IA Scholar Query: On CCS with Parametric Relabelling.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 05 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Anomaly of (2+1)-Dimensional Symmetry-Enriched Topological Order from (3+1)-Dimensional Topological Quantum Field Theory
https://scholar.archive.org/work/cxm3ab3favfc3l2xbcvucnze3i
Symmetry acting on a (2+1)D topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1)D on-site symmetry. In this paper, we develop a (3+1)D topological quantum field theory to calculate the anomaly indicators of a (2+1)D topological order with a general finite group symmetry G, which may contain anti-unitary elements and/or permute anyons. These anomaly indicators are partition functions of the (3+1)D topological quantum field theory on a specific manifold equipped with some G-bundle, and they are expressed using the data characterizing the topological order and the symmetry actions. Combined with the relative anomaly formalism, our framework actually enables us to calculate the anomaly of a given topological order with a fully general symmetry. Our framework is applied to derive the anomaly indicators for various symmetry groups, including ℤ_2×ℤ_2, ℤ_2^T×ℤ_2^T, etc, where ℤ_2 and ℤ_2^T denote a unitary and anti-unitary order-2 group, respectively.Weicheng Ye, Liujun Zouwork_cxm3ab3favfc3l2xbcvucnze3iWed, 05 Oct 2022 00:00:00 GMTCohomology in singular blocks of parabolic category 𝒪
https://scholar.archive.org/work/5ne6tpmeq5dehbisvgbr7v7bv4
We determine the dimensions of Ext-groups between simple modules and dual generalized Verma modules in singular blocks of parabolic versions of category 𝒪 for complex semisimple Lie algebras and affine Kac-Moody algebras.Jonathan Gruberwork_5ne6tpmeq5dehbisvgbr7v7bv4Wed, 05 Oct 2022 00:00:00 GMTDetection and Evaluation of Clusters within Sequential Data
https://scholar.archive.org/work/bue3nywa3nbqffglua55yjgbd4
Motivated by theoretical advancements in dimensionality reduction techniques we use a recent model, called Block Markov Chains, to conduct a practical study of clustering in real-world sequential data. Clustering algorithms for Block Markov Chains possess theoretical optimality guarantees and can be deployed in sparse data regimes. Despite these favorable theoretical properties, a thorough evaluation of these algorithms in realistic settings has been lacking. We address this issue and investigate the suitability of these clustering algorithms in exploratory data analysis of real-world sequential data. In particular, our sequential data is derived from human DNA, written text, animal movement data and financial markets. In order to evaluate the determined clusters, and the associated Block Markov Chain model, we further develop a set of evaluation tools. These tools include benchmarking, spectral noise analysis and statistical model selection tools. An efficient implementation of the clustering algorithm and the new evaluation tools is made available together with this paper. Practical challenges associated to real-world data are encountered and discussed. It is ultimately found that the Block Markov Chain model assumption, together with the tools developed here, can indeed produce meaningful insights in exploratory data analyses despite the complexity and sparsity of real-world data.Alexander Van Werde, Albert Senen-Cerda, Gianluca Kosmella, Jaron Sanderswork_bue3nywa3nbqffglua55yjgbd4Tue, 04 Oct 2022 00:00:00 GMTThe Theory of Duality and Periodicity
https://scholar.archive.org/work/53ay35j3dbdnlotkppxfr552le
Dualism is a metaphysical, philosophical concept which refers to two irreducible, heterogeneous principles. This idea is known to appear in a lot of places in the universe, however a rigorous mathematical definition and theory is not yet established in a formal way. In this paper, we develop a novel theory to represent philosophical dualism in a formal mathematical construction with the context of quantum physics, known as the "theory of duality". We will use traditional Chinese philosophical concepts in duality as the foundation as it greatly resembles to the mathematical and physical construction for our purpose. The idea of periodicity based on Taoism will also be introduced mathematically. This paper will demonstrate how to convolve metaphysical idea into mathematics and physics. Finally, we will implement the concept of duality to prove some fundamental theorems of Buddhism.B.T.T.Wongwork_53ay35j3dbdnlotkppxfr552leTue, 27 Sep 2022 00:00:00 GMTExtreme singular values of inhomogeneous sparse random rectangular matrices
https://scholar.archive.org/work/fozmrgesfvd5tp67bnzjaxtwwq
We develop a unified approach to bounding the largest and smallest singular values of an inhomogeneous random rectangular matrix, based on the non-backtracking operator and the Ihara-Bass formula for general Hermitian matrices with a bipartite block structure. Our main results are probabilistic upper (respectively, lower) bounds for the largest (respectively, smallest) singular values of a large rectangular random matrix X. These bounds are given in terms of the maximal and minimal ℓ_2-norms of the rows and columns of the variance profile of X. The proofs involve finding probabilistic upper bounds on the spectral radius of an associated non-backtracking matrix B. The two-sided bounds can be applied to the centered adjacency matrix of sparse inhomogeneous Erdős-Rényi bipartite graphs for a wide range of sparsity. In particular, for Erdős-Rényi bipartite graphs 𝒢(n,m,p) with p=ω(log n)/n, and m/n→ y ∈ (0,1), our sharp bounds imply that there are no outliers outside the support of the Marčenko-Pastur law almost surely. This result is novel, and it extends the Bai-Yin theorem to sparse rectangular random matrices.Ioana Dumitriu, Yizhe Zhuwork_fozmrgesfvd5tp67bnzjaxtwwqSun, 25 Sep 2022 00:00:00 GMTQuantitative Stability of Barycenters in the Wasserstein Space
https://scholar.archive.org/work/67sacox2hrdhja5vk2midj2ufm
Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability measures of interest are often not accessible in their entirety and the practitioner may have to deal with statistical or computational approximations instead. In this article, we quantify the effect of such approximations on the corresponding barycenters. We show that Wasserstein barycenters depend in a Hölder-continuous way on their marginals under relatively mild assumptions. Our proof relies on recent estimates that quantify the strong convexity of the dual quadratic optimal transport problem and a new result that allows to control the modulus of continuity of the push-forward operation under a (not necessarily smooth) optimal transport map.Guillaume Carlierwork_67sacox2hrdhja5vk2midj2ufmWed, 21 Sep 2022 00:00:00 GMTPartial self-testing and randomness certification in the triangle network
https://scholar.archive.org/work/llb36jxi5raglpnqb3tuew4mxm
Quantum nonlocality can be demonstrated without inputs (i.e. each party using a fixed measurement setting) in a network with independent sources. Here we consider this effect on ring networks, and show that the underlying quantum strategy can be partially characterized, or self-tested, from observed correlations. Applying these results to the triangle network allows us to show that the nonlocal distribution of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)] requires that (i) all sources produce a minimal amount of entanglement, (ii) all local measurements are entangled, and (iii) each local outcome features a minimal entropy. Hence we show that the triangle network allows for genuine network quantum nonlocality and certifiable randomness.Pavel Sekatski, Sadra Boreiri, Nicolas Brunnerwork_llb36jxi5raglpnqb3tuew4mxmTue, 20 Sep 2022 00:00:00 GMTOn the Whitney near extension problem, BMO, alignment of data, best approximation in algebraic geometry, manifold learning and their beautiful connections: A modern treatment
https://scholar.archive.org/work/ju47ae6twzbfzaxbexq5rsaute
This paper provides fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic geometry, approximation theory, complex-harmonic analysis and high dimensional data science. Modern techniques in algebraic geometry, approximation theory, computational harmonic analysis and extensions develop the first of its kind, a unified framework which allows for a simultaneous study of labeled and unlabeled near alignment data problems in of ℝ^D with the near isometry extension problem for discrete and non-discrete subsets of ℝ^D with certain geometries. In addition, the paper surveys related work on clustering, dimension reduction, manifold learning, vision as well as minimal energy partitions, discrepancy and min-max optimization. Numerous open problems are given.Steven B. Damelinwork_ju47ae6twzbfzaxbexq5rsauteSun, 18 Sep 2022 00:00:00 GMTType II Double Field Theory in Superspace
https://scholar.archive.org/work/6p4rfljj4jdq3ihvp7k3xtxple
We explore type II supersymmetric double field theory in superspace. The double supervielbein is an element of the orthosymplectic group OSp(10,10|64), which also governs the structure of generalized superdiffeomorphisms. Unlike bosonic double field theory, the local tangent space must be enhanced from the double Lorentz group in order to eliminate unphysical components of the supervielbein and to define covariant torsion and curvature tensors. This leads to an infinite hierarchy of local tangent space symmetries, which are connected to the super-Maxwell_∞ algebra. A novel feature of type II is the Ramond-Ramond sector, which can be encoded as an orthosymplectic spinor (encoding the complex of super p-forms in conventional superspace). Its covariant field strength bispinor itself appears as a piece of the supervielbein. We provide a concise discussion of the superspace Bianchi identities through dimension two and show how to recover the component supersymmetry transformations of type II DFT. In addition, we show how the democratic formulation of type II superspace may be recovered by gauge-fixing.Daniel Butterwork_6p4rfljj4jdq3ihvp7k3xtxpleThu, 15 Sep 2022 00:00:00 GMTShuffle approach towards quantum affine and toroidal algebras
https://scholar.archive.org/work/2izocjie5jbybkhwyisjk4xoay
These are detailed lecture notes of the crash-course on shuffle algebras delivered by the author at Tokyo University of Marine Science and Technology during the second week of March 2019. These notes consist of three chapters, providing a separate treatment for: the quantum loop algebras of 𝔰𝔩_n (as well as their super- and 2-parameter generalizations), the quantum toroidal algebras of 𝔤𝔩_1, and the quantum toroidal algebras of 𝔰𝔩_n. We provide the shuffle realization of the corresponding "positive" subalgebras as well as of the commutative subalgebras and some combinatorial representations for the toroidal algebras. One of the key techniques involved is that of "specialization maps". Each chapter aims to emphasize a different aspect of the theory: in the first chapter we use shuffle algebras to construct a family of new PBWD bases for type A quantum loop algebras and their integral forms; in the second chapter, we provide a geometric interpretation of the Fock modules and use shuffle description of a commutative subalgebra to construct an action of the Heisenberg algebra on the equivariant K-theory of the Hilbert schemes of points; in the last chapter, we relate vertex and combinatorial representations of quantum toroidal algebras of 𝔰𝔩_n using Miki's isomorphism and use shuffle realization to explicitly compute Bethe commutative subalgebras and their limits. The latter construction is inspired by Enriquez's work relating shuffle algebras to the correlation functions of quantum affinized algebras.Alexander Tsymbaliukwork_2izocjie5jbybkhwyisjk4xoayFri, 09 Sep 2022 00:00:00 GMTOn the Axiomatisation of Branching Bisimulation Congruence over CCS
https://scholar.archive.org/work/n6cjz5rlmbaozj6owg6m6me5fm
In this paper we investigate the equational theory of (the restriction, relabelling, and recursion free fragment of) CCS modulo rooted branching bisimilarity, which is a classic, bisimulation-based notion of equivalence that abstracts from internal computational steps in process behaviour. Firstly, we show that CCS is not finitely based modulo the considered congruence. As a key step of independent interest in the proof of that negative result, we prove that each CCS process has a unique parallel decomposition into indecomposable processes modulo branching bisimilarity. As a second main contribution, we show that, when the set of actions is finite, rooted branching bisimilarity has a finite equational basis over CCS enriched with the left merge and communication merge operators from ACP.Luca Aceto, Valentina Castiglioni, Anna Ingólfsdóttir, Bas Luttik, Bartek Klin, Sławomir Lasota, Anca Muschollwork_n6cjz5rlmbaozj6owg6m6me5fmTue, 06 Sep 2022 00:00:00 GMTFractional Hall Conductivity and Spin-c Structure in Solvable Lattice Hamiltonians
https://scholar.archive.org/work/rczn4oer35av7b6rbshzsz42km
The Kapustin-Fidkowski no-go theorem forbids U(1) symmetric topological orders with non-trivial Hall conductivity in (2+1)d from admitting commuting projector Hamiltonians, where the latter is the paradigmatic method to construct exactly solvable lattice models for topological orders. Even if a topological order would intrinsically have admitted commuting projector Hamiltonians, the theorem forbids so once its interplay with U(1) global symmetry which generates Hall conductivity is taken into consideration. Nonetheless, in this work, we show that for all (2+1)d U(1) symmetric abelian topological orders of such kind, we can construct a lattice Hamiltonian that is controllably solvable at low energies, even though not "exactly" solvable; hence, this no-go theorem does not lead to significant difficulty in the lattice study of these topological orders. Moreover, for the fermionic topological orders in our construction, we introduce the lattice notion of spin-c structure – a concept important in the continuum that has previously not been adequately introduced in the lattice context.Zhaoyu Han, Jing-Yuan Chenwork_rczn4oer35av7b6rbshzsz42kmTue, 06 Sep 2022 00:00:00 GMTNonparametric correlation-based methods with biomedical applications
https://scholar.archive.org/work/7yuavu3vdncljiaposmctalqdi
This cumulative dissertation consists of three manuscripts on nonparametric methodology, i.e., Simultaneous inference for Kendall's tau, Group sequential methods for the Mann-Whitney parameter, and The nonparametric Behrens-Fisher problem in small samples. The manuscript on Kendall's τ fully develops a nonparametric estimation theory for multiple rank correlation coefficients in terms of Kendall's τA and τB, Somers' D, as well as Kruskal and Goodman's γ, necessitating joint estimation of both the probabilities of ties occurring and the probability of concordance minus discordance. As for the second manuscript, I review and further develop group sequential methodology for the Mann-Whitney parameter. With the aid of data from a clinical trial in patients with relapse-remitting multiple sclerosis, I demonstrate how one could repeatedly estimate the Mann-Whitney parameter during an ongoing trial together with repeated confidence intervals obtained by test inversion. In addition, I give simple approximate power formulas for this group sequential setting. The last manuscript further explores how best to approximate the sampling distribution of the Mann-Whitney parameter in terms of the nonparametric Behrens-Fisher problem, an issue that has arisen from the preceding manuscript. In that regard, I explore different variance estimators and a permutation approach that have been proposed in the literature and examine some slightly modified ways as regards a small sample t approximation as well. In all three manuscripts, I carried out simulations for various settings to assess the adequacy of the proposed methods.Claus P. Nowak, Technische Universität Dortmundwork_7yuavu3vdncljiaposmctalqdiWed, 31 Aug 2022 00:00:00 GMTQuantifying Information Extraction using Generalized Quantum Measurements
https://scholar.archive.org/work/feaeu6otofenrlvxz7gxqcnm5i
Observational entropy is interpreted as the uncertainty an observer making measurements associates with a system. So far, properties that make such an interpretation possible rely on the assumption of ideal projective measurements. Here we show that the same properties hold even when considering generalized measurements, and therefore such an interpretation stands completely general. Thus, observational entropy is a well-defined quantifier to test how influential a given series of measurements is in information extraction. Our generalized measurement framework allows us to explore how the measurement probe and schemes influence information extraction, tasks impossible within the projective measurement framework. We provide theoretical lower bounds to the probe dimension and show that a qubit probe can perfectly extract information from any multi-dimensional quantum system. Furthermore, we compare the repeated measurement scheme, in which a pointer repeatedly measures the system of interest, to the repeated contact scheme, in which the pointer repeatedly stores information of the measured observable and is measured only at the end of the protocol. Here, we find that in most cases the repeated contact scheme has a higher observational entropy implying that the observer can extract the most amount of quantum information using this scheme. Finally, we discuss observational entropy as a tool for quantum state inference. Further developed, this could find wide applications in quantum information processing and help determine the best read-out procedures from quantum memories.Dominik Šafránek, Juzar Thingnawork_feaeu6otofenrlvxz7gxqcnm5iThu, 25 Aug 2022 00:00:00 GMTMinor Invertible Products Assignment and Sparse Hyperdeterminants
https://scholar.archive.org/work/3zgjyejo2zcjxcexckx5zakwxq
We consider an extension of Minor Assignment Problems derived from the determinantal expansion of matrix products, under the condition that the terms of the expansion are units of C(t). This restriction places constraints on the sparsity and the factorization properties of a family of hyperdeterminants derived from Grassmann-Pl\"ucker relations. We find minimal conditions guaranteeing that allowed assignments returning a determinantal expansion are the trivial ones, i.e., those induced by the action of a diagonal matrix of Laurent monomials on a pair of constant matrices. Counterexamples are provided when such conditions do not hold. Connections with the characterization of forbidden configurations in a different combinatorial context, as well as potential applications to statistical modeling and choice theory, are also discussed.Mario Angelelliwork_3zgjyejo2zcjxcexckx5zakwxqThu, 18 Aug 2022 00:00:00 GMTMethods of classical and free probability
https://scholar.archive.org/work/xttjetydqzgqzklt4d5nfckbee
This is a joint introduction to classical and free probability, which are twin sisters. We discuss in detail the foundations and main results of both theories, by insisting on their common features, and by using a light formalism, based on standard calculus. We include as well a brief discussion of more advanced aspects.Teo Banicawork_xttjetydqzgqzklt4d5nfckbeeTue, 16 Aug 2022 00:00:00 GMTConstructing uniform spaces
https://scholar.archive.org/work/bdsubtnyzzhjrjclacvs5i6wxu
We exhibit geometric conditions that ensure a metric space is uniform.David A. Herronwork_bdsubtnyzzhjrjclacvs5i6wxuMon, 15 Aug 2022 00:00:00 GMTOptimal and tight Bell inequalities for state-independent contextuality sets
https://scholar.archive.org/work/ttq3fpjhafd2nkxo7lktla32ji
Two fundamental quantum resources, nonlocality and contextuality, can be connected through Bell inequalities that are violated by state-independent contextuality (SI-C) sets. These Bell inequalities allow for applications requiring simultaneous nonlocality and contextuality. For existing Bell inequalities, the nonlocality produced by SI-C sets is very sensitive to noise. This precludes actual experiments. Here, we identify the Bell inequalities for which the nonlocality produced by SI-C sets is optimal, i.e., maximally robust to either noise or detection inefficiency, for the simplest SI-C [S. Yu and C. H. Oh, Phys. Rev. Lett. 108, 030402 (2012)] and Kochen-Specker sets [A. Cabello et al., Phys. Lett. A 212, 183 (1996)] and show that, in both cases, nonlocality is sufficiently resistant for experiments.Junior R. Gonzales-Ureta, Ana Predojević, Adán Cabellowork_ttq3fpjhafd2nkxo7lktla32jiMon, 15 Aug 2022 00:00:00 GMTFudan lectures on string theory
https://scholar.archive.org/work/hk2klromz5bwtbzlb4k2gmgtr4
These are lecture notes on string theory at Fudan University.Satoshi Nawata, Runkai Tao, Daisuke Yokoyamawork_hk2klromz5bwtbzlb4k2gmgtr4Mon, 15 Aug 2022 00:00:00 GMTVacuum energy in (2+1)-dimensional quantum field theory on curved spaces
https://scholar.archive.org/work/jpilhshm3zbrhem2mwc7efcxqa
Relativistic quantum degrees of freedom in their vacuum state endow geometric backgrounds with an energy, as demonstrated by the Casimir Effect. We explore the vacuum energy (or free energy at finite temperature) of (2+1)-dimensional ultrastatic relativistic quantum field theories as a functional of their spatial geometry. These theories have physical realisations as, for example, the low-energy effective description of the electronic structure of graphene: four free massless Dirac fermions. We define a UV-finite unambiguous measure of free energy for these setups: the free energy difference. We compute it for the free scalar with curvature coupling and free Dirac fermion using heat kernel methods, deriving analytic expressions for perturbative and long-wavelength deformations of maximally-symmetric two-spaces (namely the plane and the round sphere) and, using a novel numerical approach, highly-accurate estimates in the case of large (axisymmetric) deformations to the sphere. We find that for these theories, as with holographic conformal field theories (CFTs) dual to vacuum Einstein gravity with a negative cosmological constant, it is universally negative for non-trivial deformations of maximally-symmetric two-spaces and can be made arbitrarily negative as the geometry becomes singular. In fact, we find that the differenced heat kernel has a definite sign — a much stronger result. We also observe a qualitative similarity between the (appropriately normalised) vacuum energies of a conformal scalar, massless Dirac fermion and holographic CFT on deformations of the two-sphere, and a remarkably close quantitive agreement between the latter two — very dissimilar in nature — theories. Finally, we show vacuum energy negativity for all perturbative deformations to Poincaré-invariant, power-counting-renormalisable theories on the plane. Our results indicate that relativistic quantum degrees of freedom universally disfavour smooth geometries and we note this effect has the potential to be measured experimentally.Lucas Samuel Wallis, Toby Wiseman, Science And Technology Facilities Council (Great Britain)work_jpilhshm3zbrhem2mwc7efcxqaThu, 11 Aug 2022 00:00:00 GMT