IA Scholar Query: Computability of the Spectrum of Self-Adjoint Operators.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 01 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Isadore M. Singer (1924–2021) In Memoriam Part 1: Scientific Works
https://scholar.archive.org/work/aejx3oq2lvch5gdpwoqpzzlbqe
Robert Bryant, Jean-Michel Bismut, Jeff Cheeger, Phillip Griffiths, Simon Donaldson, Nigel Hitchin, H Blaine Lawson, Michail Gromov, Adam Marcus, Daniel Spielman, Nikhil Srivastava, Edward Wittenwork_aejx3oq2lvch5gdpwoqpzzlbqeSat, 01 Oct 2022 00:00:00 GMTConvergence problems in nonlocal dynamics with nonlinearity
https://scholar.archive.org/work/7t2bd2jorbf4xhlroj4ikl3rhq
We study nonlocal nonlinear dynamical systems and uncover the gradient structure to investigate the convergence of solutions. Mainly but not exclusively, we use the Lojasiewicz inequality to prove convergence results in various spaces with continuous, or discrete temporal domain, and finite, or infinite dimensional spatial domain. To be more specific, we analyze Lotka-Volterra type dynamics and concentration-dispersion dynamics. Lotka-Volterra equations describe the population dynamics of a group of species, in which individuals interact either competitively or cooperatively with each other. It is well-known that Lotka-Volterra equations form a gradient system with respect to the Shahshahani metric. The Shahshahani metric, unfortunately, becomes singular in the scenario where some species become extinct. This singular nature of the Shahshahani metric is an obstacle to the usual convergence analysis. Under the assumption that the interaction between species is symmetric, we present two different methods to derive the convergence result. One, the entropy trapping method, is to adapt the idea of Akin and Hofbauer (Math. Biosci. 61 (1982) 51{62) of using monotonicity of the energy to bound the entropy, which provides the proximal distance of the solution from the desired equilibrium. Another method, inspired by Jabin and Liu's observation in (Nonlinearity 30 (2017) 4220) is to change variables to resolve the singular nature of gradient structure. We apply this idea to show the convergence result in generalized Lokta-Volterra systems, such as regularized Lotka-Volterra systems and nonlocal semi-linear heat equations, which can be seen as an infinite dimensional Lotka-Volterra equations with mutation. Concentration-dispersion dynamics is a new type of equation that is inspired by fifixed point formulations for solitary wave shapes. The equations are designed in a way that the solution evolves to match the shape of a concentrated and dispersed version of the solution, which is an outcome of power nonlinearity and convol [...]Won Eui Hongwork_7t2bd2jorbf4xhlroj4ikl3rhqFri, 30 Sep 2022 00:00:00 GMTThe Prime Geodesic Theorem for PSL_2(ℤ[i]) and Spectral Exponential Sums
https://scholar.archive.org/work/uxwfignztjewhnfb36pvc2jaaq
This work addresses the Prime Geodesic Theorem for the Picard manifold ℳ = PSL_2(ℤ[i]) \𝔥^3, which asks for the asymptotic evaluation of a counting function for the closed geodesics on ℳ. Let E_Γ(X) be the error term in the Prime Geodesic Theorem. We establish that E_Γ(X) = O_ε(X^3/2+ε) on average as well as many pointwise bounds. The second moment bound parallels an analogous result for Γ = PSL_2(ℤ) due to Balog et al. and our innovation features the delicate analysis of sums of Kloosterman sums with an explicit manipulation of oscillatory integrals. The proof of the pointwise bounds requires Weyl-strength subconvexity for quadratic Dirichlet L-functions over ℚ(i). Moreover, an asymptotic formula for a spectral exponential sum in the spectral aspect for a cofinite Kleinian group Γ is given. Our numerical experiments visualise in particular that E_Γ(X) obeys a conjectural bound of size O_ϵ(X^1+ε).Ikuya Kanekowork_uxwfignztjewhnfb36pvc2jaaqThu, 29 Sep 2022 00:00:00 GMTReinforcement-learning-based actuator selection method for active flow control
https://scholar.archive.org/work/ybgpgmqksbcnbm3pewnqtjqiwm
This paper addresses the issue of actuator selection for active flow control by proposing a novel method built on top of a reinforcement learning agent. Starting from a pre-trained agent using numerous actuators, the algorithm estimates the impact of a potential actuator removal on the value function, indicating the agent's performance. It is applied to two test cases, the one-dimensional Kuramoto-Sivashinsky equation and a laminar bi-dimensional flow around an airfoil at Re=1000 for different angles of attack ranging from 12 to 20 degrees, to demonstrate its capabilities and limits. The proposed actuator-sparsification method relies on a sequential elimination of the least relevant action components, starting from a fully developed layout. The relevancy of each component is evaluated using metrics based on the value function. Results show that, while still being limited by this intrinsic elimination paradigm (i.e. the sequential elimination), actuator patterns and obtained policies demonstrate relevant performances and allow to draw an accurate approximation of the Pareto front of performances versus actuator budget.Romain Paris, Samir Beneddine, Julien Dandoiswork_ybgpgmqksbcnbm3pewnqtjqiwmThu, 29 Sep 2022 00:00:00 GMTCovering models of the asymmetric quantum Rabi model: η-shifted non-commutative harmonic oscillators
https://scholar.archive.org/work/y7v2luph6rde5bnw7vraaekiye
The non-commutative harmonic oscillator (NCHO) is a matrix valued differential operator introduced as a generalization of the quantum harmonic oscillator. The spectrum of the NCHO has remarkable properties, including the presence of a number theoretical structure such as modular forms, elliptic curves, Eichler cohomology observed in the special values of the associated spectral zeta function. Notably, the Heun ODE picture of the eigenvalue problem of the NCHO reveals a connection with the quantum Rabi model (QRM), a fundamental interaction model from quantum optics. In this paper we introduce an η-shifted NCHO (η-NCHO) that has an analog relation with the asymmetric quantum Rabi model (AQRM) and describe its basic properties. Despite the fact that the shift factor does not break the parity symmetry, a certain type of degeneracies appears for η∈1/2ℤ, as if mirroring the situation of the AQRM. Furthermore, we give a detailed description of the confluence process, that we call iso-parallel confluence process due to the fact that it requires a parallel transformation of the two parameters describing the spectrum of η-NCHO and representations of 𝔰𝔩_2(ℝ). In particular, we relate the eigenvalues of the two models under the iso-parallel confluence process, including how the finite eigenvalues of the η-NCHO correspond to Juddian (or quasi-exact) solutions of the AQRM. From the point of view of this confluence process, a family of η-NCHO correspond to a single AQRM, thus we may say that the η-NCHO is a covering of the AQRM. We expect the study of the η-NCHO and the AQRM from this point of view to be helpful for the clarification of several questions on the AQRM, including the number of Juddian solutions and the hidden symmetry.Cid Reyes-Bustos, Masato Wakayamawork_y7v2luph6rde5bnw7vraaekiyeThu, 29 Sep 2022 00:00:00 GMTSteklov flows on trees and applications
https://scholar.archive.org/work/7qmq6yxu25fzfa6lgoaj4rnjoa
We introduce the Steklov flows on finite trees, i.e. the flows (or currents) associated with the Steklov problem. By constructing appropriate Steklov flows, we prove the monotonicity and rigidity of the first nonzero Steklov eigenvalues on trees: for finite trees _1 and _2, the first nonzero Steklov eigenvalue of _1 is greater than or equal to that of _2, provided that _1 is a subgraph of _2. Moreover, we give the sufficient and necessary condition in which the equality holds.Zunwu He, Bobo Huawork_7qmq6yxu25fzfa6lgoaj4rnjoaThu, 29 Sep 2022 00:00:00 GMTExotic Monoidal Structures and Abstractly Automorphic Representations for GL(2)
https://scholar.archive.org/work/7hwy6rfpkvb4nofyk3itqyfswu
We use the theta correspondence to study the equivalence between Godement-Jacquet and Jacquet-Langlands L-functions for GL(2). We show that the resulting comparison is in fact an exotic symmetric monoidal structure on the category of GL(2)-modules. Moreover, this enables us to construct an Abelian category of abstractly automorphic representations, whose irreducible objects are the usual automorphic representations. We speculate that this category is a natural setting for the study of automorphic phenomena for GL(2), and demonstrate its basic properties. This paper is a part of the author's thesis.Gal Dorwork_7hwy6rfpkvb4nofyk3itqyfswuWed, 28 Sep 2022 00:00:00 GMTOn Pidduck polynomials and zeros of the Riemann zeta function
https://scholar.archive.org/work/undchxvxwbbixg6txrjukjo24y
For 1<p<∞, we prove that a necessary and sufficient condition for s to be a zero of the Riemann zeta function in the strip 0< s<1 is that ([ 1 1/3 1/5 1/7 1/9 ⋯; -s/3 1 1/3 1/5 1/7 ⋯; -s/5 -s/5 1 1/3 1/5 ⋯; -s/7 -s/7 -s/7 1 1/3 ⋯; -s/9 -s/9 -s/9 -s/9 1 ⋯; ⋮ ⋮ ⋮ ⋮ ⋮ ⋱; ])([ v_0; v_1; v_2; v_3; v_4; ⋮; ⋮; ]) =0 has a nontrivial solution (v_k)_k=0^∞ in ℓ^p. An explicit formula for v_k is constructed in terms of Pidduck polynomials. In the process, it is also shown that Pidduck polynomials form an orthogonal basis with respect to an inner product of polynomials f,g whereby we replace in a formal expression "∑_n=1^∞ (-1)^n+1n f(n^2) g(n^2)" the divergent sums "∑_n=1^∞ (-1)^n+1n^1+2k" with their zeta-function regularized values. We also discuss the modification for possible non-simple zeros and conclude with applications to the question of the simplicity of the zeros and a relation to the Hilbert-Pólya program.Ori J. Ganorwork_undchxvxwbbixg6txrjukjo24yWed, 28 Sep 2022 00:00:00 GMTStudies of quantum chromodynamics with jets at the CMS experiment at the LHC
https://scholar.archive.org/work/tl6cqxvdijhwbnd4wxd3l6sbii
Several people played a decisive role in accomplishing this thesis and helped me in dierent aspects. In Hamburg, I would like to extend my deepest gratitude to Patrick L.S. Connor for his invaluable contribution to this work and for training me to consider scientic research as a "share, help, learn, cross-check, enjoy" cycle. Besides developing the overall analysis framework, he was always reachable for help and support, making the work with him a continuous upskilling process. I am also extremely grateful to Paolo Gunnellini for his contributions to the analysis, but mainly for his crucial guidance during my rst steps in high energy physics and his availability to help whenever I needed to. At DESY, I am deeply indebted to Hannes Jung for all his hospitality and support. Apart from that, he also gave me the opportunity to work with his wonderful team, to whom I am also grateful. In particular, many thanks toParaskevas Gianneios, University Of Ioanninawork_tl6cqxvdijhwbnd4wxd3l6sbiiWed, 28 Sep 2022 00:00:00 GMTThe coarse Baum-Connes conjecture for certain relative expanders
https://scholar.archive.org/work/qnjh64mkxbgjnke767gpe7l32e
Let ( 1→ N_m→ G_m→ Q_m→ 1 )_m∈ℕ be a sequence of extensions of finite groups such that their coarse disjoint unions have bounded geometry. In this paper, we show that if the coarse disjoint unions of ( N_m )_m∈ℕ and ( Q_m )_m∈ℕ are coarsely embeddable into Hilbert space, then the coarse Baum-Connes conjecture holds for the coarse disjoint union of ( G_m )_m∈ℕ. As an application, the coarse Baum-Connes conjecture holds for the relative expanders constructed by G. Arzhantseva and R. Tessera, and the special box spaces of free groups discovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander. This enlarges the class of metric spaces known to satisfy the coarse Baum-Connes conjecture. In particular, it solves an open problem raised by G. Arzhantseva and R. Tessera on the coarse Baum-Connes conjecture for relative expanders.Jintao Deng, Qin Wang, Guoliang Yuwork_qnjh64mkxbgjnke767gpe7l32eWed, 28 Sep 2022 00:00:00 GMTMassless Fermions on a half-space: The curious case of 2+1-dimensions
https://scholar.archive.org/work/az62b7sol5gnnpdndtlcf66dma
Boundary conditions for a massless Dirac fermion in 2+1 dimensions where the space is a half-plane are discussed in detail. It is argued that linear boundary conditions that leave the Hamiltonian Hermitian generically break C P and T symmetries as well as Lorentz and conformal symmetry. We show that there is essentially one special case where a single species of fermion has CPT and the full Poincare and conformal symmetry of the boundary. We show that, with doubled fermions, there is a second special case which respects CPT but still violates Lorentz and conformal symmetry. This second special case is essentially the unique boundary condition where the Dirac operator has fermion zero mode edge states. We discuss how the edge states lead to exotic representations of scale, phase and translation symmetries and how imposing a symmetry requirement leads to edge ferromagnetism of the system. We prove that the exotic ferromagnetic representations are indeed carried by the ground states of the system perturbed by a class of interaction Hamiltonians which includes the non-relativistic Coulomb interaction.Shovon Biswas, Gordon W. Semenoffwork_az62b7sol5gnnpdndtlcf66dmaWed, 28 Sep 2022 00:00:00 GMTOn critical points of eigenvalues of the Montgomery family of quartic oscillators
https://scholar.archive.org/work/kioxkebsfvh5xi32an3z222rve
We discuss spectral properties of the family of quartic oscillators 𝔥_ℳ(α) =-d^2/dt^2 +(1/2 t^2 -α)^2 on the real line, where α∈ℝ is a parameter. This operator appears in a variety of applications coming from quantum mechanics to harmonic analysis on Lie groups, Riemannian geometry and superconductivity. We study the variations of the eigenvalues λ_j(α) of 𝔥_ℳ(α) as functions of the parameter α.We prove that for j sufficiently large, α↦λ_j(α) has a unique critical point, which is a nondegenerate minimum.We also prove that the first eigenvalue λ_1(α) enjoys the same property and give a numerically assisted proof that the same holds for the second eigenvalue λ_2(α). The proof for excited states relies on a semiclassical reformulation of the problem. In particular, we develop a method permitting to differentiate with respect to the semiclassical parameter, which may be of independent interest.Bernard Helfferwork_kioxkebsfvh5xi32an3z222rveWed, 28 Sep 2022 00:00:00 GMTA Tale of Two Saddles
https://scholar.archive.org/work/3424et3qyjbjnpyadzzyrdweai
We find a new on-shell replica wormhole in a computation of the generating functional of JT gravity coupled to matter. We show that this saddle has lower action than the disconnected one, and that it is stable under restriction to real Lorentzian sections, but can be unstable otherwise. The behavior of the classical generating functional thus may be strongly dependent on the signature of allowed perturbations. As part of our analysis, we give an LM-style construction for computing the on-shell action of replicated manifolds even as the number of boundaries approaches zero, including a type of one-step replica symmetry breaking that is necessary to capture the contribution of the new saddle. Our results are robust against quantum corrections; in fact, we find evidence that such corrections may sometimes stabilize this new saddle.Venkatesa Chandrasekaran, Netta Engelhardt, Sebastian Fischetti, Sergio Hernández-Cuencawork_3424et3qyjbjnpyadzzyrdweaiWed, 28 Sep 2022 00:00:00 GMTLearning Dissipative Dynamics in Chaotic Systems
https://scholar.archive.org/work/fvemotnu4ncstfrs3emw7ftlsm
Chaotic systems are notoriously challenging to predict because of their sensitivity to perturbations and errors due to time stepping. Despite this unpredictable behavior, for many dissipative systems the statistics of the long term trajectories are governed by an invariant measure supported on a set, known as the global attractor; for many problems this set is finite dimensional, even if the state space is infinite dimensional. For Markovian systems, the statistical properties of long-term trajectories are uniquely determined by the solution operator that maps the evolution of the system over arbitrary positive time increments. In this work, we propose a machine learning framework to learn the underlying solution operator for dissipative chaotic systems, showing that the resulting learned operator accurately captures short-time trajectories and long-time statistical behavior. Using this framework, we are able to predict various statistics of the invariant measure for the turbulent Kolmogorov Flow dynamics with Reynolds numbers up to 5000.Zongyi Li, Miguel Liu-Schiaffini, Nikola Kovachki, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumarwork_fvemotnu4ncstfrs3emw7ftlsmWed, 28 Sep 2022 00:00:00 GMTAn Algebra of Observables for de Sitter Space
https://scholar.archive.org/work/ko2bh23banczjmyjwp2zanuex4
We describe an algebra of observables for a static patch in de Sitter space, with operators gravitationally dressed to the worldline of an observer. The algebra is a von Neumann algebra of Type II_1. There is a natural notion of entropy for a state of such an algebra. There is a maximum entropy state, which corresponds to empty de Sitter space, and the entropy of any semiclassical state of the Type II_1 algebras agrees, up to an additive constant independent of the state, with the expected generalized entropy S_gen=(A/4G_N)+S_out. An arbitrary additive constant is present because of the renormalization that is involved in defining entropy for a Type II_1 algebra.Venkatesa Chandrasekaran, Roberto Longo, Geoff Penington, Edward Wittenwork_ko2bh23banczjmyjwp2zanuex4Wed, 28 Sep 2022 00:00:00 GMTDynamical comparison and 𝒵-stability for crossed products of simple C^*-algebras
https://scholar.archive.org/work/jmfprssvxzekdkxz6nnbt2cfhq
We establish 𝒵-stability for crossed products of outer actions of amenable groups on 𝒵-stable C^*-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain such result using a weak form of dynamical comparison, which we verify in great generality. We complement our results by proving that McDuffness with respect to invariant traces is automatic in many cases of interest. This is the case, for instance, for every action of an amenable group G on a classifiable C^*-algebra A whose trace space T(A) is a Bauer simplex with finite dimensional boundary ∂_e T(A), and such that the induced action G↷∂_eT(A) is free. If G = ℤ^d and the action G↷∂_eT(A) is free and minimal, then we obtain McDuffness with respect to invariant traces, and thus 𝒵-stability of the corresponding crossed product, also in case ∂_e T(A) has infinite covering dimension.Eusebio Gardella, Shirly Geffen, Petr Naryshkin, Andrea Vaccarowork_jmfprssvxzekdkxz6nnbt2cfhqWed, 28 Sep 2022 00:00:00 GMTFrom thin plates to Ahmed bodies: linear and weakly non-linear stability of rectangular prisms
https://scholar.archive.org/work/s25giw77j5dwnmnwvpm6zaggre
We study the stability of laminar wakes past three-dimensional rectangular prisms. The width-to-height ratio is set to W/H=1.2, while the length-to-height ratio 1/6<L/H<3 covers a wide range of geometries from thin plates to elongated Ahmed bodies. First, global linear stability analysis yields a series of pitchfork and Hopf bifurcations: (i) at lower Reynolds numbers Re, two stationary modes, A and B, become unstable, breaking the top/bottom and left/right planar symmetries, respectively; (ii) at larger Re, two oscillatory modes become unstable and, again, each mode breaks one of the two symmetries. The critical Re of these four modes increase with L/H, qualitatively reproducing the trend of stationary and oscillatory bifurcations in axisymmetric wakes (e.g. thin disk, sphere and bullet-shaped bodies). Next, a weakly non-linear analysis based on the two stationary modes A and B yields coupled amplitude equations. For Ahmed bodies, as Re increases state (A,0) appears first, followed by state (0,B). While there is a range of bistability of those two states, only (0,B) remains stable at larger Re, similar to the static wake deflection (across the larger base dimension) observed in the turbulent regime. The bifurcation sequence, including bistability and hysteresis, is validated with fully non-linear direct numerical simulations, and is shown to be robust to variations in W and L in the range of common Ahmed bodies.G. A. Zampogna, E. Boujowork_s25giw77j5dwnmnwvpm6zaggreWed, 28 Sep 2022 00:00:00 GMTDual braided quantum E(2) groups
https://scholar.archive.org/work/d3rw7d2by5dtjkqxpscwl2btt4
An explicit construction of the braided dual of quantum E(2) groups is described over the circle group 𝕋 with respect to a specific R-matrix R. Additionally, the corresponding bosonization is also described.Atibur Rahamanwork_d3rw7d2by5dtjkqxpscwl2btt4Wed, 28 Sep 2022 00:00:00 GMTA Hochschild-Kostant-Rosenberg theorem and residue sequences for logarithmic Hochschild homology
https://scholar.archive.org/work/r2rmazkmbnen7iv2ctjaiimd3m
This paper incorporates the theory of Hochschild homology into our program on log motives. We discuss a geometric definition of logarithmic Hochschild homology of derived pre-log rings and construct an Andr\'e-Quillen type spectral sequence. The latter degenerates for derived log smooth maps between discrete pre-log rings. We employ this to show a logarithmic version of the Hochschild-Kostant-Rosenberg theorem and that logarithmic Hochschild homology is representable in the category of log motives. Among the applications, we deduce a generalized residue sequence involving blow-ups of log schemes.Federico Binda, Tommy Lundemo, Doosung Park, Paul Arne Østværwork_r2rmazkmbnen7iv2ctjaiimd3mWed, 28 Sep 2022 00:00:00 GMTArtificial Intelligence and Advanced Materials
https://scholar.archive.org/work/tkf566mg6zf77a7xan6anloxvu
Artificial intelligence is gaining strength and materials science can both contribute to and profit from it. In a simultaneous progress race, new materials, systems and processes can be devised and optimized thanks to machine learning techniques and such progress can be turned into in-novative computing platforms. Future materials scientists will profit from understanding how machine learning can boost the conception of advanced materials. This review covers aspects of computation from the fundamentals to directions taken and repercussions produced by compu-tation to account for the origins, procedures and applications of artificial intelligence. Machine learning and its methods are reviewed to provide basic knowledge on its implementation and its potential. The materials and systems used to implement artificial intelligence with electric charges are finding serious competition from other information carrying and processing agents. The impact these techniques are having on the inception of new advanced materials is so deep that a new paradigm is developing where implicit knowledge is being mined to conceive materi-als and systems for functions instead of finding applications to found materials. How far this trend can be carried is hard to fathom as exemplified by the power to discover unheard of mate-rials or physical laws buried in data.Cefe Lópezwork_tkf566mg6zf77a7xan6anloxvuWed, 28 Sep 2022 00:00:00 GMT