Showing papers in "Osaka Journal of Mathematics in 2019"
TL;DR: In this paper, it was shown that perfect fluid spacetimes with divergence-free projective, concircular, conharmonic or quasi-conformal curvature tensors are generalized Robertson Walker (GRW) spacetimits.
Abstract: We show that $n$-dimensional perfect fluid spacetimes with diver\-gen\-ce-free conformal curvature tensor and constant scalar curvature are generalized Robertson Walker (GRW) spacetimes; as a consequence a perfect fluid Yang pure space is a GRW spacetime. We also prove that perfect fluid spacetimes with harmonic generalized curvature tensor are, under certain conditions, GRW spacetimes. As particular cases, perfect fluids with divergence-free projective, concircular, conharmonic or quasi-conformal curvature tensor are GRW spacetimes. Finally, we explore some physical consequences of such results.
27 citations
TL;DR: In this paper, the strong instability of standing waves was studied for nonlinear Schrodinger equations with an $L 2 -supercritical nonlinearity and an attractive inverse power potential, where the standing wave is strongly unstable.
Abstract: We study the strong instability of standing waves $e^{i\omega t}\phi_\omega(x)$ for nonlinear Schrodinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $\omega\in\mathbb{R}$ is a frequency, and $\phi_\omega\in H^1(\mathbb{R}^N)$ is a ground state of the corresponding stationary equation. Recently, for nonlinear Schrodinger equations with a harmonic potential, Ohta~(2018) proved that if $\partial_\lambda^2S_\omega(\phi_\omega^\lambda)|_{\lambda=1}\le0$, then the standing wave is strongly unstable, where $S_\omega$ is the action, and $\phi_\omega^\lambda(x)\mathrel{\mathop:}=\lambda^{N/2}\phi_\omega(\lambda x)$ is the scaling, which does not change the $L^2$-norm. In this paper, we prove the strong instability under the same assumption as the above-mentioned in inverse power potential case. Our proof is applicable to nonlinear Schrodinger equations with other potentials such as an attractive Dirac delta potential.
21 citations
Journal Article•
TL;DR: In this article, the authors restudies the limiting behavior of the empirical distribution but in regimes where the parameter β$ is allowed to vary as a function of the matrix size, and they show that the above Marchenko-Pastur law holds as long as βN \to 2c \in (0, \infty)$, the limiting measure is related to associated Laguerre orthogonal polynomials.
Abstract: Beta Laguerre ensembles, generalizations of Wishart and Laguerre ensembles, can be realized as eigenvalues of certain random tridiagonal matrices. Analogous to the Wishart ($\beta=1$) case and the Laguerre $(\beta = 2)$ case, for fixed $\beta$, it is known that the empirical distribution of the eigenvalues of the ensembles converges weakly to Marchenko-Pastur distributions, almost surely. The paper restudies the limiting behavior of the empirical distribution but in regimes where the parameter $\beta$ is allowed to vary as a function of the matrix size $N$. We show that the above Marchenko-Pastur law holds as long as $\beta N \to \infty$. When $\beta N \to 2c \in (0, \infty)$, the limiting measure is related to associated Laguerre orthogonal polynomials. Gaussian fluctuations around the limit are also studied.
12 citations
TL;DR: In this paper, the authors introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semi-inverse parabolic parabolic PDEs, and compare it with the standard definition of viscosity solution.
Abstract: The aim of the present work is the introduction of a viscosity type solution, called $strong$-$viscosity$ $solution$ emphasizing also a similarity with the existing notion of $strong$ $solution$ in the literature. It has the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. First, we introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semilinear parabolic partial differential equations; we compare it with the standard definition of viscosity solution. Afterwards, we extend the concept of strong-viscosity solution to the case of semilinear parabolic path-dependent partial differential equations, providing an existence and uniqueness result.
12 citations
TL;DR: In this article, the Stiefel-Whitney polynomial of a smooth manifold with an almost complex manifold was shown to have stable parallelizability when the manifold is a complex flag manifold.
Abstract: Let $X$ be a smooth manifold with a (smooth) involution $\sigma:X\to X$ such that ${\rm Fix}(\sigma)
e \emptyset$. We call the space $P(m,X):=\mathbb{S}^m\times X/\!\sim$ where $(v,x)\sim (-v,\sigma(x))$ a generalized Dold manifold. When $X$ is an almost complex manifold and the differential $T\sigma: TX\to TX$ is conjugate complex linear on each fibre, we obtain a formula for the Stiefel-Whitney polynomial of $P(m,X)$ when $H^1(X;\mathbb{Z}_2)=0$. We obtain results on stable parallelizability of $P(m,X)$ and a very general criterion for the (non) vanishing of the unoriented cobordism class $[P(m,X)]$ in terms of the corresponding properties for $X$. These results are applied to the case when $X$ is a complex flag manifold.
7 citations
TL;DR: In this paper, the authors extend some results about Follmer's pathwise Ito calculus that have only been derived for continuous paths to cadlag paths with quadratic variation.
Abstract: We extend some results about Follmer's pathwise Ito calculus that have only been derived for continuous paths to cadlag paths with quadratic variation. We study some fundamental properties of pathwise Ito integrals with respect to cadlag integrators, especially associativity and the integration by parts formula. Moreover, we study integral equations with respect to pathwise Ito integrals. We prove that some classes of integral equations, which can be explicitly solved in the usual stochastic calculus, can also be solved within the framework of Follmer's calculus.
7 citations
TL;DR: In this article, the authors studied the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms in ${\mathbb R}^{n}$.
Abstract: In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms in ${\mathbb R}^{n}$. As is pointed out by [10], in this combination, the frictional damping term is dominant for the viscoelastic one for the global dynamics of the linear equation. In this note we observe that if the initial data is small, the frictional damping term is again dominant even in the nonlinear equation case. In other words, our main result is diffusion phenomena: the solution is approximated by the heat kernel with a suitable constant. Especially, the result obtained for the $n = 3$ case is essentially new. Our proof is based on several estimates for the corresponding linear equations.
7 citations
TL;DR: In this article, it was shown that the drift vector of the h-path process with given two end point marginals is a measurable function of space, time and marginal at each time.
Abstract: We show that the solution of Schrodinger's functional equation is measurable in space, kernel and marginals. As an application, we show that the drift vector of the h-path process with given two end point marginals is a measurable function of space, time and marginal at each time. In particular, we show that the coefficients of mean field PDE systems which the marginals satisfy are measurable functions of space, time and marginal.
6 citations
TL;DR: In this article, the authors studied the asymptotic behavior of global solutions to the initial value problem for the generalized KdV-Burgers equation and showed that the solution to this equation converges to a self-similar solution to the Burgers equation.
Abstract: We study the asymptotic behavior of global solutions to the initial value problem for the generalized KdV-Burgers equation. One can expect that the solution to this equation converges to a self-similar solution to the Burgers equation, due to earlier works related to this problem. Actually, we obtain the optimal asymptotic rate similar to those results and the second asymptotic profile for the generalized KdV-Burgers equation.
5 citations
TL;DR: In this paper, the first examples of flat fiber type contractions of Fano manifolds onto varieties that are not weak Fano were given, and it was shown that these morphisms are Fano conic bundles.
Abstract: We give the first examples of flat fiber type contractions of Fano manifolds onto varieties that are not weak Fano, and we prove that these morphisms are Fano conic bundles. We also review some known results about the interaction between the positivity properties of anticanonical divisors of varieties of contractions.
5 citations
TL;DR: In this paper, the moduli space of Lagrangian submanifolds in a strict nearly Kahler 6-manifold has been studied and a formula for the second variation of the volume of a Lagrangians in a nearly-Kahler manifold has been derived.
Abstract: Lagrangian submanifolds in strict nearly Kahler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in $G_2$-manifolds. We prove that the mean curvature of a Lagrangian submanifold $L$ in a nearly Kahler manifold $(M, J, g)$ is symplectically dual to the Maslov 1-form on $L$. Using relative calibrations, we derive a formula for the second variation of the volume of a Lagrangian submanifold $L^3$ in a strict nearly Kahler manifold $(M^6, J, g)$ and compare it with McLean's formula for special Lagrangian submanifolds. We describe a finite dimensional local model of the moduli space of compact Lagrangian submanifolds in a strict nearly Kahler 6-manifold. We show that there is a real analytic atlas on $(M^6, J, g)$ in which the strict nearly Kahler structure $(J, g)$ is real analytic. Furthermore, w.r.t. an analytic strict nearly Kahler structure the moduli space of Lagrangian submanifolds of $M^6$ is a real analytic variety, whence infinitesimal Lagrangian deformations are smoothly obstructed if and only if they are formally obstructed. As an application, we relate our results to the description of Lagrangian submanifolds in the sphere $S^6$ with the standard nearly Kahler structure described in [34].
TL;DR: In this article, the authors studied the initial boundary value problem for a time-dependent Ginzburg-Landau model in superconductivity and proved the uniform boundedness of strong solutions with respect to diffusion coefficient 0 < ϵ < 1.
Abstract: We study the initial boundary value problem for a time-dependent Ginzburg-Landau model in superconductivity. First, we prove the uniform boundedness of strong solutions with respect to diffusion coefficient 0 < $\epsilon$ < 1 in the case of Coulomb gauge. Our second result is the global existence and uniqueness of the weak solutions to the limit problem when $\epsilon=0$.
TL;DR: In this paper, the authors used Blanchfield forms to study knots that can be turned into an unknot using a single $\overline{t}_{2k}$ move.
Abstract: In this note we use Blanchfield forms to study knots that can be turned into an unknot using a single $\overline{t}_{2k}$ move.
TL;DR: In this paper, an algorithm for computing the Teichmuller polynomial for a certain class of fibered alternating links associated to trees is presented, and a mutant pair of such links is distinguished by the TPE.
Abstract: We give an algorithm for computing the Teichmuller polynomial for a certain class of fibered alternating links associated to trees. Furthermore, we exhibit a mutant pair of such links distinguished by the Teichmuller polynomial.
TL;DR: In this article, various approximation results of solutions of equations $f(x,Y)=0$ were studied, where some components of the solutions do not depend on all the variables $x_j$.
Abstract: We study various approximation results of solutions of equations $f(x,Y)=0$ where $f(x,Y)\in\mathbb{K}[\![x]\!][Y]^r$ and $x$ and $Y$ are two sets of variables, and where some components of the solutions $y(x)\in\mathbb{K}[\![x]\!]^m$ do not depend on all the variables $x_j$. These problems were highlighted by M. Artin.
TL;DR: In this article, the authors give a sufficient condition for a weakly reducible Heegaard splitting to be uncritical and unstabilized, and also give a necessary condition for it to be critical.
Abstract: In this paper, we give a sufficient condition for a (weakly reducible) Heegaard splitting to be unstabilized and uncritical. We also give a sufficient condition for a Heegaard splitting to be critical.
TL;DR: In this paper, the authors provide the algebraic counterpart of the mixed Reidemeister moves for virtual knots and links, with semi-arcs labeled by a given set, and define invariants with values in groups using 2-cocycles.
Abstract: Given a set $X$, we provide the algebraic counterpart of the (mixed) Reidemeister moves for virtual knots and links, with semi-arcs labeled by $X$: we define (commutative and noncommutative) invariants with values in groups, using ``2-cocycles", and we also introduce a universal group $U_{nc}^{fg}(X)$ and functions $\pi_f, \pi_g\colon X\times X\to U_{nc}^{fg}(X)$ governing all 2-cocycles in $X$. We exhibit examples of computations -of the group and their invariants- achieved using GAP [7].
TL;DR: In this paper, the theory of standard pentads is used to embed an arbitrary reductive Lie algebra and its finite-dimensional completely reducible representation into some larger graded Lie algebra.
Abstract: Using the theory of standard pentads, we can embed an arbitrary finite-dimensional reductive Lie algebra and its finite-dimensional completely reducible representation into some larger graded Lie algebra. However, it is not easy to find the structure of the ``larger graded Lie algebra'' from the definition in general cases. Under these, the first aim of this paper is to show that the ``larger graded Lie algebra'' is isomorphic to some PC Lie algebra, which are Lie algebras corresponding to special standard pentads called pentads of Cartan type. The second aim is to find the structure of a PC Lie algebra.
TL;DR: In this paper, the authors address the question when two rooted trees have the same plucking polynomial and give an algebraic and a geometric solution to the problem, and give a criterion for a sequence of nonnegative integers to be realized as a rooted tree.
Abstract: In this paper we address the following question: When do two rooted trees have the same plucking polynomial? The solution provided in the present paper has an algebraic version (Theorem 2.5) and a geometric version (Theorem 1.2). Furthermore, we give a criterion for a sequence of non-negative integers to be realized as a rooted tree.
TL;DR: In this paper, the authors prove versions of Alexander and Markov's theorems for extended welded braids and links, following Kamada's approach to the case of welded objects.
Abstract: Extended welded links are a generalization of Fenn, Rimanyi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of Alexander and Markov's theorems for extended welded braids and links, following Kamada's approach to the case of welded objects.
TL;DR: In this paper, it was shown that the Thurston seminorms of all finite covers of an aspherical 3-manifold can determine whether it is a graph manifold, a mixed 3-Manifold, or hyperbolic.
Abstract: We show that the Thurston seminorms of all finite covers of an aspherical 3-manifold determine whether it is a graph manifold, a mixed 3-manifold or hyperbolic.
TL;DR: In this paper, the authors extend the results of [7] on the topology of real Bott manifold to generalized real Bott manifolds and give a characterization for it to be abelian.
Abstract: Generalized Bott manifolds (over $\mathbb C$ and $\mathbb R$) have been defined by Choi, Masuda and Suh in [4]. In this article we extend the results of [7] on the topology of real Bott manifolds to generalized real Bott manifolds. We give a presentation of the fundamental group, prove that it is solvable and give a characterization for it to be abelian. We further prove that these manifolds are aspherical only in the case of real Bott manifolds and compute the higher homotopy groups. Furthermore, using the presentation of the cohomology ring with $\mathbb Z_2$-coefficients, we derive a combinatorial characterization for orientablity and spin structure.
TL;DR: In this article, it was shown that the cohomology groups of such a space can be computed via some Hochster's type formula, which generalizes the usual Hochsters' formula for moment-angle complexes, and the stable decomposition of momentangle complexes can also be extended to such spaces.
Abstract: Any finite simplicial complex $\mathcal{K}$ and a partition of the vertex set of $\mathcal{K}$ determines a canonical quotient space of the moment-angle complex of ${\mathcal K}$. We prove that the cohomology groups of such a space can be computed via some Hochster's type formula, which generalizes the usual Hochster's formula for the cohomology groups of moment-angle complexes. In addition, we show that the stable decomposition of moment-angle complexes can also be extended to such spaces. This type of spaces include all the quasitoric manifolds that are pullback from the linear models. And we prove that the moment-angle complex associated to a finite simplicial poset is always homotopy equivalent to one of such spaces.
TL;DR: In this article, it was shown that any abelian subgroup of the mapping class group of a compact connected non-orientable surface with boundary components whose Euler characteristic is negative is finitely generated and the maximal torsion-free rank of it is
Abstract: One of the basic and important problems to study algebraic structures of the mapping class groups is finding abelian subgroups included in the mapping class groups. Birman-Lubotzky-McCarthy gave the answer of this question for the orientable surfaces, namely, they proved that any abelian subgroup of the mapping class groups for orientable surfaces of genus $g$ with $b$ boundary components and $c$ connected components is finitely generated and the maximal torsion-free rank of it is $3g+b-3c$. In the present paper, we prove that any abelian subgroup of the mapping class group of a compact connected non-orientable surface $N$ of genus $g\geq 1$ with $n\geq 0$ boundary components whose Euler characteristic is negative is finitely generated and the maximal torsion-free rank of it is $\frac{3}{2}(g-1)+n-2$ if $g$ is odd and $\frac{3}{2}g+n-3$ if $g$ is even.
TL;DR: In this article, the trace map associated to a family of 1D discrete Dirac operators with Sturmian potentials was established. But the invariants of the trace maps were not defined.
Abstract: We establish invariants for the trace map associated to a family of 1D discrete Dirac operators with Sturmian potentials. Using these invariants we prove that the operators have purely singular continuous spectrum of zero Lebesgue measure, uniformly on the mass and parameters that define the potentials. For rotation numbers of bounded density we prove that these Dirac operators have purely $\alpha$-continuous spectrum, as to the Schrodinger case, for some $\alpha \in (0,1)$. To the Sturmian Schrodinger and Dirac models we establish a comparison between invariants of the trace maps, which allows to compare the numbers $\alpha$'s and lower bounds on transport exponents.
TL;DR: For a positive Hopf plumbed arborescent Seifert surface, this article studied the set of Hopf bands up to homology and up to the action of the monodromy.
Abstract: For a positive Hopf plumbed arborescent Seifert surface $S$, we study the set of Hopf bands $H\subset S$, up to homology and up to the action of the monodromy. The classification of Seifert surfaces for which this set is finite is closely related to the classification of finite Coxeter groups.
TL;DR: In this article, a classification of non-flat totally geodesic surfaces in compact Riemannian symmetric spaces of classical type is given, where the surfaces are defined in terms of a set of classes.
Abstract: We give a classification of non-flat totally geodesic surfaces in compact Riemannian symmetric spaces of classical type.
TL;DR: In this article, the non-periodic component of a variant of the stable Auslander--Reiten quiver for the category of $A$-lattices that contain ''Heller lattices'' was determined.
Abstract: We consider the Kronecker algebra $A=\mathcal{O}[X,Y]/(X^2,Y^2)$, where $\mathcal{O}$ is a complete discrete valuation ring. Since $A\otimes\kappa$ is a special biserial algebra, where $\kappa$ is the residue field of $\mathcal{O}$, one can compute a complete list of indecomposable $A\otimes \kappa$-modules. For each indecomposable $A\otimes \kappa$-module, we obtain a special kind of $A$-lattices called ``Heller lattices''. In this paper, we determine the non-periodic component of a variant of the stable Auslander--Reiten quiver for the category of $A$-lattices that contains ``Heller lattices''.
TL;DR: In this article, it was shown that for odd and even Clifford indices, the analogies hold for such curves of even Clifford index $c = 2c+3$ resp.
Abstract: We say that a curve $X$ of genus $g$ has maximally computed Clifford index if the Clifford index $c$ of $X$ is, for $c>2$, computed by a linear series of the maximum possible degree $d$ < $g$; then $d = 2c+3$ resp. $d = 2c+4$ for odd resp. even $c$. For odd $c$ such curves have been studied in [6]. In this paper we analyze if/how far analoguous results hold for such curves of even Clifford index $c$.
TL;DR: In this article, for a Brauer tree algebra and a star-shaped Brauer algebra, the authors give operations on the two-sided tilting complex (D_T$) constructed in [3] which is isomorphic to the Rickard tree-to-star complex (RTCT$).
Abstract: In this note, for a Brauer tree algebra $A$ and a star-shaped Brauer tree algebra $B$ which is derived equivalent to $A$, we give operations on the two-sided tilting complex $D_T$ of $A\otimes B^{op}$-modules constructed in [3] which is isomorphic to the Rickard tree-to-star complex $T$ constructed in [5] in $D^b(A)$, and we show that the operations on $D_T$ correspond to operations called $foldings$ on the Rickard tree-to-star complex $T$ given in [7].