Abstract:
Let $\RR_S$ denote the expansion of the real ordered field by a family of real-valued functions $S$, where each function in $S$ is defined on a compact box and is a member of some quasianalytic class which is closed under the operations of function composition, division by variables, and implicitly defined functions. It is shown that the first order theory of $\RR_S$ is decidable if and only if two oracles, called the approximation and precision oracles for $S$, are decidable. Loosely stated, the approximation oracle for $S$ allows one to approximate any partial derivative of any function in $S$ to within any given error, and the precision oracle for $S$ allows one to decide when a manifold $M\subseteq\RR^n$ is contained in a coordinate hyperplane $\{x\in\RR^n : x_i = 0\}$ when one is given $i\in\{1,\ldots,n\}$ and a system of equations which defines $M$ nonsingularly, where the functions occurring in the equations are rational polynomials of the coordinate variables $x = (x_1,\ldots,x_n)$ and the partial derivatives of the functions in $S$. A key component of the proof is the development of a local resolution of singularities procedure which is effective in the approximation and precision oracles for $\S$, and in the course of proving our main theorem, numerous theorems about the model theory of such structures $\RR_S$ are also proven.

Abstract:
Let $\RR_S$ denote the expansion of the real ordered field by a family of real-valued functions $S$, where each function in $S$ is defined on a compact box and is a member of some quasianalytic class which is closed under the operations of function composition, division by variables, and extraction of implicitly defined functions. It is shown that if the family $S$ is generic (which is a certain technically defined transcendence condition), then the theory of $\RR_S$ is decidable if and only if $S$ is computably $C^\infty$ (which means that all the partial derivatives of the functions in $S$ may be effectively approximated). It is also shown that, in a certain topological sense, many generic, computably $C^\infty$ families $S$ exist.

Abstract:
One-third of stroke and transient ischemic attack (TIA) are cryptogenic, and paroxysmal atrial fibrillation (PAF) has been suggested as a possible cause for these cryptogenic strokes. Multiple studies have recently evaluated long-term cardiac rhythm monitoring with good yield for PAF. The duration of monitoring varies between studies as well as the qualifying event definition. Moreover, the clinical significance of very brief atrial fibrillation events is unclear in the literature. This paper provides an overview of current advances in the detection of paroxysmal atrial fibrillation, the clinical and genetic factors predictive of arrhythmia detection, and the therapeutic dilemma concerning this approach. 1. Introduction One-third of stroke and transient ischemic attack (TIA) are cryptogenic requiring additional investigation and intervention [1]. Occult paroxysmal atrial fibrillation (PAF) has been suggested as a possible cause for these cryptogenic strokes [2]. Atrial fibrillation has been long associated with high risk of stroke, but most of this knowledge is derived from patient data from chronic atrial fibrillation. It has been suggested that PAF is more prevalent than persistent atrial fibrillation in stroke and TIA patients [3]. Anticoagulation therapy initiated after detection of atrial fibrillation (AF) provides an additional 40% risk reduction of stroke as compared to antiplatelet therapy alone [4]. Therefore, it is important to diagnose AF after an ischemic stroke to provide maximal stroke prevention therapy. Current standard of care dictates an admission electrocardiogram (ECG) and at least 24？h of continuous telemetry monitoring [5]. However, brief asymptomatic paroxysmal atrial fibrillation events may remain undetected by traditional methods of screening. Recent technological advances have made it possible to perform long-term cardiac rhythm monitoring up to months or even years after a stroke. 2. Definition Paroxysmal atrial fibrillation is not clearly defined in the literature. There is controversy over the duration and morphology of the ECG data in defining an event qualifying for atrial fibrillation. Studies evaluating the incidence of PAF in stroke and TIA patient populations have used different definitions adding confusion about the true incidence. In our paper, we have highlighted the need for a rigorous definition of paroxysmal atrial fibrillation especially in the light of widely used advanced rhythm monitoring devices. 3. Epidemiology Atrial fibrillation prevalence is associated with age with 0.5% at 50–59 years of age increasing

Abstract:
We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation with , , a positive integer, and (note that gives the Fibonacci numbers). Let . We find simple closed form continued fraction expansions for for any integer by exploiting elementary properties of the recurrence relation and continued fractions. 1. Introduction In [1], van der Poorten wrote that the elementary nature and simplicity of the theory of continued fractions is mostly well disguised in the literature. This makes one reluctant to quote sources when making a remark on the subject and seems to necessitate redeveloping the theory ab initio. As our work is an outgrowth of [1], we happily refer the reader to that paper for some basic background information on continued fractions, and to the books [2, 3] for proofs. Briefly, every real number has a continued fraction expansion where each is an integer (and a positive integer unless ). The ’s are called the partial quotients. For brevity we often write If we truncate the expansion at , we obtain the th partial quotient The ’s and ’s satisfy the very important relation Continued fractions encode much useful information about the algebraic structure of a number and frequently arise in approximation theory and dynamical systems. Clearly, is rational if and only if its continued fraction is finite, and a beautiful theorem of Lagrange states that is a quadratic irrational if and only if the continued fraction expansion is periodic. In this paper, we explore the continued fraction expansions of powers of quadratic surds. Recall that a quadratic surd is an irrational number of the form , where and is a nonsquare integer. By Lagrange’s theorem we know these numbers and their powers have periodic continued fractions, which suggests many questions, such as what is the length of the period as well as what are the entries. Some special cases were done by van der Poorten in [1]. He studied solutions to Pell’s equation , ( a nonsquare integer). Using the solution, he derived expansions for the continued fraction of and (with ) and then for the expansions of some simple functions of these numbers as well as numbers related to Diophantine equations similar to Pell’s equation. Another technique that shows promise in manipulating continued fractions comes from an unfinished paper of Gosper [4]. He develops a set of algorithms for finding closed form expressions of simple functions of a given quadratic irrational, as well as for more complicated functions combining quadratic

Abstract:
We use the Millennium Simulation to quantify the statistical accuracy and precision of the escape velocity technique for measuring cluster-sized halo masses at z~0.1. We show that in 3D, one can measure nearly unbiased (<4%) halo masses (>1.5x10^14 M_solar h^-1) with 10-15% scatter. Line-of-sight projection effects increase the scatter to ~25%, where we include the known velocity anisotropies. The classical "caustic" technique incorporates a calibration factor which is determined from N-body simulations. We derive and test a new implementation which eliminates the need for calibration and utilizes only the observables: the galaxy velocities with respect to the cluster mean v, the projected positions r_p, an estimate of the Navarro-Frenk-White (NFW) density concentration and an estimate of the velocity anisotropies, beta. We find that differences between the potential and density NFW concentrations induce a 10% bias in the caustic masses. We also find that large (100%) systematic errors in the observed ensemble average velocity anisotropies and concentrations translate to small (5%-10%) biases in the inferred masses.

Abstract:
We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. For any $q > 0$ and constructible functions $f$ and $\mu$ on $E\times\RR^n$, we prove a theorem describing the structure of the set of all $(x,p)$ in $E \times (0,\infty]$ for which $y \mapsto f(x,y)$ is in $L^p(|\mu|_{x}^{q})$, where $|\mu|_{x}^{q}$ is the positive measure on $\RR^n$ whose Radon-Nikodym derivative with respect to the Lebesgue measure is $y\mapsto |\mu(x,y)|^q$. We also prove a closely related preparation theorem for $f$ and $\mu$. These results relate analysis (the study of $L^p$-spaces) with geometry (the study of zero loci).

Abstract:
We call a function "constructible" if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. Our main theorem gives uniform bounds on the decay of parameterized families of oscillatory integrals with a constructible amplitude function and a globally subanalytic phase function, assuming that the amplitude function is integrable and that the phase function satisfies a certain natural condition called the hyperplane condition. As a simple application of this theorem, we also show that any continuous, integrable, constructible function of a single variable has an integrable Fourier transform.

Abstract:
We study Lebesgue integration of sums of products of globally subanalytic functions and their logarithms, called constructible functions. Our first theorem states that the class of constructible functions is stable under integration. The second theorem treats integrability conditions in Fubini-type settings, and the third result gives decay rates at infinity for constructible functions. Further, we give preparation results for constructible functions related to integrability conditions.

Abstract:
We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions in the family of all characters modulo $q$, with $Q/2 < q\leq Q$. For test functions whose Fourier transform is supported in $(-3/2, 3/2)$, we calculate this quantity beyond the square-root cancellation expansion arising from the $L$-function Ratios Conjecture of Conrey, Farmer and Zirnbauer. We discover the existence of a new lower-order term which is not predicted by this powerful conjecture. This is the first family where the 1-level density is determined well enough to see a term which is not predicted by the Ratios Conjecture, and proves that the exponent of the error term $Q^{-\frac 12 +\epsilon}$ in the Ratios Conjecture is best possible. We also give more precise results when the support of the Fourier Transform of the test function is restricted to the interval $[-1,1]$. Finally we show how natural conjectures on the distribution of primes in arithmetic progressions allow one to extend the support. The most powerful conjecture is Montgomery's, which implies that the Ratios Conjecture's prediction holds for any finite support up to an error $Q^{-\frac 12 +\epsilon}$.