In lecture 19, we saw an LP relaxation based algorithm to solve the sparsest cut problem with an approximation guarantee of O(logn). In this lecture, we will show that the integrality gap of the LP relaxation is O(logn) and hence this is the best approximation factor one can get via the LP relaxation. We will also start developing an SDP relaxation based algorithm which provides an O( √ log n) ...

In this paper, block distance matrices are introduced. Suppose F is a square block matrix in which each block is a symmetric matrix of some given order. If F is positive semidefinite, the block distance matrix D is defined as a matrix whose (i, j)-block is given by Dij = Fii+Fjj−2Fij . When each block in F is 1 × 1 (i.e., a real number), D is a usual Euclidean distance matrix. Many interesting ...

The design of grating-based instruments to handle and condition coherent ultrafast pulses in the extreme-ultraviolet is discussed. The main application of such instruments is the monochromatization of high-order laser harmonics and free-electron-laser pulses in the femtosecond time scale. Broad-band monochromators require the use of diffraction gratings at grazing incidence. A grating can be us...

The coefficients of dispersion, Dij , and the dispersivity, aijkl , appear in the expression for the flux of a solute in saturated flow through porous media. We present a detailed analysis of these tensors in an axially symmetric porous medium and show that in such a medium, the dispersivity is governed by 6 independent moduli. We present also the constraints that have to be satisfied by these ...

A much simpler proof of Theorem 1 from [1] is presented below, using notation and formulas numeration of [1]. The text below replaces the subsection General case from §4 of [1, p. 11]. General case. In the general case for some ω we are interested in a pairs (xi,xj) with dij = ωn. But there may exist a pairs (xk,xl) with dkl < ωn. Using the “cleaning” procedure [2] we show that the influence of...

The Zeldovich approximation (ZA) predicts the formation of a web of singularities. While these singularities may only exist in the most formal interpretation of the ZA, they provide a powerful tool for the analysis of initial conditions. We present a novel method to find the skeleton of the resulting cosmic web based on singularities in the primordial deformation tensor and its higher order der...

Lithium containing droplet and cluster targets irradiated by laser pulses are proposed as prospective source of for soft x-ray lithography. Analytical model and simulations show that laser with repetition rate of several MHz with energy of several mJ and pulse duration 10 ps is required.

We show that according to the GRB internal shock model the pulse duration and the interval between neighboring pulses are correlated and have a similar distribution. We analyze a sample of bright long bursts and find that the pulses duration have a lognormal distribution while the intervals has an excess of long intervals (relative to lognormal distribution). This excess can be explained by the...

We extend Duration Calculus to a logic which allows description of Discrete Processes where several steps of computation can occur at the same time point. The resulting logic is called Duration Calculus of Weakly Monotonic Time (WDC). It allows effects such as true synchrony and digitisation to be modelled. As an example of this, we formulate a novel semantics of Timed CSP assuming that the com...