arXiv Math @arxiv_math@qoto.org

We review an algorithm developed for parameter estimation within the Continuous Data Assimilation (CDA) approach. We present an alternative derivation for the algorithm presented in a paper by Carlson, Hudson, and Larios (CHL, 2021). This derivation relies on the same assumptions as the previous derivation but frames the problem as a finite dimensional root-finding problem. Within the approach we develop, the algorithm developed in (CHL, 2021) is simply a realization of Newton's method. We then consider implementing other derivative based optimization algorithms; we show that the Levenberg Maqrquardt algorithm has similar performance to the CHL algorithm in the single parameter estimation case and generalizes much better to fitting multiple parameters. We then implement these methods in three example systems: the Lorenz '63 model, the two-layer Lorenz '96 model, and the Kuramoto-Sivashinsky equation.

Among all metrics based on p-norms, the Manhattan (p=1), euclidean (p=2) and Chebyshev distances (p=infinity) are the most widely used for their interpretability, simplicity and technical convenience. But these are not the only arguments for the ubiquity of these three p-norms. This article proves that there is a volume-surface correspondence property that is unique to them. More precisely, it is shown that sampling uniformly from the volume of an n-dimensional p-ball and projecting to its surface is equivalent to directly sampling uniformly from its surface if and only if p is 1, 2 or infinity. Sampling algorithms and their implementations in Python are also provided.

Symplectic integrators offer vastly superior performance over traditional numerical techniques for conservative dynamical systems, but their application to \emph{dissipative} systems is inherently difficult due to dissipative systems' lack of symplectic structure. Leveraging the intrinsic variational structure of higher-order dynamics, this paper presents a general technique for applying existing symplectic integration schemes to dissipative systems, with particular emphasis on viscous fluids modeled by the Navier-Stokes equations. Two very simple such schemes are developed here. Not only are these schemes unconditionally stable for dissipative systems, they also outperform traditional methods with a similar degree of complexity in terms of accuracy for a given time step. For example, in the case of viscous flow between two infinite, flat plates, one of the schemes developed here is found to outperform both the implicit Euler method and the explicit fourth-order Runge-Kutta method in predicting the velocity profile. To the authors' knowledge, this is the very first time that a symplectic integration scheme has been applied successfully to the Navier-Stokes equations. We interpret the present success as direct empirical validation of the canonical Hamiltonian formulation of the Navier-Stokes problem recently published by Sanders~\emph{et al.} More sophisticated symplectic integration schemes are expected to exhibit even greater performance. It is hoped that these results will lead to improved numerical methods in computational fluid dynamics.

Model order reduction (MOR) is essential in integrated circuit design, particularly when dealing with large-scale electromagnetic models extracted from complex designs. The numerous passive elements introduced in these models pose significant challenges in the simulation process. MOR methods based on balanced truncation (BT) help address these challenges by producing compact reduced-order models (ROMs) that preserve the original model's input-output port behavior. In this work, we present an extended Krylov subspace-based BT approach with a frequency-aware convergence criterion and efficient implementation techniques for reducing large-scale models. Experimental results indicate that our method generates accurate and compact ROMs while achieving up to x22 smaller ROMs with similar accuracy compared to ANSYS RaptorX ROMs for large-scale benchmarks.

This research thesis presents a novel higher-order spectral element method (SEM) formulated in cylindrical coordinates for analyzing electromagnetic fields in waveguides filled with complex anisotropic media. In this study, we consider a large class of cylindrical waveguides: radially-bounded and radially-unbounded domains; homogeneous and inhomogeneous waveguides; concentric and non-concentric geometries; Hermitian and non-Hermitian anisotropic media tensors. This work explores different wave equation formulations for one-layer eccentric and multilayer cylindrical waveguides. For the first case, we can define a new normalized scalar Helmholtz equation for decoupling TM and TE modes, and for the second, a vectorial Helmholtz equation for hybrid modes in multilayered anisotropic structures. Additionally, we formulate a transformation optics (TO) framework to include non-symmetric and non-Hermitian media tensors for non-concentric multilayer waveguides. Lastly, we model excitation sources for logging sensors applied in geophysical problems using the fields obtained by SEM. We validate the proposed approach against analytical solutions, perturbation-based and mode-matching-based methods, finite-elements, and finite-integration numerical methods. Our technique obtains accurate results with fewer elements and degrees of freedom (DoF) than Cartesian-based SEM and ordinary finite-element approaches. To this end, we use higher-order two-dimensional basis functions associated with the zeros of the completed Lobatto polynomial to model the fields in each reference element. The convergence analysis demonstrates the absence of the Runge effect as the expansion order increases. Numerical results show that our formulation is efficient and accurate for modeling cylindrical waveguided geometries filled with complex media.

We consider a radiation solution $ψ$ for the Helmholtz equation in an exterior region in $\mathbb R^2$. We show that $ψ$ in the exterior region is uniquely determined by its imaginary part $Im(ψ)$ on an interval of a line $L$ lying in the exterior region. This result has holographic prototype in the recent work [Nair, Novikov, arXiv:2408.08326]. Some other curves for measurements instead of the lines $L$ are also considered. Applications to the Gelfand-Krein-Levitan inverse problem and passive imaging are also indicated.

Probability distribution theory helps in studying the impact of various dimensions in life while the Mittag-Leffler function and bicomplex are used in electromagnetism, quantum mechanics, and signal theory. Considering the importance of both, the purpose of this paper is to introduce bicomplex Mittag-Leffler distribution theory with the help of the bicomplex Mittag-Leffler function. Moreover, it also tells us about the moment-generating function, the first four moments, the mean, and the variance of this endeavor.

A class of linear parabolic equations is considered. We derive a framework for the a posteriori error analysis of time discretisations by Richardson extrapolation of arbitrary order combined with finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator. The crucial point in the analysis is the design of suitable polynomial reconstructions in time from approximations that are given only in the mesh points.

We formulate and prove $\textit{a priori}$ bounds for the renormalization of Hénon-like maps (under certain regularity assumptions). This provides a certain uniform control on the small-scale geometry of the dynamics, and ensures pre-compactness of the renormalization sequence. In a sequel to this paper, a priori bounds are used in the proof of the main results, including renormalization convergence, finite-time checkability of the required regularity conditions and regular unicriticality of the dynamics.

Diestel, Hundertmark and Lemanczyk asked whether every $k$-tangle in a graph is induced by a set of vertices by majority vote. We reduce their question to graphs whose size is bounded by a function in $k$. Additionally, we show that if for any fixed $k$ this problem has a positive answer, then every $k$-tangle is induced by a vertex set whose size is bounded in $k$. More generally, we prove for all $k$ that every $k$-tangle in a graph $G$ is induced by a weight function $V(G) \to \mathbb{N}$ whose total weight is bounded in $k$. As the key step of our proofs, we show that any given $k$-tangle in a graph $G$ is the lift of a $k$-tangle in some topological minor of $G$ whose size is bounded in $k$.

This article introduces a weak pseudo-inverse of a monotone function, which is applied to prove that the associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=t^{[-1]}(F(t(x),t(y)))$ where $F:[0,\infty]^2\rightarrow[0,\infty]$ is an associative function with neutral element in $[0,\infty]$, $t: [0,1]\rightarrow [0,\infty]$ is a left continuous monotone function and $t^{[-1]}:[0,\infty]\rightarrow[0,1]$ is the weak pseudo-inverse of $t$ depends only on properties of the range of $t$.

It is conjectured since long that for any convex body $P\subset \mathbb{R}^n$ there exists a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $P$. The conjecture is proven for $n=2,3,4$. We treat the same problem for convex polytopes in $\mathbb{R}^3$ and prove that each generic simple polytope has a point in its interior with $10$ normals to the boundary. This is an exact bound: there exists a tetrahedron with no more than $10$ normals from a point in its interior.

Let $Δ= \sum_{m=0}^\infty q^{(2m+1)^2} \in \mathbb{F}_2[[q]]$ be the reduction mod 2 of the $Δ$ series. A modular form $f$ modulo $2$ of level 1 is a polynomial in $Δ$. If $p$ is an odd prime, then the Hecke operator $T_p$ transforms $f$ in a modular form $T_p(f)$ which is a polynomial in $Δ$ whose degree is smaller than the degree of $f$, so that $T_p$ is nilpotent. The order of nilpotence of $f$ is defined as the smallest integer $g=g(f)$ such that, for every family of $g$ odd primes $p_1,p_2,\ldots,p_g$, the relation $T_{p_1}T_{p_2}\ldots T_{p_g}(f)=0$ holds. We show how one can compute explicitly $g(f)$; if $f$ is a polynomial of degree $d\geqslant 1$ in $Δ$, one finds that $g(f) < \frac 32 \sqrt d$.

We study the Ollivier-Ricci curvature and its modification introduced by Lin, Lu, and Yau on graphs. We provide a complete characterization of all graphs with Lin-Lu-Yau curvature at least one. We then explore the relationship between the Lin-Lu-Yau curvature and the Ollivier-Ricci curvature with vanishing idleness on regular graphs. An exact formula for the difference between these two curvature notions is established, along with an equality condition. This condition allows us to characterize edges that are bone-idle in regular graphs. Furthermore, we demonstrate the non-existence of 3-regular bone-idle graphs and present a complete characterization of all 4-regular bone-idle graphs. We also show that there exist no 5-regular bone-idle graphs that are symmetric or a Cartesian product of a 3-regular and a 2-regular graph.

We extend two known constructions that relate regular subdivisions to initial degenerations of projective toric varieties and Grassmannians. We associate a point configuration $A$ with any homogeneous ideal $I$. We obtain upper and lower bounds on each initial ideal of $I$ in terms of regular subdivisions of $A$. Both bounds can be interpreted categorically via limits over face posets of subdivisions. We also investigate when these bounds are exact. This is an extended abstract of a forthcoming paper.

The Aldous-Hoover Theorem concerns an infinite matrix of random variables whose distribution is invariant under finite permutations of rows and columns. It states that, up to equality in distribution, each random variable in the matrix can be expressed as a function only depending on four key variables: one common to the entire matrix, one that encodes information about its row, one that encodes information about its column, and a fourth one specific to the matrix entry. We state and prove the theorem within a category-theoretic approach to probability, namely the theory of Markov categories. This makes the proof more transparent and intuitive when compared to measure-theoretic ones. A key role is played by a newly identified categorical property, the Cauchy--Schwarz axiom, which also facilitates a new synthetic de Finetti Theorem. We further provide a variant of our proof using the ordered Markov property and the d-separation criterion, both generalized from Bayesian networks to Markov categories. We expect that this approach will facilitate a systematic development of more complex results in the future, such as categorical approaches to hierarchical exchangeability.

A famous conjecture of Chowla on the least primes in arithmetic progressions implies that the abscissa of convergence of the Weil representation zeta function for a procyclic group $G$ only depends on the set $S$ of primes dividing the order of $G$ and that it agrees with the abscissa of the Dedekind zeta function of $\mathbb{Z}[p^{-1}\mid p \not\in S]$. Here we show that these consequences hold unconditionally for random procyclic groups in a suitable model. As a corollary, every real number $1 \leq β\leq 2$ is the Weil abscissa of some procyclic group.

In this paper we prove a reverse Hölder inequality for the variable exponent Muckenhoupt weights $\mathcal{A}_{p(\cdot)}$, introduced by the first author, Fiorenza, and Neugeabauer. All of our estimates are quantitative, showing the dependence of the exponent function on the $\mathcal{A}_{p(\cdot)}$ characteristic. As an application, we use the reverse Hölder inequality to prove that the matrix $\mathcal{A}_{p(\cdot)}$ weights, introduced in our previous paper, have both a right and left-openness property. This result is new even in the scalar case.

We establish the independence of multipliers for polynomial endomorphisms of $\mathbb C^n$ and endomorphisms of $\mathbb P^n.$ This allows us to extend results about the bifurcation measure and the critical height obtained in \cite{arXiv:2305.02246} to the case of polynomial endomorphisms of $\mathbb C^n$ for $n\geq 3$. An important step in the proof is the irreducibility of the spaces of endomorphisms with $N$ marked periodic points, which is of independent interest.

In this expository article, we study and discuss invariants of vector fields and holomorphic foliations that intertwine the theories of complex analytic singular varieties and singular holomorphic foliations on complex manifolds: two different settings with many points in common.