Showing papers in "Journal of The London Mathematical Society-second Series in 2003"
TL;DR: In this paper, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equation is a positive integer, which is probably the first time critical point theory has been applied to deal with the presence of periodic solutions of difference systems.
Abstract: By critical point theory, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equation is a positive integer. This is probably the first time critical point theory has been applied to deal with the existence of periodic solutions of difference systems.This project is supported by TRATOYT of China and by the State Education Commission Trans-Century Training Program Foundation for the Talents.
193 citations
TL;DR: In this article, a complete classification of escaping points is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays.
Abstract: The points which converge to ∞ under iteration of the maps z↦λexp(z) for λ ∈ C/{0} are investigated. A complete classification of such ‘escaping points’ is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter λ.
It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpinska for specific choices of λ.
148 citations
TL;DR: By means of generalized Riccati transformation techniques and generalized exponential functions, some oscillation criteria are given for the nonlinear dynamic equation in this article, and sufficient conditions are obtained for the oscillation of all solutions.
Abstract: By means of generalized Riccati transformation techniques and generalized exponential functions, some oscillation criteria are given for the nonlinear dynamic equation \[ (p(t)x^{\Delta} (t))^{\Delta}+q(t)(f\circ x^{\sigma})=0 \] on time scales. The results are also applied to linear and nonlinear dynamic equations with damping, and some sufficient conditions are obtained for the oscillation of all solutions.
140 citations
TL;DR: In this paper, a detailed analysis of the set of divergence points is performed and it is shown that it has a surprisingly rich structure and can be partitioned into an uncountable family of pairwise disjoint sets each with full dimension.
Abstract: Let and be the self-similar set and the self-similar measure associated with an IFS (iterated function system) with probabilities satisfying the open set condition. Let denote the full shift space and let denote the natural projection. The (symbolic) local dimension of at is defined by , where for . A point for which the limit does not exist is called a divergence point. In almost all of the literature the limit is assumed to exist, and almost nothing is known about the set of divergence points. In the paper a detailed analysis is performed of the set of divergence points and it is shown that it has a surprisingly rich structure. For a sequence , let denote the set of accumulation points of . For an arbitrary subset of , the Hausdorff and packing dimension of the set
\[ \left\{\omega\in\Sigma\left\vert {\sf A}\left(\frac{\log\mu K_{\omega\mid n}}{\log\hbox{ diam }K_{\omega\mid n}}\right)\right.=I\right\} \]and related sets is computed. An interesting and surprising corollary to this result is that the set of divergence points is extremely ‘visible’; it can be partitioned into an uncountable family of pairwise disjoint sets each with full dimension.In order to prove the above statements the theory of normal and non-normal points of a self-similar set is formulated and developed in detail. This theory extends the notion of normal and non-normal numbers to the setting of self-similar sets and has numerous applications to the study of the local properties of self-similar measures including a detailed study of the set of divergence points.
111 citations
108 citations
TL;DR: In this paper, the authors studied the interplay between cotorsion and torsion pairs in finite linear algebra, and showed that every cotorion pair for mod Λ induces an acyclic pair for Mod Λ. This has some interesting applications, even for the category of finitely presented modules.
Abstract: Fo ra na rtin algebra Λ, cotorsion pairs are studied for the category mod Λ of finitely presented Λ-modules and for the category Mod Λ of all Λ-modules. It is shown that every cotorsion pair for mod Λ induces ac otorsion pair for Mod Λ. This has some interesting applications, even for the category of finitely presented modules. Another theme of the paper is the interplay between cotorsion and torsion pairs. This leads to a conjecture which is an analogue of the telescope conjecture in stable homotopy theory.
92 citations
TL;DR: In this article, the flag curvature of a Finsler metric with isotropic S-curvature is studied and the curvature is partially determined when certain non-Riemannian quantities such as Cartan torsion, Landsberg curvature and S-Curvature vanish.
Abstract: The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In the paper, Finsler metrics of scalar curvature (that is, the flag curvature is a scalar function on the slit tangent bundle) are studied and the flag curvature is partially determined when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, locally projectively flat Randers metrics with isotropic S-curvature are classified.
90 citations
TL;DR: In this paper, the so-called Weiss conjecture on admissible operators for bounded analytic semigroups with generator was shown to hold even for non-Hilbertian Banach spaces.
Abstract: New results concerning the so-called Weiss conjecture on admissible operators for bounded analytic semigroups are given. Let \[ \left(T_t\right)_{t\geqslant 0} \] be a bounded analytic semigroup with generator . This holds in particular if \[ \left(T_t\right)_{t\geqslant 0}\] is a contractive (analytic) semigroup on Hilbert space. In the converse direction, it is shown that this may happen for a bounded analytic semigroup on Hilbert space that is not similar to a contractive one. Applications in non-Hilbertian Banach spaces are also given.
86 citations
TL;DR: In this article, it was shown that there exists a primitive normal basis over over, all of whose members are primitive and free, and that such a basis can be found for every additive generator.
Abstract: Given , a power of a prime , denote by the finite field of order , and, for a given positive integer , by its extension of degree . A primitive element of is a generator of the cyclic group . Additively too, the extension is cyclic when viewed as an -module, being the Galois group of over . The classical form of this result – the normal basis theorem – is that there exists an element (an additive generator) whose conjugates form a basis of over is a free element of over , and a basis like this is a normal basis over . The core result linking additive and multiplicative structure is that there exists , simultaneously primitive and free over . This yields a primitive normal basis over , all of whose members are primitive and free. Existence of such a basis for every extension was demonstrated by Lenstra and Schoof [5] (completing work by Carlitz [1, 2] and Davenport [4]).
75 citations
TL;DR: In this paper, it was shown that every special Lagrangian cone in C 3 determines, and is determined by, a primitive harmonic surface in the 6- symmetric space SU3/SO2.
Abstract: In this article I show that every special Lagrangian cone in C 3 determines, and is determined by, a primitive harmonic surface in the 6- symmetric space SU3/SO2. For cones over tori, this allows us to use the classification theory of harmonic tori to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some of the examples found recently by Joyce are put into this context.
73 citations
TL;DR: In this article, the self-similar measure for a linear function system, µ, is shown to be differentiable on (0, ∞) and satisfies the multifractal formalism.
Abstract: Let µ be the self-similar measure for a linear function system Sj x = ρx + bj (j =1 , 2 ,...,m )o n the real line with the probability weight {pj } m=1 .U nder the condition that {Sj } m=1 satisfies the finite type condition, the L q -spectrum τ(q )o fµ is shown to be differentiable on (0, ∞); as an application, µ is exact dimensional and satisfies the multifractal formalism.
TL;DR: A conjecture due to Holroyd and Johnson that an analogue of the Erdős-Ko-Rado theorem holds for $k$ -separated sets is proved in this paper.
Abstract: A set $A\subseteq \{1,2,\ldots,n\}$ is said to be $k$ - separated if, when considered on the circle, any two elements of $A$ are separated by a gap of size at least $k$ . A conjecture due to Holroyd and Johnson that an analogue of the Erdős–Ko–Rado theorem holds for $k$ -separated sets is proved. In particular, the result holds for the vertex-critical subgraph of the Kneser graph identified by Schrijver, the collection of separated sets. A version of the Erdős–Ko–Rado theorem for weighted $k$ -separated sets is also given.
TL;DR: In this paper, it was proved that the measure algebra M(G) of a locally compact group G is Connes-amenable if and only if G is amenable.
Abstract: It is proved that the measure algebra M(G) of a locally compact group G is Connes-amenable if and only if G is amenable.
TL;DR: In this article, conditions under which perturbations of certain base potentials are uniquely determined from the location of eigenvalues and resonances in the context of a Schrodinger operator on a half line are given.
Abstract: A new technique is presented which gives conditions under which perturbations of certain base potentials are uniquely determined from the location of eigenvalues and resonances in the context of a Schrodinger operator on a half line. The method extends to complex-valued potentials and certain potentials whose first moment is not integrable.
TL;DR: In this paper, dualizing differential graded (DGDG) modules over DG algebras was studied and the main result was that a finite DG module is dualizing precisely when its Bass number is 1.
Abstract: The paper explores dualizing differential graded (DG) modules over DG algebras. The focus is on DG algebras that are commutative local, and finite. One of the main results established is that, for this class of DG algebras, a finite DG module is dualizing precisely when its Bass number is 1. As a corollary, one obtains that the Avramov–Foxby notion of Gorenstein DG algebras coincides with that due to Frankild and Jorgensen. One other key result is that, under suitable hypotheses, any two dualizing DG modules are quasiisomorphic up to a suspension. In addition, it is established that a number of naturally occurring DG algebras possess dualizing DG modules.
TL;DR: In this article, it was shown that all permutation groups with a transitive minimal normal subgroup that have no fixed point free elements of prime order are not 2-closed and substantial progress was made towards asserting the truth of the polycirculant conjecture.
Abstract: The paper determines all permutation groups with a transitive minimal normal subgroup that have no fixed point free elements of prime order. All such groups are primitive and are wreath products in a product action involving $M_{11}$
in its action on 12 points. These groups are not 2-closed and so substantial progress is made towards asserting the truth of the polycirculant conjecture that every 2-closed transitive permutation group has a fixed point free element of prime order. All finite simple groups $T$
with a proper subgroup meeting every ${\rm Aut}(T)$
-conjugacy class of elements of $T$
of prime order are also determined.
TL;DR: In this paper, it was shown that the rigidity property of linearly recurrent Cantor systems remains true for the class of linear recurrent systems with substitution subshifts and some odometers.
Abstract: The class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitution subshifts and odometers measure--theoretical and continuous eigenvalues are the same. It is natural to ask whether this rigidity property remains true for the class of linearly recurrent Cantor systems. We give partial answers to this question.
TL;DR: In this article, the authors studied analyticity properties of the pressure function of non-recurrent maps in the presence of singularities on the Julia set and proposed a tower-based approach for the analysis.
Abstract: The pressure function p(t )o fa non-recurrent map is real analytic on some interval (0 ,t ∗ )w it ht∗ strictly greater than the dimension of the Julia set. The proof is an adaptation of the well known tower techniques to the complex dynamics situation. In general, p(t )n eed not be analytic on the whole positive axis. In this paper we study analyticity properties of the pressure function of non-recurrent maps. Our approach is based on the well known tower techniques adapted to the complex dynamics situation. The pressure function p(t), which is defined in terms of the Poincar´ es erie s( see (1.4)), carries essential information about ergodic and dimensional properties of the maximal measure. In particular, it characterizes the dimension spectrum of harmonic measure on the Julia set in the case of a polynomial dynamics. According to the classical theory of Sinai, Ruelle and Bowen, p(t )i s real analytic if the dynamics is hyperbolic ,t hat is, expanding on the Julia set. This fact is closely related to the so called ‘spectral gap’ phenomenon, which also implies other important features of hyperbolic dynamics such as the existence of equilibrium states, exponential decay of correlations, etc. The problem of extending (some parts of) the classical theory to the non-hyperbolic case has become one of the central themes in the ergodic theory of conformal dynamics. In the first part of this work [8], we provided a detailed analysis of the negative part t 0 is substantially more complicated (and more important). The main difficulty arises from the presence of singularities (critical points) on the Julia set. To circumvent this difficulty, we propose to use a tower construction which forces the dynamics to be expanding on some auxiliary space. The tower method has been widely used in the general theory of dynamical systems with some degree of hyperbolicity (see especially [12]), and in particular in 1-dimensional real dynamics, where the construction is known as Hof bauer’s tower .T oa pply this method in the complex case, it is natural to use some basic elements of the Yoccoz jigsaw puzzle structure (see [9]). We will discuss only the simplest type of non-hyperbolic behavior: the case of nonrecurrent dynamics (every critical point in the Julia set is non-recurrent, that is, is away from its iterates) without parabolic cycles (see [2 ]f orvarious characterizations
TL;DR: In this paper, the authors give necessary and sufficient conditions for a Banach space operator with the single valued extension property (SVEP) to satisfy Weyl's theorem and a-Weyl theorem.
Abstract: We give necessary and sufficient conditions for a Banach space operator with the single valued extension property (SVEP) to satisfy Weyl’s theorem and a-Weyl’s theorem. We show that if T or T ∗ has SVEP and T is transaloid, then Weyl’s theorem holds for f(T ) for every f ∈ H(σ(T )). When T ∗ has SVEP, T is transaloid and T is a-isoloid, then a-Weyl’s theorem holds for f(T ) for every f ∈ H(σ(T )). We also prove that if T or T ∗ has SVEP, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.
TL;DR: In this paper, the coefficients in asymptotics of regularized traces and associated trace distributions for Schrodinger operators with short and long range potentials are derived, and a connection with non-commutative Taylor formulas is established.
Abstract: The coefficients in asymptotics of regularized traces and associated trace (spectral) distributions for Schrodinger operators with short and long range potentials are computed. A kernel expansion for the Schrodinger semigroup is derived, and a connection with non-commutative Taylor formulas is established.
TL;DR: In this paper, strong and weak type estimates for the Mehler kernel operator with respect to the region of holomorphy of the Ornstein-Uhlenbeck semigroup were obtained.
Abstract: For each p in [1, ∞) let Ep denote the closure of the region of holomorphy of the Ornstein-Uhlenbeck semigroup {Ht : t> 0} on L p with respect to the Gaussian measure. We prove sharp weak type and strong type estimates for the maximal operator f �→ H ∗ f = sup{|Hzf | : z ∈ Ep} and for a class of related operators. As a consequence of our methods, we give a new and simpler proof of the weak type 1 estimate for the maximal operator associated to the Mehler kernel.
TL;DR: A new characterization of the full upper and lower Gaussian or sub‐Gaussian heat kernel estimates is given in terms of a certain Sobolev inequality on a manifold with polynomial volume growth satisfying Gaussian upper bounds of the heat kernel.
Abstract: On a manifold with polynomial volume growth satisfying Gaussian upper bounds of the heat kernel, a simple characterization of the matching lower bounds is given in terms of a certain Sobolev inequality. The method also works in the case of so-called sub-Gaussian or sub-diffusive heat kernels estimates, which are typical of fractals. Together with previously known results, this yields a new characterization of the full upper and lower Gaussian or sub-Gaussian heat kernel estimates.
TL;DR: For each Artin group, the reduced l 2 -cohomology of its Salvetti complex is computed as discussed by the authors, which is a CW-complex which is conjectured to be a model for the classifying space of the artin group.
Abstract: For each Artin group we compute the reduced l^2-cohomology of its `Salvetti complex'. This is a CW-complex which is conjectured to be a model for the classifying space of the Artin group. When this conjecture is known to hold our calculation describes the l^2-cohomology of the Artin group.
TL;DR: In this paper, the authors consider the problem of finite distortion mappings in the theory of nonlinear elasticity and show that the connection with deformations of elastic bodies and regularity of energy minimizers can be found in many applications, particularly in holomorphic dynamics and also in the study of first order degenerate elliptic systems.
Abstract: Recently, systematic studies of mappings of finite distortion have emerged as a key area in geometric function theory. The connection with deformations of elastic bodies and regularity of energy minimizers in the theory of nonlinear elasticity is perhaps a primary motivation for such studies, but there are many other applications as well, particularly in holomorphic dynamics and also in the study of first order degenerate elliptic systems, for instance the Beltrami systems we consider here.
TL;DR: A construction of the Monster simple group is described implicitly as matrices over the field of 3 elements in this article, where the construction is expressed as a linear combination of matrices.
Abstract: A construction of the Monster simple group is described implicitly as matrices over the field of 3 elements.
TL;DR: Hosokawa, Izuchi, and Zheng as discussed by the authors showed that any asymptotic interpolating sequence in a uniform algebra eventually is interpolating, and they also showed that the concept of type 1 interpolating sequences in the unit disk can be verified.
Abstract: T. Hosokawa, K. Izuchi and D. Zheng recently introduced the concept of asymptotic interpolating sequences (of type 1) in the unit disk for on arbitrary domains is verified. It is shown that any asymptotic interpolating sequence in a uniform algebra eventually is interpolating.
TL;DR: In this article, it was shown that the best bound on the limitation of the rotations is 0 = arccos( 1=4), which is the same bound for the twist property of a periodic solution.
Abstract: In most cases, the third order approximation of a scalar Newtonian equation can lead to the Lyapunov stability of a periodic solution through the obtaining of a nonzero twist coecient. Recently, Ortega obtained the twist property of a periodic solution when the second order coecient does not change sign and the third one is negative under a crucial limitation to the rotation of the linearization equation. The paper nds that the best bound on the limitation of the rotations is 0 = arccos( 1=4).
TL;DR: In this article, an almost-orthogonality principle for maximal operators over arbitrary sets of directions in R 2 was proved for a general class of maximal operators, where the maximal functions associated to a certain partition of the set of directions and the particular structure of this partition are given.
Abstract: In this paper we prove an almost-orthogonality principle for maximal operators over arbitrary sets of directions in R 2 . Namely, we obtain L p -bounds for an operator of this type from the corresponding L p -bounds of the maximal functions associated to a certain partition of the set of directions, and from the particular structure of this partition. We give applications to several types of maximal operators.
TL;DR: In this article, the Brauer tensor product of two symbol algebras is defined over a rational extension of a finite-dimensional division algebra and the trace form of the algebra can be defined over any algebra of degree 4.
Abstract: Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$ . $A$ is defined over a subfield $F_0$ if there exists an $F_0$ -algebra $A_0$ such that $A = A_0 \bigotimes_{F_0} F$ . The following are shown. (i) In many cases $A$ can be defined over a rational extension of $k$ . (ii) If $A$ has odd degree $n \geq 5$ , then $A$ is defined over a field $F_0$ of transcendence degree $\leq {\frac{1}{2}}(n-1)(n-2)$ over $k$ . (iii) If $A$ is a $\mathbb{Z}/m \times \mathbb{Z}/2$ -crossed product for some $m \geq 2$ (and in particular, if $A$ is any algebra of degree 4) then $A$ is Brauer equivalent to a tensor product of two symbol algebras. Consequently, ${\rm M}_m(A)$ can be defined over a field $F_0$ such that ${\rm trdeg}_k(F_0) \leq 4$ . (iv) If $A$ has degree 4 then the trace form of $A$ can be defined over a field $F_0$ of transcendence degree $\leq 4$ . (In (i), (iii) and (iv) it is assumed that the center of $A$ contains certain roots of unity.)