IA Scholar Query: Algebraic and Euclidean Lattices: Optimal Lattice Reduction and Beyond.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgMon, 28 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Turbulence as Clebsch Confinement
https://scholar.archive.org/work/qrlmjshh65cfddfvw4x3lbhb44
We argue that in the strong turbulence phase, as opposed to the weak one, the Clebsch variables compactify to the sphere S_2 and are not observable as wave excitations. Various topologically nontrivial configurations of this confined Clebsch field are responsible for vortex sheets. Stability equations (CVS) for closed vortex surfaces (bubbles of Clebsch field) are derived and investigated. The exact non-compact solution for the stable vortex sheet family is presented. Compact solutions are proven not to exist by De Lellis and Brué. Asymptotic conservation of anomalous dissipation on stable vortex surfaces in the turbulent limit is discovered. We derive an exact formula for this anomalous dissipation as a surface integral of the square of velocity gap times the square root of minus local normal strain. Topologically stable time-dependent solutions, which we call Kelvinons, are introduced. They have a conserved velocity circulation around static loop; this makes them responsible for asymptotic PDF tails of velocity circulation, perfectly matching numerical simulations. The loop equation for circulation PDF as functional of the loop shape is derived and studied. This equation is exactly equivalent to the Schrödinger equation in loop space, with viscosity ν playing the role of Planck's constant. This equivalence opens the way for direct numerical simulation of turbulence on quantum computers. Kelvinons are fixed points of the loop equation at turbulent limit ν→ 0. Area law and the asymptotic scaling law for mean circulation at a large area are derived. The representation of the solution of the loop equation in terms of a singular stochastic equation for momentum loop trajectory is presented.Alexander Migdalwork_qrlmjshh65cfddfvw4x3lbhb44Mon, 28 Nov 2022 00:00:00 GMTSimulation Intelligence: Towards a New Generation of Scientific Methods
https://scholar.archive.org/work/rfujm43y4ngcnml5emnvjksbjy
The original "Seven Motifs" set forth a roadmap of essential methods for the field of scientific computing, where a motif is an algorithmic method that captures a pattern of computation and data movement. We present the "Nine Motifs of Simulation Intelligence", a roadmap for the development and integration of the essential algorithms necessary for a merger of scientific computing, scientific simulation, and artificial intelligence. We call this merger simulation intelligence (SI), for short. We argue the motifs of simulation intelligence are interconnected and interdependent, much like the components within the layers of an operating system. Using this metaphor, we explore the nature of each layer of the simulation intelligence operating system stack (SI-stack) and the motifs therein: (1) Multi-physics and multi-scale modeling; (2) Surrogate modeling and emulation; (3) Simulation-based inference; (4) Causal modeling and inference; (5) Agent-based modeling; (6) Probabilistic programming; (7) Differentiable programming; (8) Open-ended optimization; (9) Machine programming. We believe coordinated efforts between motifs offers immense opportunity to accelerate scientific discovery, from solving inverse problems in synthetic biology and climate science, to directing nuclear energy experiments and predicting emergent behavior in socioeconomic settings. We elaborate on each layer of the SI-stack, detailing the state-of-art methods, presenting examples to highlight challenges and opportunities, and advocating for specific ways to advance the motifs and the synergies from their combinations. Advancing and integrating these technologies can enable a robust and efficient hypothesis-simulation-analysis type of scientific method, which we introduce with several use-cases for human-machine teaming and automated science.Alexander Lavin, David Krakauer, Hector Zenil, Justin Gottschlich, Tim Mattson, Johann Brehmer, Anima Anandkumar, Sanjay Choudry, Kamil Rocki, Atılım Güneş Baydin, Carina Prunkl, Brooks Paige, Olexandr Isayev, Erik Peterson, Peter L. McMahon, Jakob Macke, Kyle Cranmer, Jiaxin Zhang, Haruko Wainwright, Adi Hanuka, Manuela Veloso, Samuel Assefa, Stephan Zheng, Avi Pfefferwork_rfujm43y4ngcnml5emnvjksbjySun, 27 Nov 2022 00:00:00 GMTA variational principle in the parametric geometry of numbers
https://scholar.archive.org/work/2aouitjuz5ahtmagbkatb52puy
We extend the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of m linear forms in n variables, and establish a new connection to the metric theory via a variational principle that computes fractal dimensions of a variety of sets of number-theoretic interest. The proof relies on two novel ingredients: a variant of Schmidt's game capable of computing the Hausdorff and packing dimensions of any set, and the notion of templates, which generalize Roy's rigid systems. In particular, we compute the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of Dani's correspondence principle, the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Other applications include quantitative strengthenings of theorems due to Cheung and Moshchevitin, which originally resolved conjectures due to Starkov and Schmidt respectively; as well as dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions, completing partial results due to Baker, Bugeaud, Cheung, Chevallier, Dodson, Laurent and Rynne.Tushar Das, Lior Fishman, David Simmons, Mariusz Urbańskiwork_2aouitjuz5ahtmagbkatb52puySat, 26 Nov 2022 00:00:00 GMTQuantum Gravity in 30 Questions
https://scholar.archive.org/work/mllid5xuefabziu5uy4uz7avs4
Quantum gravity is the missing piece in our understanding of the fundamental interactions today. Given recent observational breakthroughs in gravity, providing a quantum theory for what lies beyond general relativity is more urgent than ever. However, the complex history of quantum gravity and the multitude of available approaches can make it difficult to get a grasp of the topic and its main challenges and opportunities. We provide a guided tour of quantum gravity in the form of 30 questions, aimed at a mixed audience of learners and practitioners. The issues covered range from basic motivational and background material to a critical assessment of the status quo and future of the subject. The emphasis is on structural issues and our current understanding of quantum gravity as a quantum field theory of dynamical geometry beyond perturbation theory. We highlight the identification of quantum observables and the development of effective numerical tools as critical to future progress.Renate Loll, Giuseppe Fabiano, Domenico Frattulillo, Fabian Wagnerwork_mllid5xuefabziu5uy4uz7avs4Wed, 23 Nov 2022 00:00:00 GMTEnd-to-end resource analysis for quantum interior point methods and portfolio optimization
https://scholar.archive.org/work/qqjpiwjbavd7dmso7ufco4m674
We study quantum interior point methods (QIPMs) for second-order cone programming (SOCP), guided by the example use case of portfolio optimization (PO). We provide a complete quantum circuit-level description of the algorithm from problem input to problem output, making several improvements to the implementation of the QIPM. We report the number of logical qubits and the quantity/depth of non-Clifford T-gates needed to run the algorithm, including constant factors. The resource counts we find depend on instance-specific parameters, such as the condition number of certain linear systems within the problem. To determine the size of these parameters, we perform numerical simulations of small PO instances, which lead to concrete resource estimates for the PO use case. Our numerical results do not probe large enough instance sizes to make conclusive statements about the asymptotic scaling of the algorithm. However, already at small instance sizes, our analysis suggests that, due primarily to large constant pre-factors, poorly conditioned linear systems, and a fundamental reliance on costly quantum state tomography, fundamental improvements to the QIPM are required for it to lead to practical quantum advantage.Alexander M. Dalzell, B. David Clader, Grant Salton, Mario Berta, Cedric Yen-Yu Lin, David A. Bader, Nikitas Stamatopoulos, Martin J. A. Schuetz, Fernando G. S. L. Brandão, Helmut G. Katzgraber, William J. Zengwork_qqjpiwjbavd7dmso7ufco4m674Tue, 22 Nov 2022 00:00:00 GMTPreparation for Quantum Simulation of the 1+1D O(3) Non-linear σ-Model using Cold Atoms
https://scholar.archive.org/work/w4ghmixcovaf7p73yoocuccefa
The 1+1D O(3) non-linear σ-model is a model system for future quantum lattice simulations of other asymptotically-free theories, such as non-Abelian gauge theories. We find that utilizing dimensional reduction can make efficient use of two-dimensional layouts presently available on cold atom quantum simulators. A new definition of the renormalized coupling is introduced, which is applicable to systems with open boundary conditions and can be measured using analog quantum simulators. Monte Carlo and tensor network calculations are performed to determine the quantum resources required to reproduce perturbative short-distance observables. In particular, we show that a rectangular array of 48 Rydberg atoms with existing quantum hardware capabilities should be able to adiabatically prepare low-energy states of the perturbatively-matched theory. These states can then be used to simulate non-perturbative observables in the continuum limit that lie beyond the reach of classical computers.Anthony N. Ciavarella, Stephan Caspar, Hersh Singh, Martin J. Savagework_w4ghmixcovaf7p73yoocuccefaTue, 22 Nov 2022 00:00:00 GMTCovering and packing with homothets of limited capacity
https://scholar.archive.org/work/uc5owpu35nbdjhcfxstve5zxti
This work revolves around the two following questions: Given a convex body C⊂ℝ^d, a positive integer k and a finite set S⊂ℝ^d (or a finite μ Borel measure in ℝ^d), how many homothets of C are required to cover S if no homothet is allowed to cover more than k points of S (or have measure more than k)? how many homothets of C can be packed if each of them must cover at least k points of S (or have measure at least k)? We prove that, so long as S is not too degenerate, the answer to both questions is Θ_d(|S|/k), where the hidden constant is independent of d, this is clearly best possible up to a multiplicative constant. Analogous results hold in the case of measures. Then we introduce a generalization of the standard covering and packing densities of a convex body C to Borel measure spaces in ℝ^d and, using the aforementioned bounds, we show that they are bounded from above and below, respectively, by functions of d. As an intermediate result, we give a simple proof the existence of weak ϵ-nets of size O(1/ϵ) for the range space induced by all homothets of C. Following some recent work in discrete geometry, we investigate the case d=k=2 in greater detail. We also provide polynomial time algorithms for constructing a packing/covering exhibiting the Θ_d(|S|/k) bound mentioned above in the case that C is an Euclidean ball. Finally, it is shown that if C is a square then it is NP-hard to decide whether S can be covered by |S|/4 squares containing 4 points each.Oriol Solé Piwork_uc5owpu35nbdjhcfxstve5zxtiThu, 17 Nov 2022 00:00:00 GMTSpin foams, Refinement limit and Renormalization
https://scholar.archive.org/work/r667zvp2t5blniwhxgmph3fpxq
Spin foams provide path integrals for quantum gravity, which employ discretizations as regulator. To obtain regulator independent predictions, we must remove these fiducial structures in a suitable refinement limit. In this chapter we present the current state of research: We begin with a discussion on the role of diffeomorphism symmetries in discrete systems, the notion of scale in background independent theories and how we can consistently improve theories via renormalization to reduce regulator dependence. We present the consistent boundary formulation, which provides a renormalization framework for background independent theories, and discuss tensor network methods and restricted spin foams, which provide concrete renormalization algorithms aiming at the construction of consistent boundary amplitudes for spin foams. We furthermore discuss effective spin foams, which have allowed for the construction of a perturbative refinement limit and an associated effective continuum action.Seth K. Asante, Bianca Dittrich, Sebastian Steinhauswork_r667zvp2t5blniwhxgmph3fpxqThu, 17 Nov 2022 00:00:00 GMTA new lattice Boltzmann scheme for linear elastic solids: periodic problems
https://scholar.archive.org/work/sbatqhxbtbe57nqgby7ujqje54
We propose a new second-order accurate lattice Boltzmann scheme that solves the quasi-static equations of linear elasticity in two dimensions. In contrast to previous works, our formulation solves for a single distribution function with a standard velocity set and avoids any recourse to finite difference approximations. As a result, all computational benefits of the lattice Boltzmann method can be used to full capacity. The novel scheme is systematically derived using the asymptotic expansion technique and a detailed analysis of the leading-order error behavior is provided. As demonstrated by a linear stability analysis, the method is stable for a very large range of Poisson's ratios. We consider periodic problems to focus on the governing equations and rule out the influence of boundary conditions. The analytical derivations are verified by numerical experiments and convergence studies.Oliver Boolakee and Martin Geier and Laura De Lorenziswork_sbatqhxbtbe57nqgby7ujqje54Wed, 16 Nov 2022 00:00:00 GMTA machine learning route between band mapping and band structure
https://scholar.archive.org/work/k5ychvqgozfp3f45w2inroz5x4
Electronic band structure (BS) and crystal structure are the two complementary identifiers of solid state materials. While convenient instruments and reconstruction algorithms have made large, empirical, crystal structure databases possible, extracting quasiparticle dispersion (closely related to BS) from photoemission band mapping data is currently limited by the available computational methods. To cope with the growing size and scale of photoemission data, we develop a pipeline including probabilistic machine learning and the associated data processing, optimization and evaluation methods for band structure reconstruction, leveraging theoretical calculations. The pipeline reconstructs all 14 valence bands of a semiconductor and shows excellent performance on benchmarks and other materials datasets. The reconstruction uncovers previously inaccessible momentum-space structural information on both global and local scales, while realizing a path towards integration with materials science databases. Our approach illustrates the potential of combining machine learning and domain knowledge for scalable feature extraction in multidimensional data.Rui Patrick Xian, Vincent Stimper, Marios Zacharias, Maciej Dendzik, Shuo Dong, Samuel Beaulieu, Bernhard Schölkopf, Martin Wolf, Laurenz Rettig, Christian Carbogno, Stefan Bauer, Ralph Ernstorferwork_k5ychvqgozfp3f45w2inroz5x4Tue, 15 Nov 2022 00:00:00 GMTAspects of scaling and scalability for flow-based sampling of lattice QCD
https://scholar.archive.org/work/zjsp4xuugrhdtlb2esc5wckciy
Recent applications of machine-learned normalizing flows to sampling in lattice field theory suggest that such methods may be able to mitigate critical slowing down and topological freezing. However, these demonstrations have been at the scale of toy models, and it remains to be determined whether they can be applied to state-of-the-art lattice quantum chromodynamics calculations. Assessing the viability of sampling algorithms for lattice field theory at scale has traditionally been accomplished using simple cost scaling laws, but as we discuss in this work, their utility is limited for flow-based approaches. We conclude that flow-based approaches to sampling are better thought of as a broad family of algorithms with different scaling properties, and that scalability must be assessed experimentally.Ryan Abbott, Michael S. Albergo, Aleksandar Botev, Denis Boyda, Kyle Cranmer, Daniel C. Hackett, Alexander G. D. G. Matthews, Sébastien Racanière, Ali Razavi, Danilo J. Rezende, Fernando Romero-López, Phiala E. Shanahan, Julian M. Urbanwork_zjsp4xuugrhdtlb2esc5wckciyMon, 14 Nov 2022 00:00:00 GMTInvestigating strongly interacting multicritical field theories using numerical conformal bootstrap
https://scholar.archive.org/work/ry5sypox4zd4zbkobjawb5xar4
In strongly correlated quantum systems, conventional techniques for understanding physical behaviour near multicritical points break down. Numerical conformal bootstrap is a promising new method that has proven demonstrably useful in investigating critical phenomena. The method uses the enhanced conformal symmetry exhibited by quantum field theories describing criticality in order to definitively rule out theories based on fundamental symmetry grounds alone. By requiring only that physically plausible conformal field theories (CFTs) satisfy a mathematical self-consistency relation, it allows one to rule out vast amounts of violating CFT data, allowing a rigorous mapping to made of 'allowed' CFTs. Such a mapping is powerful due to the concept of universality, whereby the critical behaviour of many disparate physical models is identical and is captured by the same set of critical exponents and the same CFT. This thesis aims to explore the effectiveness of numerical bootstrap. We begin by motivating the bootstrap in a condensed matter context, before reviewing necessary results in the conformal field theory literature and establishing the bootstrap method. A large part of the thesis will be an overview of the working of the method and the approximations made, as well as its applications to simple symmetry groups. We will demonstrate the validity of the method by replicating flagship results, before describing the novel work we have carried out in exploring more complicated global symmetry groups, where more than one fixed point is predicted. Finally, we will conclude and comment on the limitations of the thesis and lines of possible future work.Matthew Dowens, Chris Hooleywork_ry5sypox4zd4zbkobjawb5xar4Mon, 14 Nov 2022 00:00:00 GMTScalarization functionals in mathematical finance and vector optimization : a new view on acceptance sets and risk measures
https://scholar.archive.org/work/ycmszwabfbbsnmzxof2mhvo5iu
Risikomaße und Akzeptanzmengen sind wichtige Untersuchungsobjekte, die auf der von Artzner et al. 1999 eingeführten Axiomatik beruhen. Im aktuellen Umfeld steigender regulatorischer Anforderungen an Finanzinstitute hinsichtlich ihrer Kapital- und Risikopositionen sind optimale Finanzentscheidungen wesentlich. Diese Dissertation leistet einen wichtigen Beitrag zur Bestimmung kostenminimaler Investitionsentscheidungen zur Erfüllung solcher regulatorischen Vorgaben. Im Mittelpunkt steht dabei die Untersuchung der Eigenschaften des zugehörigen Risikofunktionals, die daraufhin wesentlich für die Bestimmung kostenminimaler Lösungen des Vektoroptimierungsproblems sind. Ferner werden die (schwach) effizienten Punkte der Akzeptanzmenge bestimmt und Zusammenhänge zur Löosung des beschriebenen Optimierungsproblems abgeleitet. Mittels eines besonders allgemeinen Finanzmarktmodells wird eine weitreichende Anwendbarkeit in Theorie und Praxis gewährleistet.Marcel Marohn, Universitäts- Und Landesbibliothek Sachsen-Anhalt, Martin-Luther Universität, Christiane Tammer, Gemayqzel Bouza Allendework_ycmszwabfbbsnmzxof2mhvo5iuFri, 11 Nov 2022 00:00:00 GMTTowards near-term quantum simulation of materials
https://scholar.archive.org/work/pzwomhk5nvcylcmk7cimxgt2om
Simulation of materials is one of the most promising applications of quantum computers. On near-term hardware the crucial constraint on these simulations is circuit depth. Many quantum simulation algorithms rely on a layer of unitary evolutions generated by each term in a Hamiltonian. This appears in time-dynamics as a single Trotter step, and in variational quantum eigensolvers under the Hamiltonian variational ansatz as a single ansatz layer. We present a new quantum algorithm design for materials modelling where the depth of a layer is independent of the system size. This design takes advantage of the locality of materials in the Wannier basis and employs a tailored fermionic encoding that preserves locality. We analyse the circuit costs of this approach and present a compiler that transforms density functional theory data into quantum circuit instructions – connecting the physics of the material to the simulation circuit. The compiler automatically optimises circuits at multiple levels, from the base gate level to optimisations derived from the physics of the specific target material. We present numerical results for materials spanning a wide structural and technological range. Our results demonstrate a reduction of many orders of magnitude in circuit depth over standard prior methods that do not consider the structure of the Hamiltonian. For example our results improve resource requirements for Strontium Vanadate (SrVO_3) from 864 to 180 qubits for a 3×3×3 lattice, and the circuit depth of a single Trotter or variational layer from 7.5× 10^8 to depth 884. Although this is still beyond current hardware, our results show that materials simulation may be feasible on quantum computers without necessarily requiring scalable, fault-tolerant quantum computers, provided quantum algorithm design incorporates understanding of the materials and applications.Laura Clinton, Toby Cubitt, Brian Flynn, Filippo Maria Gambetta, Joel Klassen, Ashley Montanaro, Stephen Piddock, Raul A. Santos, Evan Sheridanwork_pzwomhk5nvcylcmk7cimxgt2omThu, 10 Nov 2022 00:00:00 GMTRandom Walks, Spectral Gaps, and Khintchine's Theorem on Fractals
https://scholar.archive.org/work/glxjwr3iuzdlvigjt2gxijmuvy
This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle 1/3 set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of ℝ^d (for any d≥ 1) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle 1/5 Cantor set"; i.e. the set of numbers whose base 5 expansions miss a single digit. The key new ingredient is an effective equidistribution theorem for certain fractal measures on the homogeneous space ℒ_d+1 of unimodular lattices; a result of independent interest. The latter is established via a new technique involving the construction of S-arithmetic operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. As a consequence of our methods, we show that spherical averages of certain random walks naturally associated to the fractal measures effectively equidistribute on ℒ_d+1.Osama Khalil, Manuel Luethiwork_glxjwr3iuzdlvigjt2gxijmuvyThu, 10 Nov 2022 00:00:00 GMTMultiple Packing: Lower Bounds via Infinite Constellations
https://scholar.archive.org/work/klhyxug4zjeg3aixk4b74u7aku
We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let N>0 and L∈ℤ_≥2. A multiple packing is a set 𝒞 of points in ℝ^n such that any point in ℝ^n lies in the intersection of at most L-1 balls of radius √(nN) around points in 𝒞. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. In this paper, we derive the best known lower bounds on the optimal density of list-decodable infinite constellations for constant L under a stronger notion called average-radius multiple packing. To this end, we apply tools from high-dimensional geometry and large deviation theory.Yihan Zhang, Shashank Vatedkawork_klhyxug4zjeg3aixk4b74u7akuWed, 09 Nov 2022 00:00:00 GMTBounds for Multiple Packing and List-Decoding Error Exponents
https://scholar.archive.org/work/xlrfu6cqbjfhpgj6zutsmngkua
We revisit the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let N>0 and L∈ℤ_≥2. A multiple packing is a set 𝒞 of points in ℝ^n such that any point in ℝ^n lies in the intersection of at most L-1 balls of radius √(nN) around points in 𝒞. We study the multiple packing problem for both bounded point sets whose points have norm at most √(nP) for some constant P>0 and unbounded point sets whose points are allowed to be anywhere in ℝ^n. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. In this paper, we derive various bounds on the largest possible density of a multiple packing in both bounded and unbounded settings. A related notion called average-radius multiple packing is also studied. Some of our lower bounds exactly pin down the asymptotics of certain ensembles of average-radius list-decodable codes, e.g., (expurgated) Gaussian codes and (expurgated) Poisson Point Processes. To this end, we apply tools from high-dimensional geometry and large deviation theory. Some of our lower bounds on the optimal multiple packing density are the best known lower bounds. These bounds are obtained via a proxy known as error exponent. The latter quantity is the best exponent of the probability of list-decoding error when the code is corrupted by a Gaussian noise. We establish a curious inequality which relates the error exponent, a quantity of average-case nature, to the list-decoding radius, a quantity of worst-case nature. We derive various bounds on the error exponent in both bounded and unbounded settings which are of independent interest beyond multiple packing.Yihan Zhang, Shashank Vatedkawork_xlrfu6cqbjfhpgj6zutsmngkuaWed, 09 Nov 2022 00:00:00 GMTAsymptotic properties of approximate Bayesian computation
https://scholar.archive.org/work/e3o46qzetbh2bd2hwft4f6zcii
Approximate Bayesian computation is becoming an accepted tool for statistical analysis in models withDavid T. Frazier, Gael M. Martin, Christian P. Robert, Judith Rousseauwork_e3o46qzetbh2bd2hwft4f6zciiWed, 09 Nov 2022 00:00:00 GMTMultiple Packing: Lower and Upper Bounds
https://scholar.archive.org/work/xkthqa66qvfoveymjeauwtzzji
We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let N>0 and L∈ℤ_≥2. A multiple packing is a set 𝒞 of points in ℝ^n such that any point in ℝ^n lies in the intersection of at most L-1 balls of radius √(nN) around points in 𝒞. We study the multiple packing problem for both bounded point sets whose points have norm at most √(nP) for some constant P>0 and unbounded point sets whose points are allowed to be anywhere in ℝ^n. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. In this paper, we derive various bounds on the largest possible density of a multiple packing in both bounded and unbounded settings. A related notion called average-radius multiple packing is also studied. Some of our lower bounds exactly pin down the asymptotics of certain ensembles of average-radius list-decodable codes, e.g., (expurgated) Gaussian codes and (expurgated) spherical codes. In particular, our lower bound obtained from spherical codes is the best known lower bound on the optimal multiple packing density and is the first lower bound that approaches the known large L limit under the average-radius notion of multiple packing. To derive these results, we apply tools from high-dimensional geometry and large deviation theory.Yihan Zhang, Shashank Vatedkawork_xkthqa66qvfoveymjeauwtzzjiWed, 09 Nov 2022 00:00:00 GMTA Panorama Of Physical Mathematics c. 2022
https://scholar.archive.org/work/5ic2soxwonefdcg6gr2bi7r7gy
What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forward-looking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1].Ibrahima Bah, Daniel S. Freed, Gregory W. Moore, Nikita Nekrasov, Shlomo S. Razamat, Sakura Schafer-Namekiwork_5ic2soxwonefdcg6gr2bi7r7gyWed, 09 Nov 2022 00:00:00 GMT