We consider uniformly elliptic and parabolic second-order equations with bounded zeroth-order and bounded VMO leading coefficients and possibly growing first-order coefficients. We look for solutions which are summable to the p-th power with respect to the usual Lebesgue measure along with their first and second-order derivatives with respect to the spatial variable.

Developing foundational notions in the theory of Riemann surfaces, we prove the Riemann-Roch Theorem and Abel’s Theorem. These notions include sheaf cohomology, with particular focus on the zeroth and first cohomology groups, exact cohomology sequences induced by short exact sequences of sheaves, divisors, the Jacobian variety, and the Abel-Jacobi map. The general method of proof involves basic...

In this communication, we present a theoretical analysis, design, and fabrication of limited-diffractive planar Bessel beam launcher, that exhibits zeroth-order profile in the transverse electric field component with respect to z-propagation axis. The launcher is designed by synthesizing finite zeroth-order, first kind Hankel aperture distribution, polarized along fixed polarization unit vector...

Relying on a recurrence relation for the hypersums of powers of integers put forward recently, we develop an iterative procedure which allows us to express a hypersum of arbitrary order in terms of ordinary (zeroth order) power sums. Then, we derive the coefficients of the hypersum polynomial as a function of the Bernoulli numbers and the Stirling numbers of the first kind.

We formulate, under general conditions, the problem of maximisation of the total entropy of the system, assumed to be in a composable form, for fixed total value of the constrained quantity. We derive the general form of the composability function and also point out the criterion which leads to a violation of the zeroth law of thermodynamics. ∗e-mail: [email protected]

Using two methods we show that a quantized discrete breather in a 1D lattice is stable. One method uses path integrals and compares correlations for a (linear) local mode with those of the quantum breather. The other takes a local mode as the zeroth order system relative to which numerical, cutoff-insensitive diagonalization of the Hamiltonian is performed.

We apply persistent homology to four collaboration networks. We show that the intervals for the zeroth and first Betti numbers correspond to tangible features of the structure of these networks. Finally, we use persistent homology to distinguish collaboration networks from similar random networks.

Isolated horizons [1], much like Killing horizons [2], were introduced in order to deal with the more practical problems involving black holes. Both formulations are local, in contrast with the global nature of event horizons [3]. However, isolated horizons, unlike Killing horizons, do not require a Killing vector field in their neighbourhood. Thus, isolated horizons are characterized by a weak...

We show that the central charge $k$ reduction of universal extension elliptic Hall algebra is isomorphic to trace, or zeroth Hochschild homology, quantum Heisenberg category $k$. As an application, we construct large families representations algebra.

the present study has a cognitive approach to empathy and character engagement in cinema. in discussing how through empathy and engagement with real and fictional characters, we extend our minds and enhance our cognitive abilities, the present paper studies bahram tavakkoli’s wandering in the fog (2010), to show how andy clark and david chalmers’s idea of the extended mind as joined with murray...