AI for Science

We prove the method of power iteration for matrices with at most finite entries from the Levi-Civita field $\mathcal C$ under the assumption that there exists an eigenvalue with the strictly largest in absolute value complex part. In this case the weak convergence of a start vector to the eigenvector, that corresponds to the largest eigenvalue, is proven. Further, we prove that the Rayleigh quotient of the largest eigenvector also converges weakly to the corresponding eigenvalue. As a corollary, the same holds for matrices and polynomials over the Puiseux series field. In addition to that, we deliver an implementation of our method in Python.

We use the measure theoretical entropy from Vivas-Sirvent and the topological entropies from Dinh-Sibony and Kelly-Tennant to prove that the equidistribution measures $μ_-$ and $μ_+$ found by the author maximize entropy for the $1$-parameter family $\lbrace\mathcal{F}_a\rbrace_a$ studied by Bullett-Lomonaco. We find as well a measure of maximal entropy for the composition of two deleted covering correspondences studied by Bullett.

Associated to any root datum, there is an elliptic affine Hecke algebra defined by Ginzburg, Kapranov, and Vasserot. In this study, we establish a Fourier-Mukai functor from the representation category of the elliptic affine Hecke algebra to the corresponding category associated with the Langlands dual root datum. To achieve this connection, we employ the elliptic Hecke algebra with dynamical parameters as an intermediary.

We prove that a smooth, complex plane curve $C$ of odd degree can be defined by a polynomial with real coefficients if and only if $C$ is isomorphic to its complex conjugate. Counterexamples are known for curves of even degree. More generally, we prove that a plane curve $C$ over an algebraically closed field $K$ of characteristic $0$ with field of moduli $k_{C}\subset K$ is defined by a polynomial with coefficients in $k'$, where $k'/k_{C}$ is an extension with $[k':k_{C}]\le 3$ and $[k':k_{C}]\mid \operatorname{deg} C$.

We investigate properties of spectrum of normalized Laplacian $\mathcal L$ for finite graphs over non-Archimedean ordered fields. We prove a Cheeger's inequality for first non-zero eigenvalue. Then we describe properties of the operator $\mathcal P=I-\mathcal L$, which is a generalization of transition operator. We show that Cheeger estimate $α_1\preceq \sqrt{1-h^2}$ for the second largest eigenvalue of $\mathcal P$ is crucial for investigation of the convergence of analogue of random walk to equilibrium over a non-Archimedean ordered fields. We consider examples over the Levi-Civita field.

Let $f$ be a primitive form of weight $2k+j-2$ for $SL_2(Z)$, and let $P$ be a prime ideal of the Hecke field of $f$. We denote by $Sp_m(Z)$ the Siegel modular group of degree $m$. Suppose that $k$ is congruent to $0$ modulo $4$, $j$ is congruent to $0$ modulo $4$, and that $P$ divides the algebraic part of $L(k+j,f)$. Put ${\bf k}=(k+j/2,k+j/2,j/2+4,j/2+4)$. Then under certain easily checkable conditions, we prove that there exists a Hecke eigenform $F$ in the space of modular forms of weight $(k+j,k)$ for $Sp_2(Z)$ such that $[I_2(f)]^{\bf k}$ is congruent to $A^{(I)}_4(F)$ modulo $P$. Here, $[I_2(f)]^{\bf k}$ is the Klingen-Eisenstein lift of the Saito-Kurokawa lift $I_2(f)$ of $f$ to the space of modular forms of weight ${\bf k}$ for $Sp_4(Z)$, and $A^{(I)}_4(F)$ is a certain lift of $F$ to the space of cusp forms of weight ${\bf k}$ for $Sp_4(Z)$. As an application, we prove Harder's conjecture on the congruence between the Hecke eigenvalues of $F$ and some quantities related to the Hecke eigenvalues of $f$. This version gives proofs of Lemmas 7.2 and 7.3 and Corollaries 7.4 and 7.5 in the paper arXiv:2306.07582v2.

Many seismic imaging methods use wave field extrapolation operators to redatum sources and receivers from the surface into the subsurface. We discuss wave field extrapolation operators that account for internal multiple reflections, in particular propagator matrices, transfer matrices and Marchenko focusing functions. A propagator matrix is a square matrix that `propagates' a wave-field vector from one depth level to another. It accounts for primaries and multiples and holds for propagating and evanescent waves. A Marchenko focusing function is a wave field that focuses at a designated point in space at zero time. Marchenko focusing functions are useful for retrieving the wave field inside a heterogeneous medium from the reflection response at its surface. By expressing these focusing functions in terms of the propagator matrix, the usual approximations (such as ignoring evanescent waves) are avoided. While a propagator matrix acts on the full wave-field vector, a transfer matrix (according to the definition employed in this paper)`transfers' a decomposed wave-field vector (containing downgoing and upgoing waves) from one depth level to another. It can be expressed in terms of decomposed Marchenko focusing functions. We present propagator matrices, transfer matrices and Marchenko focusing functions in a consistent way and discuss their mutual relations. In the main text we consider the acoustic situation and in the appendices we discuss other wave phenomena. Understanding these mutual connections may lead to new developments of Marchenko theory and its applications in wave field focusing, Green's function retrieval and imaging.

We prove that a smooth, complex plane curve of odd degree can be defined by a polynomial with coefficients in $\mathbb{R}$ if and only if it is isomorphic to its complex conjugate; there are counterexamples in even degree. Over arbitrary base fields of characteristic $0$, we prove that a smooth plane curve of degree prime with $6$ can be defined by a polynomial with coefficients in the field of moduli. We also prove results about fields of moduli of algebraic cycles in $\mathbb{P}^{2}$. In particular, these apply to singular plane curves of arbitrary degree, too.

We study the behavior of the pressure function for Hölder continuous potentials on mixing subshifts of finite type. The classical theory of thermodynamic formalism shows that such pressure functions are convex, analytic and have slant asymptotes. We provide a sharp exponential lower bound on how fast the pressure function approaches its asymptotes. As a counterpart, we also show that there is no corresponding upper bound by exhibiting systems for which the convergence is arbitrarily slow. However, we prove that the exponential upper bound still holds for a generic Hölder potential. In addition, we determine that the pressure function satisfies a coarse uniform convexity property. Asymptotic bounds and quantitative convexity estimates are the first additional general properties of the pressure function obtained in the settings of Bowen and Ruelle since their groundbreaking work more than 40 years ago.

This paper introduces the trivial fiber topology on schemes. For one-dimensional base schemes, we use it to describe fibrant replacements in the stable motivic homotopy category and motivic infinite loop spaces. We also extend the Garkusha-Panin and Voevodsky strict $\mathbb{A}^{1}$-invariance theorems to one-dimensional base schemes. The trivial fiber topology plays a central role in the proof of refined localization results for motivic homotopy categories. Moreover, we extend Morel's $\mathbb{A}^{1}$-connectivity theorem on Nisnevich sheaves of stable motivic homotopy groups. These results open new vistas for computations of motivic invariants over deeper base schemes of arithmetic interest.

A method of constructing algebraic-geometric codes with many automorphisms arising from Galois points for algebraic curves is presented.

In Part I, the present authors introduced the notion of a quasi-Galois point, for investigating the automorphism groups of plane curves. In this second part, the number of quasi-Galois points for smooth plane curves is described. In particular, sextic or quartic curves with many quasi-Galois points are characterized.