Tootfinder

The family of $a$-floor quotient partial orders
Jeffrey C. Lagarias, David Harry Richman
https://arxiv.org/abs/2403.04342 https://arx…

The family of $a$-floor quotient partial orders
An approximate divisor order is a partial order on the positive integers $\mathbb{N}^+$ that refines the divisor order and is refined by the additive total order. A previous paper studied such a partial order on $\mathbb{N}^+$, produced using the floor function. A positive integer $d$ is a floor quotient of $n$, denoted $d \,\preccurlyeq_{1}\, n$, if there is a positive integer $k$ such that $d = \lfloor{n / k}\rfloor$. The floor quotient relation defines a partial order on the positive integer…

Solve arbitrary one-loop reduction with generating function
Tingfei Li, Yuekai Song, Liang Zhang
https://arxiv.org/abs/2404.04644 https://

Solve arbitrary one-loop reduction with generating function
Recently, the concept of generating function has been employed in one-loop reduction. For one-loop integrals encompassing arbitrary tensor ranks and higher-pole contributions, the generating function can be decomposed into a tensor part and a higher-pole part. While the tensor component has been thoroughly addressed in recent studies, there remains a lack of satisfactory investigations regarding the higher-pole part. In this work, we completely solve the problem. We first establish the partial …

Joint effective equidistribution of partial lattices in positive characteristic
Tal Horesh, Fr\'ed\'eric Paulin
https://arxiv.org/abs/2404.04368 ht…

Joint effective equidistribution of partial lattices in positive characteristic
Let $ν$ be a place of a global function field $K$ over a finite field, with associated affine function ring $R_ν$ and completion $K_ν$, and let $1 \leq \mathfrak{m}<\textbf{d}$. The aim of this paper is to prove an effective triple joint equidistribution result for primitive partial $R_ν$-lattices $Λ$ of rank $\mathfrak{m}$ in $K_ν^{\;\textbf{d}}$ as their covolume tends to infinity: of their $K_ν$-linear span $V_Λ$ in the rank-$\mathfrak{m}$ Grassmannian space of $K_ν^{\;\textbf{d}}$;…

This https://arxiv.org/abs/2209.13104 has been replaced.
link: https://scholar.google.com/scholar?q=a

A Neural Network Approach for Stochastic Optimal Control
We present a neural network approach for approximating the value function of high-dimensional stochastic control problems. Our training process simultaneously updates our value function estimate and identifies the part of the state space likely to be visited by optimal trajectories. Our approach leverages insights from optimal control theory and the fundamental relation between semi-linear parabolic partial differential equations and forward-backward stochastic differential equations. To focus …

The $\bar\partial$-problem on $Z(q)$-domains
Debraj Chakrabarti, Phillip S. Harrington, Andrew Raich
https://arxiv.org/abs/2403.04756 https://

The $\bar\partial$-problem on $Z(q)$-domains
Given a complex manifold containing a relatively compact $Z(q)$ domain, we give sufficient geometric conditions on the domain so that its $L^2$-cohomology in degree $(p,q)$ (known to be finite dimensional) vanishes. The condition consists of the existence of a smooth weight function in a neighborhood of the closure of the domain, where the complex Hessian of the weight has a prescribed number of eigenvalues of a particular sign, along with good interaction at the boundary of the Levi form with …

This https://arxiv.org/abs/2402.08239 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_sta…

Interaction Decomposition of prediction function
This paper discusses the foundation of methods for accurately grasping interaction effects. The partial dependence (PD) and accumulated local effects (ALE) methods, which capture interaction effects as terms, are known as global model-agnostic methods in the interpretable machine learning field. ALE provides a functional decomposition of the prediction function. In the present study, we propose and mathematically formalize the requirements of an interaction decomposition (ID) that decomposes a …

Dynamic Gaussian Graph Operator: Learning parametric partial differential equations in arbitrary discrete mechanics problems
Chu Wang, Jinhong Wu, Yanzhi Wang, Zhijian Zha, Qi Zhou
https://arxiv.org/abs/2403.02810

Dynamic Gaussian Graph Operator: Learning parametric partial differential equations in arbitrary discrete mechanics problems
Deep learning methods have access to be employed for solving physical systems governed by parametric partial differential equations (PDEs) due to massive scientific data. It has been refined to operator learning that focuses on learning non-linear mapping between infinite-dimensional function spaces, offering interface from observations to solutions. However, state-of-the-art neural operators are limited to constant and uniform discretization, thereby leading to deficiency in generalization on …

Stochastic partial differential equations for superprocesses in random environments
Jieliang Hong, Jie Xiong
https://arxiv.org/abs/2403.03638 https://

Stochastic partial differential equations for superprocesses in random environments
Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment described by a Gaussian noise $W^g=\{W^g(t,x), t\geq 0, x\in \mathbb{R}^d\}$ white in time and colored in space with correlation kernel $g(x,y)$. We show that when $d=1$, $X_t$ admits a jointly continuous density function $X_t(x)$ that is a unique in law solution to a stochastic partial differential equation \begin{align*} \frac{\partial }{\partial t}X_t(x)=\fracΔ{2} X_t(x)+\sqrt{X_t(x)} \dot{V}(t,x)+X_t(x)\dot{W}^g(t, x) , \qu…

Some observations regarding the RBF-FD approximation accuracy dependence on stencil size
Andrej Kolar-Po\v{z}un, Mitja Jan\v{c}i\v{c}, Miha Rot, Gregor Kosec
https://arxiv.org/abs/2404.03793

Some observations regarding the RBF-FD approximation accuracy dependence on stencil size
When solving partial differential equations on scattered nodes using the Radial Basis Function-generated Finite Difference (RBF-FD) method, one of the parameters that must be chosen is the stencil size. Focusing on Polyharmonic Spline RBFs with monomial augmentation, we observe that it affects the approximation accuracy in a particularly interesting way - the solution error oscillates under increasing stencil size. We find that we can connect this behaviour with the spatial dependence of the si…

Introduction to the monodromy conjecture
Willem Veys
https://arxiv.org/abs/2403.03343 https://arxiv.org/pdf/2403.03343

Introduction to the monodromy conjecture
The monodromy conjecture is a mysterious open problem in singularity theory. Its original version relates arithmetic and topological/geometric properties of a multivariate polynomial $f$ over the integers, more precisely, poles of the $p$-adic Igusa zeta function of $f$ should induce monodromy eigenvalues of $f$. The case of interest is when the zero set of $f$ has singular points. We first present some history and motivation. Then we expose a proof in the case of two variables, and partial res…

This https://arxiv.org/abs/1912.12057 has been replaced.
link: https://scholar.google.com/scholar?q=a

Existence of Schrodinger Evolution with Absorbing Boundary Condition
Consider a non-relativistic quantum particle with wave function inside a region $Ω\subset \mathbb{R}^3$, and suppose that detectors are placed along the boundary $\partial Ω$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the p…

This https://arxiv.org/abs/2311.00369 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Direct System Identification of Dynamical Networks with Partial Measurements: a Maximum Likelihood Approach
This paper introduces a novel direct approach to system identification of dynamic networks with missing data based on maximum likelihood estimation. Dynamic networks generally present a singular probability density function, which poses a challenge in the estimation of their parameters. By leveraging knowledge about the network's interconnections, we show that it is possible to transform the problem into a more tractable form by applying linear transformations. This results in a nonsingular pro…

Estimation and Inference in Ultrahigh Dimensional Partially Linear Single-Index Models
Shijie Cui, Xu Guo, Zhe Zhang
https://arxiv.org/abs/2404.04471 https…

Estimation and Inference in Ultrahigh Dimensional Partially Linear Single-Index Models
This paper is concerned with estimation and inference for ultrahigh dimensional partially linear single-index models. The presence of high dimensional nuisance parameter and nuisance unknown function makes the estimation and inference problem very challenging. In this paper, we first propose a profile partial penalized least squares estimator and establish the sparsity, consistency and asymptotic representation of the proposed estimator in ultrahigh dimensional setting. We then propose an $F$-t…

Analytic conjugation between planar differential systems and potential systems
F. J. S. Nascimento
https://arxiv.org/abs/2404.02036 https://

Analytic conjugation between planar differential systems and potential systems
In this article it is proved that an analytical planar vector field with a non-degenerate center at $(0,0)$ is analytically conjugate, in a neighborhood of $(0,0)$, to a Hamiltonian vector field of the form $y\frac{\partial}{\partial x}-V'(x)\frac{\partial}{\partial y}$, where $V$ is an analytic function defined in a neighborhood of the origin such that $V(0)=V'(0)=0$ and $V''(0)>0.$

This https://arxiv.org/abs/2402.16575 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Optimal shapes for positivity preserving
We are looking for an optimal convex domain on which the boundary value problem $$\left\{\begin{array}{cc}(-Δ)^2 u_γ-γΔu_γ= f,& \mbox{ in }Ω\\ u_γ=\partial_νu_γ=0,& \mbox{ on }\partialΩ\end{array}\right.$$ admits a nonnegative solution for the most $γ$, if $f$ is a given nonnegative function.

This https://arxiv.org/abs/2311.00369 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Direct System Identification of Dynamical Networks with Partial Measurements: a Maximum Likelihood Approach
This paper introduces a novel direct approach to system identification of dynamic networks with missing data based on maximum likelihood estimation. Dynamic networks generally present a singular probability density function, which poses a challenge in the estimation of their parameters. By leveraging knowledge about the network's interconnections, we show that it is possible to transform the problem into a more tractable form by applying linear transformations. This results in a nonsingular pro…

Unique continuation of Schr\"odinger-type equations for $\bar\partial$
Yifei Pan, Yuan Zhang
https://arxiv.org/abs/2404.01947 https://

Unique continuation of Schrödinger-type equations for $\bar\partial$
The purpose of this paper is to study the unique continuation property for a Schrödinger-type equation $ \bar\partial u = Vu$ on a domain in $\mathbb C^n$, where the solution $u$ may be a scalar function, or a vector-valued function. While simple examples show that the unique continuation property fails in general if the potential $V\in L^{p}, p<2n$, we first prove that, in the case when $u $ is a scalar function, the unique continuation property holds when $V\in L_{loc}^{2n}$ and is $\bar\par…

This https://arxiv.org/abs/2208.08626 has been replaced.
link: https://scholar.google.com/scholar?q=a

CP-PINNs: Data-Driven Changepoints Detection in PDEs Using Online Optimized Physics-Informed Neural Networks
We investigate the inverse problem for Partial Differential Equations (PDEs) in scenarios where the parameters of the given PDE dynamics may exhibit changepoints at random time. We employ Physics-Informed Neural Networks (PINNs) - universal approximators capable of estimating the solution of any physical law described by a system of PDEs, which serves as a regularization during neural network training, restricting the space of admissible solutions and enhancing function approximation accuracy. …

Convergence of a Finite Volume Scheme for Compactly Heterogeneous Scalar Conservation Laws
Abraham Sylla
https://arxiv.org/abs/2405.02203 https://

Convergence of a Finite Volume Scheme for Compactly Heterogeneous Scalar Conservation Laws
We build a finite volume scheme for the scalar conservation law $\partial_t u + \partial_x (H(x, u)) = 0$ with bounded initial condition for a wide class of flux function $H$, convex with respect to the second variable. The main idea for the construction of the scheme is to use the theory of discontinuous flux. We prove that the resulting approximating sequence converges boundedly almost everywhere on $\mathopen]0, +\infty\mathclose[$ to the entropy solution.

Computable domains of a Halting Function
Abel Luis Peralta
https://arxiv.org/abs/2404.09484 https://arxiv.org/pdf/2404.09484

Computable domains of a Halting Function
We discuss the possibility of constructing a function that validates the definition or not definition of the partial recursive functions of one variable. This is a topic in computability theory, which was first approached by Alan M. Turing in 1936 in his foundational work "On Computable Numbers". Here we face it using the Model of computability of the recursive functions instead of the Turing's machines, but the results are transferable from one to another paradigm with ease. Recursive function…

This https://arxiv.org/abs/2403.08531 has been replaced.
link: https://scholar.google.com/scholar?q=a

Free fermions, neutrality and modular transformations
With a view towards higher-spin applications, we study the partition function of a free complex fermion in 2d CFT, restricted to the neutral (zero fermion number) sector. This restriction leads to a partial theta function with a combinatoric interpretation in terms of Dyson's crank of a partition. More crucially, this partition function can be expressed in terms of a q-hypergeometric function with quantum modular properties. This allows us to find its high-temperature asymptotics, including sub…

On a conjecture of Gross, Mansour and Tucker for $\Delta$-matroids
Remi Cocou Avohou
https://arxiv.org/abs/2404.13839 https://arxiv.o…

On a conjecture of Gross, Mansour and Tucker for $Δ$-matroids
Gross, Mansour, and Tucker introduced the partial-duality polynomial of a ribbon graph [Distributions, European J. Combin. 86, 1--20, 2020], the generating function enumerating partial duals by the Euler genus. Chmutov and Vignes-Tourneret wondered if this polynomial and its conjectured properties would hold for general delta-matroids, which are combinatorial abstractions of ribbon graphs. Yan and Jin contributed to this inquiry by identifying a subset of delta-matroids-specifically, even norma…

LQ Control of Traffic Flow Models via Variable Speed Limits
Brian Block, Stephanie Stockar
https://arxiv.org/abs/2403.02507 https://a…

LQ Control of Traffic Flow Models via Variable Speed Limits
In this paper, an extension of a linear control design for hyperbolic linear partial differential equations is presented for a first-order traffic flow model. Starting from the Lighthill-Whitham-Richards (LWR) model, variable speed limit control (VSL) is applied through a modification of Greenshield's equilibrium flow model. Then, an optimal linear quadratic (LQ) controller is designed on the linear LWR model. The LQ state feedback function is found via the solution of a Riccati differential eq…

Non-Linear Traces on Semifinite Factors and Generalized Singular Numbers
Masaru Nagisa, Yasuo Watatani
https://arxiv.org/abs/2404.18339 https://

Non-Linear Traces on Semifinite Factors and Generalized Singular Numbers
We introduce non-linear traces of the Choquet type and Sugeno type on a semifinite factor $\mathcal{M}$ as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need weighted dimension function $p \mapsto α(τ(p))$ for projections $p \in \mathcal{M}$, which is an analog of a monotone measure. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and Sugeno type respect…

Euler Products at the Centre and Applications to Chebyshev's Bias
Arshay Sheth
https://arxiv.org/abs/2405.01512 https://arxiv.org…

Euler Products at the Centre and Applications to Chebyshev's Bias
Let $π$ be an irreducible cuspidal automorphic representation of $\text{GL}_n(\mathbb A_\mathbb Q)$ with associated $L$-function $L(s, π)$. We study the behaviour of the partial Euler product of $L(s, π)$ at the center of the critical strip. Under the assumption of the Generalized Riemann Hypothesis for $L(s, π)$ and assuming the Ramanujan--Petersson conjecture when necessary, we establish an asymptotic, off a set of finite logarithmic measure, for the partial Euler product at the central p…

Limitless FaaS: Overcoming serverless functions execution time limits with invoke driven architecture and memory checkpoints
Rodrigo Landa Andraca, Mahdi Zareei
https://arxiv.org/abs/2402.09377

Limitless FaaS: Overcoming serverless functions execution time limits with invoke driven architecture and memory checkpoints
Function-as-a-Service (FaaS) allows to directly submit function code to a cloud provider without the burden of managing infrastructure resources. Each cloud provider establishes execution time limits to their FaaS offerings, which impose the risk of spending computation time without achieving partial results. In this work, a framework that enables limitless execution time in FaaS, with little to no modifications to the user-provided function code, is presented. After a thorough literature and t…

This https://arxiv.org/abs/2202.04573 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the Uniqueness and Stability of the Equilibrium Price in Quasi-Linear Economies
In this paper, we show that if every consumer in an economy has a quasi-linear utility function, then the normalized equilibrium price is unique, and it is locally stable with respect to the tâtonnement process. If the dimension of the consumption space is two, then this result can be expressed by the corresponding partial equilibrium model. Our study can be seen as that extends the results in partial equilibrium theory to economies with more than two dimensional consumption space.

This https://arxiv.org/abs/2305.17079 has been replaced.
link: https://scholar.google.com/scholar?q=a

Complete Multiparty Session Type Projection with Automata
Multiparty session types (MSTs) are a type-based approach to verifying communication protocols. Central to MSTs is a projection operator: a partial function that maps protocols represented as global types to correct-by-construction implementations for each participant, represented as a communicating state machine. Existing projection operators are syntactic in nature, and trade efficiency for completeness. We present the first projection operator that is sound, complete, and efficient. Our proj…

Derivative-enhanced Deep Operator Network
Yuan Qiu, Nolan Bridges, Peng Chen
https://arxiv.org/abs/2402.19242 https://arxiv.org/pdf/2…

Derivative-enhanced Deep Operator Network
Deep operator networks (DeepONets), a class of neural operators that learn mappings between function spaces, have recently been developed as surrogate models for parametric partial differential equations (PDEs). In this work we propose a derivative-enhanced deep operator network (DE-DeepONet), which leverages the derivative information to enhance the prediction accuracy, and provide a more accurate approximation of the derivatives, especially when the training data are limited. DE-DeepONet inco…

Output Feedback Periodic Event-triggered Control of Coupled $2\times 2$ Linear Hyperbolic PDEs
Eranda Somathilake, Bhathiya Rathnayake, Mamadou Diagne
https://arxiv.org/abs/2404.02298

Output Feedback Periodic Event-triggered Control of Coupled $2\times 2$ Linear Hyperbolic PDEs
This article introduces an observer-based periodic event-triggered control (PETC) strategy for boundary control of a system characterized by $2\times2$ linear hyperbolic partial differential equations (PDEs). An anti-collocated actuation and sensing configuration is considered, and an exponentially convergent observer for state estimation from boundary data is designed. Initially, a continuous-time dynamic event-triggering mechanism requiring constant monitoring of the triggering function is de…

This https://arxiv.org/abs/2309.04310 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On polynomial completeness properties of finite Mal'cev algebras
Polynomial completeness results aim at characterizing those functions that are induced by polynomials. Each polynomial function is congruence preserving, but the opposite need not be true. A finite algebraic structure $\mathbf{A}$ is called strictly 1-affine complete if every unary partial function from a subset of $A$ to $A$ that preserves the congruences of $\mathbf{A}$ can be interpolated by a polynomial function of $\mathbf{A}$. The problem of characterizing strictly 1-affine complete finit…

This https://arxiv.org/abs/2402.12316 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A note on the smash product and regular associativity
This note is about the smash product of pointed topological spaces, without relying on some convenient subcategory. We deal with its partial associativity properties and their connection with the function spaces, introducing a property of regular associativity related to the colax monoidal structure of the product. We also study a large class of triples of pointed spaces where associativity fails. Lax and colax monoidal structures are unusual and interesting, in category theory. Some parts of…

Limits on Inferring T-cell Specificity from Partial Information
James Henderson, Yuta Nagano, Martina Milighetti, Andreas Tiffeau-Mayer
https://arxiv.org/abs/2404.12565

Limits on Inferring T-cell Specificity from Partial Information
A key challenge in molecular biology is to decipher the mapping of protein sequence to function. To perform this mapping requires the identification of sequence features most informative about function. Here, we quantify the amount of information (in bits) that T-cell receptor (TCR) sequence features provide about antigen specificity. We identify informative features by their degree of conservation among antigen-specific receptors relative to null expectations. We find that TCR specificity syne…

This https://arxiv.org/abs/2310.12917 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_grqc_…

Exact wave-optical imaging of a Kerr-de Sitter black hole using Heun's equation
Spacetime perturbations due to scalar, vector, and tensor fields on a fixed background geometry can be described in the framework of Teukolsky's equation. In this work, wave scattering is treated analytically, using the Green's function method and solutions to the separated radial and angular differential equations in combination with a partial wave technique for a scalar and monochromatic perturbation. The results are applied to analytically describe wave-optical imaging via Kirchhoff-Fresnel …

LQ Control of Traffic Flow Models via Variable Speed Limits
Brian Block, Stephanie Stockar
https://arxiv.org/abs/2403.02507 https://a…

LQ Control of Traffic Flow Models via Variable Speed Limits
In this paper, an extension of a linear control design for hyperbolic linear partial differential equations is presented for a first-order traffic flow model. Starting from the Lighthill-Whitham-Richards (LWR) model, variable speed limit control (VSL) is applied through a modification of Greenshield's equilibrium flow model. Then, an optimal linear quadratic (LQ) controller is designed on the linear LWR model. The LQ state feedback function is found via the solution of a Riccati differential eq…

A Bayesian approach with Gaussian priors to the inverse problem of source identification in elliptic PDEs
Matteo Giordano
https://arxiv.org/abs/2402.19214 …

A Bayesian approach with Gaussian priors to the inverse problem of source identification in elliptic PDEs
We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on Gaussian priors, leading to convenient conjugate formulae for posterior inference. We review recent results providing theoretical guarantees on the quality of the resulting posterior-based estimation and uncertainty quantification, and we discuss the applicat…

Twisted Adiabatic Limit for Complex Structures
Dan Popovici
https://arxiv.org/abs/2404.06908 https://arxiv.org/pdf/2404.06908<…

Twisted Adiabatic Limit for Complex Structures
Given a complex manifold $X$ and a smooth positive function $η$ thereon, we perturb the standard differential operator $d=\partial + \bar\partial$ acting on differential forms to a first-order differential operator $D_η$ whose principal part is $η\partial + \bar\partial$. The role of the zero-th order part is to force the integrability property $D_η^2=0$ that leads to a cohomology isomorphic to the de Rham cohomology of $X$, while the components of types $(0,\,1)$ and $(1,\,0)$ of $D_η$ in…

This https://arxiv.org/abs/2309.00826 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Metrical properties of finite product of partial quotients in arithmetic progressions
We investigate the dynamics of continued fractions and explore the ergodic behaviour of the products of mixed partial quotients in continued fractions of real numbers. For any function $Φ:\mathbb N\to [2,+\infty)$ and any integer $d\geq 1$, we determine the Lebesgue measure and Hausdorff dimension of the set of real numbers for which the product of partial quotients in arithmetic progressions satisfy $a_n(x)a_{2n}(x)\cdots a_{dn}(x)\geq Φ(n)$ for infinitely many positive integers $n$. Our f…

This https://arxiv.org/abs/2311.04893 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qu…

Maximum Entropy Principle as Postulate of Quantum Mechanics
Even a century after the formulation of Quantum Mechanics (QM), the wave function collapse (WFC) remains a contentious aspect of the theory. Environment-induced decoherence has offered a partial resolution by illustrating how unitary evolution in an open quantum system can lead to effective WFC within its components. However, this approach by itself does not lead to a fully self-consistent reformulation of QM. We introduce a modified set of QM postulates, which exclude both WFC and Born's proba…

Surface profile recovery from electromagnetic field with physics--informed neural networks
Yuxuan Chen, Ce Wang, Yuan Hui, Mark Spivack
https://arxiv.org/abs/2404.14984

Surface profile recovery from electromagnetic field with physics--informed neural networks
Physics--informed neural networks (PINN) have shown their potential in solving both direct and inverse problems of partial differential equations. In this paper, we introduce a PINN-based deep learning approach to reconstruct one-dimensional rough surfaces from field data illuminated by an electromagnetic incident wave. In the proposed algorithm, the rough surface is approximated by a neural network, with which the spatial derivatives of surface function can be obtained via automatic differenti…

This https://arxiv.org/abs/2403.12394 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Calculus of variations on hypergraphs
We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the maximum principle on hypergraphs have also been proven. As applications, we demonstrated how these can be used to study partial differential equations on hypergraphs.

This https://arxiv.org/abs/2307.15819 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Small-time controllability for the nonlinear Schrödinger equation on $\mathbb{R}^N$ via bilinear electromagnetic fields
We address the small-time controllability problem for a nonlinear Schrödinger equation (NLS) on $\mathbb{R}^N$ in the presence of magnetic and electric external fields. We choose a particular framework where the equation becomes $i\partial_t ψ= [-Δ+u_0(t)h_{\vec{0}}+\langle u(t), P\rangle +κ|ψ|^{2p}]ψ$. Here, the control operators are defined by the zeroth Hermite function $h_{\vec{0}}(x)$ and the momentum operator $P=i\nabla$. In detail, we study when it is possible to control …

This https://arxiv.org/abs/2403.06156 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_nli…

Nonlinear Schrödinger equation in terms of elliptic and hyperelliptic $σ$ functions
It is known that the elliptic function solutions of the nonlinear Schrödinger equation are reduced to the algebraic differential relation in terms of the Weierstrass sigma function, $\displaystyle{ \left[-\sqrt{-1}\frac{\partial}{\partial t} -α\frac{\partial}{\partial u}\right]Ψ+\frac{1}{2} \frac{\partial^2}{\partial u^2}Ψ+(Ψ^* Ψ) Ψ= \frac12 (2β+\wp(v)+λ_2-α^2)Ψ}$, where $Ψ(u;v, t):=\mathrm{e}^{αu+\sqrt{-1}βt+c}$ $\displaystyle{\frac{\mathrm{e}^{-ζ(v)u}σ(u+v)}{σ(u)σ(v)}}$, its…

A superconvergence result in the RBF-FD method
Andrej Kolar-Po\v{z}un, Mitja Jan\v{c}i\v{c}, Gregor Kosec
https://arxiv.org/abs/2404.03393 https://<…

A superconvergence result in the RBF-FD method
Radial Basis Function-generated Finite Differences (RBF-FD) is a meshless method that can be used to numerically solve partial differential equations. The solution procedure consists of two steps. First, the differential operator is discretised on given scattered nodes and afterwards, a global sparse matrix is assembled and inverted to obtain an approximate solution. Focusing on Polyharmonic Splines as our Radial Basis Functions (RBFs) of choice, appropriately augmented with monomials, it is we…

This https://arxiv.org/abs/2306.04514 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Grading of affine Weyl semi-groups of Kac-Moody type
For any Kac-Moody root data $\mathcal D$, D. Muthiah and D. Orr have defined a partial order on the semi-direct product $W^+$ of the integral Tits cone with the vectorial Weyl group of $\mathcal D$, and a strictly compatible $\mathbb Z$-valued length function. We classify covers for this order and show that this length function defines a $\mathbb Z$-grading of $W^+$, generalizing the case of affine ADE root systems and giving a positive answer to a conjecture of Muthiah and Orr.

This https://arxiv.org/abs/2210.00309 has been replaced.
link: https://scholar.google.com/scholar?q=a

Some quantitative estimates for the nonlinear Fourier Transform
Recently, Alexei Poltoratski proved pointwise convergence of the non-linear Fourier transform giving a partial answer to a question of Muscalu, Tao and Thiele. We quantify his techniques and, in particular, give an alternative proof of a weaker version of maximal Hausdorff-Young estimate that follows from the work of Christ and Kiselev. Meanwhile, we obtain quantitative estimates for the de Branges function associated to the NLFT through its zeros and the Hardy-Littlewood maximal function of th…

Single-channel, single-energy partial-wave analysis with continuity improved through minimal phase constraints
A. Svarc, R. W. Workman
https://arxiv.org/abs/2404.11990

Single-channel, single-energy partial-wave analysis with continuity improved through minimal phase constraints
Single-energy partial-wave analysis has often been applied as a way to fit data with minimal model dependence. However, remaining unconstrained, partial waves at neighboring energies will vary discontinuously because the overall amplitude phase cannot be determined through single-channel measurements. This problem can be mitigated through the use of a constraining penalty function based on an associated energy-dependent fit. However, the weight given to this constraint results in a biased fit t…

This https://arxiv.org/abs/2309.04310 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On polynomial completeness properties of finite Mal'cev algebras
Polynomial completeness results aim at characterizing those functions that are induced by polynomials. Each polynomial function is congruence preserving, but the opposite need not be true. A finite algebraic structure $\mathbf{A}$ is called strictly 1-affine complete if every unary partial function from a subset of $A$ to $A$ that preserves the congruences of $\mathbf{A}$ can be interpolated by a polynomial function of $\mathbf{A}$. The problem of characterizing strictly 1-affine complete finit…

Mean values of multiplicative functions and applications to residue-class distribution
Paul Pollack, Akash Singha Roy
https://arxiv.org/abs/2402.16266 http…

Mean values of multiplicative functions and applications to residue-class distribution
We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi-Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo $q$, where $q$ varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $ϕ(n)$ among the coprime residues to the "co…

This https://arxiv.org/abs/2201.02323 has been replaced.
link: https://scholar.google.com/scholar?q=a

Distributed Nash Equilibrium Seeking over Time-Varying Directed Communication Networks
We study distributed algorithms for finding a Nash equilibrium (NE) in a class of non-cooperative convex games under partial information. Specifically, each agent has access only to its own smooth local cost function and can receive information from its neighbors in a time-varying directed communication network. To this end, we propose a distributed gradient play algorithm to compute a NE by utilizing local information exchange among the players. In this algorithm, every agent performs a gradie…

Twist polynomial as a weight system for set systems
Qingying Deng, Xian'an Jin, Qi Yan
https://arxiv.org/abs/2404.10216 https://a…

Twist polynomial as a weight system for set systems
Recently, Chmutov proved that the partial-dual polynomial considered as a function on chord diagrams satisfies the four-term relation. Deng et al. then proved that this function on framed chord diagrams also satisfies the four-term relation, i.e., is a framed weight system. In this paper, we extend their results to the twist polynomial of a set system by proving that the twist polynomial on set systems satisfies the four-term relation.

Multiplicity results for fully nonlinear elliptic equations with natural gradient growth
Mohan Mallick, Ram Baran Verma
https://arxiv.org/abs/2404.19042 https://arxiv.org/pdf/2404.19042
arXiv:2404.19042v1 Announce Type: new
Abstract: In this paper, we prove a theorem concerning the existence of three solutions for the following boundary value problem: \begin{equation*} -\mathcal{M}_{\lambda,\Lambda}^ (D^2u)-\Gamma|Du|^2=f(u)~~~\text{in}\ \Omega, u=0~~~\text{on}\ \partial\Omega, \end{equation*} where $f:[0,\infty]\to[0,\infty]$ is a $C^{\alpha}$ function and $\Omega$ denotes a bounded, smooth domain in $\mathbb{R}^N$. By constructing two ordered pairs of sub and supersolutions for a specific class of $f$ exhibiting sublinear growth, we further establish the existence of three positive solutions to the aforementioned boundary value problem.

This https://arxiv.org/abs/2402.13692 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Reconfigurable Intelligent Surface assisted Integrated Communication, Sensing, and Computation Systems
This paper studies a mobile edge computing (MEC) assisted integrated sensing and communication (ISAC), where reconfigurable intelligent surface (RIS) is used to alleviate the attenuation of communication links during computational offloading. In this paradigm, the dual function radar and communication (DFRC)-enabled user equipments (UEs) simultaneously perform radar sensing and communication tasks, facilitating the offloading of partial computational tasks to an edge computing server (ECS) thro…

This https://arxiv.org/abs/2305.00608 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_sta…

Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regression
We study the properties of differentiable neural networks activated by rectified power unit (RePU) functions. We show that the partial derivatives of RePU neural networks can be represented by RePUs mixed-activated networks and derive upper bounds for the complexity of the function class of derivatives of RePUs networks. We establish error bounds for simultaneously approximating $C^s$ smooth functions and their derivatives using RePU-activated deep neural networks. Furthermore, we derive improv…

This https://arxiv.org/abs/2404.04644 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hepp…

Solve arbitrary one-loop reduction with generating function
Recently, the concept of generating function has been employed in one-loop reduction. For one-loop integrals encompassing arbitrary tensor ranks and higher-pole contributions, the generating function can be decomposed into a tensor part and a higher-pole part. While the tensor component has been thoroughly addressed in recent studies, there remains a lack of satisfactory investigations regarding the higher-pole part. In this work, we completely solve the problem. We first establish the partial …

This https://arxiv.org/abs/2011.14747 has been replaced.
link: https://scholar.google.com/scholar?q=a

Asymptotic bias reduction of maximum likelihood estimates via penalized likelihoods with differential geometry
A procedure for asymptotic bias reduction of maximum likelihood estimates of generic estimands is developed. The estimator is realized as a plug-in estimator, where the parameter maximizes the penalized likelihood with a penalty function that satisfies a quasi-linear partial differential equation of the first order. The integration of the partial differential equation with the aid of differential geometry is discussed. Applications to generalized linear models, linear mixed-effects models, and …

Mean values of multiplicative functions and applications to residue-class distribution
Paul Pollack, Akash Singha Roy
https://arxiv.org/abs/2402.16266 http…

Mean values of multiplicative functions and applications to residue-class distribution
We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi-Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo $q$, where $q$ varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $ϕ(n)$ among the coprime residues to the "co…

This https://arxiv.org/abs/2306.16314 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Summation-by-parts operators for general function spaces: The second derivative
Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other appro…

Nonlinear Schr\"odinger equation in terms of elliptic and hyperelliptic $\sigma$ functions
Shigeki Matsutani
https://arxiv.org/abs/2403.06156 https://…

Nonlinear Schrödinger equation in terms of elliptic and hyperelliptic $σ$ functions
It is known that the elliptic function solutions of the nonlinear Schrödinger equation are reduced to the algebraic differential relation in terms of the Weierstrass sigma function, $\displaystyle{ \left[-\sqrt{-1}\frac{\partial}{\partial t} -α\frac{\partial}{\partial u}\right]Ψ+\frac{1}{2} \frac{\partial^2}{\partial u^2}Ψ+(Ψ^* Ψ) Ψ= \frac12 (2β+\wp(v)+λ_2-α^2)Ψ}$, where $Ψ(u;v, t):=\mathrm{e}^{αu+\sqrt{-1}βt+c}$ $\displaystyle{\frac{\mathrm{e}^{-ζ(v)u}σ(u+v)}{σ(u)σ(v)}}$, its…

A note on the smash product and regular associativity
Marco Grandis
https://arxiv.org/abs/2402.12316 https://arxiv.org/pdf/2402.12316…

A note on the smash product and regular associativity
This note is about the smash product of pointed topological spaces, without relying on some convenient subcategory. We deal with its partial associativity properties and their connection with the function spaces, introducing a property of regular associativity related to the colax monoidal structure of the product. We also study a large class of triples of pointed spaces where associativity fails. Lax and colax monoidal structures are unusual and interesting, in category theory. Some parts of…

This https://arxiv.org/abs/2307.05415 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Reliable optimal controls for SEIR models in epidemiology
We present and compare two different optimal control approaches applied to SEIR models in epidemiology, which allow us to obtain some policies for controlling the spread of an epidemic. The first approach uses Dynamic Programming to characterise the value function of the problem as the solution of a partial differential equation, the Hamilton-Jacobi-Bellman equation, and derive the optimal policy in feedback form. The second is based on Pontryagin's maximum principle and directly gives open-loo…

Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green's function, and intermediate asymptotics for a finite thin film with elastic resistance
Nectarios C. Papanicolaou, Ivan C. Christov
https://arxiv.org/abs/2402.18740

Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green's function, and intermediate asymptotics for a finite thin film with elastic resistance
A linear sixth-order partial differential equation (PDE) of ``parabolic'' type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order Sturm--Liouville eigenvalue value problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, …

This https://arxiv.org/abs/2402.07671 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qu…

Piecewise Polynomial Tensor Network Quantum Feature Encoding
This work introduces a novel method for embedding continuous variables into quantum circuits via piecewise polynomial features, utilizing low-rank tensor networks. Our approach, termed Piecewise Polynomial Tensor Network Quantum Feature Encoding (PPTNQFE), aims to broaden the applicability of quantum algorithms by incorporating spatially localized representations suited for numerical applications like solving partial differential equations and function regression. We demonstrate the potential o…

This https://arxiv.org/abs/1802.06294 has been replaced.
link: https://scholar.google.com/scholar?q=a

Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations
We consider the generalized Korteweg-de Vries equation $\partial_t u = -\partial_x(\partial_x^2 u + f(u))$, where $f(u)$ is an odd function of class $C^3$. Under some assumptions on $f$, this equation admits \emph{solitary waves}, that is solutions of the form $u(t, x) = Q_v(x - vt - x_0)$, for $v$ in some range $(0, v_*)$. We study pure two-solitons in the case of the same limit speed, in other words global solutions $u(t)$ such that \begin{equation} \label{eq:abstract} \tag{$\ast$} \lim_{t\to…

This https://arxiv.org/abs/2403.03638 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Stochastic partial differential equations for superprocesses in random environments
Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment described by a Gaussian noise $W^g=\{W^g(t,x), t\geq 0, x\in \mathbb{R}^d\}$ white in time and colored in space with correlation kernel $g(x,y)$. We show that when $d=1$, $X_t$ admits a jointly continuous density function $X_t(x)$ that is a unique in law solution to a stochastic partial differential equation \begin{align*} \frac{\partial }{\partial t}X_t(x)=\fracΔ{2} X_t(x)+\sqrt{X_t(x)} \dot{V}(t,x)+X_t(x)\dot{W}^g(t, x) , \qu…

This https://arxiv.org/abs/2311.16930 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

About a Proposition on Escobar's paper "The Geometry of the First Non-zero Stekloff Eigenvalue"
Let $(M^{2},g_{0})$ be a compact manifold with boundary, and let $g$ and $g_{0}$ be conformally related by $g=e^{2f}g_{0}$. We show that the inequality $$ν_{1}(g)\geq\Big(\max_{x\in\partial M}e^{-f(x)}\Big)ν_{1}(g_{0})$$ stated in Proposition 2 in [1], is only possible when the equality is achieved. In order to achieve such equality, it is required that the function $f$ be constant on $\partial M$, as it is mentioned in Remark 3 also in [1]. Hence, the scope of this inequality is less broad t…

This https://arxiv.org/abs/2201.02323 has been replaced.
link: https://scholar.google.com/scholar?q=a

Distributed Nash Equilibrium Seeking over Time-Varying Directed Communication Networks
We study distributed algorithms for finding a Nash equilibrium (NE) in a class of non-cooperative convex games under partial information. Specifically, each agent has access only to its own smooth local cost function and can receive information from its neighbors in a time-varying directed communication network. To this end, we propose a distributed gradient play algorithm to compute a NE by utilizing local information exchange among the players. In this algorithm, every agent performs a gradie…

Oscillatory integrals and weighted gradient flows
Michael Greenblatt
https://arxiv.org/abs/2403.12751 https://arxiv.org/pdf/2403.1275…

Oscillatory integrals and weighted gradient flows
We investigate estimating scalar oscillatory integrals by integrating by parts in directions based on $(x_1 \partial_{x_1} f(x) ,..., x_n \partial_{x_n}f(x))$, where $f(x)$ is the phase function. We prove a theorem which provides estimates that are uniform with respect to linear perturbations of the phase and investigate some consequences. When the phase function is quasi-homogeneous the theorem gives estimates for the associated surface measure Fourier transforms that are generally not too far…

Logarithm laws for BCZ map
Yiming Li
https://arxiv.org/abs/2403.15160 https://arxiv.org/pdf/2403.15160…

Logarithm laws for BCZ map
We present the logarithm laws for partial sum of itinerary function over non-periodic BCZ orbit using the even and odd diophantine exponent defined by Athreya-Margulis. We give the detailed description of BCZ map and introduce its excursion.

This https://arxiv.org/abs/2401.16573 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the Hörmander's estimate
The motivation of the note is to obtain a Hörmander-type $L^2$ estimate for $\bar\partial$ equation. The feature of the new estimate is that the constant is independent of the weight function. Moreover, our estimate can be used for non-plurisubharmonic weight function.

This https://arxiv.org/abs/2202.04573 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the Uniqueness and Stability of the Equilibrium Price in Quasi-Linear Economies
In this paper, we show that if every consumer in an economy has a quasi-linear utility function, then the normalized equilibrium price is unique, and is locally stable with respect to the tâtonnement process. Our study can be seen as that extends the results in partial equilibrium theory to economies with more than two dimensional consumption space. Moreover, we discuss the surplus analysis in such economies.

This https://arxiv.org/abs/1802.06294 has been replaced.
link: https://scholar.google.com/scholar?q=a

Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations
We consider the generalized Korteweg-de Vries equation $\partial_t u = -\partial_x(\partial_x^2 u + f(u))$, where $f(u)$ is an odd function of class $C^3$. Under some assumptions on $f$, this equation admits \emph{solitary waves}, that is solutions of the form $u(t, x) = Q_v(x - vt - x_0)$, for $v$ in some range $(0, v_*)$. We study pure two-solitons in the case of the same limit speed, in other words global solutions $u(t)$ such that \begin{equation} \label{eq:abstract} \tag{$\ast$} \lim_{t\to…

Which statistical hypotheses are afflicted with false confidence?
Ryan Martin
https://arxiv.org/abs/2404.16228 https://arxiv.org/pdf/…

Which statistical hypotheses are afflicted with false confidence?
The false confidence theorem establishes that, for any data-driven, precise-probabilistic method for uncertainty quantification, there exists (non-trivial) false hypotheses to which the method tends to assign high confidence. This raises concerns about the reliability of these widely-used methods, and shines new light on the consonant belief function-based methods that are provably immune to false confidence. But an existence result alone is insufficient. Towards a partial answer to the title q…

Piecewise Polynomial Tensor Network Quantum Feature Encoding
Mazen Ali, Matthias Kabel
https://arxiv.org/abs/2402.07671 https://arxiv…

Piecewise Polynomial Tensor Network Quantum Feature Encoding
This work introduces a novel method for embedding continuous variables into quantum circuits via piecewise polynomial features, utilizing low-rank tensor networks. Our approach, termed Piecewise Polynomial Tensor Network Quantum Feature Encoding (PPTNQFE), aims to broaden the applicability of quantum algorithms by incorporating spatially localized representations suited for numerical applications like solving partial differential equations and function regression. We demonstrate the potential o…

This https://arxiv.org/abs/2311.12296 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A property of holomorphic functions related to plurisubharmonic functions
Let $φ$ be a negative plurisubharmonic function in a pseudoconvex domain $Ω$ in $\mathbb{C}^{n}$ and $f$ be a bounded holomorphic function belonging to $L^{2}(Ω, φ)$. For all negative plurisubharmonic functions $ψ$ satisfying that $ψ$ is close to $φ$ in the $L^1(Ω)$ norm. Using the recent results about $L^2-$estimate for $\overline{\partial}-$equation and strong openness conjecture, we prove that there exists a holomorphic function $g$ belonging to $L^{2}( Ω, ψ)$ such that $g$ is clos…

This https://arxiv.org/abs/2306.16593 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_sta…

Autoregressive with Slack Time Series Model for Forecasting a Partially-Observed Dynamical Time Series
This study delves into the domain of dynamical systems, specifically the forecasting of dynamical time series defined through an evolution function. Traditional approaches in this area predict the future behavior of dynamical systems by inferring the evolution function. However, these methods may confront obstacles due to the presence of missing variables, which are usually attributed to challenges in measurement and a partial understanding of the system of interest. To overcome this obstacle, …

This https://arxiv.org/abs/2208.02767 has been replaced.
link: https://scholar.google.com/scholar?q=a

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration
We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both…

Second-order Information Promotes Mini-Batch Robustness in Variance-Reduced Gradients
Sachin Garg, Albert S. Berahas, Micha{\l} Derezi\'nski
https://arxiv.org/abs/2404.14758 <…

Second-order Information Promotes Mini-Batch Robustness in Variance-Reduced Gradients
We show that, for finite-sum minimization problems, incorporating partial second-order information of the objective function can dramatically improve the robustness to mini-batch size of variance-reduced stochastic gradient methods, making them more scalable while retaining their benefits over traditional Newton-type approaches. We demonstrate this phenomenon on a prototypical stochastic second-order algorithm, called Mini-Batch Stochastic Variance-Reduced Newton ($\texttt{Mb-SVRN}$), which com…

This https://arxiv.org/abs/2308.00326 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Safety Control of Uncertain MIMO Systems Using Dynamic Output Feedback Barrier Pairs
Safety control of dynamical systems using barrier functions relies on knowing the full state information. This paper introduces a novel approach for safety control in uncertain MIMO systems with partial state information. The proposed method combines the synthesis of a vector norm barrier function and a dynamic output feedback safety controller to ensure robust safety enforcement. The safety controller guarantees the invariance of the barrier function under uncertain dynamics and disturbances. …

This https://arxiv.org/abs/2403.12394 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Calculus of variations on hypergraphs
We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the maximum principle on hypergraphs have also been proven. As applications, we demonstrated how these can be used to study partial differential equations on hypergraphs.

Which statistical hypotheses are afflicted with false confidence?
Ryan Martin
https://arxiv.org/abs/2404.16228 https://arxiv.org/pdf/…

Which statistical hypotheses are afflicted with false confidence?
The false confidence theorem establishes that, for any data-driven, precise-probabilistic method for uncertainty quantification, there exists (non-trivial) false hypotheses to which the method tends to assign high confidence. This raises concerns about the reliability of these widely-used methods, and shines new light on the consonant belief function-based methods that are provably immune to false confidence. But an existence result alone is insufficient. Towards a partial answer to the title q…

On the covariant coefficients of geodesic sprays on Finsler manifolds
S. G. Elgendi, A. Soleiman, Nabil L. Youssef
https://arxiv.org/abs/2404.07995 https:/…

On the covariant coefficients of geodesic sprays on Finsler manifolds
For a Finsler metric $F$, we introduce the notion of $F$-covariant coefficients $H_i$ of the geodesic spray of $F$ (Def. 3.1). We study some geometric consequences concerning the objects $H_i$. If the $F$-covariant coefficients $H_i$ are written in the form $H_i={\dot{\partial}}_iH$, for some smooth function $H$ on ${\mathcal T\hspace{-1pt}M}$, positively 3-homogeneous in y, then $H$ is called spray scalar or simply $S$-scalar. We prove that if the $S$-scalar exists, then it is of the form $H=\…

This https://arxiv.org/abs/2307.10917 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qu…

Efficient quantum amplitude encoding of polynomial functions
Loading functions into quantum computers represents an essential step in several quantum algorithms, such as quantum partial differential equation solvers. Therefore, the inefficiency of this process leads to a major bottleneck for the application of these algorithms. Here, we present and compare two efficient methods for the amplitude encoding of real polynomial functions on $n$ qubits. This case holds special relevance, as any continuous function on a closed interval can be uniformly approxim…

This https://arxiv.org/abs/2203.01360 has been replaced.
link: https://scholar.google.com/scholar?q=a

Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations
Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires informative training data that are often challenging to collect in science and engineering applications. This work proposes Neural Galerkin schemes based on deep learning that generate training data with active learning for numerically solving high-dimensional partial differential equations. Neural Galerkin schemes build on the Dirac-Frenkel variation…

This https://arxiv.org/abs/2309.13350 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Generalized impedance boundary conditions with vanishing or sign-changing impedance
We consider a Laplace type problem with a generalized impedance boundary condition of the form $\partial_νu=-\partial_x(g\partial_xu)$ on a flat part $Γ$ of the boundary. Here $ν$ is the outward unit normal vector to $\partialΩ$, $g$ is the impedance function and $x$ is the coordinate along $Γ$. Such problems appear for example in the modelling of small perturbations of the boundary. In the literature, the cases $g=1$ or $g=-1$ have been investigated. In this work, we address situations wh…

This https://arxiv.org/abs/2401.16344 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A $\mathrm{L}^2$-maximum principle for circular arcs on the disk
In this article, we prove a novel $\mathrm{L}^2$-maximum principle for harmonic functions on the disk with respect to circular arcs. More precisely, we prove that for any harmonic function $u$ on a disk $Ω$ with non-tangential maximal function in $\mathrm{L}^2(\partial Ω)$, the supremum of $\lVert u \rVert_{\mathrm{L}^2 (Γ)}$ over circular arcs $Γ\subset \overlineΩ$ is attained at the boundary $Γ= \partial Ω$. We achieve this through a sharp geometry-dependent estimate on the norm $\lVer…

This https://arxiv.org/abs/2403.15160 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Logarithm laws for BCZ map
We present the logarithm laws for the partial sum of the itinerary function over non-periodic BCZ orbits, utilizing the even and odd Diophantine exponent defined by Athreya-Margulis. We also give a detailed description of the BCZ map and its excursions.

The distance problem on measured metric spaces
David J. Aldous, Guillaume Blanc, Nicolas Curien
https://arxiv.org/abs/2403.10926 https://

The distance problem on measured metric spaces
What distributions arise as the distribution of the distance between two typical points in some measured metric space? This seems to be a surprisingly subtle problem. We conjecture that every distribution with a density function whose support contains $0$ does arise in this way, and give some partial results in that direction.

This https://arxiv.org/abs/2302.05013 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Compactness of Toeplitz operators with continuous symbols on pseudoconvex domains in $\mathbb{C}^n$
Let $Ω$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ with Lipschitz boundary and $ϕ$ be a continuous function on $\overlineΩ$. We show that the Toeplitz operator $T_ϕ$ with symbol $ϕ$ is compact on the weighted Bergman space if and only if $ϕ$ vanishes on the boundary of $Ω$. We also show that compactness of the Toeplitz operator $T^{p,q}_ϕ$ on $\overline{\partial}$-closed $(p,q)$-forms for $0\leq p\leq n$ and $q\geq 1$ is equivalent to $ϕ=0$ on $Ω$.

This https://arxiv.org/abs/2302.05013 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Compactness of Toeplitz operators with continuous symbols on pseudoconvex domains in $\mathbb{C}^n$
Let $Ω$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ with Lipschitz boundary and $ϕ$ be a continuous function on $\overlineΩ$. We show that the Toeplitz operator $T_ϕ$ with symbol $ϕ$ is compact on the weighted Bergman space if and only if $ϕ$ vanishes on the boundary of $Ω$. We also show that compactness of the Toeplitz operator $T^{p,q}_ϕ$ on $\overline{\partial}$-closed $(p,q)$-forms for $0\leq p\leq n$ and $q\geq 1$ is equivalent to $ϕ=0$ on $Ω$.

This https://arxiv.org/abs/2111.09765 has been replaced.
link: https://scholar.google.com/scholar?q=a

Boundary $C^{2, α}$ Regularity for the Oblique Boundary Value Problem of Monge-Ampère Equations
We study the good shape property of boundary sections of convex solutions of the oblique boundary value problem for Monge-Ampère equations $$\det D^2u =f(x) \text{ in } Ω, \quad D_βu = ϕ(x) \text{ on } \partial Ω.$$ In the two-dimensional case, we prove the global $C^{2,α}$ estimate for the solution. When the dimension $n \geq 3$, we show that this estimate still holds if the solution is bounded from above by a quadratic function in the tangent direction. We also obtain an existence resul…

This https://arxiv.org/abs/2112.00994 has been replaced.
link: https://scholar.google.com/scholar?q=a

Transforming Radical Differential Equations to Algebraic Differential Equations
In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial relations among them by means of a rational change of variables. The solutions of the given equation and its transformation correspond one-to-one. This work can be seen as a generalization of previous work on reparametrization of ODEs and PDEs with radical c…

Calculus of variations on hypergraphs
Mengqiu Shao, Yulu Tian, Liang Zhao
https://arxiv.org/abs/2403.12394 https://arxiv.org/pdf/2403…

Calculus of variations on hypergraphs
We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the maximum principle on hypergraphs have also been proven. As applications, we demonstrated how these can be used to study partial differential equations on hypergraphs.

Dirichlet problems associated to abstract nonlocal space-time differential operators
Joshua Willems
https://arxiv.org/abs/2404.11289 https://

Dirichlet problems associated to abstract nonlocal space-time differential operators
Let the abstract fractional space--time operator $(\partial_t + A)^s$ be given, where $s \in (0,\infty)$ and $-A \colon \mathsf{D}(A) \subseteq X \to X$ is a linear operator generating a uniformly bounded strongly measurable semigroup $(S(t))_{t\ge0}$ on a complex Banach space $X$. We consider the corresponding Dirichlet problem of finding a function $u \colon \mathbb{R} \to X$ such that $(\partial_t + A)^s u(t) = 0$ on $(t_0, \infty)$ and $u(t) = g(t)$ on $(-\infty, t_0]$, for given $t_0 \in \…

This https://arxiv.org/abs/2309.14184 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Linearly implicit exponential integrators for damped Hamiltonian PDEs
Structure-preserving linearly implicit exponential integrators are constructed for Hamiltonian partial differential equations with linear constant damping. Linearly implicit integrators are derived by polarizing the polynomial terms of the Hamiltonian function and portioning out the nonlinearly of consecutive time steps. They require only a solution of one linear system at each time step. Therefore they are computationally more advantageous than implicit integrators. We also construct an expone…

This https://arxiv.org/abs/2308.16129 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms
In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -Δ_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $Ω$,} \newline u\geq 0 & \text{in $Ω$,} \newline u=0 & \text{on $\partial Ω$,} \ \end{cases} \end{equation*} in a domain $Ω\subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p

Quantitative estimates in almost periodic homogenization of parabolic systems
Jun Geng, Bojing Shi
https://arxiv.org/abs/2403.07672 https://

Quantitative estimates in almost periodic homogenization of parabolic systems
We consider a family of second-order parabolic operators $\partial_t+\mathcal{L}_\varepsilon$ in divergence form with rapidly oscillating, time-dependent and almost-periodic coefficients. We establish uniform interior and boundary Hölder and Lipschitz estimates as well as convergence rate. The estimates of fundamental solution and Green's function are also established. In contrast to periodic case, the main difficulty is that the corrector equation $ (\partial_s+\mathcal{L}_1)(χ^β_{j})=-\mat…

This https://arxiv.org/abs/2310.02026 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On Harnack inequality to the homogeneous nonlinear degenerate parabolic equations
In this paper, the Harnack inequality result are established for a new class of the homogeneous nonlinear degenerate parabolic equations \begin{align*} div A(t,x,u,\nabla_x u)-\partial_t \vert u\vert^{p-2}u=0 \end{align*} on a bounded domain $ D \subset R^{n+1}. $ Let $A(t,x,ξ,η)$ be measurable function on $R\times R^n\times R\times R^n\to R^n$ that satisfies the Caratheodory conditions for $ \, \text{arbitrary } \, (t,x)\in D$ and $(ξ,η)\in R^{1}\times R^n.$ The following growth conditions…

This https://arxiv.org/abs/2303.15458 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Stochastic Method for Solving Time-Fractional Differential Equations
We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resorting to a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function. This is accomplished through the approximation of the expected value of a suitable multiplicative functional of a st…

The fourth-order Schr\"{o}dinger equation on lattices
Jiawei Cheng
https://arxiv.org/abs/2403.07445 https://arxiv.org/pdf/2403.0…

The fourth-order Schrödinger equation on lattices
In this paper, we study the fourth-order Schrödinger equation \begin{equation*} i \partial_t u + Δ^2 u - γΔu = \pm |u|^{s-1}u \end{equation*} on the lattice $\mathbb{Z}^d$ with dimensions $d=1,2$ and parameter $γ\in \mathbb{R}$. In order to establish sharp dispersive estimates, we consider the fundamental solution as an oscillatory integral and analyze the Newton polyhedron of its phase function. Furthermore, we prove Strichartz estimates which yield the existence of global sol…