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We study integral operators on the space of square-integrable functions from a compact set, $X$, to a separable Hilbert space, $H$. The kernel of such an operator takes values in the ideal of Hilbert-Schmidt operators on $H$. We establish regularity conditions on the kernel under which the associated integral operator is trace class. First, we extend Mercer's theorem to operator-valued kernels by proving that a continuous, nonnegative-definite, Hermitian symmetric kernel defines a trace class integral operator on $L^2(X;H)$ under an additional assumption. Second, we show that a general operator-valued kernel that is defined on a compact set and that is H\"older continuous with H\"older exponent greater than a half is trace class provided that the operator-valued kernel is essentially bounded as a mapping into the space of trace class operators on $H$. Finally, when $\dim H < \infty$, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds., Comment: 27 pages

We consider the existence and spectral stability of nonlinear discrete localized solutions representing light pulses propagating in a twisted multi-core optical fiber. By considering an even number, $N$, of waveguides, we derive asymptotic expressions for solutions in which the bulk of the light intensity is concentrated as a soliton-like pulses confined to a single waveguide. The leading order terms obtained are in very good agreement with results of numerical computations. Furthermore, as in the model without temporal dispersion, when the twist parameter, $\phi$, is given by $\phi = \pi/N$, these standing waves exhibit optical suppression, in which a single waveguide remains unexcited, to leading order. Spectral computations and numerical evolution experiments suggest that these standing wave solutions are stable for values of the coupling parameter less than a critical value, at which point a spectral instability results from the collision of an internal eigenvalue with the eigenvalues at the origin. This critical value has a maximum when $\phi = \pi/N$., Comment: 20 pages, 9 figures

The Stochastic Weighted Particle Method (SWPM) is a Monte Carlo technique developed by Rjasanow and Wagner that generalizes Bird's Direct Simulation Monte Carlo (DSMC) method for solving the Boltzmann equation. To reduce computational cost due to the gradual increase in the number of stochastic particles in the SWPM, Rjasanow and Wagner proposed several particle reduction schemes designed to preserve specified moments of the velocity distribution. Here, we introduce an improved particle reduction scheme that preserves all moments of the velocity distribution up to the second order, as well as the raw and central heat flux both within each group of particles to be reduced and for the entire system. Furthermore, we demonstrate that with the new reduction scheme the scalar fourth-order moment can be computed more accurately at a reduced computational cost., Comment: 26 pages

We apply the spectral-Lagrangian method of Gamba and Tharkabhushanam for solving the homogeneous Boltzmann equation to compute the low probability tails of the velocity distribution function, $f$, of a particle species. This method is based on a truncation, $Q^{\operatorname{tr}}(f,f)$, of the Boltzmann collision operator, $Q(f,f)$, whose Fourier transform is given by a weighted convolution. The truncated collision operator models the situation in which two colliding particles ignore each other if their relative speed exceeds a threshold, $g_{\text{tr}}$. We demonstrate that the choice of truncation parameter plays a critical role in the accuracy of the numerical computation of $Q$. Significantly, if $g_{\text{tr}}$ is too large, then accurate numerical computation of the weighted convolution integral is not feasible, since the decay rate and degree of oscillation of the convolution weighting function both increase as $g_{\text{tr}}$ increases. We derive an upper bound on the pointwise error between $Q$ and $Q^{\text{tr}}$, assuming that both operators are computed exactly. This bound provides some additional theoretical justification for the spectral-Lagrangian method, and can be used to guide the choice of $g_{\text{tr}}$ in numerical computations. We then demonstrate how to choose $g_{\text{tr}}$ and the numerical discretization parameters so that the computation of the truncated collision operator is a good approximation to $Q$ in the low probability tails. Finally, for several different initial conditions, we demonstrate the feasibility of accurately computing the time evolution of the velocity pdf down to probability density levels ranging from $10^{-5}$ to $10^{-9}$., Comment: 32 pages

We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic-quintic complex Ginzburg-Landau equation about a periodically-stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg-Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically-stationary pulses in ultrafast fiber lasers., Comment: 16 pages, comments welcome!! Reviewer comments incorporated!

The theory of differential characters is developed completely from a de Rham - Federer viewpoint. Characters are defined as equivalence classes of special currents, called sparks, which appear naturally in the theory of singular connections. There are many different spaces of currents which yield the character groups. A multiplication of de Rham-Federer characters is defined using transversality results for flat and rectifiable currents. There is a natural equivalence of ring functors from deRham - Federer characters to Cheeger-Simons characters. Differential characters have a topology and natural smooth Pontrjagin duals (introduced here). A principal result is the formulation and proof of duality for characters on oriented manifolds. It is shown that the pairing (a,b) --> a*b([X]) given by multiplication and evaluation on the fundamental cycle, gives an isomorphism of the group of differential characters of degree k with the dual to characters in degree n-k-1 where n = dim(X). It is also shown that there are natural Thom homomorphisms for differential characters. Gysin maps are also defined for differential characters. Many examples including Morse sparks, Hodge sparks and characteristic sparks are examined in detail.

A canonically defined mod 2 linear dependency current is associated to each collection of m sections of a real rank n vector bundle. This current is supported on the linear dependency set of the collection of sections. It is defined whenever the collection satisfies a weak measure theoretic condition called "atomicity". Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2, closed, locally integrally flat current of degree q= n-m+1 and hence determines a mod 2 cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover it is the q-th Stielfel-Whitney class of the vector bundle. More is true if q is odd or q=n. In this case a linear dependency current which is twisted by the orientation of the bundle can be associated to the collection. The cohomology class of the linear dependency current is 2-torsion and is the q-th twisted integral Stiefel-Whitney class of the bundle. In addition, higher dependency and general degeneracy currents of bundle maps are studied, together with applications to singularities of projections and maps. These results rely on a theorem of Federer which states that the complex of integrally flat currents mod p computes cohomology mod p. An alternate approach to Federer's theorem is offered in an appendix. This approach is simpler and is via sheaf theory., Comment: 39 pages, amstex file, to appear in The Journal of Geometric Analysis