Efficient implementation of discrete Pascal transform using difference operators

An alternative way is provided to define the discrete Pascal transform using difference operators to reveal the fundamental concept of the transform, in both one- and two-dimensional cases, which is extended to cover non-square two-dimensional applications. Efficient modularised implementations are proposed.

The Fundamental Gap For Hyperbolic Triangles

In 1983, M. van den Berg made his Fundamental Gap Conjecture about the difference between the first two Dirichlet eigenvalues (the fundamental gap) of any convex domain in the Euclidean plane. Recently, progress has been made in the case where the domains are polygons and, in particular, triangles. We examine the conjecture for triangles in hyperbolic geometry, though we seek an for an upper bound for the fundamental gap rather than a lower bound.

Zeros of Weakly Holomorphic Modular Forms of Levels 2 and 3

Let M-k(#)(N) be the space of weakly holomorphic modular forms for Gamma(0)(N) that are holomorphic at all cusps except possibly at infinity. We study a canonical basis for M-k(#)(2) and M-k(#)(3) and prove that almost all modular forms in this basis have the property that the majority of their zeros in a fundamental domain lie on a lower boundary arc of the fundamental domain.

SEARCH FOR LONG-LIVED WEAKLY INTERACTING PARTICLES USING THE PIERRE AUGER OBSERVATORY

Data collected with the surface detector array of the Pierre Auger Observatory during the period from January 1, 2004 to March 17, 2015 was examined for evidence of production of long-lived weakly interacting particles in interactions of ultra high energy cosmic rays in the atmosphere. The search was performed using extensive air showers with primary energies more than 10 EeV and arrival directions in the range of 57.5◦ to 77.5◦. There was no evidence of significant production of such particles. An upper limit on the fraction of extensive air showers in which such particles are produced was set.

TV Signal Delivery to Cable Operators and DTH Platform Operators

The paper discusses new business models of transmission of television programs in the context of definitions of broadcasting and retransmission. Typically the whole process of supplying content to the end user has two stages: a media service provider supplies a signal assigned to a given TV channel to the cable operators and satellite DTH platform operators (dedicated transmission), and cable operators and satellite DTH platform operators transmit this signal to end users. In each stage the signals are encoded and are not available for the general public without the intervention of cable/platform operators. The services relating to the supply and transmission of the content are operated by different business entities: each earns money separately and each uses the content protected by copyright. We should determine how to define the actions of the entity supplying the signal with the TV program directly to the cable/digital platform operator and the actions of the entity providing the end user with the signal. The author criticizes the approach presented in the Chellomedia and Norma rulings, arguing that they lead to a significant level of legal uncertainty, and poses the basic questions concerning the notion of “public” in copyright.

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows

In this article, we develop the a priori and a posteriori error analysis of hp-version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain Ω ⊂ ℝd, d = 2, 3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm, which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp-adaptive refinement algorithm.