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46599c3a77eea6b3f9e2b454f3d22b5843710550391de0fa43938f810e0eaab4
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2026-01-21T00:00:00-05:00
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Three-chromatic geometric hypergraphs
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arXiv:2112.01820v2 Announce Type: replace Abstract: We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erd\H{o}s-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.
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https://arxiv.org/abs/2112.01820
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1292a1a7995f2eaa30cdb69c33550864ef43da44f19b929929a682dc823ab76e
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2026-01-21T00:00:00-05:00
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p-adic adelic metrics and Quadratic Chabauty I
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arXiv:2112.03873v4 Announce Type: replace Abstract: We give a new construction of $p$-adic heights on varieties over number fields using $p$-adic Arakelov theory. In analogy with Zhang's construction of real-valued heights in terms of adelic metrics, these heights are given in terms of $p$-adic adelic metrics on line bundles. In particular, we describe a construction of canonical $p$-adic heights on abelian varieties and we show that we recover the canonical Mazur--Tate height and, for Jacobians, the height constructed by Coleman and Gross. Our main application is a new and simplified approach to the Quadratic Chabauty method for the computation of rational points on certain curves over the rationals, by pulling back the canonical height on the Jacobian with respect to a carefully chosen line bundle. We show that our construction allows us to reprove, without using $p$-adic Hodge theory or arithmetic fundamental groups, several results due to Balakrishnan and Dogra. Our method also extends to primes $p$ of bad reduction. One consequence of our work is that for any canonical height ($p$-adic or $\mathbb{R}$-valued) on an abelian variety (and hence on pull-backs to other varieties), the local contribution at a finite prime $q$ can be constructed using $q$-analytic methods.
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https://arxiv.org/abs/2112.03873
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f7c1a414253a575eaff78a5bd08dc1b28bceda7dc84df513bc335844a758c991
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2026-01-21T00:00:00-05:00
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Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{\"o}dinger equation
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arXiv:2201.09809v2 Announce Type: replace Abstract: We study an inverse problem related to the dynamical Schr{\"o}dinger equation in a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linear Schr\"odinger equation possesses a trivial solution, we linearize the equation around the trivial solution. Demonstrating the well-posedness of the direct problem under appropriate conditions on initial and boundary data, it is observed that the solution admits $\eps$-expansion. By taking into account the fact that the terms $\Oh(|\nabla u(t,x)|^3)$ are negligible in this context, we shall reconstruct the time-dependent coefficients such as electric potential and vector-valued function associated with quadratic nonlinearity from the knowledge of input-output map using the geometric optics solution and Fourier inversion.
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https://arxiv.org/abs/2201.09809
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b6172b8e6882899c44e19f13f9a68b6c97ac6f8676c79909c0316e28643b5992
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2026-01-21T00:00:00-05:00
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Improvements in $L^2$ Restriction bounds for Neumann Data along closed curves
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arXiv:2203.01208v2 Announce Type: replace Abstract: We seek to improve the restriction bounds of Neumann data of Laplace eigenfunctions $u_h$ by studying the $L^2$ restriction bounds of Neumann data and their $L^2$ concentration as measured by defect measures. Let $\gamma$ be a closed smooth curve with unit exterior normal $\nu$. We can show that $\| h \partial_\nu u_{h} \|_{L^2(\Gamma)}=o(1)$ if $\{u_h\}$ is tangentially concentrated with respect to $\gamma$. As a key ingredient of the proof, we give a detailed analysis of the $L^2$ norms over $\gamma$ of the Neumann data $h\partial_\nu u_h$ when mircolocalized away the cotangential direction.
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https://arxiv.org/abs/2203.01208
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dcc33839cf9503d98073ce93b96b7b7dc0cb7ba950056b3ca324a398bf58c656
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2026-01-21T00:00:00-05:00
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Holomorphic foliations of degree two and arbitrary dimension
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arXiv:2207.12880v5 Announce Type: replace Abstract: We prove a complete classification of degree-$2$ foliations on $\mathbb{P}^n$ in any dimension, assuming they are not algebraically integrable. If $\mathcal{F}$ is such a foliation, then either $\mathcal{F}$ is the linear pull-back of a degree-$2$ foliation by curves on $\mathbb{P}^{n-k+1}$, or a logarithmic foliation of type $(1^{n-k+1},2)$, or a logarithmic foliation of type $(1^{n-k+3})$, or the linear pull-back of a degree-$2$ foliation of dimension $2$ on $\mathbb{P}^{n-k+2}$ tangent to an action of the Lie algebra $\mathfrak{aff}(\mathbb{C})$. Meanwhile, we prove that any $2$-dimensional foliation tangent to a global vector field must satisfy that its tangent sheaf is either not locally free or has a direct summand isomorphic to $\mathcal{O}_{\mathbb{P}^{n}}(a)$, with $a\in\{0,1\}$. As a byproduct of our classification, we describe the geometry of Poisson structures on $\mathbb{P}^{4}$ with generic rank two.
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https://arxiv.org/abs/2207.12880
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cfd66bb6fc279cc060e04c7d5d6c0f665b94217d8db454c02ee41c342f8e8d46
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2026-01-21T00:00:00-05:00
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Tame class field theory over local fields
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arXiv:2209.02953v2 Announce Type: replace Abstract: For a quasi-projective scheme $X$ admitting a smooth compactification over a local field of residue characteristic $p > 0$, we construct a continuous reciprocity homomorphism from a tame class group to the abelian tame etale fundamental group of $X$. We describe the prime-to-$p$ parts of its kernel and cokernel. This generalizes the higher dimensional unramified class field theory over local fields by Jannsen-Saito and Forre. We also prove a finiteness theorem for the geometric part of the abelian tame etale fundamental group, generalizing the results of Grothendieck and Yoshida for the unramified fundamental group.
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https://arxiv.org/abs/2209.02953
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13950b65070c5a28283c31d0e5f05e23356f2f531d7ad578e74bec1f03777fd9
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2026-01-21T00:00:00-05:00
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Functional dimension of feedforward ReLU neural networks
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arXiv:2209.04036v2 Announce Type: replace Abstract: It is well-known that the parameterized family of functions representable by fully-connected feedforward neural networks with ReLU activation function is precisely the class of piecewise linear functions with finitely many pieces. It is less well-known that for every fixed architecture of ReLU neural network, the parameter space admits positive-dimensional spaces of symmetries, and hence the local functional dimension near any given parameter is lower than the parametric dimension. In this work we carefully define the notion of functional dimension, show that it is inhomogeneous across the parameter space of ReLU neural network functions, and continue an investigation - initiated in [14] and [5] - into when the functional dimension achieves its theoretical maximum. We also study the quotient space and fibers of the realization map from parameter space to function space, supplying examples of fibers that are disconnected, fibers upon which functional dimension is non-constant, and fibers upon which the symmetry group acts non-transitively.
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https://arxiv.org/abs/2209.04036
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52f816a6c6084ed10900c5578a9774083fe01205cd1b37170340fab4e283ac3b
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2026-01-21T00:00:00-05:00
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Nondivergence of Reductive group action on Homogeneous Spaces
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arXiv:2209.06463v2 Announce Type: replace Abstract: Let $X=G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its non-cocompact arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $X$. We give several equivalent algebraic conditions on $H$ for the existence of a fixed compact set in $X$ intersecting \textit{every} $H$-orbit. This generalizes previous results concerning certain special reductive group action on $X$ in this setting. When $G$ is of real rank one, $\Gamma$ is a non-cocompact lattice of $G$ and $H<G$ is an algebraic group, we also obtain an algebraic condition on $H$ which is equivalent to the return of \textit{every} $H$-orbit to a single compact set in $X$. This complements our results in the case of arithmetic lattice.
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https://arxiv.org/abs/2209.06463
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9bb860686b17f2a163782183b07f48d281da850e0afc20ad2fcf50c1512dfa10
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2026-01-21T00:00:00-05:00
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Prime Solutions of Diagonal Diophantine Systems
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arXiv:2209.06934v2 Announce Type: replace Abstract: An asymptotic formula for the number of prime solutions of a general diagonal system of Diophantine equations is established, contingent on the existence of an appropriate mean value bound and on local solvability. In conjunction with the Vinogradov Mean Value Theorem this yields an asymptotic formula for solutions of Vinogradov systems and in conjunction with Hooley's work on seven cubes this yields a conditional result for the Waring-Goldbach problem on seven cubes of primes, contingent on Hooley's form of the Riemann hypothesis.
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https://arxiv.org/abs/2209.06934
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c26dfff9c795db1cf97b85f36818334c9002e310a868c751188e25328c242cd1
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2026-01-21T00:00:00-05:00
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Expander graphs are globally synchronizing
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arXiv:2210.12788v4 Announce Type: replace Abstract: The Kuramoto model is fundamental to the study of synchronization. It consists of a collection of oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with sufficient expansion must be globally synchronizing, meaning that a homogeneous Kuramoto model of identical oscillators on such a graph will converge to the fully synchronized state with all the oscillators having the same phase, for every initial state up to a set of measure zero. In particular, we show that for any $\varepsilon > 0$ and $p \geq (1 + \varepsilon) (\log n) / n$, the homogeneous Kuramoto model on the Erd\H{o}s-R\'enyi random graph $G(n, p)$ is globally synchronizing with probability tending to one as $n$ goes to infinity. This improves on a previous result of Kassabov, Strogatz, and Townsend and solves a conjecture of Ling, Xu, and Bandeira. We also show that the model is globally synchronizing on any $d$-regular Ramanujan graph, and on typical $d$-regular graphs, for large enough degree $d$.
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https://arxiv.org/abs/2210.12788
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be1828c50a391d0a748796d39e871bcfa81f9c7fff4800a3262def58535c9504
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2026-01-21T00:00:00-05:00
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Turing meets Moore-Penrose: Computing the Pseudoinverse on Turing Machines
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arXiv:2212.02940v2 Announce Type: replace Abstract: The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra and, thereby, in many different fields. Despite its proven existence, an algorithmic approach is typically necessary to obtain the pseudoinverse in practical applications. Therefore, we analyze if and to what degree the pseudoinverse can be computed on perfect digital hardware platforms modeled as Turing machines. For this, we utilize the notion of an effective algorithm that describes a provably correct computation: upon an input of any error parameter, the algorithm provides an approximation within the given error bound with respect to the unknown solution. We prove that a universal effective algorithm for computing the pseudoinverse of any matrix with a finite error bound does not exist on Turing machines. However, for specific classes of matrices, we show that provably correct algorithms exist and obtain a characterization of the properties of the input set, leading to the effective computability breakdown.
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https://arxiv.org/abs/2212.02940
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c4e02f4134d4133771c88e2eaa3a3f5e376e780f51641288cca4785607e1b109
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2026-01-21T00:00:00-05:00
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Diophantine Criterion for Non-trivial Shafarevich-Tate Groups
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arXiv:2301.03486v3 Announce Type: replace Abstract: The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with \textit{generalized Fermat's equation}. In this work, we are interested in the integer solutions of a similar quartic equation $pu^{2} = v^{2}+w^{2}$. For a particular form of $u,v$, and $w$, we prove that the elliptic curves $E_p: y^2 = x(x-1)(x+p^2)$, for primes $p \equiv 1 \pmod{8}$ where $q = (p^2+1)/2$ is also prime, exhibit a sharp dichotomy based on the solution of the aforementioned Diophantine equation: either $\mathrm{rank}(E_p(\mathbb{Q})) = 2$ with trivial Shafarevich-Tate group or $\mathrm{rank} = 0$ with $\Sha(E_p/\mathbb{Q})[2] \cong (\mathbb{Z}/2\mathbb{Z})^2$.
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https://arxiv.org/abs/2301.03486
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0d2dd4c856fd1d4986b36e120ae7d6ad96056c2272329cf39b52ed295a01b252
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2026-01-21T00:00:00-05:00
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Relations between e, $\pi$, golden ratios and $\sqrt{2}$
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arXiv:2301.09643v4 Announce Type: replace Abstract: We write out relations between the base of natural logarithms ($e$), the ratio of the circumference of a circle to its diameter ($\pi$), the golden ratios ($\Phi_p$) of the additive $p$-sequences, and the ratio of the diagonal of a square to its side ($\sqrt{2}$). An additive $p$-sequence is a natural extension of the Fibonacci sequence in which every term is the sum of $p$-previous terms given $p \ge 1$ initial values called seeds.
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https://arxiv.org/abs/2301.09643
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6a0513ebd92d2c4553690a9b185578bdddd7fd2cb514f823aa5024728779d4f1
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2026-01-21T00:00:00-05:00
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A relative Nadel-type vanishing theorem
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arXiv:2302.11080v3 Announce Type: replace Abstract: Let $f:X\rightarrow Y$ be a K\"{a}hler fibration from a complex manifold $X$ to an analytic space $Y$. We show several relative Nadel-type vanishing theorems.
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https://arxiv.org/abs/2302.11080
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f30ff57baf1f7c9ac5202effbacdcfb8041e2dadb7b6fdceabca13650a34550d
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2026-01-21T00:00:00-05:00
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Controlled Interacting Branching Diffusion Processes: Relaxed Formulation in the Mean-Field Regime
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arXiv:2304.07064v3 Announce Type: replace Abstract: The focus of this article is studying an optimal control problem for branching diffusion processes. Initially, we introduce the problem in its strong formulation and expand it to include linearly growing drifts. Then, we present a relaxed formulation that provides a suitable characterization based on martingale measures. Considering weak controls, we prove they are equivalent to strong controls in the relaxed setting, and establish the equivalence between the strong and relaxed problem, under a Filippov--type convexity condition. Furthermore, by defining control rules, we can restate the problem as the minimization of a lower semi-continuous function over a compact set, leading to the existence of optimal controls both for the relaxed problem and the strong one. Finally, with a useful embedding technique, we show that the value function solves a system of HJB equations, establishing a verification theorem. We then apply it to a linear-quadratic example and a kinetic one.
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https://arxiv.org/abs/2304.07064
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a2def47fd29ae770eb70826347af973e47b941533d7f485d3020272bf9f5d06c
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2026-01-21T00:00:00-05:00
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Continuous-time extensions of discrete-time cocycles
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arXiv:2305.07338v2 Announce Type: replace Abstract: We consider linear cocycles taking values in $\textup{SL}_d(\mathbb{R})$ driven by homeomorphic transformations of a smooth manifold, in discrete and continuous time. We show that any discrete-time cocycle can be extended to a continuous-time cocycle, while preserving its characteristic properties. We provide a necessary and sufficient condition under which this extension is canonical in the sense that the base is extended to an associated suspension flow and that the discrete-time cocycle is recovered as the time-1 map of the continuous-time cocycle. Further, we refine our general result for the case of (quasi-)periodic driving. We use our findings to construct a non-uniformly hyperbolic continuous-time cocycle in $\SL{2}$ over a uniquely ergodic driving.
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https://arxiv.org/abs/2305.07338
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76ac07abb1734a170b617d779076eed9fa33d82ed07d74bfdf884079d40f38de
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2026-01-21T00:00:00-05:00
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Quasimaps to quivers with potentials
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arXiv:2306.01302v2 Announce Type: replace Abstract: This paper is concerned with a non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential, and prove a gluing formula in the formalism of cohomological field theories. The main examples studied in this paper is when the above setting arises from quivers with potentials, where the above construction gives quantum correction to the equivariant Chow homology of the critical locus. Following similar ideas as in quasimaps to Nakajima quiver varieties studied by the Okounkov school, we analyse vertex functions in several examples, including Hilbert schemes of points on $\mathbb{C}^3$, moduli spaces of perverse coherent systems on the resolved conifold, and a quiver which defines higher $\mathfrak{sl}_2$-spin chains. Bethe equations are calculated in these cases. The construction in the present paper is based on the theory of gauged linear sigma models as well as shifted symplectic geometry of Pantev, To\"en, Vaquie and Vezzosi, and uses the virtual pullback formalism of symmetric obstruction theory of Park, which arises from the recent development of Donaldson-Thomas theory of Calabi-Yau 4-folds.
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https://arxiv.org/abs/2306.01302
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d77e3d67b96e6e5c4ce93763cab9d951fd28a61b1d9ba3c92dc5b2588312cb8c
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2026-01-21T00:00:00-05:00
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The distribution of Ridgeless least squares interpolators
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arXiv:2307.02044v2 Announce Type: replace Abstract: The Ridgeless minimum $\ell_2$-norm interpolator in overparametrized linear regression has attracted considerable attention in recent years in both machine learning and statistics communities. While it seems to defy conventional wisdom that overfitting leads to poor prediction, recent theoretical research on its $\ell_2$-type risks reveals that its norm minimizing property induces an `implicit regularization' that helps prediction in spite of interpolation. This paper takes a further step that aims at understanding its precise stochastic behavior as a statistical estimator. Specifically, we characterize the distribution of the Ridgeless interpolator in high dimensions, in terms of a Ridge estimator in an associated Gaussian sequence model with positive regularization, which provides a precise quantification of the prescribed implicit regularization in the most general distributional sense. Our distributional characterizations hold for general non-Gaussian random designs and extend uniformly to positively regularized Ridge estimators. As a direct application, we obtain a complete characterization for a general class of weighted $\ell_q$ risks of the Ridge(less) estimators that are previously only known for $q=2$ by random matrix methods. These weighted $\ell_q$ risks not only include the standard prediction and estimation errors, but also include the non-standard covariate shift settings. Our uniform characterizations further reveal a surprising feature of the commonly used generalized and $k$-fold cross-validation schemes: tuning the estimated $\ell_2$ prediction risk by these methods alone lead to simultaneous optimal $\ell_2$ in-sample, prediction and estimation risks, as well as the optimal length of debiased confidence intervals.
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https://arxiv.org/abs/2307.02044
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ecc0da5df116ece8296a1034b142fa055aa613f61c368ffbb872c4e635593324
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2026-01-21T00:00:00-05:00
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A First-Order Algorithm for Decentralised Min-Max Problems
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arXiv:2308.11876v2 Announce Type: replace Abstract: In this work, we consider a connected network of finitely many agents working cooperatively to solve a min-max problem with convex-concave structure. We propose a decentralised first-order algorithm which can be viewed as a non-trivial combination of two algorithms: PG-EXTRA for decentralised minimisation problems and the forward reflected backward method for (non-distributed) min-max problems. In each iteration of our algorithm, each agent computes the gradient of the smooth component of its local objective function as well as the proximal operator of its nonsmooth component, following by a round of communication with its neighbours. Our analysis shows that the sequence generated by the method converges under standard assumptions with non-decaying stepsize.
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https://arxiv.org/abs/2308.11876
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2c03a50debcf8e109c2df19fcadbc185114b21a43c7fe678d5fad04aad2f3c75
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2026-01-21T00:00:00-05:00
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Fast and Inverse-Free Algorithms for Deflating Subspaces
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arXiv:2310.00193v4 Announce Type: replace Abstract: This paper explores a key question in numerical linear algebra: how can we compute projectors onto the deflating subspaces of a regular matrix pencil $(A,B)$, in particular without using matrix inversion or defaulting to an expensive Schur decomposition? We focus specifically on spectral projectors, whose associated deflating subspaces correspond to sets of eigenvalues/eigenvectors. In this work, we present a high-level approach to computing these projectors, which combines rational function approximation with an inverse-free arithmetic of Benner and Byers [Numerische Mathematik 2006]. The result is a numerical framework that captures existing inverse-free methods, generates an array of new options, and provides straightforward tools for pursuing efficiency on structured problems (e.g., definite pencils). To exhibit the efficacy of this framework, we consider a handful of methods in detail, including Implicit Repeated Squaring and iterations based on the matrix sign function. In an appendix, we demonstrate that recent, randomized divide-and-conquer eigensolvers -- which are built on fast methods for individual projectors -- can be adapted to produce the generalized Schur form of any matrix pencil in nearly matrix multiplication time.
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https://arxiv.org/abs/2310.00193
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221ed213f19c4821363331881fde7c69561b24568d9f35b04927fe5dfde2206b
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2026-01-21T00:00:00-05:00
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Global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction
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arXiv:2310.13461v2 Announce Type: replace Abstract: The classical Fourier's law, which states that the heat flux is proportional to the temperature gradient, induces the paradox of infinite propagation speed for heat conduction. To accurately simulate the real physical process, the hyperbolic model of heat conduction named Cattaneo's law was proposed, which leads to the finite speed of heat propagation. A natural question is that whether the large-time behavior of the heat flux for compressible flow would be different for these two laws. In this paper, we aim to address this question by studying the global well-posedness and optimal time-decay rates of classical solutions to the compressible Navier-Stokes system with Cattaneo's law. By designing a new method, we obtain the optimal time-decay rates for the highest derivatives of the heat flux, which cannot be derived for the system with Fourier's law by Matsumura and Nishida [Proc. Japan Acad. Ser. A Math. Sci., 55(9):337-342, 1979]. In this sense, our results first reveal the essential differences between the two laws.
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https://arxiv.org/abs/2310.13461
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ce9175122c4f4ae8cd631fdff10ad602368e4bb0c490649ed6bb55ab8a510669
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2026-01-21T00:00:00-05:00
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The asymptotic behavior of rarely visited edges of the simple random walk
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arXiv:2310.16657v2 Announce Type: replace Abstract: In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $\alpha(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(\alpha(n))$, show that $n\to \mathbb{E}(\alpha(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(\alpha(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (\alpha(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(1/32,1/2]$ such that $\limsup\limits_{n\to+\infty}\frac{\alpha(n)}{(\log n)^2}=C$ almost surely.
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https://arxiv.org/abs/2310.16657
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abbe22bc7372d1be620bd429b96b8da51ebd44f0824febd146b411955e204df6
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2026-01-21T00:00:00-05:00
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Factorization structures, cones, and polytopes
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arXiv:2311.07328v4 Announce Type: replace Abstract: Factorization structures occur in toric differential and discrete geometry, and can be viewed in multiple ways, e.g., as objects determining substantial classes of explicit toric Sasaki and K\"ahler geometries, as special coordinates on such, or as an apex generalisation of cyclic polytopes featuring a generalised Gale's evenness condition. This article presents a comprehensive study of factorization structures. It establishes their structure theory and introduces their use in the geometry of cones and polytopes. The article explains the construction of polytopes and cones compatible with a given factorization structure, and exemplifies it for product Segre-Veronese and Veronese factorization structures, where the latter case includes cyclic polytopes. Further, it derives the generalised Gale's evenness condition for compatible cones, polytopes and their duals, and explicitly describes faces of these. Factorization structures naturally provide generalised Vandermonde identities, which relate normals of any compatible polytope, and which are used for Veronese factorization structure to find examples of Delzant and rational Delzant compatible polytopes. The article offers a myriad of factorization structure examples, which are later characterised to be precisely factorization structures with decomposable curves, and raises the question if these encompass all factorization structures, i.e., the existence of an indecomposable factorization curve.
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https://arxiv.org/abs/2311.07328
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196588446301f41d97ea49bb8e2633e214ac9260e624c90d85c1715f01cb3d56
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2026-01-21T00:00:00-05:00
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Abelian gauge-like groups of $L_\infty$-algebras
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arXiv:2311.08512v3 Announce Type: replace Abstract: Given a finite type degree-wise nilpotent $L_\infty$-algebra, we construct an abelian group that acts on the set of Maurer-Cartan elements of the given $L_\infty$-algebra so that the quotient by this action becomes the moduli space of equivalence classes of Maurer-Cartan elements. Specializing this to degree-wise nilpotent dg Lie algebras, we find that the associated ordinary gauge group of the dg Lie algebra with the Baker-Campbell-Hausdorff multiplication might be substituted by the underlying additive group. This additive group acts on the Maurer-Cartan elements, and the quotient by this action yields the moduli space of gauge-equivalence classes of Maurer-Cartan elements.
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https://arxiv.org/abs/2311.08512
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54571ac0725232239005205972256b2907e831d4083941e2ff8ad00c8370eac7
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2026-01-21T00:00:00-05:00
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Quaternion-Valued Wavelets on the Plane: A Construction via the Douglas-Rachford Approach
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arXiv:2311.12614v2 Announce Type: replace Abstract: This paper presents a reformulation of the construction of nonseparable multiresolution quaternion-valued wavelets on the plane as a feasibility problem. The constraint sets in the feasibility problem are derived from the standard conditions of smoothness, compact support, and orthonormality. To solve the resulting feasibility problems, we employ a product space formulation of the Douglas-Rachford algorithm. This approach yields novel examples of nonseparable, multiresolution, compactly supported, smooth, and orthonormal quaternion-valued wavelets on the plane. Additionally, by introducing a symmetry-promoting constraint, we construct symmetric quaternion-valued scaling functions on the plane.
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https://arxiv.org/abs/2311.12614
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691c0004c71fec5d0a97bce209532a0c4ea2f2f7d6673bbc9c3258242ac86866
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2026-01-21T00:00:00-05:00
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On the equivalence of static and dynamic weak optimal transport
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arXiv:2311.13872v3 Announce Type: replace Abstract: We show that there is a PDE formulation in terms of Fokker-Planck equations for weak optimal transport problems. The main novelty is that we introduce a minimization problem involving Fokker-Planck equations in the extended sense of measure-valued solutions and prove that it is equal to the associated weak transport problem.
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https://arxiv.org/abs/2311.13872
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4f3b41662208e6d8d8bbf1ebc3e3597b28fd5efb3d41a40ffce3d42dc1e715b2
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2026-01-21T00:00:00-05:00
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Ancient mean curvature flows from minimal hypersurfaces
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arXiv:2311.15278v4 Announce Type: replace Abstract: For $n\geq 2$, we construct $I$-dimensional family of embedded ancient solutions to mean curvature flow arise from an unstable minimal hypersurface $\Sigma$ with finite total curvature in $\mathbb{R}^{n+1}$, where $I$ is the Morse index of the Jacobi operator on $\Sigma$.
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https://arxiv.org/abs/2311.15278
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8fa76dba51c90644a82723828ce62154381ca3c2fe0f7e3e7ed5b36a4c13cc8a
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2026-01-21T00:00:00-05:00
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Propagating front solutions in a time-fractional Fisher-KPP equation
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arXiv:2311.15651v5 Announce Type: replace Abstract: In this paper, we treat the Fisher-KPP equation with a Caputo-type time fractional derivative and discuss the propagation speed of the solution. The equation is a mathematical model that describes the processes of sub-diffusion, proliferation, and saturation. We first consider a traveling wave solution to study the propagation of the solution, but we cannot define it in the usual sense due to the time fractional derivative in the equation. We therefore assume that the solution asymptotically approaches a traveling wave solution, and the asymptotic traveling wave solution is formally introduced as a potential asymptotic form of the solution. The existence and the properties of the asymptotic traveling wave solution are discussed using a monotone iteration method. Finally, the behavior of the solution is analyzed by numerical simulations based on the result for asymptotic traveling wave solutions.
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https://arxiv.org/abs/2311.15651
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db2332ac75ab82a080cd39896857bf962641f5055d0dd54397bc8918b8d9e248
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2026-01-21T00:00:00-05:00
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Non-vanishing of Kolyvagin systems and Iwasawa theory
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arXiv:2312.09301v2 Announce Type: replace Abstract: Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime. In 1991 Kolyvagin conjectured that the system of cohomology classes for torsion quotients of the $p$-adic Tate module of $E$ derived from Heegner points over ring class fields of a suitable imaginary quadratic field $K$ (i.e., the Heegner point Kolyvagin system of $E/K$) is non-trivial. In this paper we prove Kolyvagin's conjecture when $p$ is a prime of good ordinary reduction for $E$ that splits in $K$. In particular, our results cover many cases where $p$ is an Eisenstein prime for $E$, complementing Wei Zhang's earlier results on the conjecture by a different approach. Our methods also yield a proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of $E$, as conjectured by Wei Zhang in 2014, as well as proofs of analogous results for the Kolyvagin system obtained from Kato's Euler system.
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https://arxiv.org/abs/2312.09301
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d53292e1b63d3035673790e0aafe2cf2367a9cbf570d73878d5cea506163feaa
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2026-01-21T00:00:00-05:00
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On the mixed monotonicity of polynomial functions
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arXiv:2312.15517v2 Announce Type: replace Abstract: In this paper, it is shown that every polynomial function is mixed monotone globally with a polynomial decomposition function. For univariate polynomials, the decomposition functions can be constructed from the Gram matrix representation of polynomial functions. The tightness of polynomial decomposition functions is discussed. Several examples are provided. An example is provided to show that polynomial decomposition functions, in addition to being global decomposition functions, can be much tighter than local decomposition functions constructed using local Jacobian bounds. Furthermore, an example is provided to demonstrate the application to reachable set over-approximation.
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https://arxiv.org/abs/2312.15517
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54d06df007db9ae668deacd33f3ca3d719530a14b8d88d5e93390c737a009cb2
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2026-01-21T00:00:00-05:00
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Virtual Holonomic and Nonholonomic Constraints on Lie groups
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arXiv:2312.17531v2 Announce Type: replace Abstract: This paper develops a geometric framework for virtual constraints on Lie groups, with emphasis on mechanical systems modeled as affine connection systems. Virtual holonomic and virtual nonholonomic constraints, including linear and affine nonholonomic constraints, are formulated directly at the level of the Lie algebra and characterized as feedback--invariant manifolds. For each class of constraint, we establish existence and uniqueness conditions for enforcing feedback laws and show that the resulting closed--loop trajectories evolve as the dynamics of mechanical systems endowed with induced constrained connections, generalizing classical holonomic and nonholonomic reductions. Beyond stabilization, the framework enables the systematic generation of low--dimensional motion primitives on Lie groups by enforcing invariant, possibly affine, manifolds and shaping nontrivial dynamical regimes. The approach is illustrated through representative examples, including quadrotor UAVs and a rigid body with an internal rotor, where classical control laws are recovered as special cases and affine constraint--induced motion primitives are obtained.
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https://arxiv.org/abs/2312.17531
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99818173f6d8c37be74f6439e8a58e2bb8794476a738092a6457ce948f2472c7
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2026-01-21T00:00:00-05:00
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Separable homology of graphs and the separability complex
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arXiv:2401.01320v2 Announce Type: replace Abstract: We introduce the separability complex, a one-complex associated to a finite regular cover of the rose and show that it is connected if and only if the fundamental group of the associated cover is generated by its intersection with the set of elements in proper free factors of $\mathbf{F}_n$. The separability complex admits an action of $\mathrm{Out}(\mathbf{F}_n)$ by isometries if the associated cover corresponds to a characteristic subgroup of $\mathbf{F}_n$. We prove that the separability complex of the rose has infinite diameter and is nonhyperbolic, implying it is not quasi-isometric to the free splitting complex or the free factor complex. As a consequence, we obtain that the Cayley graph of $\mathbf{F}_n$ with generating set consisting of all primitive elements of $\mathbf{F}_n$ is nonhyperbolic.
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https://arxiv.org/abs/2401.01320
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b06621019b69332ecc149a37d7b231dc9ebda1fcfca53f142850ca78af09cc3f
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2026-01-21T00:00:00-05:00
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A classification of neighborhoods around leaves of a singular foliation
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arXiv:2401.05966v3 Announce Type: replace Abstract: We classify singular foliations admitting a given leaf and a given transverse singular foliation.
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https://arxiv.org/abs/2401.05966
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a2771c7e13f132df1a7d1b5b27d785e5a9bc0239f6222b705edcbd303d70cbec
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2026-01-21T00:00:00-05:00
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Local dimension spectrum for dominated planar self-affine sets
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arXiv:2401.13626v2 Announce Type: replace Abstract: The local dimension spectrum provides a framework for quantifying the fractal properties of a measure, and it is well understood for non-overlapping self-similar measures. In this article, we study the local dimension spectrum for dominated self-affine measures. By analyzing exact dimensionality, we obtain deterministic results that extend the scope of the local dimension spectrum beyond the almost-sure setting.
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https://arxiv.org/abs/2401.13626
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b02fff27c51cd32d86e18f5535ac454f521729ceb1fb9b7a85f6fa894b86c338
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2026-01-21T00:00:00-05:00
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On generalized Beauville decompositions
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arXiv:2402.08861v2 Announce Type: replace Abstract: Motivated by the Beauville decomposition of an abelian scheme and the "Perverse = Chern" phenomenon for a compactified Jacobian fibration, we study in this paper splittings of the perverse filtration for compactified Jacobian fibrations. On the one hand, we prove for the Beauville-Mukai system associated with an irreducible curve class on a $K3$ surface the existence of a Fourier-stable multiplicative splitting of the perverse filtration, which extends the Beauville decomposition for the nonsingular fibers. Our approach is to construct a Lefschetz decomposition associated with a Fourier-conjugate $\mathfrak{sl}_2$-triple, which relies heavily on recent work concerning the interaction between derived equivalences and LLV algebras for hyper-K\"ahler varieties. Motivic lifting and connections to the Beauville-Voisin conjectures are also discussed. On the other hand, we construct for any $g\geq 2$ a compactified Jacobian fibration of genus $g$ curves such that each curve is integral with at worst simple nodes and the (multiplicative) perverse filtration does not admit a multiplicative splitting. Our argument relies on the recently established universal double ramification cycle relations. This shows that in general an extension of the Beauville decomposition cannot exist for compactified Jacobian fibrations even when the simplest singular point appears.
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https://arxiv.org/abs/2402.08861
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70c4473f24a4f0464b439c7a2cf64144d963fdf1c998fd8da7cbaf1ddb588c0e
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2026-01-21T00:00:00-05:00
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Uniform bounds for bilinear symbols with linear K-quasiconformally embedded singularity
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arXiv:2402.11661v2 Announce Type: replace Abstract: We prove bounds in the strict local $L^{2}(\mathbb{R}^{d})$ range for trilinear Fourier multiplier forms with a $d$-dimensional singular subspace. Given a fixed parameter $K \ge 1$, we treat multipliers with non-degenerate singularity that are push-forwards by $K$-quasiconformal matrices of suitable symbols. As particular applications, our result recovers the uniform bounds for the one-dimensional bilinear Hilbert transforms in the strict local $L^{2}$ range, and it implies the uniform bounds for two-dimensional bilinear Beurling transforms, which are new, in the same range.
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https://arxiv.org/abs/2402.11661
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d5c19cae4a15de550c76019e6955e8f8e9dabd13a4499c77ec2c64056e3e27c9
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2026-01-21T00:00:00-05:00
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On transverse-universality of twist knots
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arXiv:2402.12585v3 Announce Type: replace Abstract: In the search for transverse-universal knots in the standard contact structure on $\mathbb{S}^3$, we present a classification of the transverse twist knots with maximal self-linking number, that admit only overtwisted contact branched covers. As a direct consequence, we obtain an infinite family of transverse knots in $(\mathbb{S}^3,\xi_{std})$ that are not transverse-universal, although they are universal in the topological sense.
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https://arxiv.org/abs/2402.12585
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9b8ebd580dd38b0cfcb3444d95dfcf7762ec972cd78691babf68eaa6139408c9
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2026-01-21T00:00:00-05:00
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Hyperuniformity and optimal transport of point processes
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arXiv:2402.13705v4 Announce Type: replace Abstract: We examine optimal matchings or transport between two stationary random measures. It covers allocation from the Lebesgue measure to a point process and matching a point process to a regular (shifted) lattice. The main focus of the article is the impact of hyperuniformity(reduced variance fluctuations in point processes) to optimal transport: in dimension 2, we show that the typical matching cost has finite second moment under a mild logarithmic integrability condition on the reduced pair correlation measure, showing that most planar hyperuniform point processes are L2-perturbed lattices. Our method also retrieves known sharp bounds in finite windows for neutral integrable systems such as Poisson processes, and also applies to hyperfluctuating systems. Further, in three dimensions onwards, all point processes with an integrable pair correlation measure are L2-perturbed lattices without requiring hyperuniformity.
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https://arxiv.org/abs/2402.13705
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7ed55fa5bf0aed7b00f3f92303908adbccd159fd1df3a064bc71c4e92eefa3e5
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2026-01-21T00:00:00-05:00
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Bilinear Rough Singular Integrals near the Critical Integrability via Sharp Fourier Multiplier Criteria
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arXiv:2402.15785v2 Announce Type: replace Abstract: We establish boundedness results for bilinear singular integral operators with rough homogeneous kernels whose restriction to the unit sphere belongs to the Orlicz space $L(\log L)^\alpha$. This improves the previously best known condition for boundedness of such bilinear operators obtained in the paper of the first and third authors, and provides estimates close to the conjectured endpoint of integrability suggested by the linear theory. The proof is based on a new sharp boundedness criterion for bilinear Fourier multiplier operators associated with sums of dyadic dilations of a fixed symbol $m_0$, compactly supported away from the origin. This criterion admits the best possible behavior with respect to a modulation of $m_0$ and is intimately connected with sharp shifted square function estimates.
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https://arxiv.org/abs/2402.15785
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ebbc1cf2abbbdb9dfa10d852d36114fbf671daa63241c6eec4313040471c9515
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2026-01-21T00:00:00-05:00
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Counting rationals and diophantine approximation in missing-digit Cantor sets
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arXiv:2402.18395v2 Announce Type: replace Abstract: We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture of those authors about the corresponding intrinsic diophantine approximation problem. Moreover, we make further progress towards conjectures of Bugeaud--Durand and Levesley--Salp--Velani on the distribution of diophantine exponents in missing-digit sets. A key tool in our study is Fourier $\ell^1$ dimension introduced by the last named author in [H. Yu, Rational points near self-similar sets, arXiv:2101.05910]. An important technical contribution of the paper is a method to compute this quantity.
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https://arxiv.org/abs/2402.18395
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d6ce7ff7335c9a03deb82faca24b957db8f620e6769956db755fed8319045bcb
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2026-01-21T00:00:00-05:00
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A restricted additive smoother for finite cell flow problems
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arXiv:2403.11636v2 Announce Type: replace Abstract: In this work, we propose an adaptive geometric multigrid method for the solution of large-scale finite cell flow problems. The finite cell method seeks to circumvent the need for a boundary-conforming mesh through the embedding of the physical domain in a regular background mesh. As a result of the intersection between the physical domain and the background computational mesh, the resultant systems of equations are typically numerically ill-conditioned, rendering the appropriate treatment of cutcells a crucial aspect of the solver. To this end, we propose a smoother operator with favorable parallel properties and discuss its memory footprint and parallelization aspects. We propose three cache policies that offer a balance between cached and on-the-fly computation and discuss the optimization opportunities offered by the smoother operator. It is shown that the smoother operator, on account of its additive nature, can be replicated in parallel exactly with little communication overhead, which offers a major advantage in parallel settings as the geometric multigrid solver is consequently independent of the number of processes. The convergence and scalability of the geometric multigrid method is studied using numerical examples. It is shown that the iteration count of the solver remains bounded independent of the problem size and depth of the grid hierarchy. The solver is shown to obtain excellent weak and strong scaling using numerical benchmarks with more than 665 million degrees of freedom. The presented geometric multigrid solver is, therefore, an attractive option for the solution of large-scale finite cell problems in massively parallel high-performance computing environments.
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https://arxiv.org/abs/2403.11636
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9ce4cf0c9c71ac7daf34e2fbc31bf5b42e385adbc6ccc91317c57fd1e64e9252
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What Is a Good Imputation Under MAR Missingness?
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arXiv:2403.19196v5 Announce Type: replace Abstract: Missing values pose a persistent challenge in modern data science. Consequently, there is an ever-growing number of publications introducing new imputation methods in various fields. The present paper attempts to take a step back and provide a more systematic analysis. Starting from an in-depth discussion of the Missing at Random (MAR) condition for nonparametric imputation, we first investigate whether the widely used fully conditional specification (FCS) approach indeed identifies the correct conditional distributions. Based on this analysis, we propose three essential properties an ideal imputation method should meet, thus enabling a more principled evaluation of existing methods and more targeted development of new methods. In particular, we introduce a new imputation method, denoted mice-DRF, that meets two out of the three criteria. We also discuss ways to compare imputation methods, based on distributional distances. Finally, numerical experiments illustrate the points made in this discussion.
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https://arxiv.org/abs/2403.19196
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251adc3f0c082036eac9b8e6b1efeb878794fefbdc403fc4c667142f700dec2c
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2026-01-21T00:00:00-05:00
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Analytic holonomicity of real C$^{{\mathrm{exp}}}$-class distributions
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arXiv:2403.20167v2 Announce Type: replace Abstract: We introduce a notion of distributions on $\mathbb{R}^n$, called distributions of C$^{{\mathrm{exp}}}$-class, based on wavelet transforms of distributions and the theory from Cluckers, Comte, Miller, Rolin, Servi (2018) about C$^{{\mathrm{exp}}}$-class functions. We prove that the framework of C$^{{\mathrm{exp}}}$-class distributions is closed under natural operations, like push-forward, pull-back, derivation and anti-derivation, and, in the tempered case, Fourier transforms. Our main result is the (real analytic) holonomicity of all distributions of C$^{{\mathrm{exp}}}$-class.
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https://arxiv.org/abs/2403.20167
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e8563bdf0ce79798edc81202403f7cf6c55365556f45315866d26a72c478c803
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2026-01-21T00:00:00-05:00
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Constructive proofs for some semilinear PDEs on $H^2(e^{|x|^2/4},\mathbb{R}^d)$
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arXiv:2404.04054v2 Announce Type: replace Abstract: We develop computer-assisted tools to study semilinear equations of the form \begin{equation*} -\Delta u -\frac{x}{2}\cdot \nabla{u}= f(x,u,\nabla u) ,\quad x\in\mathbb{R}^d. \end{equation*} Such equations appear naturally in several contexts, and in particular when looking for self-similar solutions of parabolic PDEs. We develop a general methodology, allowing us not only to prove the existence of solutions, but also to describe them very precisely. We introduce a spectral approach based on an eigenbasis of $\mathcal{L}:= -\Delta -\frac{x}{2}\cdot \nabla$ in spherical coordinates, together with a quadrature rule allowing to deal with nonlinearities, in order to get accurate approximate solutions. We then use a Newton-Kantorovich argument, in an appropriate weighted Sobolev space, to prove the existence of a nearby exact solution. We apply our approach to nonlinear heat equations, to nonlinear Schr\"odinger equations and to a generalised viscous Burgers equation, and obtain both radial and non-radial self-similar profiles.
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https://arxiv.org/abs/2404.04054
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f453c32c9c826d72e03379cf76098b4358120f54f4d5b14c774dcce32e1d74fb
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2026-01-21T00:00:00-05:00
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Monochromatic polynomial sumset structures on $\mathbb{N}$
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arXiv:2404.05226v4 Announce Type: replace Abstract: In the paper, we search for monochromatic infinite additive structures involving polynomials over $\mathbb{N}$. It is proved that for any $r\in \mathbb{N}$, any two distinct natural numbers $a,b$, and any $2$-coloring of $\mathbb{N}$, there exist two sets $B,C\subset \mathbb{N}$ with $|B|=r$ and $|C|=\infty$ such that there exists some color containing $B+aC$ and $B+bC$.
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https://arxiv.org/abs/2404.05226
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5b715e73f1c0f14a3b3104d6aeee7d2c8c9b32db473650976f0f0a16cf66fc48
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2026-01-21T00:00:00-05:00
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Cohen-Macaulay representations of Artin-Schelter Gorenstein algebras of dimension one
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arXiv:2404.05925v4 Announce Type: replace Abstract: Tilting theory is one of the central tools in modern representation theory, in particular in the study of Cohen-Macaulay representations. We study Cohen-Macaulay representations of $\mathbb N$-graded Artin-Schelter Gorenstein algebras $A$ of dimension one, without assuming the connectedness condition. This framework covers a broad class of noncommutative Gorenstein rings, including classical $\mathbb N$-graded Gorenstein orders. We prove that the stable category $\underline{\mathsf{CM}}_0^{\mathbb Z}A$ admits a silting object if and only if $A_0$ has finite global dimension. In this case we give such a silting object explicitly. Assuming that $A$ is ring-indecomposable, we further show that $\underline{\mathsf{CM}}_0^{\mathbb Z}A$ admits a tilting object if and only if either $A$ is Artin-Schelter regular or the average Gorenstein parameter of $A$ is non-positive. These results generalize those of Buchweitz, Iyama, and Yamaura. We give two proofs of the second result: one via Orlov-type semiorthogonal decompositions, and the other via a direct calculation. As an application, we show that for a Gorenstein tiled order $A$, the category $\underline{\mathsf{CM}}^{\mathbb Z}A$ is equivalent to the derived category of the incidence algebra of an explicitly constructed poset. We also apply our results and Koszul duality to study smooth noncommutative projective quadric hypersurfaces $\mathsf{qgr}\,B$ of arbitrary dimension. We prove that $\mathsf{D}^{\mathrm b}(\mathsf{qgr}\,B)$ admits an explicitly constructed tilting object, which contains the tilting object of $\underline{\mathsf{CM}}^{\mathbb Z}B$ due to Smith and Van den Bergh as a direct summand via Orlov's semiorthogonal decomposition.
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https://arxiv.org/abs/2404.05925
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4d87b700b116b1d364c31d9e95cb131baab6cf99c046fffe7c10b33cd3bed60e
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2026-01-21T00:00:00-05:00
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Lower bound for the first eigenvalue of $p-$Laplacian and applications in asymptotically hyperbolic Einstein manifolds
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arXiv:2405.02669v2 Announce Type: replace Abstract: This paper investigates the first Dirichlet eigenvalue for the $p$-Laplacian in Riemannian manifolds. Firstly, we establish a lower bound for this eigenvalue under the condition that the domain includes a specific function which fulfills certain criteria related to divergence and gradient conditions. In the subsequent section, we introduce an enhanced lower bound for the eigenvalue, which is linked to the distance function defined in the domain. As a practical application, we provide an estimation for the first Dirichlet eigenvalue of geodesic balls with large radius in asymptotically hyperbolic Einstein manifolds.
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https://arxiv.org/abs/2405.02669
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053d8e06e4ed5f6fa616403d16fd91d545f6d2d7b73c4e9d7c4ba83275796183
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2026-01-21T00:00:00-05:00
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Fast Two-Time-Scale Stochastic Gradient Method with Applications in Reinforcement Learning
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arXiv:2405.09660v4 Announce Type: replace Abstract: Two-time-scale optimization is a framework introduced in Zeng et al. (2024) that abstracts a range of policy evaluation and policy optimization problems in reinforcement learning (RL). Akin to bi-level optimization under a particular type of stochastic oracle, the two-time-scale optimization framework has an upper level objective whose gradient evaluation depends on the solution of a lower level problem, which is to find the root of a strongly monotone operator. In this work, we propose a new method for solving two-time-scale optimization that achieves significantly faster convergence than the prior arts. The key idea of our approach is to leverage an averaging step to improve the estimates of the operators in both lower and upper levels before using them to update the decision variables. These additional averaging steps eliminate the direct coupling between the main variables, enabling the accelerated performance of our algorithm. We characterize the finite-time convergence rates of the proposed algorithm under various conditions of the underlying objective function, including strong convexity, Polyak-Lojasiewicz condition, and general non-convexity. These rates significantly improve over the best-known complexity of the standard two-time-scale stochastic approximation algorithm. When applied to RL, we show how the proposed algorithm specializes to novel online sample-based methods that surpass or match the performance of the existing state of the art. Finally, we support our theoretical results with numerical simulations in RL.
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https://arxiv.org/abs/2405.09660
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a13b7a791de0da39d12b69f87ef50f0eb7ead34fa4df21c9c2898cc1dbe9fec8
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2026-01-21T00:00:00-05:00
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On a Conjecture by Hayashi on Finite Connected Quandles
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arXiv:2405.11660v2 Announce Type: replace Abstract: A quandle is an algebraic structure whose binary operation is idempotent, right-invertible and right self-distributive. Right-invertibility ensures right translations are permutations and right self-distributivity ensures further they are automorphisms. For finite connected quandles, all right translations have the same cycle structure, called the profile of the connected quandle. Hayashi conjectured that the longest length in the profile of a finite connected quandle is a multiple of the remaining lengths. We prove that this conjecture is true for profiles with at most five lengths.
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https://arxiv.org/abs/2405.11660
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03dd82c0605b4cfb2e6d3c1aefa9195df6b278839dc021f7a68edab48399e5ee
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2026-01-21T00:00:00-05:00
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Rate Optimality and Phase Transition for User-Level Local Differential Privacy
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arXiv:2405.11923v3 Announce Type: replace Abstract: Most of the literature on differential privacy considers the item-level case where each user has a single observation, but a growing field of interest is that of user-level privacy where each of the $n$ users holds $T$ observations and wishes to maintain the privacy of their entire collection. In this paper, we derive a general minimax lower bound, which shows that, for locally private user-level estimation problems, the risk cannot, in general, be made to vanish for a fixed number of users even when each user holds an arbitrarily large number of observations. We then derive matching, up to logarithmic factors, lower and upper bounds for univariate and multidimensional mean estimation, sparse mean estimation and non-parametric density estimation. In particular, with other model parameters held fixed, we observe phase transition phenomena in the minimax rates as $T$ the number of observations each user holds varies. In the case of (non-sparse) mean estimation and density estimation, we see that, for $T$ below a phase transition boundary, the rate is the same as having $nT$ users in the item-level setting. Different behaviour is however observed in the case of $s$-sparse $d$-dimensional mean estimation, wherein consistent estimation is impossible when $d$ exceeds the number of observations in the item-level setting, but is possible in the user-level setting when $T \gtrsim s \log (d)$, up to logarithmic factors. This may be of independent interest for applications as an example of a high-dimensional problem that is feasible under local privacy constraints.
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https://arxiv.org/abs/2405.11923
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165cede06af6d169f27f02a99a029f8a705e64c2e66ae78863ba04f5aeb51288
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2026-01-21T00:00:00-05:00
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Distribution Steering for Discrete-Time Uncertain Ensemble Systems
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arXiv:2405.12415v2 Announce Type: replace Abstract: Ensemble systems appear frequently in many engineering applications and, as a result, they have become an important research topic in control theory. These systems are best characterized by the evolution of their underlying state distribution. Despite the work to date, few results exist dealing with the problem of directly modifying (i.e., ``steering'') the distribution of an ensemble system. In addition, in most existing results, the distribution of the states of an ensemble of discrete-time systems is assumed to be Gaussian. However, in case the system parameters are uncertain, it is not always realistic to assume that the distribution of the system follows a Gaussian distribution, thus complicating the solution of the overall problem. In this paper, we address the general distribution steering problem for first-order discrete-time ensemble systems, where the distributions of the system parameters and the states are arbitrary with finite first few moments. Linear system dynamics are considered using the method of power moments to transform the original infinite-dimensional problem into a finite-dimensional one. We also propose a control law for the ensuing moment system, which allows us to obtain the power moments of the desired control inputs. Finally, we solve the inverse problem to obtain the feasible control inputs from their corresponding power moments. We provide a numerical example to validate our theoretical developments.
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https://arxiv.org/abs/2405.12415
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a6a4600c7090af8882369439897fe9f5dbf3c661d7c13a8a883595a0d4f6a706
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2026-01-21T00:00:00-05:00
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On non-topologizable semigroups
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arXiv:2405.16992v3 Announce Type: replace Abstract: We find anti-isomorphic submonoids $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ of the bicyclic monoid $\mathscr{C}(a,b)$ with the following properties: every Hausdorff left-continuous (right-continuous) topology on $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) is discrete and there exists a compact Hausdorff topological monoid $S$ which contains $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) as a submonoid. Also, we construct a non-discrete right-continuous (left-continuous) topology $\tau_p^+$ ($\tau_p^-$) on the semigroup $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) which is not left-continuous (right-continuous).
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https://arxiv.org/abs/2405.16992
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8e961179b12718fc596f1dd8c10da1697131ded3d4f3a006c2c80bb750b58029
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2026-01-21T00:00:00-05:00
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CHANI: Correlation-based Hawkes Aggregation of Neurons with bio-Inspiration
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arXiv:2405.18828v2 Announce Type: replace Abstract: The present work aims at proving mathematically that a neural network inspired by biology can learn a classification task thanks to local transformations only. In this purpose, we propose a spiking neural network named CHANI (Correlation-based Hawkes Aggregation of Neurons with bio-Inspiration), whose neurons activity is modeled by Hawkes processes. Synaptic weights are updated thanks to an expert aggregation algorithm, providing a local and simple learning rule. We were able to prove that our network can learn on average and asymptotically. Moreover, we demonstrated that it automatically produces neuronal assemblies in the sense that the network can encode several classes and that a same neuron in the intermediate layers might be activated by more than one class, and we provided numerical simulations on synthetic dataset. This theoretical approach contrasts with the traditional empirical validation of biologically inspired networks and paves the way for understanding how local learning rules enable neurons to form assemblies able to represent complex concepts.
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https://arxiv.org/abs/2405.18828
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fbd147f5d6c13a9d94244f33892351180cd25a42b894abb4e7c28dbd16ad991b
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2026-01-21T00:00:00-05:00
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Uniform Resolvent Estimates for Subwavelength Resonators: The Minnaert Bubble Case
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arXiv:2406.02192v4 Announce Type: replace Abstract: Subwavelength resonators are small scaled objects that exhibit contrasting medium properties (eigher in intensity or sign) while compared to the ones of a uniform background. Such contrasts allow them to resonate at specific frequencies. There are two ways to mathematically define these resonances. First, as the frequencies for which the related system of integral equations is not injective. Second, as the frequencies for which the related resolvent operator of the natural Hamiltonian, given by the wave-operator, has a pole. In this work, we consider, as the subwavelength resonator, the Minneart bubble. We show that these two mentioned definitions are equivalent. Most importantly, 1. we derive the related resolvent estimates which are uniform in terms of the size/contrast of the resonators. As a by product, we show that the resolvent operators have no scattering resonances in the upper half complex plane while they exhibit two scattering resonances in the lower half plane which converge to the real axis, as the size of the bubble tends to zero. As these resonances are poles of the natural Hamiltonian, given by the wave-operator, and have the Minnaert frequency as their dominating real part, this justifies calling them Minnaert resonances. 2. we derive the asymptotic estimates of the generated scattered fields which are uniform in terms of the incident frequency and which are valid everywhere in space (i.e. inside or outside the bubble). The dominating parts, for both the resolvent operator and the scattered fields, are given by the ones of the point-scatterer supported at the location of the bubble. In particular, these dominant parts are non trivial (not the same as those of the background medium) if and only if the used incident frequency identifies with the Minnaert one.
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https://arxiv.org/abs/2406.02192
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b98724b7b1d4d86a25948c82b57a67b8c42daca79aabe8fcb450c8b192bcb37c
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2026-01-21T00:00:00-05:00
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Solutions to the exercises from the book "Albert algebras over commutative rings"
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arXiv:2406.02933v3 Announce Type: replace Abstract: This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024, as well as errata and addenda.
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https://arxiv.org/abs/2406.02933
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818b1805ca8a8c949498d769498ad0be61e2562045100b597f0d8ecf39c18666
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2026-01-21T00:00:00-05:00
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Exact formulae for ranks of partitions
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arXiv:2406.06294v2 Announce Type: replace Abstract: In 2009, Bringmann arXiv:0708.0691 [math.NT] used the circle method to prove an asymptotic formula for the Fourier coefficients of rank generating functions. In this paper, we prove that Bringmann's formula, when summing up to infinity and in the case of prime modulus, gives a Rademacher-type exact formula involving sums of vector-valued Kloosterman sums. As a corollary, in another paper arXiv:2406.07469 [math.NT], we will provide a new proof of Dyson's conjectures by showing that the certain Kloosterman sums vanish.
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https://arxiv.org/abs/2406.06294
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405a2d53e29f9d71430c6accd4f1fce661e28671e4f3c31ff316dd3269d6578a
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2026-01-21T00:00:00-05:00
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Multivariate extreme values for dynamical systems
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arXiv:2406.14807v3 Announce Type: replace Abstract: We establish a theory for multivariate extreme value analysis of dynamical systems. Namely, we provide conditions adapted to the dynamical setting which enable the study of dependence between extreme values of the components of $\R^d$-valued observables evaluated along the orbits of the systems. We study this cross-sectional dependence, which results from the combination of a spatial and a temporal dependence structures. We give several illustrative applications, where concrete systems and dependence sources are introduced and analysed.
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https://arxiv.org/abs/2406.14807
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1de7dcc42a8a1d7192d28c625686cf86f00d765268f0610fdc8034d0433f9f69
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2026-01-21T00:00:00-05:00
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Geometric structures for maximal representations and pencils
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arXiv:2407.01254v2 Announce Type: replace Abstract: We study fibrations of the projective model for the symmetric space associated with $\text{SL}(2n,\mathbb{R})$ by codimension $2$ projective subspaces, or pencils of quadrics. In particular we show that if such a smooth fibration is equivariant with respect to a representation of a closed surface group, the representation is quasi-isometrically embedded, and even Anosov if the pencils in the image contain only non-degenerate quadrics. We use this to characterize maximal representations among representations of a closed surface group into $\text{Sp}(2n,\mathbb{R})$ by the existence of an equivariant continuous fibration of the associated symmetric space, satisfying an additional technical property. These fibrations extend to fibrations of the projective structures associated to maximal representations by bases of pencils of quadrics.
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https://arxiv.org/abs/2407.01254
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a729d7fdc8acc5167e45fb3928b9db004f90b82b5222f23508d8adc4072a3666
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2026-01-21T00:00:00-05:00
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Heights of Ceresa and Gross-Schoen cycles
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arXiv:2407.01304v4 Announce Type: replace Abstract: We study the Beilinson-Bloch heights of Ceresa and Gross-Schoen cycles in families. We construct that for any $g\ge 3$, a Zariski open dense subset $\mathcal{M}_g^{\mathrm{amp}}$ of $\mathcal{M}_g$, the coarse moduli of curves of genus $g$ over $\mathbb{Q}$, such that the heights of Ceresa cycles and Gross-Schoen cycles over $\mathcal{M}_g^{\mathrm{amp}}$ have a lower bound and satisfy the Northcott property.
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https://arxiv.org/abs/2407.01304
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6ee5772d80cb1cfe324e392101030b9d23d7384f36d21ce3e12db1e287e49c9d
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2026-01-21T00:00:00-05:00
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A uniform-in-time nonlocal approximation of the standard Fokker-Planck equation
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arXiv:2407.03870v4 Announce Type: replace Abstract: We study a nonlocal approximation of the Fokker-Planck equation in which we can estimate the speed of convergence to equilibrium in a way which does not degenerate as we approach the local limit of the equation. This uniform estimate cannot be easily obtained with standard inequalities or entropy methods, but can be obtained through the use of Harris's theorem, finding interesting links to quantitative versions of the central limit theorem in probability. As a consequence one can prove that solutions of this nonlocal approximation converge to solutions of the usual Fokker-Planck equation uniformly in time-hence we show the approximation is asymptotic-preserving in this sense. The associated equilibrium has some interesting tail and regularity properties, which we also study.
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https://arxiv.org/abs/2407.03870
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e5c360c4a257a1dd6263a16027957a9c0160b63aedcadce677cd878afcd3a6a5
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2026-01-21T00:00:00-05:00
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Lipschitz regularity for solutions to an orthotropic $q$-Laplacian-type equation in the Heisenberg group
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arXiv:2407.07548v3 Announce Type: replace Abstract: We establish the local Lipschitz regularity for solutions to an orthotropic q-Laplacian-type equation within the Heisenberg group. Our approach is largely inspired by the works of X. Zhong, who investigated the q-Laplacian in the same setting and proved the H\"older regularity for the gradient of solutions. Due to the degeneracy of the current equation, such regularity for the gradient of solutions is not even known in the Euclidean setting for dimensions greater than 2, where only boundedness is expected.
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https://arxiv.org/abs/2407.07548
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78d0ba9e932e7e6af11bf98db34909f045d1a15b9a29944e90646b278c41be68
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2026-01-21T00:00:00-05:00
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An Adaptive Proximal ADMM for Nonconvex Linearly Constrained Composite Programs
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arXiv:2407.09927v3 Announce Type: replace Abstract: This paper develops an adaptive proximal alternating direction method of multipliers (ADMM) for solving linearly constrained, composite optimization problems under the assumption that the smooth component of the objective is weakly convex, while the non-smooth component is convex and block-separable. The proposed method is adaptive to all problem parameters, including smoothness and weak convexity constants, and allows each of its block proximal subproblems to be inexactly solved. Each iteration of our adaptive proximal ADMM consists of two steps: the sequential solution of each block proximal subproblem; and adaptive tests to decide whether to perform a full Lagrange multiplier and/or penalty parameter update(s). Without any rank assumptions on the constraint matrices, it is shown that the adaptive proximal ADMM obtains an approximate first-order stationary point of the constrained problem in a number of iterations that matches the state-of-the-art complexity for the class of proximal ADMM's. The three proof-of-concept numerical experiments that conclude the paper suggest our adaptive proximal ADMM enjoys significant computational benefits.
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https://arxiv.org/abs/2407.09927
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a96e7e8684860eeb40b25cbe4469c84ee6a9857a6bf1ab609e1afe3fe8ea62bc
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2026-01-21T00:00:00-05:00
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A short nonstandard proof of the Spectral Theorem for unbounded self-adjoint operators
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arXiv:2407.16136v3 Announce Type: replace Abstract: By nonstandard analysis, a very short and elementary proof of the Spectral Theorem for unbounded self-adjoint operators is given.
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https://arxiv.org/abs/2407.16136
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a2a07d523065306e0e006e136dcc6f08d8fba2eaa897d8c6138bb81ee52b95a5
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2026-01-21T00:00:00-05:00
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Winners with Confidence: Discrete Argmin Inference with an Application to Model Selection
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arXiv:2408.02060v4 Announce Type: replace Abstract: We study the problem of finding the index of the minimum value of a vector from noisy observations. This problem is relevant in population/policy comparison, discrete maximum likelihood, and model selection. We develop an asymptotically normal test statistic, even in high-dimensional settings and with potentially many ties in the population mean vector, by integrating concepts and tools from cross-validation and differential privacy. The key technical ingredient is a central limit theorem for globally dependent data. We also propose practical ways to select the tuning parameter that adapts to the signal landscape. Numerical experiments and data examples demonstrate the ability of the proposed method to achieve a favorable bias-variance trade-off in practical scenarios.
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https://arxiv.org/abs/2408.02060
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0459f75c5c630213448a67b309af75ced50a24f0cd5e7b93c152987a351a8fc2
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2026-01-21T00:00:00-05:00
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Long-time behaviour of a multidimensional age-dependent branching process with a singular jump kernel modelling telomere shortening
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arXiv:2408.02476v2 Announce Type: replace Abstract: In this article, we investigate the ergodic behaviour of a multidimensional age-dependent branching process with a singular jump kernel, motivated by studying the phenomenon of telomere shortening in cell populations. Our model tracks individuals evolving within a continuous-time framework indexed by a binary tree, characterised by age and a multidimensional trait. Branching events occur with rates dependent on age, where offspring inherit traits from their parent with random increase or decrease in some coordinates, while the most of them are left unchanged. Exponential ergodicity is obtained at the cost of an exponential normalisation, despite the fact that we have an unbounded age-dependent birth rate that may depend on the multidimensional trait, and a non-compact transition kernel. These two difficulties are respectively treated by stochastically comparing our model to Bellman-Harris processes, and by using a weak form of a Harnack inequality. We conclude this study by giving examples where the assumptions of our main result are verified.
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https://arxiv.org/abs/2408.02476
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d2a76996468dcd69cc77493ace72910f4c1f68013c2c2e22b811cceaf78f65c1
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2026-01-21T00:00:00-05:00
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Poisson Approximation of prime divisors of shifted primes
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arXiv:2408.03803v4 Announce Type: replace Abstract: We develop an analog for shifted primes of the Kubilius model of prime factors of integers. We prove a total variation distance estimate for the difference between the model and actual prime factors of shifted primes, and apply it to show that the prime factors of shifted primes in disjoint sets behave like independent Poisson variables. As a consequence, we establish a transference principle between the anatomy of random integers up to x and of random shifted primes p+a with p < x.
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https://arxiv.org/abs/2408.03803
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ebbda67ef2ecf50f61e2ebdab3faf6301c2b0d0a6425ce09c042a710b5c4f67b
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2026-01-21T00:00:00-05:00
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On a family of arithmetic series related to the M\"obius function
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arXiv:2409.02754v5 Announce Type: replace Abstract: Let $P^-(n)$ denote the smallest prime factor of a natural integer $n>1$. Furthermore let $\mu$ and $\omega$ denote respectively the M\"obius function and the number of distinct prime factors function. We show that, given any set ${{\scr P}}$ of prime numbers with a natural density, we have $\sum_{P^-(n)\in \scr P}\mu(n)\omega(n)/n=0$ and provide a effective estimate for the rate of convergence. This extends a recent result of Alladi and Johnson, who considered the case when ${\scr P}$ is an arithmetic progression.
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https://arxiv.org/abs/2409.02754
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557f55f8fd931e205cb0d7980633291e4fa58decf8b69ef7e127aab271e47a7d
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2026-01-21T00:00:00-05:00
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Counting points on generic character varieties
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arXiv:2409.04735v3 Announce Type: replace Abstract: We count points on character varieties associated with punctured surfaces and regular semisimple generic conjugacy classes in reductive groups. We find that the number of points are palindromic polynomials. This suggests a $P=W$ conjecture for these varieties. We also count points on the corresponding additive character varieties and find that the number of points are also polynomials, which we conjecture have non-negative coefficients. These polynomials can be considered as the reductive analogues of the Kac polynomials of comet-shaped quivers.
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https://arxiv.org/abs/2409.04735
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594cecd9a68830e7f828e14da47d0c510529020fed1f0a647deb4a76fb106faf
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2026-01-21T00:00:00-05:00
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A Complexity Dichotomy for Temporal Valued Constraint Satisfaction Problems
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arXiv:2409.07285v2 Announce Type: replace Abstract: We study the complexity of the valued constraint satisfaction problem (VCSP) for every valued structure with the domain ${\mathbb Q}$ that is preserved by all order-preserving bijections. Such VCSPs will be called temporal, in analogy to the (classical) constraint satisfaction problem: a relational structure is preserved by all order-preserving bijections if and only if all its relations have a first-order definition in $({\mathbb Q};<)$, and the CSPs for such structures are called temporal CSPs. Many optimization problems that have been studied intensively in the literature can be phrased as a temporal VCSP. We prove that a temporal VCSP is in P, or NP-complete. Our analysis uses the concept of fractional polymorphisms. This is the first dichotomy result for VCSPs over infinite domains which is complete in the sense that it treats all valued structures with a given automorphism group.
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https://arxiv.org/abs/2409.07285
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ab3401596963694fae26a0d4e580eaba4890a3e5ca2d7910aaf9a929a721f6fb
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2026-01-21T00:00:00-05:00
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Matrix perturbation analysis of methods for extracting singular values from approximate singular subspaces
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arXiv:2409.09187v2 Announce Type: replace Abstract: Given (orthonormal) approximations $\tilde{U}$ and $\tilde{V}$ to the left and right subspaces spanned by the leading singular vectors of a matrix $A$, we discuss methods to approximate the leading singular values of $A$ and study their accuracy. In particular, we focus our analysis on the generalized Nystr\"om approximation, as surprisingly, it is able to obtain significantly better accuracy than classical methods, namely Rayleigh-Ritz and (one-sided) projected SVD. A key idea of the analysis is to view the methods as finding the exact singular values of a perturbation of $A$. In this context, we derive a matrix perturbation result that exploits the structure of such $2\times2$ block matrix perturbation. Furthermore, we extend it to block tridiagonal matrices. We then obtain bounds on the accuracy of the extracted singular values. This leads to sharp bounds that predict well the approximation error trends and explain the difference in the behavior of these methods. Finally, we present an approach to derive an a-posteriori version of those bounds, which are more amenable to computation in practice.
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https://arxiv.org/abs/2409.09187
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cf4a9748a861277433e2939c1d6ccb7b7002ab7b7d9904103adbd930c81ed48b
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2026-01-21T00:00:00-05:00
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Averaging theory and catastrophes
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arXiv:2409.11054v2 Announce Type: replace Abstract: When a dynamical system is subject to a periodic perturbation, the averaging method can be applied to obtain an autonomous leading order "guiding system", placing the time dependence at higher orders. Recent research focused on investigating invariant structures in non-autonomous differential systems arising from hyperbolic structures in the guiding system, such as periodic orbits and invariant tori. Complementarily, the effect that bifurcations in the guiding system have on the original non-autonomous one has also been recently explored, albeit less frequently. This paper extends this study by providing a broader description of the dynamics that can emerge from non-hyperbolic structures of the guiding system. Specifically, we prove here that $\mathcal{K}$-universal bifurcations in the guiding system `persist' in the original non-autonomous one, while non-versal bifurcations, such as the transcritical and pitchfork, do not. We illustrate the results on examples of a fold, a transcritical, a pitchfork, and a saddle-focus.
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https://arxiv.org/abs/2409.11054
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604334c839c3de3a952e59b9c0a681938ef297e55a58c23233a14a22198c72f0
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2026-01-21T00:00:00-05:00
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QMC integration based on arbitrary (t,m,s)-nets yields optimal convergence rates on several scales of function spaces
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arXiv:2409.12879v2 Announce Type: replace Abstract: We study the integration problem over the $s$-dimensional unit cube on four types of Banach spaces of integrands. First we consider Haar wavelet spaces, consisting of functions whose Haar wavelet coefficients exhibit a certain decay behavior measured by a parameter $\alpha >0$. We study the worst case error of integration over the norm unit ball and provide upper error bounds for quasi-Monte Carlo (QMC) cubature rules based on arbitrary $(t,m,s)$-nets as well as matching lower error bounds for arbitrary cubature rules. These results show that using arbitrary $(t,m,s)$-nets as sample points yields the best possible rate of convergence. Afterwards we study spaces of integrands of fractional smoothness $\alpha \in (0,1)$ and state a sharp Koksma-Hlawka-type inequality. More precisely, we show that on those spaces the worst case error of integration is equal to the corresponding fractional discrepancy. Those spaces can be continuously embedded into tensor product Bessel potential spaces, also known as Sobolev spaces of dominated mixed smoothness, with the same set of parameters. The latter spaces can be embedded into suitable Besov spaces of dominating mixed smoothness $\alpha$, which in turn can be embedded into the Haar wavelet spaces with the same set of parameters. Therefore our upper error bounds on Haar wavelet spaces for QMC cubatures based on $(t,m,s)$-nets transfer (with possibly different constants) to the corresponding spaces of integrands of fractional smoothness and to Sobolev and Besov spaces of dominating mixed smoothness. Moreover, known lower error bounds for periodic Sobolev and Besov spaces of dominating mixed smoothness show that QMC integration based on arbitrary $(t,m,s)$-nets yields the best possible convergence rate on periodic as well as on non-periodic Sobolev and Besov spaces of dominating smoothness.
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https://arxiv.org/abs/2409.12879
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c68c5683637690d0058e79be120cece814c1c3ebdd069364f210b47c2ad54801
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2026-01-21T00:00:00-05:00
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D2D Coded Caching from Two Classes of Optimal DPDAs using Cross Resolvable Designs
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arXiv:2409.14350v2 Announce Type: replace Abstract: Device to device (D2D) communication is one of the most promising techniques for fifth-generation and beyond wireless communication systems. This paper considers coded caching in a wireless D2D network, in which a central server initially places the data in the user cache memories, and all user demands are served through inter-user coded multicast transmissions. D2D placement delivery array (DPDA) was proposed as a tool for designing coded caching schemes with reduced subpacketization levels in a D2D network. In this paper, we first constructed three classes of DPDAs using a cross resolvable design, a group divisible design, and a newly developed block design. The resulting D2D schemes achieve low subpacketization levels while meeting the known lower bound on the transmission load of a DPDA. These classes of constructed DPDAs either simplify or generalize all existing DPDA constructions that achieve the known lower bound and have low subpacketization levels. Furthermore, a new lower bound on the transmission load of a DPDA is proposed. Two new classes of DPDAs are then constructed using a cross resolvable design and a newly developed block design, respectively. These constructions yield low-subpacketization D2D schemes and achieve the proposed lower bound on the transmission load. Compared to existing schemes with the same system parameters as those obtained from the proposed DPDAs, the proposed schemes have an advantage in either transmission load or subpacketization level or both.
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https://arxiv.org/abs/2409.14350
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cc20b8b58e0f301040c9096d47ee6cc27e353bfe14f7f2aa8af5baa4fa90dd72
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2026-01-21T00:00:00-05:00
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Crystallinity for syntomic cohomology, \'etale cohomology, and algebraic $K$-theory
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arXiv:2409.20543v2 Announce Type: replace Abstract: We prove for $n\geq c-1$ that the functor taking an animated ring $R$ to its mod $(p^c,v_1^{p^n})$ syntomic cohomology factors through the functor $R \mapsto R/p^{c(n+2)}$, a phenomenon we term crystallinity for mod $(p^c,v_1^{p^n})$ syntomic cohomology. As an application, we completely and explicitly compute the mod $(p,v_1 ^{p^{n}-1})$ algebraic $K$-theory of $\mathbb Z/p^{k}$ whenever $k \geq n+2$ and $p>2$. As a second application, we deduce crystallinity for the mod $p^c$ syntomic complexes associated to smooth $p$-adic formal schemes, and in particular for the Galois equivariant mod $p^c$ \'etale cohomologies of their adic generic fibers. Finally, we strengthen known $p$-adic convergence theorems for the topological Hochschild homology of ring spectra, and as a result relate crystallinity for algebraic $K$-theory to Lichtenbaum--Quillen theorems.
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https://arxiv.org/abs/2409.20543
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d7666bdf2688df4b94a90644400173fee799c518cc0433fb00918b512040e3bc
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2026-01-21T00:00:00-05:00
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Entropy contraction of the Gibbs sampler under log-concavity
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arXiv:2410.00858v2 Announce Type: replace Abstract: The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm which iteratively samples from the conditional distributions of a probability measure $\pi$ of interest. Under the assumption that $\pi$ is strongly log-concave, we show that the random scan Gibbs sampler contracts in relative entropy and provide a sharp characterization of the associated contraction rate. Assuming that evaluating conditionals is cheap compared to evaluating the joint density, our results imply that the number of full evaluations of $\pi$ needed for the Gibbs sampler to mix grows linearly with the condition number and is independent of the dimension. If $\pi$ is non-strongly log-concave, the convergence rate in entropy degrades from exponential to polynomial. Our techniques are versatile and extend to Metropolis-within-Gibbs schemes and the Hit-and-Run algorithm. A comparison with gradient-based schemes and the connection with the optimization literature are also discussed.
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https://arxiv.org/abs/2410.00858
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7da4827fbd09637366e4a543b273c0f57e7bec915dad9f076f685902cc89309b
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2026-01-21T00:00:00-05:00
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The Conjecture of Dixmier for the first Weyl algebra is true
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arXiv:2410.06959v5 Announce Type: replace Abstract: Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove that the Dixmier conjecture for the first Weyl algebra is true, i.e. each algebra endomorphism of the algebra $A_1$ is an automorphism.
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https://arxiv.org/abs/2410.06959
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890c2cd35b93813ee4ddc487e33fd60f7146d4562e7f8f53921abc8300bb6ed7
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2026-01-21T00:00:00-05:00
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Uniform Space and Time Behavior for Acoustic Resonators
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arXiv:2410.09630v2 Announce Type: replace Abstract: We deal with the time-domain acoustic wave propagation in the presence of a subwavelength resonator given by a Minneart bubble. This bubble is small scaled and enjoys high contrasting mass density and bulk modulus. It is well known that, under certain regimes between these scales, such a bubble generates a single low-frequency (or subwavelength) resonance called Minnaert resonance. In this paper, we study the wave propagation governed by Minnaert resonance effects in time domain. We derive the point-approximation expansion of the wave field. The dominant part is a sum of two terms. 1. The first one, which we call the primary wave, is the wave field generated in the absence of the bubble. 2. The second one, which we call the resonant wave, is generated by the interaction between the bubble and the background. It is related to a Dirac-source, in space, that is modulated, in time, with a coefficient which is a solution of a $1$D Cauchy problem, for a second order differential equation, having as propagation and attenuation parameters the real and the imaginary parts, respectively, of the Minnaert resonance. We show that the evolution of the resonant wave remains valid for a large time of the order $\epsilon^{-1}$, where $\epsilon$ is the radius of the bubble, after which it collapses by exponentially decaying. Precisely, we confirm that such resonant wave have life-time inversely proportional to the imaginary part of the related subwavelength resonances, which is in our case given by the Minnaert one. In addition, the real part of this resonance fixes the period of the wave.
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https://arxiv.org/abs/2410.09630
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ae00040ad34421ee4d1e7ea34b0c1c00da4101aa60630f6311d1685d1d5326a3
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2026-01-21T00:00:00-05:00
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Safety on the Fly: Constructing Robust Safety Filters via Policy Control Barrier Functions at Runtime
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arXiv:2410.11157v3 Announce Type: replace Abstract: Control Barrier Functions (CBFs) have proven to be an effective tool for performing safe control synthesis for nonlinear systems. However, guaranteeing safety in the presence of disturbances and input constraints for high relative degree systems is a difficult problem. In this work, we propose the Robust Policy CBF (RPCBF), a practical approach for constructing robust CBF approximations online via the estimation of a value function. We establish conditions under which the approximation qualifies as a valid CBF and demonstrate the effectiveness of the RPCBF-safety filter in simulation on a variety of high relative degree input-constrained systems. Finally, we demonstrate the benefits of our method in compensating for model errors on a hardware quadcopter platform by treating the model errors as disturbances. Website including code: www.oswinso.xyz/rpcbf/
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https://arxiv.org/abs/2410.11157
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Academic Papers
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b372d9e799309ccde34a4256c0c6024408d8a635ba3d09bab0510dbcd435a917
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2026-01-21T00:00:00-05:00
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Subspace method based on neural networks for eigenvalue problems
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arXiv:2410.13358v2 Announce Type: replace Abstract: In this paper, we propose a subspace method based on neural networks for eigenvalue problems with high accuracy and low cost. We first construct a neural network-based orthogonal basis by some deep learning method and dimensionality reduction technique, and then calculate the Galerkin projection of the eigenvalue problem onto the subspace spanned by the orthogonal basis and obtain an approximate solution. Numerical experiments show that we can obtain approximate eigenvalues and eigenfunctions with very high accuracy but low cost.
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https://arxiv.org/abs/2410.13358
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Academic Papers
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9c5847bfe22097083f7bfd73e87345164e14d5e9b91c3cb351bf7e720c2c2081
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2026-01-21T00:00:00-05:00
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Two-stage Online Reusable Resource Allocation: Reservation, Overbooking and Confirmation Call
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arXiv:2410.15245v2 Announce Type: replace Abstract: We study a two-stage online reusable resource allocation problem over T days involving advance reservations and walk-ins. Each day begins with a reservation stage (Stage I), where reservation requests arrive sequentially. When service starts (Stage II), both reserved and walk-in customers arrive to check in and occupy resources for several days. Reserved customers can cancel without penalty before or during a confirmation call initiated by the decision maker (DM) before day's end. The DM must immediately accept or reject each booking or check-in request, potentially overbooking by accepting more reservations than capacity. An overbooking loss occurs if a reserved customer's check-in is rejected in Stage II; a reward is obtained for each occupied resource unit daily. Our goal is to develop an online policy that controls bookings and check-ins to maximize total revenue over the T-day horizon. We show that due to cancellation uncertainties and complex correlations between occupancy durations, any online policy incurs a regret of \Omega(T) compared to the offline optimal policy when the \textit{busy season} assumption does not hold. To address this, we introduce decoupled adaptive safety stocks, which use only single-day information to hedge against overbooking risks and reduce resource idling. Under the busy season condition, our policy decouples the overall offline optimal into single-day offline optimal policies. Consequently, the regret between our policy and the offline optimal decays exponentially with the time between the confirmation call and day's end, suggesting the DM can delay confirmation calls while maintaining near-optimal performance. We validate our algorithm through sythetic experiments and empirical data from an Algarve resort hotel.
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https://arxiv.org/abs/2410.15245
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4d240dcb25339b0115e5d03e190e01366114c7594f53a1346d5462ee3ad39736
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2026-01-21T00:00:00-05:00
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Entrance boundary for standard processes with no negative jumps and its application to exponential convergence to the Yaglom limit
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arXiv:2410.15447v3 Announce Type: replace Abstract: We study standard processes with no negative jumps under the entrance boundary condition. Similarly to one-dimensional diffusions, we show that the process can be made into a Feller process by attaching the boundary point to the state space. We investigate the spectrum of the infinitesimal generator in detail via the scale function, characterizing it as the zeros of an entire function. As an application, we prove that under the strong Feller property, the convergence to the Yaglom limit of the process killed on hitting the boundary is exponentially fast.
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https://arxiv.org/abs/2410.15447
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984e44617fe2e5c64355fdf20026c01f5ce2218aacd3ccf3b4410249b6b9b0a1
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2026-01-21T00:00:00-05:00
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Information-Based Martingale Optimal Transport
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arXiv:2410.16339v3 Announce Type: replace Abstract: Randomised arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between any given ordered set of random variables, at fixed pre-specified times. Utilising these processes as generators of partial information, a class of continuous-time martingale -- the filtered arcade martingales (FAMs) -- are constructed. FAMs interpolate through a sequence of target random variables, which form a discrete-time martingale. The research presented in this paper relaxes the FAM setting to the interpolation between probability measures instead and treats the problem of selecting the worst martingale coupling for given, convexly ordered, probability measures contingent on the paths of FAMs that are constructed using the martingale coupling. This optimisation problem, that we term the information-based martingale optimal transport problem (IB-MOT), can be viewed from different perspectives. It can be understood as a model-free construction of FAMs, in the case where the coupling is not determined a priori. It can also be considered from the vantage point of optimal transport (OT), where the problem is concerned with introducing a noise factor in martingale optimal transport, similarly to how the entropic regularisation of optimal transport introduces noise in OT. The IB-MOT problem is static in its nature, since its aim is to find a coupling. However, a corresponding dynamical solution can be found by considering the FAM constructed with the identified optimal coupling. The existence and uniqueness of its solution are shown and an algorithm for empirical measures is proposed.
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https://arxiv.org/abs/2410.16339
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faad20a8ec7d1d64537abcb94124d2b1ef3d3b3af1d5aebf91607d6fa00d1bde
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2026-01-21T00:00:00-05:00
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Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Approximations and regularity
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arXiv:2410.20370v3 Announce Type: replace Abstract: We study various regularization operators on plurisubharmonic functions that preserve Lelong classes with growth given by certain compact convex sets. The purpose is to show that the weighted Siciak-Zakharyuta functions associated with these Lelong classes are lower semicontinuous. These operators are given by integral, infimal, and supremal convolutions. Continuity properties of the logarithmic supporting function are studied and a precise description is given of when it is uniformly continuous. This gives a contradiction to published results about the H\"older continuity of these Siciak-Zakharyuta functions.
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https://arxiv.org/abs/2410.20370
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9b1c324ce891e6b4c78f879fef3f812f3b68b13794d9947e2e1dd9d5dcf44b98
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2026-01-21T00:00:00-05:00
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Erd\H{o}s-P\'osa property of $A$-paths in unoriented group-labelled graphs
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arXiv:2411.05372v2 Announce Type: replace Abstract: We characterize the obstructions to the Erd\H{o}s-P\'osa property of $A$-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group $\Gamma$ and for every subset $\Lambda$ of $\Gamma$, the family of $\Gamma$-labelled $A$-paths whose lengths are in $\Lambda$ satisfies the half-integral Erd\H{o}s-P\'osa property. Moreover, we give a characterization of such $\Gamma$ and $\Lambda\subseteq\Gamma$ for which the same family of $A$-paths satisfies the full Erd\H{o}s-P\'osa property.
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https://arxiv.org/abs/2411.05372
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45473decdfc9503d84d528f7d58fa42291c27691cbf27e2a5dc34d82a3b71111
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2026-01-21T00:00:00-05:00
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Torsion and semi-degeneracy of second-order maximally superintegrable systems
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arXiv:2411.06994v3 Announce Type: replace Abstract: The isotropic harmonic oscillator and the Kepler-Coulomb system are pivotal models in the Sciences. They are two examples of second-order (maximally) superintegrable (Hamiltonian) systems. These systems are classified in dimension two. A partial classification exists in dimension three. In this paper, our focus is on second-order superintegrable systems with a $(n+1)$-parameter potential with $n\geq3$. We find that these systems are underpinned by an information-geometric structure, namely the structure of a statistical manifold with torsion. We obtain a necessary and sufficient condition for such systems to extend to non-degenerate systems, i.e. to admit a maximal family of compatible potentials. The condition is geometric: we show that a $(n+1)$-parameter potential is the restriction of a non-degenerate potential if and only if a certain trace-free tensor field vanishes. We interpret this condition as the requirement that a certain affine connection has vectorial torsion. We also show that the condition for a system to be extendable is conformally invariant, allowing us to extend our results to second-order conformally superintegrable systems with a $(n+1)$-parameter potential.
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https://arxiv.org/abs/2411.06994
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51e9459a87db9efbb2b50100e371086a380bdfa9bd5e65be0f5eda88b7c3a7f4
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2026-01-21T00:00:00-05:00
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Accelerating Benders decomposition for solving a sequence of sample average approximation replications
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arXiv:2411.09091v2 Announce Type: replace Abstract: Sample average approximation (SAA) is a technique for obtaining approximate solutions to stochastic programs that uses the average from a random sample to approximate the expected value that is being optimized. Since the outcome from solving an SAA is random, statistical estimates on the optimal value of the true problem can be obtained by solving multiple SAA replications with independent samples. We study techniques to accelerate the solution of this set of SAA replications, when solving them sequentially via Benders decomposition. We investigate how to exploit similarities in the problem structure, as the replications just differ in the realizations of the random samples. Our extensive computational experiments provide empirical evidence that our techniques for using information from solving previous replications can significantly reduce the solution time of later replications.
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https://arxiv.org/abs/2411.09091
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3d2bf155f37135aebf41fd6a98b67e9a155d91b28cd310655b85538f4b70cb5c
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2026-01-21T00:00:00-05:00
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Harmonic forms on ALE Ricci-flat 4-manifolds
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arXiv:2411.09561v4 Announce Type: replace Abstract: In this paper, we compute the expansion of some harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
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https://arxiv.org/abs/2411.09561
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30e2635a414d5b91876c3502cd348a4866913263b12053a063af2f4f81965881
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2026-01-21T00:00:00-05:00
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Equivalent spectral theory for fundamental graph cut problems
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arXiv:2411.11077v2 Announce Type: replace Abstract: We introduce and develop equivalent spectral graph theory for several fundamental graph cut problems including maxcut, mincut, Cheeger cut, anti-Cheeger cut, dual Cheeger problem and their useful variants. A specified strategy for achieving an equivalent eigenproblem is proposed for a general graph cut problem via the set-pair Lov\'asz extension and the Dinkelbach scheme. For a class of 2-cut and 3-cut problems, we reveal the intrinsic difference-of-submodularity for the fractional formulations and show that their set-pair Lov\'asz extensions yield equivalent difference-of-convex structures. Building on the Dinkelbach scheme, we finally establish a unified research roadmap for nonlinear spectral theory that provides a one-to-one correspondence between certain eigenpairs and the optimal graph cut problems. The finer structure of the eigenvectors, the Courant nodal domain theorem and the graphic feature of eigenvalues are studied systematically in the setting of these new nonlinear eigenproblems.
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https://arxiv.org/abs/2411.11077
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ac0b10721f2ef0af9036feb12f84bb525195d77180ec87e8a96c033969361607
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2026-01-21T00:00:00-05:00
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A star is born: Explosive Crump-Mode-Jagers branching processes
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arXiv:2411.18749v2 Announce Type: replace Abstract: We study a family of Crump--Mode--Jagers branching processes in random environment that explode, i.e. that grow infinitely large in finite time with positive probability. Building on recent work of the author and Iyer (``On the structure of genealogical trees associated with explosive Crump--Mode--Jagers branching processes", arXiv:2311.14664, 2023), we weaken certain assumptions required to prove that the branching process, at the time of explosion, contains a (unique) individual with infinite offspring. We then apply these results to super-linear preferential attachment models. In particular, we fill gaps in some of the cases analysed in Appendix A of the work of the author and Iyer and study a large range of previously unattainable cases.
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https://arxiv.org/abs/2411.18749
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Academic Papers
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a50104d1742cdc75153aeaca8553b177419be51ba2540a55d322f61cbaf60cb0
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2026-01-21T00:00:00-05:00
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Characterization of Trees with Maximum Security
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arXiv:2411.19188v3 Announce Type: replace Abstract: The rank (also known as protection number or leaf-height) of a vertex in a rooted tree is the minimum distance between the vertex and any of its leaf descendants. We consider the sum of ranks over all vertices (known as the security) in binary trees, and produce a classification of families of binary trees for which the security is maximized. In addition, extremal results relating to maximum rank among all vertices in families of trees is discussed.
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https://arxiv.org/abs/2411.19188
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e3a71464e325e7df4a875be521170b18bc216efb7c09970853474173bc4e2381
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2026-01-21T00:00:00-05:00
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Arithmetic level raising theorem for some unitary Shimura varieties mod $p$
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arXiv:2412.03519v4 Announce Type: replace Abstract: Let $F$ be a real quadratic field in which a fixed prime $p$ is inert, and $E_0$ be an imaginary quadratic field in which $p$ splits; put $E=E_0 F$. Let ${{\rm Sh}}_{1,n-1}$ be the special fiber over $\mathbb{F}_{p^2}$ of the Shimura variety for $G(U(1,n-1)\times U(n-1,1))$ with hyperspecial level structure at $p$ for some integer $n\geq 2$. Let ${{\rm Sh}}_{1,n-1}(K_{\mathfrak{p}}^{1})$ be the special fiber over $\mathbb{F}_{p^2}$ of a Shimura variety for $G(U(1,n-1)\times U(n-1,1))$ with parahoric level structure at $p$ for some integer $n\geq 2$. We exhibit elements in the higher Chow group of the supersingular locus of ${{\rm Sh}}_{1,n-1}$ and study the stratification of ${{\rm Sh}}_{1,n-1}.$ Moreover, we study the geometry of ${{\rm Sh}}_{1,n-1}(K_{\mathfrak{p}}^{1})$ and prove a form of Ihara lemma. With Ihara lemma, we prove the the arithmetic level raising map is surjective for $n=2,3.$
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https://arxiv.org/abs/2412.03519
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Academic Papers
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13dd3dbc2e64f0f7aaacfb74ffea61f7f798b6acc6e59e2e4418261cd7664f21
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2026-01-21T00:00:00-05:00
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Rational First Integrals and Relative Killing Tensors
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arXiv:2412.04151v2 Announce Type: replace Abstract: We relate rational integrals of the geodesic flow of a (pseudo-)Riemannian metric to relative Killig tensors, describe the spaces they span and discuss upper bounds on their dimensions.
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https://arxiv.org/abs/2412.04151
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Academic Papers
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233c2ade977ea7e338aac5aa1e54272d7238f9ce009a7270fe7b5725c4550618
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2026-01-21T00:00:00-05:00
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MPAX: Mathematical Programming in JAX
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arXiv:2412.09734v3 Announce Type: replace Abstract: We present MPAX (Mathematical Programming in JAX), an open-source first-order solver for large-scale linear programming (LP) and convex quadratic programming (QP) built natively in JAX. The primary goal of MPAX is to exploit modern machine learning infrastructure for large-scale mathematical programming, while also providing advanced mathematical programming algorithms that are easy to integrate into machine learning workflows. MPAX implements two PDHG variants, r2HPDHG for LP and rAPDHG for QP, together with diagonal preconditioning, adaptive restarts, adaptive step sizes, primal-weight updates, infeasibility detection, and feasibility polishing. Leveraging JAX's compilation and parallelization ecosystem, MPAX provides across-hardware portability, batched solving, distributed optimization, and automatic differentiation. We evaluate MPAX on CPUs, NVIDIA GPUs, and Google TPUs, observing substantial GPU speedups over CPU baselines and competitive performance relative to GPU-based codebases on standard LP/QP benchmarks. Our numerical experiments further demonstrate MPAX's capabilities in high-throughput batched solving, near-linear multi-GPU scaling for dense LPs, and efficient end-to-end differentiable training. The solver is publicly available at https://github.com/MIT-Lu-Lab/MPAX.
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https://arxiv.org/abs/2412.09734
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Academic Papers
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8a6cb5844705bc50fdb43b5159e39283687bec86d52a5fdb6a1a26c20f80f654
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2026-01-21T00:00:00-05:00
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Finite type as fundamental objects even non-single-valued and non-continuous
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arXiv:2412.11675v5 Announce Type: replace Abstract: In this paper, inspired by the elegant work of Good and Meddaugh \cite{GM} and the graph models for zero-dimensional systems developed by several authors, like Gambaudo and Martens \cite{GM06}, Shimomura \cite{Sh14}. We try to discover a connection among some objects, such as finite directed graph, shift of finite type and shadowing property by employing the Closed Graph Theorem for multivalued maps. From the perspective of structure theorems, we demonstrate that every closed relation (multivalued map) on a compact, totally disconnected space is represented as an inverse limit of finite directed graph homomorphisms satisfying the Mittag-Leffler condition. Moreover, from dichotomy-theorem point of view, we prove that an inverse limit of finite directed graph homomorphisms possesses the shadowing property if and only if its induced space of infinite graph walks (as a shift of finite type) satisfies the Mittag-Leffler condition. As an application, a question raised by Boro\'nski, Bruin and Kucharski \cite{BBK} is also concerned. Furthermore, we show that under a multivalued dynamical system, the resulting dynamical behaviors exhibit greater diversity and counterintuitively compared to those observed in single-valued continuous systems.
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https://arxiv.org/abs/2412.11675
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Academic Papers
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a9caf223f639cd5cf00bb3c6c8b9ef7c6dc32b973199063966cddc0793bbac23
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2026-01-21T00:00:00-05:00
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Rare events statistics for $\mathbb Z^d$ map lattices coupled by collision
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arXiv:2412.12803v3 Announce Type: replace Abstract: Understanding the statistics of collisions among locally confined gas particles poses a major challenge. In this work we investigate $\mathbb Z^d$-map lattices coupled by collision with simplified local dynamics that offer significant insights for the above challenging problem. We obtain a first order approximation for the first collision rate at a site $\textbf{p}^*\in \mathbb Z^d$ and we prove a distributional convergence for the first collision time to an exponential, with sharp error term. Moreover, we prove that the number of collisions at site $\textbf{p}^*$ converge in distribution to a compound Poisson distributed random variable. Key to our analysis in this infinite dimensional setting is the use of transfer operators associated with the decoupled map lattice at site $\textbf{p}^*$.
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https://arxiv.org/abs/2412.12803
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Academic Papers
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dd1dd82c5c073b5726ef7802fd603fb9d669cc64c69734b7d4109a1b22aac9a5
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2026-01-21T00:00:00-05:00
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On some Sobolev and P\'olya-Szeg\"o type inequalities with weights and applications
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arXiv:2412.15490v3 Announce Type: replace Abstract: We are motivated by studying a boundary-value problem for a class of semilinear degenerate elliptic equations \begin{align}\tag{P}\label{P} \begin{cases} - \Delta_x u - |x|^{2\alpha} \dfrac{\partial^2 u}{\partial y^2} = f(x,y,u), & \textrm{in } \Omega, u = 0, & \textrm{on } \partial \Omega, \end{cases} \end{align} where $x = (x_1, x_2) \in \mathbb{R}^2$, $\Omega$ is a bounded smooth domain in $\mathbb{R}^3$, $(0,0,0) \in \Omega $, and $\alpha > 0$. In this paper, we will study this problem by establishing embedding theorems for weighted Sobolev spaces. To this end, we need a new P\'olya-Szeg\"o type inequality, which can be obtained by studying an isoperimetric problem for the corresponding weighted area. Our results then extend the existing ones in \cite{nga, Luyen2} to the three-dimensional context.
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https://arxiv.org/abs/2412.15490
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bbf6b846d7a321919d7e466981468ab8393d96ae08d48e5c2544e8a862d2812e
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2026-01-21T00:00:00-05:00
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On the ergodicity of anti-symmetric skew products with singularities and its applications
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arXiv:2412.21067v2 Announce Type: replace Abstract: We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lema\'nczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles.
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https://arxiv.org/abs/2412.21067
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Academic Papers
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95a81a680e0773a11d5554743245da879486fb840dc3acdc89880a2ff8d6196f
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2026-01-21T00:00:00-05:00
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Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Crystallographic Shifts
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arXiv:2501.02130v2 Announce Type: replace Abstract: In this thesis we consider crystal groups in dimension $n$ and their natural unitary representation on $L^2(\mathbb{R}^n)$. We show that this representation is unitarily equivalent to a direct integral of factor representations, and use this to characterize the subspaces of $L^2(\mathbb{R}^n)$ invariant under crystal symmetry shifts. Finally, by giving an explicit unitary equivalence of the natural crystal group representation, we find the \textit{central decomposition} guaranteed by direct integral theory.
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https://arxiv.org/abs/2501.02130
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9a83075ffb772c411b836351555b3bd62c7ab010f4ddeda3e415d870e2f141ab
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2026-01-21T00:00:00-05:00
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Towards a constructive framework for control theory
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arXiv:2501.02267v2 Announce Type: replace Abstract: This work presents a framework for control theory based on constructive analysis to account for discrepancy between mathematical results and their implementation in a computer, also referred to as computational uncertainty. In control engineering, the latter is usually either neglected or considered submerged into some other type of uncertainty, such as system noise, and addressed within robust control. However, even robust control methods may be compromised when the mathematical objects involved in the respective algorithms fail to exist in exact form and subsequently fail to satisfy the required properties. For instance, in general stabilization using a control Lyapunov function, computational uncertainty may distort stability certificates or even destabilize the system despite robustness of the stabilization routine with regards to system, actuator and measurement noise. In fact, battling numerical problems in practical implementation of controllers is common among control engineers. Such observations indicate that computational uncertainty should indeed be addressed explicitly in controller synthesis and system analysis. The major contribution here is a fairly general framework for proof techniques in analysis and synthesis of control systems based on constructive analysis which explicitly states that every computation be doable only up to a finite precision thus accounting for computational uncertainty. A series of previous works is overviewed, including constructive system stability and stabilization, approximate optimal controls, eigenvalue problems, Caratheodory trajectories, measurable selectors. Additionally, a new constructive version of the Danskin's theorem, which is crucial in adversarial defense, is presented.
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https://arxiv.org/abs/2501.02267
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34a66b600952fd1b4cbc683f6421b6b993be1053458d9d1294e9e083787d99ec
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2026-01-21T00:00:00-05:00
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The{N/D}-Conjecture for Nonresonant Hyperplane Arrangements
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arXiv:2501.05189v3 Announce Type: replace Abstract: This paper studies Bernstein--Sato polynomials $b_{f,0}$ for homogeneous polynomials $f$ of degree $d$ with $n$ variables. It is open to know when $-{n\over d}$ is a root of $b_{f,0}$. For essential indecomposable hyperplane arrangements, this is a conjecture by Budur, Musta\c{t}\u{a} and Teitler and implies the strong topological monodromy conjecture for arrangements. Walther gave a sufficient condition that a certain differential form does not vanish in the top cohomology group of Milnor fiber. We use Walther's result to verify the $n\over d$-conjecture for weighted hyperplane arrangements satisfying the nonresonant condition.
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https://arxiv.org/abs/2501.05189
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