Showing papers in "Tohoku Mathematical Journal in 2022"

Waring rank of binary forms, harmonic cross-ratio and golden ratio

Abstract: We discuss the Waring rank of binary forms of degree 4 and 5, without multiple factors, and point out unexpected relations to the harmonic cross-ratio, j-invariants and the golden ratio. These computations of ranks for binary forms are used to show that the combinatorics of a line arrangement in the complex projective plane does not determine the Waring rank of the defining equation even in very simple situations.

Stokes matrices for Airy equations

Abstract: We compute Stokes matrices for generalised Airy equations and prove that they are regular unipotent (up to multiplication with the formal monodromy). This class of differential equations was defined by Katz and includes the classical Airy equation. In addition, it includes differential equations which are not rigid. Our approach is based on the topological computation of Stokes matrices of the enhanced Fourier-Sato transform of a perverse sheaf due to D'Agnolo, Hien, Morando and Sabbah.

Counting rooted spanning forests for circulant foliation over a graph

Abstract: In this paper, we present a new method to produce explicit formulas for the number of rooted spanning forests $f(n)$ for the infinite family of graphs $H_n=H_n(G_1,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph $H$ on $m$ vertices with fibers $G_1,G_2,\ldots,G_m.$ Each fiber $G_i=C_n(s_{i,1},s_{i,2},\ldots,s_{i,k_i})$ of this foliation is the circulant graph on $n$ vertices with jumps $s_{i,1},s_{i,2},\ldots,s_{i,k_i}.$ This family includes the family of generalized Petersen graphs, $I$-graphs, sandwiches of circulant graphs, discrete torus graphs and others. The formulas are expressed through Chebyshev polynomials. We prove that the number of rooted spanning forests can be represented in the form $f(n)=p f(H)a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending on the number of odd elements in the set of $s_{i,j}.$ Finally, we find an asymptotic formula for $f(n)$ through the Mahler measure of the associated Laurent polynomial.

Laplacian comparison theorem on Riemannian manifolds with modified $m$-Bakry-Emery Ricci lower bounds for $m\leq1$

Abstract: In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-Émery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Bakry-Émery Ricci curvature under $m\geq n$ ([19, 21, 27, 33]) or $m=1$ ([34, 35]) if the vector field is a gradient type. When $m<1$, our results are new in the literature.

Classification of zero mean curvature surfaces of separable type in Lorentz-Minkowski space

Abstract: Consider the Lorentz-Minkowski 3-space ${\mathbb L}^3$ with the metric $dx^2+dy^2-dz^2$ in canonical coordinates $(x,y,z)$. A surface in ${\mathbb L}^3$ is said to be separable if it satisfies an equation of the form $f(x)+g(y)+h(z)=0$ for some smooth functions $f$, $g$ and $h$ defined in open intervals of the real line. In this article we classify all zero mean curvature surfaces of separable type, providing a method of construction of examples.

The variation of a functional on the Riemannian submanifolds and its applications

Abstract: The aim of this paper is to derive the first variation of a functional on an oriented submanifold in the Riemannian manifold involving an arbitrary vector field. After obtaining the first variation formula, a notion of $\sigma$-mean curvature is naturally introduced. We also find an important identity involving the $\sigma$-mean curvature and the divergence of the tangent component of the vector field. As an application, we get a general formula of the $\sigma$-mean curvature for the hypersurfaces in a class of Finsler manifolds called the general $(\alpha,\beta)$-manifolds (introduced in [38]). Hence the vanishing $\sigma$-mean curvature characterizes the minimal hypersurfaces under the Busemann-Hausdorff measure ([29]) and the Holmes-Thompson measure ([23]). We also give a general formula for the hypersurface in a Randers manifold involving the navigation data without any restriction on the vector field. In terms of the identity, we prove some nonexistence theorems of the closed orientable minimal submanifold in some special non-Minkowskian Finsler manifolds.

Heat kernel estimates on spaces with varying dimension

Abstract: We obtain sharp two-sided heat kernel estimates for some process whose regular Dirichlet form is strongly local on spaces with varying dimension, in which two spaces of general dimension are connected at one point. On these spaces, if the dimensions of the two constituent parts are different, the volume doubling property fails with respect to the measure induced by the associated Lebesgue measures. Thus the parabolic Harnack inequalities fail and the heat kernels do not enjoy Aronson type estimates. Our estimates show that the on-diagonal estimates are independent of the dimensions of the two parts of the space for small time, whereas they depend on their transience or recurrence for large time. These are multidimensional version of a space considered by Z.-Q. Chen and S. Lou (Ann. Probab. 2019), in which a 1-dimensional space and a 2-dimensional space are connected at one point.

Construction of continuum from a discrete surface by its iterated subdivisions

Abstract: Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivision method by applying the Goldberg-Coxeter subdivision and discuss the convergence of a sequence of discrete surfaces defined inductively by the subdivision. We also study the limit set as the continuum geometric object associated with the given discrete surface.

Fractional integration for irregular martingales

Abstract: We suggest two versions of the Hardy--Littlewood--Sobolev inequality for discrete time martingales. In one version, the fractional integration operator is a martingale transform, however, it may vanish if the filtration is excessively irregular; the second version lacks the martingale property while being analytically meaningful for an arbitrary filtration.

Clairaut surfaces in Euclidean three-space

Abstract: In this note, we investigate local isometric immersion of Clairaut metrics in Euclidean three-space. A Clairaut metric is determined up to isometry by a single function of one variable. We show that an isometric immersion is formally determined by two functions of one variable, uniquely up to coordinate reflection and ambient Euclidean motions. Further, if the Clairaut metric and these two functions are real-analytic, there exists a local isometric immersion realizing these data. We give a more explicit description for a finite-dimensional family of examples.

Schwarz's map for Appell's second hypergeometric system with quarter integer parameters

Abstract: We study Schwarz's map for Appell's second system ${\mathcal F}_2$ of hypergeometric differential equations in two variables with parameters $a=c_1=c_2=\frac{1}{2}$, $b_1=b_2=\frac{1}{4}$. By using theta functions with characteristics, we give a defining equation of an analytic set in ${\mathbb C}^2\times {\mathbb H}$ of its image, and express its inverse.

The second Betti number of doubly weighted homology groups of a certain pre Lie superalgebra

Abstract: We already showed that Betti numbers are all zero when w is not equal to h for (w,h)-doubly weighted homology groups of some special pre Lie superalgebra and showed the first Betti number is 0 when w = h. In this note, we show that the second Betti number is 0 when w = h.

An inverse problem for a class of lacunary canonical systems with diagonal Hamiltonian

Abstract: Hamiltonians are 2-by-2 positive semidefinite real symmetric matrix-valued functions satisfying certain conditions. In this paper, we solve the inverse problem for which recovers a Hamiltonian from the solution of a first-order system attached to a given Hamiltonian, consisting of ordinary differential equations parametrized by a set of complex numbers, under certain conditions for the solutions. This inverse problem is a generalization of the inverse problem for two-dimensional canonical systems.

On the stable reduction of hyperelliptic curves

Abstract: Let $f: S\to B$ be a surface fibration of genus $g\ge 2$ over ${\mathbb{C}}$. The semistable reduction theorem asserts there is a finite base change $\pi: B'\to B$ such that the fibration $S\times_BB'\to B'$ admits a semistable model. An interesting invariant of $f$, denoted by $N(f)$, is the minimum of $\deg(\pi)$ for all such $\pi$. In an early paper of Xiao, he gives a uniform multiplicative upper bound $N_g$ for $N(f)$ depending only on the fibre genus $g$. However, it is not known whether Xiao's bound is sharp or not. In this paper, we give another uniform upper bound $N'_g$ for $N(f)$ when $f$ is hyperelliptic. Our $N'_g$ is optimal in the sense that for every $g\ge 2$ there is a hyperelliptic fibration $f$ of genus $g$ so that $N(f)=N_g'$. In particular, Xiao's upper bound $N_g$ is optimal when $N_g=N'_g$. We show that this last equation $N_g=N_g'$ holds for infinitely many $g$.

Ramification of torsion points on a curve with superspecial reduction over an absolutely unramified base

Abstract: — Let p be a prime number, W an absolutely unramified p-adically complete discrete valuation ring with algebraically closed residue field, and X a curve over the field of fractions of W of genus greater than one. In the present paper, we study the ramification of torsion points on the curve X. A consequence of the main result of the present paper is no existence of ramified torsion point on X in the case where p is greater than three, the Jacobian variety J of X has good reduction over W , and the special fiber of the good model of J is superspecial. This consequence generalizes a theorem proved by Coleman.

Hardy-Sobolev inequality in Herz spaces

Abstract: Our aim in this paper is to establish Hardy-Sobolev inequality in the settings of Herz spaces. As an application, we show Sobolev-type integral representation for a $C^1$-function on ${\mathbb R}^N \setminus \{0\}$ which vanishes outside the unit ball.

Triple product $p$-adic $L$-function attached to $p$-adic families of modular forms

Abstract: Ming-Lun Hsieh constructed three-variable triple product p-adic L-functions attached to triples of primitive Hida families and proved interpolation formulas. We generalize his result in the unbalanced case and construct a three-variable triple product p-adic L-function attached to a primitive Hida family and two more general p-adic families of modular forms.

The axial curvature for corank 1 singular surfaces

Abstract: For singular corank 1 surfaces in $\mathbb{R^3}$, we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola, we define a new type of curvature called the axial curvature, which generalizes the singular curvature for frontal type singularities. We then study contact properties of the surface with respect to the plane orthogonal to the axial vector and show how they are related to the axial curvature. Finally, for certain fold type singularities, we relate the axial curvature with the Gaussian curvature of an appropriate blow up.

Uniform K-stability and Conformally Kähler, Einstein-Maxwell geometry on toric manifolds

Abstract: Conformally Kähler, Einstein-Maxwell metrics and $f$-extremal metrics are generalization of canonical metrics in Kähler geometry. We introduce uniform K-stability for toric Kähler manifolds, and show that uniform K-stability is necessary condition for the existence of $f$-extremal metrics on toric manifolds. Furthermore, we show that uniform K-stability is equivalent to properness of relative K-energy.

Intrinsic dimension of geometric data sets

Abstract: The curse of dimensionality is a phenomenon frequently observed in machine learning (ML) and knowledge discovery (KD). There is a large body of literature investigating its origin and impact, using methods from mathematics as well as from computer science. Among the mathematical insights into data dimensionality, there is an intimate link between the dimension curse and the phenomenon of measure concentration, which makes the former accessible to methods of geometric analysis. The present work provides a comprehensive study of the intrinsic geometry of a data set, based on Gromov's metric measure geometry and Pestov's axiomatic approach to intrinsic dimension. In detail, we define a concept of geometric data set and introduce a metric as well as a partial order on the set of isomorphism classes of such data sets. Based on these objects, we propose and investigate an axiomatic approach to the intrinsic dimension of geometric data sets and establish a concrete dimension function with the desired properties. Our model for data sets and their intrinsic dimension is computationally feasible and, moreover, adaptable to specific ML/KD-algorithms, as illustrated by various experiments.

Some achieved wedge products of positive currents

Abstract: In this paper, we study the existence of the current gT for positive plurisubharmonic currents T and unbounded plurisubharmonic functions g.

Kolmogorov operator with the vector field in Nash class

Abstract: We consider divergence-form parabolic equation with measurable uniformly elliptic matrix and the vector field in a large class containing, in particular, the vector fields in $L^p$, $p>d$, as well as some vector fields that are not even in $L_{\loc}^{2+\varepsilon}$, $\varepsilon>0$. We establish Hölder continuity of the bounded soutions, sharp two-sided Gaussian bound on the heat kernel, Harnack inequality.

New characterizations of the helicoid in a cylinder

Abstract: This paper characterizes a compact piece of the helicoid $H_C$ in a solid cylinder $C \subset \mathbb{R}^3$ from the following two perspectives. First, under reasonable conditions, $H_C$ has the smallest area among all immersed surfaces $\Sigma$ with $\partial \Sigma \subset d_1 \cup d_2 \cup S$, where $d_1$ and $d_2$ are the diameters of the top and bottom disks of $C$ and $S$ is the side surface of $C$. Second, other than $H_C$, there exists no minimal surface whose boundary consists of $d_1$, $d_2$, and a pair of rotationally symmetric curves $\gamma_1$, $\gamma_2$ on $S$ along which it meets $S$ orthogonally. We draw the same conclusion when the boundary curves on $S$ are a pair of helices of a certain height.

Torsion generators of the twist subgroup

Abstract: We show that the twist subgroup of the mapping class group of a closed connected nonorientable surface of genus $g\geq13$ can be generated by two involutions and an element of order $g$ or $g-1$ depending on whether $g$ is odd or even respectively.

Necessary and sufficient conditions for a triangle comparison theorem

Abstract: We prove a version of Topogonov's triangle comparison theorem with surfaces of revolution as model spaces. Given a model surface and a Riemannian manifold with a fixed base point, we give necessary and sufficient conditions under which every geodesic triangle in the manifold with a vertex at the base point has a corresponding Alexandrov triangle in the model. Under these conditions we also prove a version of the Maximal Radius Theorem and a Grove--Shiohama type Sphere Theorem.

Global existence for a system of multiple-speed wave equations violating the null condition

Abstract: We discuss the Cauchy problem for a system of semilinear wave equations in three space dimensions with multiple wave speeds. Though our system does not satisfy the standard null condition, we show that it admits a unique global solution for any small and smooth data. This generalizes a preceding result due to Pusateri and Shatah. The proof is carried out by the energy method involving a collection of generalized derivatives. The multiple wave speeds disable the use of the Lorentz boost operators, and our proof therefore relies upon the version of Klainerman and Sideris. Due to the presence of nonlinear terms violating the standard null condition, some of components of the solution may have a weaker decay as $t\to\infty$, which makes it difficult even to establish a mildly growing (in time) bound for the high energy estimate. We overcome this difficulty by relying upon the ghost weight energy estimate of Alinhac and the Keel-Smith-Sogge type $L^2$ weighted space-time estimate for derivatives.

On the coverings of Hantzsche-Wendt manifold

Abstract: There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable $\mathcal{G}_1,\dots,\mathcal{G}_6$ and four are non-orientable $\mathcal{B}_1,\dots,\mathcal{B}_4$. In the present paper we investigate the manifold $\mathcal{G}_6$, also known as Hantzsche-Wendt manifold; this is the unique Euclidean $3$-form with finite first homology group $H_1(\mathcal{G}_6) = \mathbb{Z}^2_4$. The aim of this paper is to describe all types of $n$-fold coverings over $\mathcal{G}_{6}$ and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group $\pi_1(\mathcal{G}_{6})$ up to isomorphism. Given index $n$, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating series for the above sequences.

Generating functions for sums of polynomial multiple zeta values

Abstract: The sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta values. We also obtain the sum formula for polynomial multiple zeta(-star) values in terms of generating functions, simultaneously generalizing the sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values.