IA Scholar Query: Hausdorff Dimension in Exponential Time.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 30 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Homologically area-minimizing surfaces with non-smoothable singularities
https://scholar.archive.org/work/dxh4mmpjyzekzo4czg2ifh2wfa
Let dimensions d≥ 3, and codimensions c≥ 3 be positive integers. Set the exceptional set E={3,4}. For d∉E,c arbitrary or d∈ E,c≤ d, we prove that for every d-dimensional integral homology class [Σ] on a compact (not necessarily orientable) d+c-dimensional smooth manifold M, there exist open sets Ω_[Σ] in the space of smooth Riemannian metrics so that all area-minimizing integral currents in [Σ] are singular for metrics in Ω_[Σ]. This settles a conjecture of White about the generic regularity of area-minimizing surfaces. The answer is sharp dimension-wise. As a byproduct, we determine the moduli space of area-minimizing currents near any area-minimizing transverse immersion of dimension d≥ 3 and codimension c≥ 3 satisfying an angle condition of asymptotically sharp order in d. Similar conclusions hold for mod 2 area-minimizing surfaces.Zhenhua Liuwork_dxh4mmpjyzekzo4czg2ifh2wfaWed, 30 Nov 2022 00:00:00 GMTPartial hyperbolicity and pseudo-Anosov dynamics
https://scholar.archive.org/work/i2rrut2zdjct5e4wyph4fjnvve
We show that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism then it also admits an Anosov flow. Moreover, we give a complete classification of partially hyperbolic diffeomorphism in hyperbolic 3-manifolds as well as partially hyperbolic diffeomorphisms in Seifert manifolds inducing pseudo-Anosov dynamics in the base. This classification is given in terms of the structure of their center (branching) foliations and the notion of collapsed Anosov flows.Sergio R. Fenley, Rafael Potriework_i2rrut2zdjct5e4wyph4fjnvveWed, 30 Nov 2022 00:00:00 GMTEvery finite graph arises as the singular set of a compact 3-d calibrated area minimizing surface
https://scholar.archive.org/work/2z6vjfkzzrfstj4f33sohbdlte
Given any (not necessarily connected) combinatorial finite graph and any compact smooth 6-manifold M^6 with the third Betti number b_3≠0, we construct a calibrated 3-dimensional homologically area minimizing surface on M equipped in a smooth metric g, so that the singular set of the surface is precisely an embedding of this finite graph. Moreover, the calibration form near the singular set is a smoothly GL(6,ℝ) twisted special Lagrangian form. The constructions are based on some unpublished ideas of Professor Camillo De Lellis and Professor Robert Bryant.Zhenhua Liuwork_2z6vjfkzzrfstj4f33sohbdlteWed, 30 Nov 2022 00:00:00 GMTFurstenberg sumset conjecture and Mandelbrot percolations
https://scholar.archive.org/work/aopn4d53arh5razp2uqz453b4i
In this paper we extend Hochman and Shmerkin's projection theorem to product measures of Mandelbrot cascades acting on ergodic measures imaged through canonical mappings of one-dimensional iterated function systems without any separation conditions. Consequently we extend Furstenberg sumset theorem to images of subshifts on symbolic spaces, and to Mandelbrot percolations on invariant sets.Catherine Bruce, Xiong Jinwork_aopn4d53arh5razp2uqz453b4iTue, 29 Nov 2022 00:00:00 GMTRegularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity
https://scholar.archive.org/work/7kjmukj4qzfdzo7zt2j73yezbm
We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm-Loewner evolution (SLE): variation regularity, modulus of continuity, and law of the iterated logarithm. For the latter two we consider the SLE with its natural parametrization. More precisely, denoting by d∈(0,2] the dimension of the curve, we show the following. 1. The optimal ψ-variation is ψ(x)=x^d(loglog x^-1)^-(d-1) in the sense that η is of finite ψ-variation for this ψ and not for any function decaying more slowly as x ↓ 0. 2. The optimal modulus of continuity is ω(s) = c s^1/d(log s^-1)^1-1/d, i.e. |η(t)-η(s)| ≤ω(t-s) for this ω, and not for any function decaying faster as s ↓ 0. 3. lim sup_t↓ 0 |η(t)| (t^1/d(loglog t^-1)^1-1/d)^-1 is a deterministic constant in (0,∞).Nina Holden, Yizheng Yuanwork_7kjmukj4qzfdzo7zt2j73yezbmTue, 29 Nov 2022 00:00:00 GMTSingular Riemannian foliations and ℐ-Poisson manifolds
https://scholar.archive.org/work/yehk4pguojavhefhrdhhj2vjoa
We recall the notion of a singular foliation (SF) on a manifold M, viewed as an appropriate submodule of 𝔛(M), and adapt it to the presence of a Riemannian metric g, yielding a module version of a singular Riemannian foliation (SRF). Following Garmendia-Zambon on Hausdorff Morita equivalence of SFs, we define the Morita equivalence of SRFs (both in the module sense as well as in the more traditional geometric one of Molino) and show that the leaf spaces of Morita equivalent SRFs are isomrophic as pseudo-metric spaces. In a second part, we introduce the category of ℐ-Poisson manifolds. Its objects and morphisms generalize Poisson manifolds and morphisms in the presence of appropriate ideals ℐ of the smooth functions on the manifold such that two conditions are satisfied: (i) The category of Poisson manifolds becomes a full subcategory when choosing ℐ=0 and (ii) there is a reduction functor from this new category to the category of Poisson algebras, which generalizes coistropic reduction to the singular setting. Every SF on M gives rise to an ℐ-Poisson manifold on T^*M and g enhances this to an SRF if and only if the induced Hamiltonian lies in the normalizer of ℐ. This perspective provides, on the one hand, a simple proof of the fact that every module SRF is a geometric SRF and, on the other hand, a construction of an algebraic invariant of singular foliations: Hausdorff Morita equivalent SFs have isomorphic reduced Poisson algebras.Hadi Nahari, Thomas Stroblwork_yehk4pguojavhefhrdhhj2vjoaTue, 29 Nov 2022 00:00:00 GMTFrom CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators
https://scholar.archive.org/work/mctin6urpfenxna7fclddwfbem
We probe the contraction from 2d relativistic CFTs to theories with Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll symmetries, using diagnostics of quantum chaos. Starting from an Ultrarelativistic limit on a relativistic scalar field theory and following through at the quantum level using an oscillator representation of states, one can show the CFT_2 vacuum evolves smoothly into a BMS_3 vacuum in the form of a squeezed state. Computing circuit complexity of this transmutation using the covariance matrix approach shows clear divergences when the BMS point is hit or equivalently when the target state becomes a boundary state. We also find similar behaviour of the circuit complexity calculated from methods of information geometry. Furthermore, we discuss the hamiltonian evolution of the system and investigate Out-of-time-ordered correlators (OTOCs) and operator growth complexity, both of which turn out to scale polynomially with time at the BMS point.Aritra Banerjee, Arpan Bhattacharyya, Priya Drashni, Srinidhi Pawarwork_mctin6urpfenxna7fclddwfbemTue, 29 Nov 2022 00:00:00 GMTGeneralisable 3D Fabric Architecture for Streamlined Universal Multi-Dataset Medical Image Segmentation
https://scholar.archive.org/work/5tt7yvwbc5f5djuc66cuommlfy
Data scarcity is common in deep learning models for medical image segmentation. Previous works proposed multi-dataset learning, either simultaneously or via transfer learning to expand training sets. However, medical image datasets have diverse-sized images and features, and developing a model simultaneously for multiple datasets is challenging. This work proposes Fabric Image Representation Encoding Network (FIRENet), a universal architecture for simultaneous multi-dataset segmentation and transfer learning involving arbitrary numbers of dataset(s). To handle different-sized image and feature, a 3D fabric module is used to encapsulate many multi-scale sub-architectures. An optimal combination of these sub-architectures can be implicitly learnt to best suit the target dataset(s). For diverse-scale feature extraction, a 3D extension of atrous spatial pyramid pooling (ASPP3D) is used in each fabric node for a fine-grained coverage of rich-scale image features. In the first experiment, FIRENet performed 3D universal bone segmentation of multiple musculoskeletal datasets of the human knee, shoulder and hip joints and exhibited excellent simultaneous multi-dataset segmentation performance. When tested for transfer learning, FIRENet further exhibited excellent single dataset performance (when pre-training on a prostate dataset), as well as significantly improved universal bone segmentation performance. The following experiment involves the simultaneous segmentation of the 10 Medical Segmentation Decathlon (MSD) challenge datasets. FIRENet demonstrated good multi-dataset segmentation results and inter-dataset adaptability of highly diverse image sizes. In both experiments, FIRENet's streamlined multi-dataset learning with one unified network that requires no hyper-parameter tuning.Siyu Liu, Wei Dai, Craig Engstrom, Jurgen Fripp, Stuart Crozier, Jason A. Dowling, Shekhar S. Chandrawork_5tt7yvwbc5f5djuc66cuommlfyTue, 29 Nov 2022 00:00:00 GMTWasserstein Stability for Persistence Diagrams
https://scholar.archive.org/work/z3jcni4xirdelcavaulfmyfmy4
The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the ∞-norm of perturbations. This has two main implications: it makes the space of persistence diagrams rather pathological and it is often provides very pessimistic bounds with respect to outliers. In this paper, we provide new stability results with respect to the p-Wasserstein distance between persistence diagrams. This includes an elementary proof for the setting of functions on sufficiently finite spaces in terms of the p-norm of the perturbations, along with an algebraic framework for p-Wasserstein distance which extends the results to wider class of modules. We also provide apply the results to a wide range of applications in topological data analysis (TDA) including topological summaries, persistence transforms and the special but important case of Vietoris-Rips complexes.Primoz Skraba, Katharine Turnerwork_z3jcni4xirdelcavaulfmyfmy4Tue, 29 Nov 2022 00:00:00 GMTCovariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems
https://scholar.archive.org/work/l2zqq6ice5herobiksi76u3se4
Dynamical vectors characterizing instability and applicable as ensemble perturbations for prediction with geophysical fluid dynamical models are analysed. The relationships between covariant Lyapunov vectors (CLVs), orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectors and finite-time normal modes (FTNMs) are examined for periodic and aperiodic systems. In the phase-space of FTNM coefficients, SVs are found to equate with unit norm FTNMs at certain times. In the long-time limit, when SVs approach OLVs, the Oseledec theorem and the relationships between OLVs and CLVs are used to connect CLVs to FTNMs in this phase-space. The covariant properties of both the CLVs, and the FTNMs, together with their phase-space independence, and the norm independence of global Lyapunov exponents and FTNM growth rates, establishes their asymptotic convergence. Conditions on the dynamical systems for the validity of these results, particularly ergodicity, boundedness and non-singular FTNM characteristic matrix and propagator, are documented. Systems with nondegenerate OLVs, and with degenerate Lyapunov spectrum as is the rule in the presence of waves such as Rossby waves, are examined, and efficient numerical methods for the calculation of leading CLVs proposed. Norm independent finite-time versions of the Kolmogorov-Sinai entropy production and Kaplan-Yorke dimension are presented.Jorgen S Frederiksenwork_l2zqq6ice5herobiksi76u3se4Tue, 29 Nov 2022 00:00:00 GMTOptimistic search: Change point estimation for large-scale data via adaptive logarithmic queries
https://scholar.archive.org/work/acjxbwmddncnhkbot76e6v6hbi
Change point estimation is often formulated as a search for the maximum of a gain function describing improved fits when segmenting the data. Searching through all candidates requires O(n) evaluations of the gain function for an interval with n observations. If each evaluation is computationally demanding (e.g. in high-dimensional models), this can become infeasible. Instead, we propose optimistic search methods with O(log n) evaluations exploiting specific structure of the gain function. Towards solid understanding of our strategy, we investigate in detail the p-dimensional Gaussian changing means setup, including high-dimensional scenarios. For some of our proposals, we prove asymptotic minimax optimality for detecting change points and derive their asymptotic localization rate. These rates (up to a possible log factor) are optimal for the univariate and multivariate scenarios, and are by far the fastest in the literature under the weakest possible detection condition on the signal-to-noise ratio in the high-dimensional scenario. Computationally, our proposed methodology has the worst case complexity of O(np), which can be improved to be sublinear in n if some a-priori knowledge on the length of the shortest segment is available. Our search strategies generalize far beyond the theoretically analyzed setup. We illustrate, as an example, massive computational speedup in change point detection for high-dimensional Gaussian graphical models.Solt Kovács, Housen Li, Lorenz Haubner, Axel Munk, Peter Bühlmannwork_acjxbwmddncnhkbot76e6v6hbiTue, 29 Nov 2022 00:00:00 GMTOn the rôle of singular functions in extending the probabilistic symbol to its most general class
https://scholar.archive.org/work/bw7vfmkycbdh3i4wam3oabmmay
The probabilistic symbol is the right-hand side derivative of the characteristic functions corresponding to the one-dimensional marginals of a stochastic process. This object, as long as the derivative exists, provides crucial information concerning the stochastic process. For a L\'evy process, one obtains the characteristic exponent while the symbol of a (rich) Feller process coincides with the classical symbol which is well known from the theory of pseudodifferential operators. Leaving these classes behind, the most general class of processes for which the symbol still exists are L\'evy-type processes. It has been an open question, whether further generalizations are possible within the framework of Markov processes. We answer this question in the present article: within the class of Hunt semimartingales, L\'evy-type processes are exactly those for which the probabilistic symbol exists. Leaving quasi-continuity behind, one can construct processes admitting a symbol. However, we show, that the applicability of the symbol might be lost for these processes. Surprisingly, in our proofs the upper and lower Dini derivatives corresponding to certain singular functions play an important r\^ole.Sebastian Rickelhoff, Alexander Schnurrwork_bw7vfmkycbdh3i4wam3oabmmayMon, 28 Nov 2022 00:00:00 GMTVolume and heat kernel fluctuations for the three-dimensional uniform spanning tree
https://scholar.archive.org/work/7yddglbrvzdczpgylhvfzjbbbu
Let 𝒰 be the uniform spanning tree on ℤ^3. We show the occurrence of log-logarithmic fluctuations around the leading order for the volume of intrinsic balls in 𝒰. As an application, we obtain similar fluctuations for the quenched heat kernel of the simple random walk on 𝒰.Daisuke Shiraishi, Satomi Watanabework_7yddglbrvzdczpgylhvfzjbbbuMon, 28 Nov 2022 00:00:00 GMTFrom Markov Processes to Semimartingales
https://scholar.archive.org/work/adwocf3y3jal7j7xl7k2csusdy
In the development of stochastic integration and the theory of semimartingales, Markov processes have been a constant source of inspiration. Despite this historical interweaving, it turned out that semimartingales should be considered the 'natural' class of processes for many concepts first developed in the Markovian framework. As an example, stochastic differential equations have been invented as a tool to study Markov processes but nowadays are treated separately in the literature. Moreover, the killing of processes has been known for decades before it made its way to the theory of semimartingales most recently. We describe, when these and other important concepts have been invented in the theory of Markov processes and how they were transferred to semimartingales. Further topics include the symbol, characteristics and generalizations of Blumenthal-Getoor indices. Some additional comments on relations between Markov processes and semimartingales round out the paper.Sebastian Rickelhoff, Alexander Schnurrwork_adwocf3y3jal7j7xl7k2csusdyMon, 28 Nov 2022 00:00:00 GMTOn the coming down from infinity of coalescing Brownian motions
https://scholar.archive.org/work/mdnskd6in5ezjfupzb4ycmxoze
Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time. Assume that there are infinitely many initial particles in the system. We give a necessary and sufficient condition for the number of particles to come down from infinity. We also identify the rate of this coming down from infinity for different initial configurations.Clayton Barnes, Leonid Mytnik, Zhenyao Sunwork_mdnskd6in5ezjfupzb4ycmxozeMon, 28 Nov 2022 00:00:00 GMTRandom clusters in the Villain and XY models
https://scholar.archive.org/work/e4num7ranrhyrhglz4wz2er6y4
In the Ising and Potts model, random cluster representations provide a geometric interpretation to spin correlations. We discuss similar constructions for the Villain and XY models, where spins take values in the circle, as well as extensions to models with different single site spin spaces. In the Villain case, we highlight natural interpretation in terms of the cable system extension of the model. We also list questions and open problems on the cluster representations obtained in this fashion.Julien Dubédat, Hugo Falconetwork_e4num7ranrhyrhglz4wz2er6y4Mon, 28 Nov 2022 00:00:00 GMTOpen Source Variational Quantum Eigensolver Extension of the Quantum Learning Machine (QLM) for Quantum Chemistry
https://scholar.archive.org/work/774zvmkbbvcxxbd3iknlxytuky
Quantum Chemistry (QC) is one of the most promising applications of Quantum Computing. However, present quantum processing units (QPUs) are still subject to large errors. Therefore, noisy intermediate-scale quantum (NISQ) hardware is limited in terms of qubits counts and circuit depths. Specific algorithms such as Variational Quantum Eigensolvers (VQEs) can potentially overcome such issues. We introduce here a novel open-source QC package, denoted Open-VQE, providing tools for using and developing chemically-inspired adaptive methods derived from Unitary Coupled Cluster (UCC). It facilitates the development and testing of VQE algorithms. It is able to use the Atos Quantum Learning Machine (QLM), a general quantum programming framework enabling to write, optimize and simulate quantum computing programs. Along with Open-VQE, we introduce myQLM-Fermion, a new open-source module (that includes the key QLM ressources that are important for QC developments (fermionic second quantization tools etc...). The Open-VQE package extends therefore QLM to QC providing: (i) the functions to generate the different types of excitations beyond the commonly used UCCSD ansätz;(ii) a new implementation of the "adaptive derivative assembled pseudo-Trotter method" (ADAPT-VQE), written in simple class structure python codes. Interoperability with other major quantum programming frameworks is ensured thanks to myQLM, which allows users to easily build their own code and execute it on existing QPUs. The combined Open-VQE/myQLM-Fermion quantum simulator facilitates the implementation, tests and developments of variational quantum algorithms towards choosing the best compromise to run QC computations on present quantum computers while offering the possibility to test large molecules. We provide extensive benchmarks for several molecules associated to qubit counts ranging from 4 up to 24.Mohammad Haidar, Marko J. Rančić, Thomas Ayral, Yvon Maday, Jean-Philip Piquemalwork_774zvmkbbvcxxbd3iknlxytukyMon, 28 Nov 2022 00:00:00 GMTThe Christoffel-Darboux kernel for topological data analysis
https://scholar.archive.org/work/uzdch5n2onaj5ba7l4t3hfk6oa
Persistent homology has been widely used to study the topology of point clouds in ℝ^n. Standard approaches are very sensitive to outliers, and their computational complexity depends badly on the number of data points. In this paper we introduce a novel persistence module for a point cloud using the theory of Christoffel-Darboux kernels. This module is robust to (statistical) outliers in the data, and can be computed in time linear in the number of data points. We illustrate the benefits and limitations of our new module with various numerical examples in ℝ^n, for n=1, 2, 3. Our work expands upon recent applications of Christoffel-Darboux kernels in the context of statistical data analysis and geometric inference (Lasserre, Pauwels and Putinar, 2022). There, these kernels are used to construct a polynomial whose level sets capture the geometry of a point cloud in a precise sense. We show that the persistent homology associated to the sublevel set filtration of this polynomial is stable with respect to the Wasserstein distance. Moreover, we show that the persistent homology of this filtration can be computed in singly exponential time in the ambient dimension n, using a recent algorithm of Basu Karisani (2022).Pepijn Roos Hoefgeest, Lucas Slotwork_uzdch5n2onaj5ba7l4t3hfk6oaMon, 28 Nov 2022 00:00:00 GMTSingular Weyl's law with Ricci curvature bounded below
https://scholar.archive.org/work/b4wd27n6wvafrdbtlqamsjvnay
We establish two surprising types of Weyl's laws for some compact RCD(K, N)/Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for RCD(K,N) spaces. Our results depends crucially on analyzing and developing important properties of the examples constructed by the last two authors, showing them isometric to the α-Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures by Cheeger-Colding and by Kapovitch-Kell-Ketterer.Xianzhe Dai, Shouhei Honda, Jiayin Pan, Guofang Weiwork_b4wd27n6wvafrdbtlqamsjvnaySun, 27 Nov 2022 00:00:00 GMTA graph discretized approximation of semigroups for diffusion with drift and killing on a complete Riemannian manifold
https://scholar.archive.org/work/ayrud35k4rc5dd4wb76vp4gznq
In the present paper, we prove that the contraction C_0-semigroup generated by a Schrödinger operator with drift on a complete Riemannian manifold is approximated by the discrete semigroups associated with a family of discrete time random walks with killing in a flow on a sequence of proximity graphs, which are constructed by partitions of the manifold. Furthermore, when the manifold is compact, we also obtain a quantitative error estimate of the convergence. Finally, we give examples of the partition of the manifold and the drift term on two typical manifolds: Euclidean spaces and model manifolds.Satoshi Ishiwata, Hiroshi Kawabiwork_ayrud35k4rc5dd4wb76vp4gznqSat, 26 Nov 2022 00:00:00 GMT