1,231 results
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152. Second Derivative Test for Intersection Bodies
- Author
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Alexander Koldobsky
- Subjects
Unit sphere ,Mathematics(all) ,Mathematics::Functional Analysis ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,01 natural sciences ,Connection (mathematics) ,010101 applied mathematics ,Combinatorics ,Intersection ,Hyperplane ,Mathematics::Metric Geometry ,0101 mathematics ,Unit (ring theory) ,Mathematics ,Counterexample ,Second derivative - Abstract
In 1956, Busemann and Petty asked whether symmetric convex bodies in Rnwith larger central hyperplane sections also have greater volume. This question was answered in the negative forn⩾5 in a series of papers giving individual counterexamples. In 1988, Lutwak introduced the concept of an intersection body and proved that every smooth nonintersection body in Rnprovides a counterexample to the Busemann–Petty problem. In this article, we use the connection between intersection bodies and positive definite distributions, established by the author in an earlier paper, to give a necessary condition for intersection bodies in terms of the second derivative of the norm. This result allows us to produce a variety of counterexamples to the Busemann–Petty problem in Rn,n⩾5. For example, the unit ball of theq-sum of any finite dimensional normed spacesXandYwithq>2, dim(X)⩾1, dim(Y)⩾4 is not an intersection body, as well as the unit balls of the Orlicz spaces ℓnM,n⩾5, withM′(0)=M″(0)=0.
- Published
- 1998
153. Connections between the Support and Linear Independence of Refinable Distributions
- Author
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Jianzhong Wang and David K. Ruch
- Subjects
Discrete mathematics ,Mathematics(all) ,Numerical Analysis ,General Mathematics ,Multiresolution analysis ,Applied Mathematics ,010102 general mathematics ,Regular polygon ,01 natural sciences ,010101 applied mathematics ,Dilation (metric space) ,Distribution (mathematics) ,If and only if ,Linear independence ,0101 mathematics ,Independence (probability theory) ,Analysis ,Mathematics ,Integer (computer science) - Abstract
The purpose of this paper is to study the relationships between the support of a refinable distributionφand the global and local linear independence of the integer translates ofφ. It has been shown elsewhere that a compactly supported distributionφhas globally independent integer translates if and only ifφhas minimal convex support. However, such a distribution may have “holes” in its support. By insisting thatφ∈L2(R) and generates a multiresolution analysis, Lemarié and Malgouyres have ensured that no such holes can occur. In this article we generalize this result to refinable distributions. We also give a result on the local linear independence of the integer translates ofφ. We work with integer dilation factorN⩾2 throughout this paper.
- Published
- 1998
- Full Text
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154. Separation of singularities for the Bergman space and application to control theory
- Author
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Marcu-Antone Orsoni, Andreas Hartmann, Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
0209 industrial biotechnology ,Pure mathematics ,General Mathematics ,Diagonal ,Boundary (topology) ,02 engineering and technology ,Space (mathematics) ,Convex polygon ,01 natural sciences ,Square (algebra) ,Mathematics - Analysis of PDEs ,020901 industrial engineering & automation ,Intersection ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Complex Variables (math.CV) ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Conjecture ,Mathematics - Complex Variables ,Mathematics::Complex Variables ,Applied Mathematics ,010102 general mathematics ,[MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV] ,Optimization and Control (math.OC) ,Bergman space ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Analysis of PDEs (math.AP) - Abstract
In this paper, we solve a separation of singularities problem in the Bergman space. More precisely, we show that if $P\subset \mathbb{C}$ is a convex polygon which is the intersection of $n$ half planes, then the Bergman space on $P$ decomposes into the sum of the Bergman spaces on these half planes. The result applies to the characterization of the reachable space of the one-dimensional heat equation on a finite interval with boundary controls. We prove that this space is a Bergman space of the square which has the given interval as a diagonal. This gives an affirmative answer to a conjecture raised in [HKT20]., Comment: Journal de Math{\'e}matiques Pures et Appliqu{\'e}es, Elsevier, 2021
- Published
- 2021
155. Pulsating waves in a dissipative medium with Delta sources on a periodic lattice
- Author
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Je Chiang Tsai, Xinfu Chen, and Xing Liang
- Subjects
Bistability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Continuum (design consultancy) ,Dirac delta function ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Classical mechanics ,Exponential stability ,symbols ,Dissipative system ,Heat equation ,Uniqueness ,0101 mathematics ,Focus (optics) ,Mathematics - Abstract
This paper studies a dissipative heat equation with Delta sources of non-linear strength located on a periodic lattice. The model arises from intracellular waves in continuum excitable media with discrete release sites. Due to the presence of Delta sources, the solution of the model has discontinuous spatial derivatives. We focus on the bistable regime of the model, determined by the decay strength parameter a and the separation distance L between release sites, in which the model admits exactly three L-periodic steady states. We establish the existence of pulsating waves spatially connecting them. For the case of waves connecting two stable L-periodic steady states, the uniqueness and global exponential stability of pulsating waves are shown. Also a new technique is introduced to find the fine structure of the tails of pulsating waves.
- Published
- 2021
156. On the mean first arrival time of Brownian particles on Riemannian manifolds
- Author
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Leo Tzou, J. C. Tzou, and Medet Nursultanov
- Subjects
Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Rigidity (psychology) ,01 natural sciences ,Arrival time ,Integral geometry ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Planar ,FOS: Mathematics ,Primary: 58J65 Secondary: 60J65, 58G15, 92C37 ,0101 mathematics ,Asymptotic expansion ,Mathematics - Probability ,Brownian motion ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We use geometric microlocal methods to compute an asymptotic expansion of mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. This paper can be seen as the Riemannian 3-manifold version of the planar result of [1] and thus enable us to see the full effect of the local extrinsic boundary geometry on the mean arrival time of the Brownian particles. Our approach also connects this question to some of the recent progress on boundary rigidity and integral geometry [21] and [18] .
- Published
- 2021
157. Waring-Hilbert problem on Cantor sets
- Author
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Yinan Guo
- Subjects
Combinatorics ,Cantor set ,General Mathematics ,010102 general mathematics ,Interval (graph theory) ,0101 mathematics ,Ternary operation ,01 natural sciences ,Complex plane ,Mathematics ,Unit interval ,Real number - Abstract
Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set C . It is shown that, for each m ≥ 3 , every real number in the unit interval [ 0 , 1 ] is the sum x 1 m + x 2 m + ⋯ + x n m with each x j in C and some n ≤ 6 m . Furthermore, every real number x in the interval [ 0 , 8 ] can be written as x = x 1 3 + x 2 3 + ⋯ + x 8 3 , the sum of eight cubic powers with each x j in C . Another Cantor set C × C is also considered. More specifically, when C × C is embedded into the complex plane ℂ , the Waring–Hilbert problem on C × C has a positive answer for powers less than or equal to 4.
- Published
- 2021
158. Global well-posedness and scattering for the Dysthe equation in L2(R2)
- Author
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Jean-Claude Saut, Razvan Mosincat, and Didier Pilod
- Subjects
Small data ,Scattering ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Bilinear interpolation ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Fourier transform ,Norm (mathematics) ,0103 physical sciences ,Bounded variation ,symbols ,Flow map ,0101 mathematics ,Schrödinger's cat ,Mathematics - Abstract
This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrodinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space L 2 ( R 2 ) . This result is sharp in view of the fact that the flow map cannot be C 3 continuous below L 2 ( R 2 ) . Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space U S 2 and its dual V S 2 of square bounded variation functions. We also prove that the initial-value problem is locally well-posed in H s ( R 2 ) , s > 0 . Our results extend to the finite depth version of the Dysthe equation.
- Published
- 2021
159. Mixed Lp projection inequality
- Author
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Zhongwen Tang and Lin Si
- Subjects
Surface (mathematics) ,Pure mathematics ,Projection (mathematics) ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Minkowski inequality ,01 natural sciences ,media_common ,Mathematics - Abstract
In this paper, the mixed L p -surface area measures are defined and the mixed L p Minkowski inequality is obtained consequently. Furthermore, the mixed L p projection inequality for mixed projection bodies is established.
- Published
- 2021
160. Lower bounds of gradient's blow-up for the Lamé system with partially infinite coefficients
- Author
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Haigang Li
- Subjects
Pointwise ,Work (thermodynamics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Linear elasticity ,Mathematical analysis ,Zero (complex analysis) ,01 natural sciences ,010101 applied mathematics ,Stress (mechanics) ,Discontinuity (linguistics) ,Arbitrarily large ,Linear form ,0101 mathematics ,Mathematics - Abstract
In composite materials, the stress may be arbitrarily large in the narrow region between two close-to-touching hard inclusions. The stress is represented by the gradient of a solution to the Lame system of linear elasticity. The aim of this paper is to establish lower bounds of the gradients of solutions of the Lame system with partially infinite coefficients as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero. Combining it with the pointwise upper bounds obtained in our previous work, the optimality of the blow-up rate of gradients is proved for inclusions with arbitrary shape in dimensions two and three. The key to show this is that we find a blow-up factor, a linear functional of the boundary data, to determine whether the blow-up will occur or not.
- Published
- 2021
161. Remarks on results by Müger and Tuset on the moments of polynomials
- Author
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Greg Markowsky and Dylan Phung
- Subjects
Combinatorics ,Polynomial ,Conjecture ,Integer ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Limit (mathematics) ,0101 mathematics ,01 natural sciences ,Methods of contour integration ,Mathematics ,Counterexample - Abstract
Let f ( x ) be a non-zero polynomial with complex coefficients, and M p = ∫ 0 1 f ( x ) p d x for p a positive integer. In a recent paper, Muger and Tuset showed that lim sup p → ∞ | M p | 1 ∕ p > 0 , and conjectured that this limit is equal to the maximum amongst the critical values of f together with the values | f ( 0 ) | and | f ( 1 ) | . We give an example that shows that this conjecture is false. It also may be natural to guess that lim sup p → ∞ | M p | 1 ∕ p is equal to the maximum of | f ( x ) | on [ 0 , 1 ] . However, we give a counterexample to this as well. We also provide a few more guesses as to the behavior of the quantity lim sup p → ∞ | M p | 1 ∕ p .
- Published
- 2021
162. On the numerical computation of orbits of dynamical systems: The higher dimensional case
- Author
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Kenneth J. Palmer and Shui-Nee Chow
- Subjects
Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,Computation ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Iterated function ,Orbit (dynamics) ,Applied mathematics ,0101 mathematics ,Finite time ,Computer Science::Databases ,Mathematics - Abstract
Chaotic dynamical systems exhibit sensitive dependence to initial conditions. So, because of round-off error, a computed orbit diverges at an exponential rate from the true orbit with the same initial condition. Nevertheless, we are able to exploit the hyperbolicity of the dynamical system to prove a “finite time” shadowing lemma, from which we deduce that a true orbit shadows the computed orbit for a large number of iterates. An algorithm for the computation of the shadowing error is given and, furthermore, the effect of round-off error on these computations is analyzed in detail. The algorithm is applied to the Hénon map. This paper is a continuation of an earlier paper on one-dimensional maps.
- Published
- 1992
163. Strong independence and the dimension of a Tverberg set
- Author
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Satya Deo and Snigdha Bharati Choudhury
- Subjects
Set (abstract data type) ,Discrete mathematics ,Euclidean space ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Independence (mathematical logic) ,Algebraic independence ,0101 mathematics ,01 natural sciences ,Geometric proof ,Mathematics - Abstract
In this expository paper, we discuss several types of independence of points in the Euclidean space R d such as algebraic independence, strong independence, weak independence, and give a detailed geometric proof of their interrelationship. We also give yet another proof of Reay’s theorem determining the dimension of a Tverberg set.
- Published
- 2021
164. Optimal convergence rate of the vanishing shear viscosity limit for compressible Navier-Stokes equations with cylindrical symmetry
- Author
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Xinhua Zhao, Huanyao Wen, Tong Yang, and Changjiang Zhu
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Prandtl number ,Boundary (topology) ,01 natural sciences ,Symmetry (physics) ,Physics::Fluid Dynamics ,010101 applied mathematics ,Boundary layer ,symbols.namesake ,Rate of convergence ,symbols ,Compressibility ,Limit (mathematics) ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
We consider the initial boundary value problem for the isentropic compressible Navier-Stokes equations with cylindrical symmetry. The existence of boundary layers is well-known when the shear viscosity vanishes. In this paper, we derive explicit Prandtl type boundary layer equations and prove the global in time stability of the boundary layer profile together with the optimal convergence rate of the vanishing shear viscosity limit without any smallness assumption on the initial and boundary data.
- Published
- 2021
165. Partial regularity for symmetric quasiconvex functionals on BD
- Author
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Franz Gmeineder
- Subjects
Pure mathematics ,Work (thermodynamics) ,Reduction (recursion theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Bounded deformation ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Maxima and minima ,Quasiconvex function ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,0101 mathematics ,Linear growth ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational work (Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 63-113), symmetric quasiconvexity is the foremost notion as to sequential weak*-lower semicontinuity of functionals on BD. The overarching main difficulty here is Ornstein's Non-Inequality, implying that the BD-case is genuinely different from the study of variational integrals on BV. In particular, this paper extends the recent work of Kristensen and the author (Partial regularity for BV-Minimizers, Arch. Ration. Mech. Anal. 232 (2019), Issue 3, 1429-1473) from the BV- to the BD-situation. Alongside, we establish partial regularity results for strongly quasiconvex functionals of superlinear growth by reduction to the full gradient case, which might be of independent interest., Comment: Version 2, 40 pages, 1 figure, final version to appear at J. Math. Pures Appl
- Published
- 2021
166. Boundary Hölder regularity for elliptic equations
- Author
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Dongsheng Li, Guanghao Hong, Yuanyuan Lian, and Kai Zhang
- Subjects
Laplace's equation ,Dirichlet problem ,Pointwise ,Laplace transform ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Nonlinear system ,Maximum principle ,0101 mathematics ,Mathematics - Abstract
This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary Holder regularity under proper geometric conditions. “Unified” means that our method is applicable for the Laplace equation, linear elliptic equations in divergence and non-divergence form, fully nonlinear elliptic equations, the p−Laplace equations and the fractional Laplace equations etc. In addition, these geometric conditions are quite general. In particular, for local equations, the measure of the complement of the domain near the boundary point concerned could be zero. The key observation in the method is that the strong maximum principle implies a decay for the solution, then a scaling argument leads to the Holder regularity. Moreover, we also give a geometric condition, which guarantees the solvability of the Dirichlet problem for the Laplace equation. The geometric meaning of this condition is more apparent than that of the Wiener criterion.
- Published
- 2020
167. Spectral multipliers without semigroup framework and application to random walks
- Author
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Peng Chen, El Maati Ouhabaz, Adam Sikora, and Lixin Yan
- Subjects
Pure mathematics ,Markov chain ,Semigroup ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Integer lattice ,Random walk ,Differential operator ,01 natural sciences ,Exponential function ,Multiplier (Fourier analysis) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Laplace operator ,Mathematics - Abstract
In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. We have two main objectives. The first one is to work outside the semigroup context. In contrast to previous works on this subject, we do not make any assumption on the semigroup. The second objective is to consider polynomial off-diagonal decay instead of exponential one. Our approach and results lead to new applications to several operators such as differential operators, pseudo-differential operators as well as Markov chains. In our general context we introduce a restriction type estimates a la Stein-Tomas. This allows us to obtain sharp spectral multiplier theorems and hence sharp Bochner-Riesz summability results in some situation. Finally, we consider the random walk on the integer lattice Z n and prove sharp Bochner-Riesz summability results similar to those known for the standard Laplacian on R n .
- Published
- 2020
168. Positive vector solutions for nonlinear Schrödinger systems with strong interspecies attractive forces
- Author
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Jinmyoung Seok, Jaeyoung Byeon, and Ohsang Kwon
- Subjects
Condensed Matter::Quantum Gases ,Interaction forces ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Classical mechanics ,symbols ,0101 mathematics ,Interspecies interaction ,Schrödinger's cat ,Mathematics - Abstract
In this paper we study the structure of positive vector solutions for nonlinear Schrodinger systems with 3 components when all interspecies interaction forces are positive and large while all intraspecies interaction forces are positive and fixed. We will show that the structure strongly depends on some relation of large interspecies interaction forces.
- Published
- 2020
169. Intersection of Siepinski gasket with its translation
- Author
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Wenxia Li and Yi Cai
- Subjects
Set (abstract data type) ,Combinatorics ,Intersection ,General Mathematics ,Gasket ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Translation (geometry) ,01 natural sciences ,Mathematics ,Sierpinski triangle - Abstract
Let E be the Sierpinski gasket, i.e., the self-similar set generated by the IFS f a ( x ) = x + a q : a ∈ { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) } . In this paper, we provide a description of the following set for 2 q 3 D q = { dim H ( E ∩ ( E + t ) ) : t ∈ T } , where T is the set of t = ( t 1 , t 2 ) with t ∈ E − E and t 1 , t 2 have unique q -expansions w.r.t − 1 , 0 , 1 .
- Published
- 2020
170. Morrey's fractional integrals in Campanato-Sobolev's space and divF = f
- Author
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Liguang Liu and Jie Xiao
- Subjects
010101 applied mathematics ,Sobolev space ,Pure mathematics ,Mean curvature ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Mathematics - Abstract
The purpose of this paper is three-fold: the first is to determine the Campanato-Sobolev space I N ( L p , κ ) by means of ∑ | α | = N ‖ D α f ‖ L p , κ - the sum of the Campanato norms of the derivatives D α f with | α | = N ; the second is to characterize Morrey's fractional integrals { T f : f ∈ L p , κ } in the Campanato-Sobolev space I s ( L p , κ ) ; the third is to find a distributional solution F ∈ ( L q , λ ) n of the mean curvature type divergence equation div F = f ∈ L p , κ (the Morrey space). And yet, the here-established three theorems and their proofs are not only novel but also nontrivial.
- Published
- 2020
171. Ground states of nonlinear Schrödinger systems with mixed couplings
- Author
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Yuanze Wu and Juncheng Wei
- Subjects
Interaction forces ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Block (permutation group theory) ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,symbols ,0101 mathematics ,Schrödinger's cat ,Mathematics ,Mathematical physics - Abstract
We consider the following k-coupled nonlinear Schrodinger systems: { − Δ u j + λ j u j = μ j u j 3 + ∑ i = 1 , i ≠ j k β i , j u i 2 u j in R N , u j > 0 in R N , u j ( x ) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , k , where N ≤ 3 , k ≥ 3 , λ j , μ j > 0 are constants and β i , j = β j , i ≠ 0 are parameters. There have been intensive studies for the above systems when k = 2 or the systems are purely attractive ( β i , j > 0 , ∀ i ≠ j ) or purely repulsive ( β i , j 0 , ∀ i ≠ j ); however very few results are available for k ≥ 3 when the systems admit mixed couplings and the components are organized into groups, i.e., there exist ( i 1 , j 1 ) and ( i 2 , j 2 ) such that β i 1 , j 1 > 0 and β i 2 , j 2 0 . In this paper we give the first systematic and an (almost) complete study on the existence of ground states when the systems admit mixed couplings and the components are organized into groups. We first divide these systems into repulsive-mixed and total-mixed cases. In the first case we prove nonexistence of ground states. In the second case we give a necessary condition for the existence of ground states and also provide estimates for Morse index. The key idea is the block decomposition of the systems (optimal block decompositions, eventual block decompositions), and the measure of total interaction forces between different blocks. Finally the assumptions on the existence of ground states are shown to be optimal in some special cases.
- Published
- 2020
172. Parabolic Minkowski convolutions and concavity properties of viscosity solutions to fully nonlinear equations
- Author
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Kazuhiro Ishige, Paolo Salani, and Qing Liu
- Subjects
Initial-boundary value problem ,Minkowski addition ,Power concavity ,Viscosity solutions ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,01 natural sciences ,Parabolic partial differential equation ,Convolution ,010101 applied mathematics ,Nonlinear system ,Operator (computer programming) ,Viscosity (programming) ,Minkowski space ,0101 mathematics ,Laplace operator ,Mathematics - Abstract
This paper is concerned with the Minkowski convolution of viscosity solutions of fully nonlinear parabolic equations. We adopt this convolution to compare viscosity solutions of initial-boundary value problems in different domains. As a consequence, we can for instance obtain parabolic power concavity of solutions to a general class of parabolic equations. Our results are applicable to the Pucci operator, the normalized q-Laplacians with 1 q ≤ ∞ , the Finsler Laplacian, and more general quasilinear operators.
- Published
- 2020
173. Dynamics, points and places at infinity, and the inversion of polynomial self-maps of R2
- Author
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Frederico Xavier and Luis Fernando Mello
- Subjects
Projectivization ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Local diffeomorphism ,Algebraic geometry ,Disjoint sets ,Jacobian conjecture ,01 natural sciences ,Inversion (discrete mathematics) ,Injective function ,Point at infinity ,0101 mathematics ,Mathematics - Abstract
In this partly expository paper we give two applications of ideas from dynamical systems to the study of the injectivity properties of a polynomial local diffeomorphism F = ( F 1 , F 2 ) : R 2 → R 2 (by the work of Pinchuk, these maps need not be globally injective). I) The Jacobian conjecture claims that all polynomial local biholomorphisms G = ( G 1 , G 2 ) : ℂ 2 → ℂ 2 must be injective. By the Abhyankar–Moh theory in algebraic geometry, G is injective if G 1 has one place at infinity. We prove that this result carries over to the real case, in the following form. It is shown that F is injective if the (possibly singular) complex curve C of { F 1 = 0 } is irreducible, its projectivization C ˜ has only one point at infinity, and the said point is covered only once by a desingularization ℛ → C ˜ . II) In our second application we show that every polynomial local diffeomorphism of R 2 into itself must be injective at least on certain large regions that contain sequences of disjoint discs of arbitrarily large radii.
- Published
- 2020
174. Lelong–Poincaré formula in symplectic and almost complex geometry
- Author
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Alexandre Sukhov and Emmanuel Mazzilli
- Subjects
Pure mathematics ,Almost complex manifold ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Vector bundle ,01 natural sciences ,General family ,symbols.namesake ,Mathematics::Algebraic Geometry ,Complex geometry ,Convergence (routing) ,Poincaré conjecture ,symbols ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Symplectic geometry ,Mathematics - Abstract
In this paper, we present two applications of the theory of singular connections developed by Harvey and Lawson (1993). The first one is a version of the Lelong–Poincare formula with estimates for sections of vector bundles over an almost complex manifold. The second one is a convergence theorem for divisors associated to a general family of symplectic submanifolds constructed by Donaldson (1996) (the case of hypersurfaces) and by Auroux in (1997) (for arbitrary dimensional submanifolds).
- Published
- 2020
175. Geometric characterization of preduals of injective Banach lattices
- Author
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Anatoly G. Kusraev and S.S. Kutatelatze
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Simplex ,General Mathematics ,010102 general mathematics ,Banach space ,Predual ,010103 numerical & computational mathematics ,Characterization (mathematics) ,Space (mathematics) ,01 natural sciences ,Injective function ,Transfer (group theory) ,Dual polyhedron ,0101 mathematics ,Mathematics - Abstract
The paper deals with the study of Banach spaces whose duals are injective Banach lattices. Davies in 1967 proved that an ordered Banach space is an L 1 -predual space if and only if it is a simplex space. In 2007 Duan and Lin proved that a real Banach space is an L 1 -predual space if and only if its every four-point subset is centerable. We prove the counterparts of these remarkable results for injectives by the new machinery of Boolean valued transfer from L 1 -spaces to injective Banach lattices.
- Published
- 2020
176. On the structure of variable exponent spaces
- Author
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Francisco L. Hernández, Mauro Sanchiz, César Ruiz, and Julio Flores
- Subjects
46E30, 47B60 ,Pure mathematics ,Variable exponent ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,010103 numerical & computational mathematics ,Disjoint sets ,Cantor function ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,Singularity ,Computer Science::Systems and Control ,FOS: Mathematics ,symbols ,0101 mathematics ,Mathematics - Abstract
The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) L p ( ⋅ ) ( Ω ) . In the second part strictly singular and disjointly strictly singular operators between spaces L p ( ⋅ ) ( Ω ) are studied. New results on the disjoint strict singularity of the inclusions L p ( ⋅ ) ( Ω ) ↪ L q ( ⋅ ) ( Ω ) are given.
- Published
- 2020
177. The Grothendieck property in Marcinkiewicz spaces
- Author
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Fedor Sukochev and B. de Pagter
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Property (philosophy) ,Concave function ,Function space ,General Mathematics ,Banach lattice ,010102 general mathematics ,Grothendieck space ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The main purpose of this paper is to fully characterize continuous concave functions ψ such that the corresponding Marcinkiewicz Banach function space M ψ is a Grothendieck space.
- Published
- 2020
178. Convergence rate for eigenvalues of the elastic Neumann–Poincaré operator in two dimensions
- Author
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Yoshihisa Miyanishi, Hyeonbae Kang, and Kazunori Ando
- Subjects
Smoothness ,Polynomial ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,01 natural sciences ,Neumann–Poincaré operator ,Domain (mathematical analysis) ,010101 applied mathematics ,Rate of convergence ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we consider the Neumann–Poincare type operator associated with the Lame system of linear elasticity. It is known that if the boundary of a planar domain is smooth enough, it has eigenvalues converging to two different points determined by Lame parameters. We show that eigenvalues converge at a polynomial rate on smooth boundaries and the convergence rate is determined by smoothness of the boundary. We also show that they converge at an exponential rate if the boundary of the domain is real analytic.
- Published
- 2020
179. A new approach for the univalence of certain integral of harmonic mappings
- Author
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Rodrigo Hernández, Hugo Arbeláez, Osvaldo Venegas, Willy Sierra, and Víctor Bravo
- Subjects
Pure mathematics ,Geometric function theory ,Mathematics::General Mathematics ,Mathematics::Complex Variables ,Mathematics - Complex Variables ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Primary: 30C45, 44A20, Secondary: 30C55 ,FOS: Mathematics ,Complex Variables (math.CV) ,0101 mathematics ,Mathematics - Abstract
The principal goal of this paper is to extend the classical problem of finding the values of α ∈ ℂ for which either f ˆ α ( z ) = ∫ 0 z ( f ( ζ ) ∕ ζ ) α d ζ or f α ( z ) = ∫ 0 z ( f ′ ( ζ ) ) α d ζ are univalent, whenever f belongs to some subclasses of univalent mappings in D , to the case of harmonic mappings, by considering the shear construction introduced by Clunie and Sheil-Small in [4] .
- Published
- 2020
180. Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions
- Author
-
Kota Uriya, Jun Ichi Segata, and Satoshi Masaki
- Subjects
Logarithm ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Gauge (firearms) ,35L71 ,01 natural sciences ,Term (time) ,010101 applied mathematics ,symbols.namesake ,Nonlinear system ,Range (mathematics) ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,symbols ,0101 mathematics ,Invariant (mathematics) ,Constant (mathematics) ,Klein–Gordon equation ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed., Comment: 25 papges, 2 figures
- Published
- 2020
181. Linear relations of Ohno sums of multiple zeta values
- Author
-
Nobuo Sato, Hideki Murahara, Tomokazu Onozuka, and Minoru Hirose
- Subjects
Power series ,Pure mathematics ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Duality (optimization) ,010103 numerical & computational mathematics ,01 natural sciences ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Relation (history of concept) ,Mathematics - Abstract
Ohno’s relation is a well-known family of relations among multiple zeta values, which can naturally be regarded as a type of duality for a certain power series which we call an Ohno sum. In this paper, we investigate Q -linear relations among Ohno sums which are not contained in Ohno’s relation. We prove two new families of such relations, and pose several further conjectural families of such relations.
- Published
- 2020
182. Invariant generalized complex structures on partial flag manifolds
- Author
-
Carlos A. B. Varea
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Differential Geometry (math.DG) ,General Mathematics ,010102 general mathematics ,Isotropy ,FOS: Mathematics ,Generalized flag variety ,010103 numerical & computational mathematics ,0101 mathematics ,Invariant (mathematics) ,01 natural sciences ,Mathematics - Abstract
The aim of this paper is to classify all invariant generalized complex structures on a partial flag manifold F Θ with at most four isotropy summands. To classify them all we proved that an invariant generalized almost complex structure on F Θ is ‘constant’ in each component of the isotropy representation.
- Published
- 2020
183. Hardy type inequalities and parametric Lamb equation
- Author
-
R. G. Nasibullin and R. V. Makarov
- Subjects
Pure mathematics ,Euclidean space ,General Mathematics ,010102 general mathematics ,Open set ,Regular polygon ,Boundary (topology) ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Domain (mathematical analysis) ,symbols.namesake ,symbols ,0101 mathematics ,Bessel function ,Parametric statistics ,Mathematics - Abstract
This paper is devoted to Hardy type inequalities with remainders for compactly supported smooth functions on open sets in the Euclidean space. We establish new inequalities with weight functions depending on the distance function to the boundary of the domain. One-dimensional L 1 and L p inequalities and their multidimensional analogues are proved. We consider spatial inequalities in open convex domains with the finite inner radius. Constants in these inequalities depend on the roots of parametric Lamb equation for the Bessel function and turn out to be sharp in some particular cases.
- Published
- 2020
184. Regularity and continuity of the multilinear strong maximal operators
- Author
-
Feng Liu, Qingying Xue, and Kôzô Yabuta
- Subjects
Mathematics::Functional Analysis ,Multilinear map ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Type (model theory) ,Weak type ,01 natural sciences ,Sobolev space ,Combinatorics ,Mathematics - Analysis of PDEs ,42B25, 47G10 ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Maximal operator ,Maximal function ,010307 mathematical physics ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Let $m\ge 1$, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operator $${\mathscr{M}}_{\mathcal{R}}(\vec{f})(x)=\sup_{\substack{R \ni x R\in\mathcal{R}}}\prod\limits_{i=1}^m\frac{1}{|R|}\int_{R}|f_i(y)|dy,$$ where $x\in\mathbb{R}^d$ and $\mathcal{R}$ denotes the family of all rectangles in $\mathbb{R}^d$ with sides parallel to the axes. When $m=1$, denote $\mathscr{M}_{\mathcal{R}}$ by $\mathcal {M}_{\mathcal{R}}$.Then, $\mathcal {M}_{\mathcal{R}}$ coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that ${\mathscr{M}}_{\mathcal{R}}$ is bounded and continuous from the Sobolev spaces $W^{1,p_1}(\mathbb{R}^d)\times \cdots\times W^{1,p_m}(\mathbb{R}^d)$ to $W^{1,p} (\mathbb{R}^d)$, from the Besov spaces $B_{s}^{p_1,q} (\mathbb{R}^d)\times\cdots\times B_s^{p_m,q}(\mathbb{R}^d)$ to $B_s^{p,q}(\mathbb{R}^d)$, from the Triebel-Lizorkin spaces $F_{s}^{p_1,q}(\mathbb{R}^d)\times\cdots\times F_s^{p_m,q}(\mathbb{R}^d)$ to $F_s^{p,q}(\mathbb{R}^d)$. As a consequence, we further showed that ${\mathscr{M}}_{\mathcal{R}}$ is bounded and continuous from the fractional Sobolev spaces $W^{s,p_1}(\mathbb{R}^d)\times \cdots\times W^{s,p_m}(\mathbb{R}^d)$ to $W^{s,p}(\mathbb{R}^d)$ for $0, Comment: 49 pages
- Published
- 2020
185. 2D stochastic Chemotaxis-Navier-Stokes system
- Author
-
Jianliang Zhai and Tusheng Zhang
- Subjects
Applied Mathematics ,General Mathematics ,Weak solution ,010102 general mathematics ,Banach space ,Fixed point ,01 natural sciences ,010101 applied mathematics ,Applied mathematics ,Uniqueness ,Navier stokes ,0101 mathematics ,Martingale (probability theory) ,Mathematics - Abstract
In this paper, we establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through introducing a new method of cutting off the stochastic system and using a fixed point argument in a carefully constructed Banach space. To get the weak solution we first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution.
- Published
- 2020
186. Kähler-Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature
- Author
-
Xiaolong Li and Lei Ni
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Curvature ,01 natural sciences ,Upper and lower bounds ,Mathematics - Abstract
In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature upper bound. New results on ancient solutions for the Ricci and Kahler-Ricci flow are obtained. Applications to Kahler manifolds with almost nonnegative orthogonal bisectional curvature are derived as consequences. The main new feature is that no curvature upper bound is assumed.
- Published
- 2020
187. Two bifurcation sets arising from the beta transformation with a hole at 0
- Author
-
Simon Baker and Derong Kong
- Subjects
Combinatorics ,Conjecture ,Transformation (function) ,General Mathematics ,Hausdorff dimension ,010102 general mathematics ,Beta (velocity) ,010103 numerical & computational mathematics ,0101 mathematics ,Characterization (mathematics) ,01 natural sciences ,Bifurcation ,Mathematics - Abstract
Given β ∈ ( 1 , 2 ] , the β -transformation T β : x ↦ β x ( mod 1 ) on the circle [ 0 , 1 ) with a hole [ 0 , t ) was investigated by Kalle et al. (2019). They described the set-valued bifurcation set E β ≔ { t ∈ [ 0 , 1 ) : K β ( t ′ ) ≠ K β ( t ) ∀ t ′ > t } , where K β ( t ) ≔ { x ∈ [ 0 , 1 ) : T β n ( x ) ≥ t ∀ n ≥ 0 } is the survivor set. In this paper we investigate the dimension bifurcation set ℬ β ≔ { t ∈ [ 0 , 1 ) : dim H K β ( t ′ ) ≠ dim H K β ( t ) ∀ t ′ > t } , where dim H denotes the Hausdorff dimension. We show that if β ∈ ( 1 , 2 ] is a multinacci number then the two bifurcation sets ℬ β and E β coincide. Moreover we give a complete characterization of these two sets. As a corollary of our main result we prove that for β a multinacci number we have dim H ( E β ∩ [ t , 1 ] ) = dim H K β ( t ) for any t ∈ [ 0 , 1 ) . This confirms a conjecture of Kalle et al. for β a multinacci number.
- Published
- 2020
188. Norm Hilbert spaces over G-modules with a convex base
- Author
-
Herminia Ochsenius and Elena Olivos
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,Hilbert space ,Banach space ,Analogy ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Norm (mathematics) ,symbols ,0101 mathematics ,Ordered subsets ,Subspace topology ,Mathematics - Abstract
By analogy with the classical definition, a Norm Hilbert space E is defined as a Banach space over a valued field K in which each closed subspace has an orthocomplement. In the rank one case (that is, the value group as well as the set of norms of the space are contained in [ 0 , ∞ ) ), they were described by van Rooij in his classical book of 1978, but the name itself was introduced in 1999 by Ochsenius and Schikhof for the case of spaces with an infinite rank valuation. Here we shall only consider spaces over fields with value groups contained in ( R + , ⋅ ) . Yet for the set of their norms we borrow, from the infinite rank case, the notion of a G -module. That structure allows for a greater complexity than what is offered by ordered subsets of R . In this paper we describe a new class of Norm Hilbert spaces, those in which the G -module has a convex base. Their characteristics will be the focus of our study.
- Published
- 2020
189. Viral diffusion and cell-to-cell transmission: Mathematical analysis and simulation study
- Author
-
Lin Wang, Zongwei Ma, Xiang-Sheng Wang, and Hongying Shu
- Subjects
Steady state ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Degenerate energy levels ,01 natural sciences ,010101 applied mathematics ,Exponential stability ,Transmission (telecommunications) ,Uniqueness ,Statistical physics ,0101 mathematics ,Diffusion (business) ,Degeneracy (mathematics) ,Basic reproduction number ,Mathematics - Abstract
We propose a general model to investigate the joint impact of viral diffusion and cell-to-cell transmission on viral dynamics. The mathematical challenge lies in the fact that the model system is partially degenerate and the solution map is not compact. While the simpler cases with only indirect transmission mode or weak cell-to-cell transmission mode have been extensively studied in the literature, it remains an open problem to understand the local and global dynamics of fully coupled viral infection model with partial degeneracy. In this paper, we identify the basic reproduction number as the spectral radius of the sum of two linear operators corresponding to direct and indirect transmission modes. It is well-known that viral mobility may induce infection in low-risk regions. However, as diffusion coefficient increases, we prove that the basic reproduction number actually decreases, which indicates that faster viral movements may result in a lower level of viral infection. By an innovative construction of Lyapunov functionals, we further demonstrate that the basic reproduction number is the threshold parameter which determines global picture of viral dynamics. In addition to the traditional dichonomy results of extinction and persistence as obtained in earlier works for many simpler models, we are able to prove global asymptotic stability of infection-free steady state and global attractiveness (as well as uniqueness) of chronic-infection steady state, depending on whether the basic reproduction number is smaller or greater than one. Numerical simulation supports our theoretical results and suggests an interesting phenomenon: boundary layer and internal layer may occur when the diffusion parameter tends to zero.
- Published
- 2020
190. Finite-time blowup in Cauchy problem of parabolic-parabolic chemotaxis system
- Author
-
Noriko Mizoguchi
- Subjects
010101 applied mathematics ,Cauchy problem ,Plane (geometry) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Time derivative ,Mathematical analysis ,Mathematics::Analysis of PDEs ,0101 mathematics ,Finite time ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with blowup in a parabolic-parabolic system describing chemotactic aggregation. In a disk, radial solutions blow up in finite time if their initial energy is less than some value. In the whole plane, the energy diverges to −∞ as time goes to +∞ for any forward selfsimilar solution. This implies that one cannot expect to get a sufficient condition for finite-time blowup using energy as in a disk. For a solution ( u , v ) , u and v denote density of cells and of chemical substance, respectively. Let τ be the coefficient of time derivative of v. We first prove that for τ > 0 there exists M ( τ ) > 0 with M ( τ ) → ∞ as τ → ∞ such that all radial solutions ( u , v ) with initial mass of u larger than M ( τ ) blow up in finite time. On the other hand, it was shown in [22] that any blowup in the system with τ = 1 is type II (not necessarily in radial case). Removing the restriction on τ, we get the conclusion for all τ > 0 .
- Published
- 2020
191. On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases
- Author
-
Tohru Ozawa, Vladimir Georgiev, and Kazumasa Fujiwara
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Type (model theory) ,01 natural sciences ,Power (physics) ,010101 applied mathematics ,Nonlinear system ,Euclidean geometry ,Strichartz estimate ,A priori and a posteriori ,0101 mathematics ,Scaling ,Well posedness ,Mathematics - Abstract
In this paper, the global well-posedness of semirelativistic equations with a power type nonlinearity on Euclidean spaces is studied. In two dimensional H s scaling subcritical case with 1 ≤ s ≤ 2 , the local well-posedness follows from a Strichartz estimate. In higher dimensional H 1 scaling subcritical case, the local well-posedness for radial solutions follows from a weighted Strichartz estimate. Moreover, in three dimensional H 1 scaling critical case, the local well-posedness for radial solutions follows from a uniform bound of solutions which may be derived by the corresponding one dimensional problem. Local solutions may be extended by a priori estimates.
- Published
- 2020
192. Concentration of nodal solutions for logarithmic scalar field equations
- Author
-
Zhi-Qiang Wang and Chengxiang Zhang
- Subjects
Sequence ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Semiclassical physics ,Function (mathematics) ,Eigenfunction ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Bound state ,0101 mathematics ,Scalar field ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper studies sign-changing solutions and their concentration behaviors of logarithmic scalar field equations in the semiclassical setting. At a local minimum of the potential function we construct an unbounded sequence of sign-changing solutions concentrating near the local minimum. This resembles a localized phenomenon of an unbounded sequence of localized bound states in the studies of nonlinear eigenvalues and eigenfunctions.
- Published
- 2020
193. On the convergence of almost minimal sets for the Hausdorff and varifold topologies
- Author
-
Yangqin Fang
- Subjects
Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hausdorff space ,Dimension function ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Geometric measure theory ,Bounded function ,Convergence (routing) ,Limit of a sequence ,0101 mathematics ,Varifold ,Mathematics - Abstract
The geometric properties of (almost) minimal sets, especially the regularity, is an interesting topic in geometric measure theory, which were often studied in the literature, for example [2] , [4] , [5] , [8] , [9] , [10] , [19] , [22] . Soap films as well as solutions to Plateau's Problem could be a typical example of such kind of minimal sets. In this paper, we will investigate the convergence of a sequence of such sets, and show that Hausdorff convergence and varifold convergence coincide on the class of almost minimal sets bounded by a uniform gauge function, and we will see that a large amount of sets, including all compact C 1 , 1 submanifolds in R n , are almost minimal.
- Published
- 2020
194. Deformations of algebraic schemes via Reedy–Palamodov cofibrant resolutions
- Author
-
Marco Manetti, Francesco Meazzini, Manetti M., and Meazzini F.
- Subjects
Noetherian ,Pure mathematics ,differential graded algebras ,model categories ,General Mathematics ,Deformation theory ,Algebraic schemes ,Field (mathematics) ,010103 numerical & computational mathematics ,Algebraic schemes, cotangent complex, deformation theory, differential graded algebras, model categories ,01 natural sciences ,Mathematics - Algebraic Geometry ,Complex space ,deformation theory ,18G55, 14D15, 16W50 ,FOS: Mathematics ,Category Theory (math.CT) ,0101 mathematics ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics ,Resolvent ,010102 general mathematics ,Mathematics - Category Theory ,Differential graded algebra ,cotangent complex ,Algebraic scheme ,Scheme (mathematics) ,Differential (mathematics) - Abstract
Let $X$ be a Noetherian separated and finite dimensional scheme over a field $\mathbb{K}$ of characteristic zero. The goal of this paper is to study deformations of $X$ over a differential graded local Artin $\mathbb{K}$-algebra by using local Tate-Quillen resolutions, i.e., the algebraic analog of the Palamodov's resolvent of a complex space. The above goal is achieved by describing the DG-Lie algebra controlling deformation theory of a diagram of differential graded commutative algebras, indexed by a direct Reedy category., Final version. To appear in Indagationes Mathematicae
- Published
- 2020
195. Injective metrizability and the duality theory of cubings
- Author
-
Dan P. Guralnik, Jared Culbertson, and Peter F. Stiller
- Subjects
Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Metric Geometry (math.MG) ,01 natural sciences ,Injective function ,Metric space ,Simplicial complex ,Mathematics - Metric Geometry ,Geometric group theory ,51K05 (primary), 57M99, 05C10 ,Metrization theorem ,Simply connected space ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
Following his discovery that finite metric spaces have injective envelopes naturally admitting a polyhedral structure, Isbell, in his pioneering work on injective metric spaces, attempted a characterization of cellular complexes admitting the structure of an injective metric space. A bit later, Mai and Tang confirmed Isbell's conjecture that a simplicial complex is injectively metrizable if and only if it is collapsible. Considerable advances in the understanding, classification and applications of injective envelopes have since been made by Dress, Huber, Sturmfels and collaborators, and most recently by Lang. Unfortunately a combination theory for injective polyhedra is still unavailable. Here we expose a connection to the duality theory of cubings -- simply connected non-positively curved cubical complexes -- which provides a more principled and accessible approach to Mai and Tang's result, providing one with a powerful tool for systematic construction of locally-compact injective metric spaces: Main Theorem. Any complete pointed Gromov--Hausdorff limit of locally-finite piecewise-$\ell_\infty$ cubings is injective. This result may be construed as a combination theorem for the simplest injective polytopes, $\ell_\infty$-parallelopipeds, where the condition for retaining injectivity is the combinatorial non-positive curvature condition on the complex. Thus it represents a first step towards a more comprehensive combination theory for injective spaces. In addition to setting the earlier work on injectively metrizable complexes within its proper context of non-positively curved geometry, this paper is meant to provide the reader with a systematic review of the results~ ---~ otherwise scattered throughout the geometric group theory literature~ ---~ on the duality theory and the geometry of cubings, which make this connection possible., Significantly expanded and revised. 40 pages, 6 figures, to appear in Expositiones Mathematicae (electronic https://doi.org/10.1016/j.exmath.2018.06.001)
- Published
- 2019
196. Concepts of curvatures in normed planes
- Author
-
Horst Martini, Emad Shonoda, and Vitor Balestro
- Subjects
Discrete mathematics ,Pure mathematics ,Fundamental theorem ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Four-vertex theorem ,Curvature ,01 natural sciences ,Constant curvature ,Fundamental theorem of curves ,Euclidean geometry ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. To complete and clarify the whole picture, we show which known concepts are equivalent, and add also a new type of curvature. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behavior of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves.
- Published
- 2019
197. The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application
- Author
-
Zhijie Chen, Ting Jung Kuo, and Chang-Shou Lin
- Subjects
Pure mathematics ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Modular form ,01 natural sciences ,Monodromy ,Mean field equation ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Flat torus ,Mathematics - Abstract
In this paper, the second in a series, we continue to study the generalized Lame equation with the Treibich-Verdier potential y ″ ( z ) = [ ∑ k = 0 3 n k ( n k + 1 ) ℘ ( z + ω k 2 | τ ) + B ] y ( z ) , n k ∈ Z ≥ 0 from the monodromy aspect. We prove the existence of a pre-modular form Z r , s n ( τ ) of weight 1 2 ∑ n k ( n k + 1 ) such that the monodromy data ( r , s ) is characterized by Z r , s n ( τ ) = 0 . This generalizes the result in [17] , where the Lame case (i.e. n 1 = n 2 = n 3 = 0 ) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δ u + e u = 16 π δ 0 and Δ u + e u = 8 π ∑ k = 1 3 δ ω k 2 on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.
- Published
- 2019
198. Equivalence of solutions to fractional p-Laplace type equations
- Author
-
Janne Korvenpää, Erik Lindgren, and Tuomo Kuusi
- Subjects
Pure mathematics ,Laplace transform ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Mathematics - Analysis of PDEs ,Bounded function ,FOS: Mathematics ,Integration by parts ,0101 mathematics ,Equivalence (measure theory) ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional $p$-Laplace type $${\rm P.V.} \int_{\mathbb R^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dy = 0.$$ Solutions are defined via integration by parts with test functions, as viscosity solutions or via comparison. Our main result states that for bounded solutions, the three different notions coincide., Comment: 21 pages, to appear in Journal de Math\'ematiques Pures et Appliqu\'ees
- Published
- 2019
199. Integral operators on BMO and Campanato spaces
- Author
-
Kwok Pun Victor Ho
- Subjects
Pure mathematics ,Hadamard transform ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,010103 numerical & computational mathematics ,0101 mathematics ,Space (mathematics) ,01 natural sciences ,Bounded mean oscillation ,Mathematics - Abstract
This paper establishes the mapping properties of integral operators on space of bounded mean oscillation and Campanato spaces. In particular, we have the Hardy’s inequality and the boundedness of the Hadamard fractional integrals on space of bounded mean oscillation and Campanato spaces.
- Published
- 2019
200. Spectra of(H1,H2)-merged subdivision graph of a graph
- Author
-
R. Rajkumar and M. Gayathri
- Subjects
Spanning tree ,Unary operation ,General Mathematics ,Subdivision graph ,010102 general mathematics ,Spectrum (functional analysis) ,010103 numerical & computational mathematics ,01 natural sciences ,Spectral line ,Combinatorics ,Adjacency list ,0101 mathematics ,Graph operations ,Laplace operator ,Mathematics - Abstract
In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, R -graph, central graph. Also, it generalizes the construction of overlay graph (Somodi et al., 2017), and consequently, Q -graph, total graph, and quasitotal graph. We denote this new graph by [ S ( G ) ] H 2 H 1 , where G is a graph and, H 1 and H 2 are suitable graphs corresponding to G . Further, we define several new unary graph operations which becomes particular cases of this construction. We determine the Adjacency and Laplacian spectra of [ S ( G ) ] H 2 H 1 for some classes of graphs G , H 1 and H 2 . From these results, we derive the L -spectrum of the graphs obtained by the unary graph operations mentioned above. As applications, these results enable us to compute the number of spanning trees and Kirchhoff index of these graphs.
- Published
- 2019
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