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The algebra and calculus of generalized differential forms are reviewed and employed to construct a class of generalized connections and to investigate their properties. The class includes generalized connections which are flat when Einstein's vacuum field equations are satisfied. Generalized Chern-Simons action principles are formulated and it is shown that certain of these have Einstein's vacuum field equations as Euler-Lagrange equations., Comment: 23 pages

Based on numerical data and a-posteriori analysis we verify rigorously the uniqueness and smoothness of global solutions to a scalar surface growth model with striking similarities to the 3D Navier--Stokes equations, for certain initial data for which analytical approaches fail. The key point is the derivation of a scalar ODE controlling the norm of the solution, whose coefficients depend on the numerical data. Instead of solving this ODE explicitly, we explore three different numerical methods that provide rigorous upper bounds for its solution

Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for generalized vector fields. Hamiltonian vector fields are discussed. Generalized affine connections and metrics are defined and the fundamental theorem of metric differential geometry is extended. The global structure of the exterior derivative of generalized forms is investigated., Comment: 26 pages

We give a simple proof of a lower bound for the Dirichlet heat kernel in terms of the Gaussian heat kernel. Using this we establish a non-existence result for semilinear heat equations with zero Dirichlet boundary conditions and initial data in $L^q(\Omega)$ when the source term $f$ is non-decreasing and $\limsup_{s\to\infty}s^{-\gamma}f(s)=\infty$ for some $\gamma>q(1+2/n)$. This allows us to construct a locally Lipschitz $f$ satisfying the Osgood condition $\int_{1}^{\infty}1/f(s)\ \,\d s =\infty$, which ensures global existence for bounded initial data, such that for every $q$ with $1\le q<\infty$ there is an initial condition $u_0\in L^q(\Om)$ for which the corresponding semilinear problem has no local-in-time solution.

Let $E$ be a linear space and suppose that $A$ is the global attractor of either (i) a homeomorphism $F:E\rightarrow E$ or (ii) a semigroup $S(\cdot)$ on $E$ that is injective on $A$. In both cases $A$ has trivial shape, and the dynamics on $A$ can be described by a homeomorphism $F:A\rightarrow A$ (in the second case we set $F=S(t)$ for some $t>0$). If the topological dimension of $A$ is finite we show that for any $\epsilon>0$ there is an embedding $e:A\rightarrow{\mathbb R}^k$, with $k\sim{\rm dim}(A)$, and a (dynamical) homeomorphism $f:\R^k\rightarrow\R^k$ such that $F$ is conjugate to $f$ on $A$ (i.e.\ $F|_A=e^{-1}\circ f\circ e$) and $f$ has an attractor $A_f$ with $e(A)\subset A_f\subset N(e(A),\epsilon)$. In other words, we show that the dynamics on $A$ is essentially finite-dimensional. We characterise subsets of ${\mathbb R}^n$ that can be the attractors of homeomorphisms as cellular sets, give elementary proofs of various topological results connected to Borsuk's theory of shape and cellularity in Euclidean spaces, and prove a controlled homeomorphism extension theorem.

We establish non-existence results for the Cauchy problem of some semilinear heat equations with non-negative initial data and locally Lipschitz, nonnegative source term $f$. Global (in time) solutions of the scalar ODE $\dot v=f(v)$ exist for $v(0)>0$ if and only if the Osgood-type condition $\int_{1}^{\infty}\frac{\dee s}{f(s)} =\infty$ holds; by comparison this ensures the existence of global classical solutions of $u_t=\Delta u+f(u)$ for bounded initial data $u_0\in L^{\infty}(\R^n)$. It is natural to ask whether the Osgood condition is sufficient to ensure that the problem still admits global solutions if the initial data is in $L^q(\R^n)$ for some $1\le q<\infty$. Here we answer this question in the negative, and in fact show that there are initial conditions for which there exists no local solution in $L^1_{\rm loc}(\R^n)$ for $t>0$., Comment: Some typos corrected from an earlier version. To appear in J. Differential Equations

We prove existence, uniqueness and regularity of weak solutions of a coupled parabolic-elliptic model in two dimensions; we consider the standard equations of magnetohydrodynamics with the advective terms removed from the velocity equation. Despite the apparent simplicity of the model, the proof requires results that are at the limit of what is available, including elliptic regularity in $L^{1}$ and a strengthened form of the Ladyzhenskaya inequality \[ \| f \|_{L^{4}} \leq c \| f \|_{L^{2,\infty}}^{1/2} \| \nabla f \|_{L^{2}}^{1/2}, \] which we derive using the theory of interpolation. The model has applications to the method of magnetic relaxation, introduced by Moffatt (J. Fluid. Mech. 159, 359-378, 1985), to construct stationary Euler flows with non-trivial topology.

Using elementary arguments based on the Fourier transform we prove that for $1 \leq q < p < \infty$ and $s \geq 0$ with $s > n(1/2-1/p)$, if $f \in L^{q,\infty}(\R^n) \cap \dot{H}^s(\R^n)$ then $f \in L^p(\R^n)$ and there exists a constant $c_{p,q,s}$ such that \[ \|f\|_{L^p} \leq c_{p,q,s} \|f\|_{L^{q,\infty}}^\theta \|f\|_{\dot H^s}^{1-\theta}, \] where $1/p = \theta/q + (1-\theta)(1/2-s/n)$. In particular, in $\R^2$ we obtain the generalised Ladyzhenskaya inequality $\|f\|_{L^4}\le c\|f\|_{L^{2,\infty}}^{1/2}\|f\|_{\dot H^1}^{1/2}$. We also show that for $s=n/2$ the norm in $\|f\|_{\dot H^{n/2}}$ can be replaced by the norm in BMO. As well as giving relatively simple proofs of these inequalities, this paper provides a brief primer of some basic concepts in harmonic analysis, including weak spaces, the Fourier transform, the Lebesgue Differentiation Theorem, and Calderon-Zygmund decompositions.

We obtain a lower bound for the period of periodic solutions of semilinear evolution equations for the full range of nonlinear terms for which standard local existence theory applies. This lower bound depends on the Lipschitz constant of the nonlinear term as an operator acting on the domain of a fractional power of the linear operator into the base space.

Let x(t) be a non-constant T-periodic solution to the ordinary differential equation x'= f(x) in a Banach space X where f is assumed to be Lipschitz continuous with constant L. Then there exists a constant c such that T L >= c, with c only depending on X. It is known that c >= 6 in any Banach space and that c = 2{\pi} in any Hilbert space, but whereas the bound of c = 2 pi is sharp in any Hilbert space, there exists only one known example of a Banach space such that c = 6 is optimal. In this paper, we show that the inequality is in fact strict in any strictly convex Banach space. Moreover, we improve the lower bound for l^p(R^n) and L^p(M, {\mu}) for a range of p close to p = 2 by using a form of Wirtinger's inequality for functions in W^{1,p}([0, T ], L^p(M, {\mu})).

A semilinear heat equation $u_{t}=\Delta u+f(u)$ with nonnegative initial data in a subset of $L^{1}(\Omega)$ is considered under the assumption that $f$ is nonnegative and nondecreasing and $\Omega\subseteq \R^{n}$. A simple technique for proving existence and regularity based on the existence of supersolutions is presented, then a method of construction of local and global supersolutions is proposed. This approach is applied to the model case $f(s)=s^{p}$, $\phi\in L^{q}(\Omega)$: new sufficient conditions for the existence of local and global classical solutions are derived in the critical and subcritical range of parameters. Some possible generalisations of the method to a broader class of equations are discussed., Comment: Expanded version of the previous submission arXiv:1111.0258v1. 14 pages

We discuss conditions for well-posedness of the scalar reaction-diffusion equation $u_{t}=\Delta u+f(u)$ equipped with Dirichlet boundary conditions where the initial data is unbounded. Standard growth conditions are juxtaposed with the no-blow-up condition $\int_{1}^{\infty}1/f(s) \d s=\infty$ that guarantees global solutions for the related ODE $\dot u=f(u)$. We investigate well-posedness of the toy PDE $u_{t}=f(u)$ in $L^{p}$ under this no-blow-up condition. An example is given of a source term $f$ and an initial condition $\psi\in L^{2}(0,1)$ such that $\int_{1}^{\infty}1/f(s)\d s=\infty$ and the toy PDE blows-up instantaneously while the reaction-diffusion equation is globally well-posed in $L^{2}(0,1)$.

We consider the two-dimensional motion of the coupled system of a viscous incompressible fluid and a rigid disc moving with the fluid, in the whole plane. The fluid motion is described by the Navier-Stokes equations and the motion of the rigid body by conservation laws of linear and angular momentum. We show that, assuming that the rigid disc is not allowed to rotate, as the radius of the disc goes to zero, the solution of this system converges, in an appropriate sense, to the solution of the Navier-Stokes equations describing the motion of only fluid in the whole plane. We also prove that the trajectory of the centre of the disc, at the zero limit of its radius, coincides with a fluid particle trajectory., Comment: 29 pages, 0 figures

Superconductivity, magnetic order, and quadrupolar order have been investigated in the filled skutterudite system Pr$_{1-x}$Nd$_{x}$Os$_4$Sb$_{12}$ as a function of composition $x$ in magnetic fields up to 9 tesla and at temperatures between 50 mK and 10 K. Electrical resistivity measurements indicate that the high field ordered phase (HFOP), which has been identified with antiferroquadruoplar order, persists to $x$ $\sim$ 0.5. The superconducting critical temperature $T_c$ of PrOs$_4$Sb$_{12}$ is depressed linearly with Nd concentration to $x$ $\sim$ 0.55, whereas the Curie temperature $T_{FM}$ of NdOs$_4$Sb$_{12}$ is depressed linearly with Pr composition to ($1-x$) $\sim$ 0.45. In the superconducting region, the upper critical field $H_{c2}(x,0)$ is depressed quadratically with $x$ in the range 0 $<$ $x$ $\lesssim$ 0.3, exhibits a kink at $x$ $\approx$ 0.3, and then decreases linearly with $x$ in the range 0.3 $\lesssim$ $x$ $\lesssim$ 0.6. The behavior of $H_{c2}(x,0)$ appears to be due to pair breaking caused by the applied magnetic field and the exhange field associated with the polarization of the Nd magnetic moments, in the superconducting state. From magnetic susceptibility measurements, the correlations between the Nd moments in the superconducting state appear to change from ferromagnetic in the range 0.3 $\lesssim$ $x$ $\lesssim$ 0.6 to antiferromagnetic in the range 0 $<$ $x$ $\lesssim$ 0.3. Specific heat measurements on a sample with $x$ $=$ 0.45 indicate that magnetic order occurs in the superconducting state, as is also inferred from the depression of $H_{c2}(x,0)$ with $x$., Comment: 7 pages, 7 figures, currently submitted to Phys. Rev. B

We discuss various issues related to the finite-dimensionality of the asymptotic dynamics of solutions of parabolic equations. In particular, we study the regularity of the vector field on the global attractor associated with these equations. We show that certain dissipative partial differential equations possess a linear term that is log-Lipschitz continuous on the attractor. We then prove that this property implies that the associated global attractor $\mathcal A$ lies within a small neighbourhood of a smooth manifold, given as a Lipschitz graph over a finite number of Fourier modes. Consequently, the global attractor $\mathcal A$ has zero Lipschitz deviation and, therefore, there are linear maps $L$ into finite-dimensional spaces, whose inverses restricted to $L\mathcal A$ are H\"older continuous with an exponent arbitrarily close to one.

Using shape theory and the concept of cellularity, we show that if $A$ is the global attractor associated with a dissipative partial differential equation in a real Hilbert space $H$ and the set $A-A$ has finite Assouad dimension $d$, then there is an ordinary differential equation in ${\mathbb R}^{m+1}$, with $m >d$, that has unique solutions and reproduces the dynamics on $A$. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor $X$ arbitrarily close to $LA$, where $L$ is a homeomorphism from $A$ into ${\mathbb R}^{m+1}$.

If $X$ is a subset of a Banach space with $X-X$ homogeneous, then $X$ can be embedded into some $\R^n$ (with $n$ sufficiently large) using a linear map $L$ whose inverse is Lipschitz to within logarithmic corrections. More precisely, $$c\,\frac{\|x-y\|}{|\,\log\|x-y\|\,|^\alpha}\le|Lx-Ly|\le c\|x-y\|$$ for all $x,y\in X$ with $\|x-y\|<\delta$ for some $\delta$ sufficiently small. A simple argument shows that one must have $\alpha>1$ in the case of a general Banach space and $\alpha>1/2$ in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved. While the argument in a general Banach space is relatively straightforward, the Hilbert space case relies on a result due to Ball (Proc. Amer. Math. Soc. 97 (1986) 465-473) which guarantees that the maximum volume of hyperplane slices of the unit cube in $\R^d$ is $\sqrt2$, in dependent of $d$.

We provide bounds on the upper box-counting dimension of negatively invariant subsets of Banach spaces, a problem that is easily reduced to covering the image of the unit ball under a linear map by a collection of balls of smaller radius. As an application of the abstract theory we show that the global attractors of a very broad class of parabolic partial differential equations (semilinear equations in Banach spaces) are finite-dimensional.

This study examined titles of successful books, i.e., those that were best sellers or won literary prizes. In all, 3,239 titles in 71 lists by decade were characterized. The single largest category was popular general fiction followed by general nonfiction. About 49% of the titles won or were nominated for awards, and 45% were best sellers. Most titles were issued in the 1950s through the 1990s. The median book title contained 3.5 words, the mode was 2.4 words, and the mean was 2.8 words. One-word titles accounted for 17% of the sample, two word titles for 34%, three word titles for 20%, and longer titles for 29%. Few titles included color words or place names, but about 16% included the name of a person. Only 2% of the tittles contained a subtitle, and the mean subtitle was 4 words long. Few titles, about 10%, contained a noun or verb word found in another title. There was little significant change over time. Attached tables show the sample distribution by category, sample distribution by decade, title length by category, and title length by decade. (Contains 18 references.) (Author/MES)

We give a simple proof of the uniqueness of fluid particle trajectories corresponding to: 1) the solution of the two-dimensional Navier Stokes equations with an initial condition that is only square integrable, and 2) the local strong solution of the three-dimensional equations with an $H^{1/2}$-regular initial condition i.e.\ with the minimal Sobolev regularity known to guarantee uniqueness. This result was proved by Chemin & Lerner (J Diff Eq 121 (1995) 314-328) using the Littlewood-Paley theory for the flow in the whole space $\R^d$, $d\ge 2$. We first show that the solutions of the differential equation $\dot{X}=u(X,t)$ are unique if $u\in L^p(0,T;H^{(d/2)-1})$ for some $p>1$ and $\sqrt{t}\,u\in L^2(0,T;H^{(d/2)+1})$. We then prove, using standard energy methods, that the solution of the Navier-Stokes equations with initial condition in $H^{(d/2)-1}$ satisfies these conditions. This proof is also valid for the more physically relevant case of bounded domains., Comment: 13 pages, 0 figures

This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be embedded into a Hilbert space using an almost bi-Lipschitz mapping (bi-Lipschitz to within logarithmic corrections). The image of this set is no longer homogeneous, but `almost homogeneous'. We therefore study the problem of embedding an almost homogeneous subset $X$ of a Hilbert space $H$ into a finite-dimensional Euclidean space. In fact we show that if $X$ is a compact subset of a Banach space and $X-X$ is almost homogeneous then, for $N$ sufficiently large, a prevalent set of linear maps from $X$ into $\Re^N$ are almost bi-Lipschitz between $X$ and its image. We are then able to use the Kuratowski embedding of $(X,d)$ into $L^\infty(X)$ to prove a similar result for compact metric spaces., Comment: Submitted to Transactions of the AMS

In their 2006 paper, Chernyshenko et al prove that a sufficiently smooth strong solution of the 3d Navier-Stokes equations is robust with respect to small enough changes in initial conditions and forcing function. They also show that if a regular enough strong solution exists then Galerkin approximations converge to it. They then use these results to conclude that the existence of a sufficiently regular strong solution can be verified using sufficiently refined numerical computations. In this paper we study the solutions with minimal required regularity to be strong, which are less regular than those considered in Chernyshenko et al (2006). We prove a similar robustness result and show the validity of the results relating convergent numerical computations and the existence of the strong solutions.

Current theoretical results for the three-dimensional Navier--Stokes equations only guarantee that solutions remain regular for all time when the initial enstrophy ($\|Du_0\|^2:=\int|{\rm curl} u_0|^2$) is sufficiently small, $\|Du_0\|^2\le\chi_0$. In fact, this smallness condition is such that the enstrophy is always non-increasing. In this paper we provide a numerical procedure that will verify regularity of solutions for any bounded set of initial conditions, $\|Du_0\|^2\le\chi_1$. Under the assumption that the equations are in fact regular we show that this procedure can be guaranteed to terminate after a finite time.

We consider the behaviour of weak solutions of the unforced three-dimensional Navier-Stokes equations, under the assumption that the initial condition has finite energy ($\|u\|^2=\int|u|^2$) but infinite enstrophy ($\|Du\|^2=\int|{\rm curl} u|^2$). We show that this has to be reflected in the solution for small times, so that in particular $\|Du(t)\|\to+\infty$ as $t\to0$. We also give some limitations on this `backwards blowup', and give an elementary proof that the upper box-counting dimension of the set of singular times can be no larger than one half. Although similar in flavour, this final result neither implies nor is implied by Scheffer's result that the 1/2-dimensional Hausdorff measure of the singular set is zero.

In this paper we consider the r\^ole that numerical computations -- in particular Galerkin approximations -- can play in problems modelled by the 3d Navier-Stokes equations, for which no rigorous proof of the existence of unique solutions is currently available. We prove a robustness theorem for strong solutions, from which we derive an {\it a posteriori} check that can be applied to a numerical solution to guarantee the existence of a strong solution of the corresponding exact problem. We then consider Galerkin approximations, and show that {\it if} a strong solution exists the Galerkin approximations will converge to it; thus if one is prepared to assume that the Navier-Stokes equations are regular one can justify this particular numerical method rigorously. Combining these two results we show that if a strong solution of the exact problem exists then this can be verified numerically using an algorithm that can be guaranteed to terminate in a finite time. We thus introduce the possibility of rigorous computations of the solutions of the 3d Navier-Stokes equations (despite the lack of rigorous existence and uniqueness results), and demonstrate that numerical investigation can be used to rule out the occurrence of possible singularities in particular examples.

In 1995, a questionnaire was sent to Tennessee public and academic libraries with book collections of more than 5,000 volumes. The response rate was 48% and the sample included 44 public, 23 four-year academic and 7 community college libraries. Forty-three (56%) of these libraries collect local government publications. While, the earliest collection began in 1890, most started in the 1980's and the 1990's. Collections are typically small--less than 100 items. Municipal codes, annual reports, financial reports, and maps are most likely to be collected. Nearly all libraries with local government collections provide reference and photo duplication service but few will lend these items. Acquisition is the most troublesome aspect of collection development. The two most common reasons for failing to collect local government materials were little local demand and the available of other nearby collections. (AEF)

This study examined the degree to which contemporary world literature is available in United States OCLC (Online Computer Library Center) libraries. The 1991 volume of "World Literature Today," reviewing 837 works, was used as the source. German, English, Spanish, and French were the most frequently encountered languages. Fiction was the literature type most likely to be reviewed in "World Literature Today." The items least likely to be held were collected works, verse, correspondence, diaries, and miscellaneous works. Major countries of publication were the United States, Germany, and France. About 17% of the reviewed works were not held in any OCLC member library. Nearly 5% were held in one OCLC library. About 78% of the contemporary world literature reviewed was held in more than one United States OCLC library. Relationships between holdings, language, literature type, and country of publication were tested. (AEF)

An increasing number of young black males are over-represented among youth who are involved in homicides, accidents, and suicides. This is being called the "new morbidity," a term coined to describe the interrelationship of self-destructive and life-threatening behaviors particularly applicable to African American males. The new morbidity refers to conditions evolving from economic, cultural, and social forces. This report describes the results of a study that uses a "public health model" to examine a set of risk factors associated with an integrated approach to the study of violence. A proposal is offered for addressing the situation from a public health perspective. The study was designed to provide a more comprehensive understanding of assault victims' risk exposure and the resources directed at those risks by using the University of Kentucky trauma program database for July 1990 through September 1996. There were 1,410 records available. The majority of victims of violence (75.8%) were male, and young people were victimized most often. Of these cases, 135 black males and 188 white males were victims. Gunshot wounds and stabbings were the most common forms of violence recorded. Data support the belief that people in central cities have higher victimization rates and more severe injuries. Many surviving victims had extremely serious injuries, a finding that suggests the importance of further research into the costs and lost quality of life due to crime victimization. Victim outcomes might be enhanced by effectively linking public health data with police crime reports. (Contains 17 references.) (SLD)