On Poincaré's isoperimetric problem for simple closed geodesics

AbstractWe show in the context of integral currents that Poincaré's isoperimetric variational problem for simple closed geodesics on ovaloids has a smooth extremal C without self-intersection. Provided the smooth Riemannian metric on the ovaloid M in question does not deviate too far from constant curvature, we further show that (i) this extremal C is connected and so is the desired non-trivial geodesic of shortest length on M and (ii) C is close (in the sense of Hausdorff distance) to a great circle

Lang's Conjecture and Sharp Height Estimates for the elliptic curves y2=x3+axy^{2}=x^{3}+ax

For elliptic curves given by the equation $E_{a}: y^{2}=x^{3}+ax$, we establish the best-possible version of Lang's conjecture on the lower bound of the canonical height of non-torsion points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.Comment: published version. Lemmas 5.1 and 6.1 now precise (with resultant refinement to Theorem 1.2). Small corrections to

On the Inverse Problem Relative to Dynamics of the w Function

In this paper we shall study the inverse problem relative to dynamics of the w function which is a special arithmetic function and shall get some results.Comment: 11 page

Anomalous Subvarieties—Structure Theorems and Applications

When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalous subvarieties. These arise when the algebraic variety meets translates of subgroups in sets larger than expected. We prove a Structure Theorem for the anomalous subvarieties, and we give some applications, emphasizing in particular the case of codimension two. We also state some related conjectures about the boundedness of absolute height on such intersections as well as their finitenes

On number fields with nontrivial subfields

What is the probability for a number field of composite degree $d$ to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if $d>6$. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this article is an estimate for the number of algebraic numbers of degree $d=e n$ and bounded height which generate a field that contains an unspecified subfield of degree $e$. If $n>\max\{e^2+e,10\}$ we get the correct asymptotics as the height tends to infinity

The theory of the exponential differential equations of semiabelian varieties

The complete first order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arises from an amalgamation-with-predimension construction in the style of Hrushovski. The theory includes necessary and sufficient conditions for a system of equations to have a solution. The necessary condition generalizes Ax's differential fields version of Schanuel's conjecture to semiabelian varieties. There is a purely algebraic corollary, the "Weak CIT" for semiabelian varieties, which concerns the intersections of algebraic subgroups with algebraic varieties.Comment: 53 pages; v3: Substantial changes, including a completely new introductio

The effect of a secondary task on kinematics during turning in Parkinson's disease with mild to moderate impairment

Patients with Parkinson's disease (PD) show typical gait asymmetries. These peculiar motor impairments are exacerbated by added cognitive and/or mechanical loading. However, there is scarce literature that chains these two stimuli. The aim of this study was to investigate the combined effects of a dual task (cognitive task) and turning (mechanical task) on the spatiotemporal parameters in mild to moderate PD. Participants (nine patients with PD and nine controls (CRs)) were evaluated while walking at their self-selected pace without a secondary task (single task), and while repeating the days of the week backwards (dual task) along a straight direction and a 60 degrees and 120 degrees turn. As speculated, in single tasking, PD patients preferred to walk with a shorter stride length (p< 0.05) but similar timing parameters, compared to the CR group; in dual tasking, both groups walked slower with shorter strides. As the turn angle increased, the speed will be reduced (p< 0.001), whereas the ground-foot contact will become greater (p< 0.001) in all the participants. We showed that the combination of a simple cognitive task and a mechanical task (especially at larger angles) could represent an important training stimulus in PD at the early stages of the pathology

A remark on the trace-map for the Silver mean sequence

In this work we study the Silver mean sequence based on substitution rules by means of a transfer-matrix approach. Using transfer-matrix method we find a recurrence relation for the traces of general transfer-matrices which characterizes electronic properties of the quasicrystal in question. We also find an invariant of the trace-map.Comment: 5 pages, minor improvements in style and presentation of calculation

Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles

The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the d-bar equation with cut-off functions and additional blowup weight functions