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arXivOn the SchmidtKolchin conjecture on differentially homogeneous polynomials. Applications to (twisted) jet differentials on projective spaces Antoine Etesse 20230206 The main goal of this paper is to prove the SchmidtKolchin conjecture. This conjecture says the following: the vector space of degree \(d\) differentially homogeneous polynomials in \((N+1)\) variables is of dimension \((N+1)^{d}\). Next, we establish a onetoone correspondance between differentially homogeneous polynomials in \((N+1)\) variables, and twisted jet differentials on projective spaces. As a byproduct of our study of differentially homogeneous polynomials, we are thus able to understand explicitly twisted jet differentials on projective spaces.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2302.02634view Article Title On the SchmidtKolchin conjecture on differentially homogeneous polynomials. Applications to (twisted) jet differentials on projective spacesAuthors Antoine EtesseAbstract The main goal of this paper is to prove the SchmidtKolchin conjecture. This conjecture says the following: the vector space of degree \(d\) differentially homogeneous polynomials in \((N+1)\) variables is of dimension \((N+1)^{d}\). Next, we establish a onetoone correspondance between differentially homogeneous polynomials in \((N+1)\) variables, and twisted jet differentials on projective spaces. As a byproduct of our study of differentially homogeneous polynomials, we are thus able to understand explicitly twisted jet differentials on projective spaces.Is Part Of 20230206 Identifier ISSN: Category math.AG math.AP math.CV math.RTLicense 
arXivReversible random number generation for adjoint Monte Carlo simulation of the heat equation Emil Løvbak, Frédéric Blondeel, Adam Lee, ... more(6) 20230206 In PDEconstrained optimization, one aims to find design parameters that minimize some objective, subject to the satisfaction of a partial differential equation. A major challenges is computing gradients of the objective to the design parameters, as applying the chain rule requires computing the Jacobian of the design parameters to the PDE's state. The adjoint method avoids this Jacobian by computing partial derivatives of a Lagrangian. Evaluating these derivatives requires the solution of a second PDE with the adjoint differential operator to the constraint, resulting in a backwardsintime simulation. Particlebased Monte Carlo solvers are often used to compute the solution to highdimensional PDEs. However, such solvers have the drawback of introducing noise to the computed results, thus requiring stochastic optimization methods. To guarantee convergence in this setting, both the constraint and adjoint Monte Carlo simulations should simulate the same particle trajectories. For large simulations, storing full paths from the constraint equation for reuse in the adjoint equation becomes infeasible due to memory limitations. In this paper, we provide a reversible extension to the family of permuted congruential pseudorandom number generators (PCG). We then use such a generator to recompute these timereversed paths for the heat equation, avoiding these memory issues.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2302.02778view Article Title Reversible random number generation for adjoint Monte Carlo simulation of the heat equationAuthors Emil Løvbak; Frédéric Blondeel; Adam Lee; Lander Vanroye; Andreas Van Barel; Giovanni SamaeyAbstract In PDEconstrained optimization, one aims to find design parameters that minimize some objective, subject to the satisfaction of a partial differential equation. A major challenges is computing gradients of the objective to the design parameters, as applying the chain rule requires computing the Jacobian of the design parameters to the PDE's state. The adjoint method avoids this Jacobian by computing partial derivatives of a Lagrangian. Evaluating these derivatives requires the solution of a second PDE with the adjoint differential operator to the constraint, resulting in a backwardsintime simulation. Particlebased Monte Carlo solvers are often used to compute the solution to highdimensional PDEs. However, such solvers have the drawback of introducing noise to the computed results, thus requiring stochastic optimization methods. To guarantee convergence in this setting, both the constraint and adjoint Monte Carlo simulations should simulate the same particle trajectories. For large simulations, storing full paths from the constraint equation for reuse in the adjoint equation becomes infeasible due to memory limitations. In this paper, we provide a reversible extension to the family of permuted congruential pseudorandom number generators (PCG). We then use such a generator to recompute these timereversed paths for the heat equation, avoiding these memory issues.Is Part Of 20230206 Identifier ISSN: Category math.NA cs.NALicense 
arXivContact problems in porous media L. Banz, F. Bertrand 20230206 The Biot problem of poroelasticity is extended by Signorini contact conditions. The resulting Biot contact problem is formulated and analyzed as a two field variational inequality problem of a perturbed saddle point structure. We present an a priori error analysis for a general as well as for a $hp$FE discretization including convergence and guaranteed convergence rates for the latter. Moreover, we derive a family of reliable and efficient residual based a posteriori error estimators, and elaborate how a simple and efficient primaldual active set solver can be applied to solve the discrete Galerkin problem. Numerical results underline our theoretical finding and show that optimal, in particular exponential, convergence rates can be achieved by adaptive schemes for two dimensional problems.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2302.02600view Article Title Contact problems in porous mediaAuthors L. Banz; F. BertrandAbstract The Biot problem of poroelasticity is extended by Signorini contact conditions. The resulting Biot contact problem is formulated and analyzed as a two field variational inequality problem of a perturbed saddle point structure. We present an a priori error analysis for a general as well as for a $hp$FE discretization including convergence and guaranteed convergence rates for the latter. Moreover, we derive a family of reliable and efficient residual based a posteriori error estimators, and elaborate how a simple and efficient primaldual active set solver can be applied to solve the discrete Galerkin problem. Numerical results underline our theoretical finding and show that optimal, in particular exponential, convergence rates can be achieved by adaptive schemes for two dimensional problems.Is Part Of 20230206 Identifier ISSN: Category math.NA cs.NALicense 
arXivThe SSL Interplay: Augmentations, Inductive Bias, and Generalization Vivien Cabannes, Bobak T. Kiani, Randall Balestriero, ... more(5) 20230206 Selfsupervised learning (SSL) has emerged as a powerful framework to learn representations from raw data without supervision. Yet in practice, engineers face issues such as instability in tuning optimizers and collapse of representations during training. Such challenges motivate the need for a theory to shed light on the complex interplay between the choice of data augmentation, network architecture, and training algorithm. We study such an interplay with a precise analysis of generalization performance on both pretraining and downstream tasks in a theory friendly setup, and highlight several insights for SSL practitioners that arise from our theory.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2302.02774view Article Title The SSL Interplay: Augmentations, Inductive Bias, and GeneralizationAuthors Vivien Cabannes; Bobak T. Kiani; Randall Balestriero; Yann LeCun; Alberto BiettiAbstract Selfsupervised learning (SSL) has emerged as a powerful framework to learn representations from raw data without supervision. Yet in practice, engineers face issues such as instability in tuning optimizers and collapse of representations during training. Such challenges motivate the need for a theory to shed light on the complex interplay between the choice of data augmentation, network architecture, and training algorithm. We study such an interplay with a precise analysis of generalization performance on both pretraining and downstream tasks in a theory friendly setup, and highlight several insights for SSL practitioners that arise from our theory.Is Part Of 20230206 Identifier ISSN: Category stat.ML cs.AI cs.LG math.ST stat.THLicense 
arXivThe semiclassical limit of a quantum Zeno dynamics Fabio Deelan Cunden, Paolo Facchi, Marilena Ligabò... more(3) 20230206 Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics setting, we study the asymptotics of a family of symbols corresponding to a truncated momentum operator, in the semiclassical limit of vanishing Planck's constant $\hbar\to0$ and large quantum number $N\to\infty$, with $\hbar N$ kept fixed. In a certain topology, the limit is the discontinuous symbol $p\chi_D(x,p)$ where $\chi_D$ is the characteristic function of the classically permitted region $D$ in the phase space. A refined analysis shows that the symbol is asymptotically close to a smooth function $p\chi_D^{(N)}(x,p)$, where $\chi_D^{(N)}$ is a smooth version of $\chi_D$ related to the integrated Airy function. We also discuss the limit from the dynamical point of view.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2302.02673view Article Title The semiclassical limit of a quantum Zeno dynamicsAuthors Fabio Deelan Cunden; Paolo Facchi; Marilena LigabòAbstract Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics setting, we study the asymptotics of a family of symbols corresponding to a truncated momentum operator, in the semiclassical limit of vanishing Planck's constant $\hbar\to0$ and large quantum number $N\to\infty$, with $\hbar N$ kept fixed. In a certain topology, the limit is the discontinuous symbol $p\chi_D(x,p)$ where $\chi_D$ is the characteristic function of the classically permitted region $D$ in the phase space. A refined analysis shows that the symbol is asymptotically close to a smooth function $p\chi_D^{(N)}(x,p)$, where $\chi_D^{(N)}$ is a smooth version of $\chi_D$ related to the integrated Airy function. We also discuss the limit from the dynamical point of view.Is Part Of 20230206 Identifier ISSN: Category mathph math.CA math.MP quantphLicense 
arXivNonholonomic systems with inequality constraints Alexandre Anahory Simoes, Leonardo Colombo 20230206 In this paper we derive the equations of motion for nonholonomic systems subject to inequality constraints, both, in continuoustime and discretetime. The last is done by discretizing the continuous timevariational principle which defined the equations of motion for a nonholonomic system subject to inequality constraints. An example is show to illustrate the theoretical results.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2302.02616view Article Title Nonholonomic systems with inequality constraintsAuthors Alexandre Anahory Simoes; Leonardo ColomboAbstract In this paper we derive the equations of motion for nonholonomic systems subject to inequality constraints, both, in continuoustime and discretetime. The last is done by discretizing the continuous timevariational principle which defined the equations of motion for a nonholonomic system subject to inequality constraints. An example is show to illustrate the theoretical results.Is Part Of 20230206 Identifier ISSN: Category math.OC mathph math.MPLicense 
arXivJacobianfree implicit MDRK methods for stiff systems of ODEs Jeremy Chouchoulis, Jochen Schütz 20230206 In this work, an approximate family of implicit multiderivative RungeKutta (MDRK) time integrators for stiff initial value problems is presented. The approximation procedure is based on the recent Approximate Implicit Taylor method (Baeza et al. in Comput. Appl. Math. 39:304, 2020). As a Taylor method can be written in MDRK format, the novel family constitutes a multistage generalization. Two different alternatives are investigated for the computation of the higher order derivatives: either directly as part of the stage equation, or either as a separate formula for each derivative added on top of the stage equation itself. From linearizing through Newton's method, it turns out that the conditioning of the Newton matrix behaves significantly different for both cases. We show that direct computation results in a matrix with a conditioning that is highly dependent on the stiffness, increasing exponentially in the stiffness parameter with the amount of derivatives. Adding separate formulas has a more favorable behavior, the matrix conditioning being linearly dependent on the stiffness, regardless of the amount of derivatives. Despite increasing the Newton system significantly in size, through several numerical results it is demonstrated that doing so can be considerably beneficial.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2302.02882view Article Title Jacobianfree implicit MDRK methods for stiff systems of ODEsAuthors Jeremy Chouchoulis; Jochen SchützAbstract In this work, an approximate family of implicit multiderivative RungeKutta (MDRK) time integrators for stiff initial value problems is presented. The approximation procedure is based on the recent Approximate Implicit Taylor method (Baeza et al. in Comput. Appl. Math. 39:304, 2020). As a Taylor method can be written in MDRK format, the novel family constitutes a multistage generalization. Two different alternatives are investigated for the computation of the higher order derivatives: either directly as part of the stage equation, or either as a separate formula for each derivative added on top of the stage equation itself. From linearizing through Newton's method, it turns out that the conditioning of the Newton matrix behaves significantly different for both cases. We show that direct computation results in a matrix with a conditioning that is highly dependent on the stiffness, increasing exponentially in the stiffness parameter with the amount of derivatives. Adding separate formulas has a more favorable behavior, the matrix conditioning being linearly dependent on the stiffness, regardless of the amount of derivatives. Despite increasing the Newton system significantly in size, through several numerical results it is demonstrated that doing so can be considerably beneficial.Is Part Of 20230206 Identifier ISSN: Category math.NA cs.NALicense 
arXivNarasimhanRamanan branes and wobbly Higgs bundles Emilio Franco, Peter B. Gothen, André G. Oliveira, ... more(4) 20230206 NarasimhanRamanan branes were introduced by the authors in a previous article. They consist of a family of $BBB$branes inside the moduli space of Higgs bundles, and a family of complex Lagrangian subvarieties. It was conjectured that these complex Lagrangian subvarieties support the $BAA$branes that are mirror dual to the NarasimhanRamanan $BBB$branes. In this article we show that the support of these branes intersects nontrivially the locus of wobbly Higgs bundles.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2302.02736view Article Title NarasimhanRamanan branes and wobbly Higgs bundlesAuthors Emilio Franco; Peter B. Gothen; André G. Oliveira; Ana PeónNietoAbstract NarasimhanRamanan branes were introduced by the authors in a previous article. They consist of a family of $BBB$branes inside the moduli space of Higgs bundles, and a family of complex Lagrangian subvarieties. It was conjectured that these complex Lagrangian subvarieties support the $BAA$branes that are mirror dual to the NarasimhanRamanan $BBB$branes. In this article we show that the support of these branes intersects nontrivially the locus of wobbly Higgs bundles.Is Part Of 20230206 Identifier ISSN: Category math.AGLicense 
arXivCodes for Correcting $t$ LimitedMagnitude Sticky Deletions Shuche Wang, Van Khu Vu, Vincent Y. F. Tan... more(3) 20230206 Codes for correcting sticky insertions/deletions and limitedmagnitude errors have attracted significant attention due to their applications of flash memories, racetrack memories, and DNA data storage systems. In this paper, we first consider the error type of $t$sticky deletions with $\ell$limitedmagnitude and propose a nonsystematic code for correcting this type of error with redundancy $2t(11/p)\cdot\log(n+1)+O(1)$, where $p$ is the smallest prime larger than $\ell+1$. Next, we present a systematic code construction with an efficient encoding and decoding algorithm with redundancy $\frac{\lceil2t(11/p)\rceil\cdot\lceil\log p\rceil}{\log p} \log(n+1)+O(\log\log n)$, where $p$ is the smallest prime larger than $\ell+1$.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2302.02754view Article Title Codes for Correcting $t$ LimitedMagnitude Sticky DeletionsAuthors Shuche Wang; Van Khu Vu; Vincent Y. F. TanAbstract Codes for correcting sticky insertions/deletions and limitedmagnitude errors have attracted significant attention due to their applications of flash memories, racetrack memories, and DNA data storage systems. In this paper, we first consider the error type of $t$sticky deletions with $\ell$limitedmagnitude and propose a nonsystematic code for correcting this type of error with redundancy $2t(11/p)\cdot\log(n+1)+O(1)$, where $p$ is the smallest prime larger than $\ell+1$. Next, we present a systematic code construction with an efficient encoding and decoding algorithm with redundancy $\frac{\lceil2t(11/p)\rceil\cdot\lceil\log p\rceil}{\log p} \log(n+1)+O(\log\log n)$, where $p$ is the smallest prime larger than $\ell+1$.Is Part Of 20230206 Identifier ISSN: Category cs.IT math.ITLicense 
arXivKontsevich's Characteristic Classes as Topological Invariants of Configuration Space Bundles Xujia Chen 20230206 Kontsevich's characteristic classes are invariants of framed smooth fiber bundles with homology sphere fibers. It was shown by Watanabe that they can be used to distinguish smooth $S^4$bundles that are all trivial as topological fiber bundles. In this article we show that this ability of Kontsevich's classes is a manifestation of the following principle: the ``real blowup'' construction on a smooth manifold essentially depends on its smooth structure and thus, given a smooth manifold (or smooth fiber bundle) $M$, the topological invariants of spaces constructed from $M$ by real blowups could potentially differentiate smooth structures on $M$. The main theorem says that Kontsevich's characteristic classes of a smooth framed bundle $\pi$ are determined by the topology of the 2point configuration space bundle of $\pi$ and framing data.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2302.03021view Article Title Kontsevich's Characteristic Classes as Topological Invariants of Configuration Space BundlesAuthors Xujia ChenAbstract Kontsevich's characteristic classes are invariants of framed smooth fiber bundles with homology sphere fibers. It was shown by Watanabe that they can be used to distinguish smooth $S^4$bundles that are all trivial as topological fiber bundles. In this article we show that this ability of Kontsevich's classes is a manifestation of the following principle: the ``real blowup'' construction on a smooth manifold essentially depends on its smooth structure and thus, given a smooth manifold (or smooth fiber bundle) $M$, the topological invariants of spaces constructed from $M$ by real blowups could potentially differentiate smooth structures on $M$. The main theorem says that Kontsevich's characteristic classes of a smooth framed bundle $\pi$ are determined by the topology of the 2point configuration space bundle of $\pi$ and framing data.Is Part Of 20230206 Identifier ISSN: Category math.GTLicense