume that sensorimotor representations are shaped by the natural behaviour statistics, and based on this assumption, we introduce machine learning models of topological neural coding and manifold learning. Our topological neural coding models, with neurons arranged in a 2D topological map, are optimised towards the optimal representational power under p(x). Applying topological neural coding to natural arm movement data reproduces features of proprioceptive representations observed in macaque S1. Our manifold learning models aims to explore the hidden structure of p(x) while learning a non-linear low-dimensional manifold embedded in the high-dimensional space. [...]Yufei Wu, Aldo Faisalwork_dqnab2gp7vbyvomttpeezkqpbuWed, 03 Aug 2022 00:00:00 GMTOn the Chow and cohomology rings of moduli spaces of stable curves
https://scholar.archive.org/work/g74bquw6szdzrdtkfadrbfvspe
In this paper, we ask: for which (g, n) is the rational Chow or cohomology ring of ℳ_g,n generated by tautological classes? This question has been fully answered in genus 0 by Keel (the Chow and cohomology rings are tautological for all n) and genus 1 by Belorousski (the rings are tautological if and only if n ≤ 10). For g ≥ 2, work of van Zelm shows the Chow and cohomology rings are not tautological once 2g + n ≥ 24, leaving finitely many open cases. Here, we prove that the Chow and cohomology rings of ℳ_g,n are isomorphic and generated by tautological classes for g = 2 and n ≤ 9 and for 3 ≤ g ≤ 7 and 2g + n ≤ 14. For such (g, n), this implies that the tautological ring is Gorenstein and ℳ_g,n has polynomial point count.Samir Canning, Hannah Larsonwork_g74bquw6szdzrdtkfadrbfvspeWed, 03 Aug 2022 00:00:00 GMTVariational problem on a metric-affine almost product manifold
https://scholar.archive.org/work/vkt5h5rqtnhcbmg2nxmop2x5hi
We study a variational problem on a smooth manifold with a decomposition of the tangent bundle into k>2 subbundles, namely, we consider the integrated sum of mixed scalar curvatures as a functional of adapted pseudo-Riemannian metric (keeping the pairwise orthogonality of the distributions) and contorsion tensor, defining a linear connection. This functional allows us to extend the class of Einstein metrics: if all distributions are one-dimensional, then it coincides with the geometrical part of Einstein-Hilbert action restricted to adapted metrics. We prove that metrics in critical pairs metric-contorsion make all distributions totally umbilical. We obtain examples and obstructions to existence of those critical pairs in some special cases: twisted products with statistical connections; semi-symmetric connections and 3-Sasaki manifolds with metric-compatible connections.Vladimir Rovenski, Tomasz Zawadzkiwork_vkt5h5rqtnhcbmg2nxmop2x5hiWed, 03 Aug 2022 00:00:00 GMTA Convolutional Persistence Transform
https://scholar.archive.org/work/zd7r6hgdczhytnm5jk6djuy4m4
We consider a new topological feauturization of d-dimensional images, obtained by convolving images with various filters before computing persistence. Viewing a convolution filter as a motif within an image, the persistence diagram of the resulting convolution describes the way the motif is distributed throughout that image. This pipeline, which we call convolutional persistence, extends the capacity of topology to observe patterns in image data. Indeed, we prove that (generically speaking) for any two images one can find some filter for which they produce different persistence diagrams, so that the collection of all possible convolutional persistence diagrams for a given image is an injective invariant. This is proven by showing convolutional persistence to be a special case of another topological invariant, the Persistent Homology Transform. Other advantages of convolutional persistence are improved stability and robustness to noise, greater flexibility for data-dependent vectorizations, and reduced computational complexity for convolutions with large stride vectors. Additionally, we have a suite of experiments showing that convolutions greatly improve the predictive power of persistence on a host of classification tasks, even if one uses random filters and vectorizes the resulting diagrams by recording only their total persistences.Elchanan Solomon, Paul Bendichwork_zd7r6hgdczhytnm5jk6djuy4m4Wed, 03 Aug 2022 00:00:00 GMTLocal Index Formulae on Noncommutative Orbifolds and Equivariant Zeta Functions for the Affine Metaplectic Group
https://scholar.archive.org/work/v4voqtqbofhjpmtp6z5fltcihq
We consider the algebra A of bounded operators on L^2(ℝ^n) generated by quantizations of isometric affine canonical transformations. The algebra A includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple (A, H, D) with H=L^2(ℝ^n, Λ(ℝ^n)) and the Euler operator D, a first order differential operator of index 1. We show that this spectral triple has simple dimension spectrum: For every operator B in the algebra Ψ(A,H,D) generated by the Shubin type pseudodifferential operators and the elements of A, the zeta function ζ_B(z) = Tr (B|D|^-2z) has a meromorphic extension to ℂ with at most simple poles. Our main result then is an explicit algebraic expression for the Connes-Moscovici cyclic cocycle. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds.Anton Savin, Elmar Schrohework_v4voqtqbofhjpmtp6z5fltcihqWed, 03 Aug 2022 00:00:00 GMTMatrix Decomposition and Applications
https://scholar.archive.org/work/fizclfz2ovdb7afrmolgtbuhu4
In 1954, Alston S. Householder published Principles of Numerical Analysis, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields.Jun Luwork_fizclfz2ovdb7afrmolgtbuhu4Wed, 03 Aug 2022 00:00:00 GMTAutomorphic Hamiltonians, Epstein Zeta Functions, and Kronecker Limit Formulas
https://scholar.archive.org/work/eq7yhear7zbyngazg7pmizqutm
First, we recount a history of how certain methods using natural self-adjoint operators have, thus far, failed to prove the Riemann Hypothesis. In Section 2, we set the analytical context necessary to have genuine proofs in later sections, rather than attractive heuristics. In Section 3, we recall the utility of designed pseudo-Laplacians by reproving meromorphic continuation of certain Eisenstein series and proving a spacing result for zeros of ζ_k(s) for k a complex quadratic field with negative determinant. In Section 4, we construct an automorphic Hamiltonian which has purely discrete spectrum on L^2(SL_r(ℤ)\ SL_r (ℝ)/SO(r, ℝ)), identify its ground state, and show how it can characterize a nuclear Fréchet automorphic Schwartz space.Adrienne Sandswork_eq7yhear7zbyngazg7pmizqutmWed, 03 Aug 2022 00:00:00 GMTSingular Boundary Conditions for Sturm–Liouville Operators via Perturbation Theory
https://scholar.archive.org/work/mp3v5qgnyvfkpe2yv3qumux32e
We show that all self-adjoint extensions of semi-bounded Sturm–Liouville operators with general limit-circle endpoint(s) can be obtained via an additive singular form bounded self-adjoint perturbation of rank equal to the deficiency indices, say d∈{1,2}. This characterization generalizes the well-known analog for semi-bounded Sturm–Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as A_Θ=A_0+ BΘ B^*, where A_0 is a distinguished self-adjoint extension and Θ is a self-adjoint linear relation in ℂ^d. The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form bounded with respect to A_0, i.e. it belongs to ℋ_-1(A_0). The construction of a boundary triple and compatible boundary pair for the symmetric operator ensure that the perturbation is well-defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations Θ. As an example, self-adjoint extensions of the classical symmetric Jacobi differential equation (which has two limit-circle endpoints) are obtained and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.Michael Bush, Dale Frymark, Constanze Liawwork_mp3v5qgnyvfkpe2yv3qumux32eTue, 02 Aug 2022 00:00:00 GMTRobust Change-Point Detection for Functional Time Series Based on U-Statistics and Dependent Wild Bootstrap
https://scholar.archive.org/work/pz4i5ddidvgnvjii3p5q32kc4y
The aim of this paper is to develop a change-point test for functional time series that uses the full functional information and is less sensitive to outliers compared to the classical CUSUM test. For this aim, the Wilcoxon two-sample test is generalized to functional data. To obtain the asymptotic distribution of the test statistic, we proof a limit theorem for a process of U-statistics with values in a Hilbert space under weak dependence. Critical values can be obtained by a newly developed version of the dependent wild bootstrap for non-degenerate 2-sample U-statistics.Lea Wegner, Martin Wendlerwork_pz4i5ddidvgnvjii3p5q32kc4yTue, 02 Aug 2022 00:00:00 GMTThe resonances of the Capelli operators for small split orthosymplectic dual pairs
https://scholar.archive.org/work/kllh6olkajb3zpxturkgw2jrce
Let (G,G') be a reductive dual pair in Sp(W) with rank G≤ rank G' and G' semisimple. The image of the Casimir element of the universal enveloping algebra of G' under the Weil representation ω is a Capelli operator. It is a hermitian operator acting on the smooth vectors of the representation space of ω. We compute the resonances of a natural multiple of a translation of this operator for small split orthosymplectic dual pairs. The corresponding resonance representations turn out to be GG'-modules in Howe's correspondence. We determine them explicitly.Roberto Bramati, Angela Pasquale, Tomasz Przebindawork_kllh6olkajb3zpxturkgw2jrceTue, 02 Aug 2022 00:00:00 GMTNew examples of entangled states on ℂ^3 ⊗ℂ^3
https://scholar.archive.org/work/z55toe4vxbc33j6fn6erdgjncu
We build apon our previous work, the Buckley-method for simultaneous construction of families of positive maps on 3 × 3 self-adjoint matrices by prescribing a set of complex zeros to the associated forms. Positive maps that are not completely positive can be used to prove (witness) that certain mixed states are entangled. We obtain entanglement witnesses that are indecomposable and belong to extreme rays of the cone of positive maps. Consequently our semidefinite program returns new examples of entangled states whose entanglement cannot be certified by the transposition map nor by other well-known positive maps. The constructed states as well as the method of their construction offer some valuable insights for quantum information theory, in particular into the geometry of positive cones.Anita Buckleywork_z55toe4vxbc33j6fn6erdgjncuTue, 02 Aug 2022 00:00:00 GMTQuantum Computing: Lecture Notes
https://scholar.archive.org/work/2pcfo6u7jzg25alp6mv6fq3w2y
This is a set of lecture notes suitable for a Master's course on quantum computation and information from the perspective of theoretical computer science. The first version was written in 2011, with many extensions and improvements in subsequent years. The first 10 chapters cover the circuit model and the main quantum algorithms (Deutsch-Jozsa, Simon, Shor, Hidden Subgroup Problem, Grover, quantum walks, Hamiltonian simulation and HHL). They are followed by 3 chapters about complexity, 4 chapters about distributed ("Alice and Bob") settings, a chapter about quantum machine learning, and a final chapter about quantum error correction. Appendices A and B give a brief introduction to the required linear algebra and some other mathematical and computer science background. All chapters come with exercises, with some hints provided in Appendix C.Ronald de Wolfwork_2pcfo6u7jzg25alp6mv6fq3w2yTue, 02 Aug 2022 00:00:00 GMTDixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori
https://scholar.archive.org/work/ecfrbhnxtvhjpn7kemeavcffby
In a previous paper we established Cwikel-type estimates on noncommutative tori and used them to get analogues in this setting of the Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators. In this paper, we focus on "curved" NC tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The Cwikel-type estimates of our previous paper are extended to pseudodifferential operators and powers of Laplace-Beltrami operators. There are several applications of these estimates. First, we get L_p-versions of the usual formula for the trace of on NC tori, i.e., for combinations of with L_p-position operators. Next, we get L_p-versions of the analogues for NC tori Connes' trace theorem and Connes' integration formula. They give formulas for the NC integrals (a.k.a. Dixmier traces) of products of L_p-position operators with or powers of the Laplace-Beltrami operators. Moreover, by combining our Cwikel-type estimates with suitable versions of the Birman-Schwinger principle we get versions of the CLR and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators associated with powers of Laplace-Beltrami operators and L_p-potentials. As in the original Euclidean case the Lieb-Thirring inequalities imply a dual Sobolev inequality for orthonormal families. Finally, we discuss spectral asymptotics and semiclassical Weyl's laws for the our classes of operators on curved NC tori. This superseded a previous conjecture in our previous paper.Edward McDonald, Raphael Pongework_ecfrbhnxtvhjpn7kemeavcffbyTue, 02 Aug 2022 00:00:00 GMTIntegrability in the chiral model of magic angles
https://scholar.archive.org/work/esgeiyzbi5fthixng6qmpxkyqe
Magic angles in the chiral model of twisted bilayer graphene are parameters for which the chiral version of the Bistritzer--MacDonald Hamiltonian exhibits a flat band at energy zero. We compute the sums over powers of (complex) magic angles and use that to show that the set of magic angles is infinite. We also provide a new proof of the existence of the first real magic angle, showing also that the corresponding flat band has minimal multiplicity for the simplest possible choice of potentials satisfying all symmetries. These results indicate (though not prove) a hidden integrability of the chiral model.Simon Becker, Tristan Humbert, Maciej Zworskiwork_esgeiyzbi5fthixng6qmpxkyqeTue, 02 Aug 2022 00:00:00 GMT