This paper investigates the late-time behaviour of certain cosmological models where oscillations play an essential role. Rigorous results are proved on the asymptotics of homogeneous and isotropic spacetimes with a linear massive scalar field as source. Various generalizations are obtained for nonlinear massive scalar fields, k-essence models and f (R) gravity. The effect of adding ordinary ma...

This paper investigates the late-time behaviour of certain cosmological models where oscillations play an essential role. Rigorous results are proved on the asymptotics of homogeneous and isotropic spacetimes with a linear massive scalar field as source. Various generalizations are obtained for nonlinear massive scalar fields, k-essence models and f(R) gravity. The effect of adding ordinary mat...

In 1953 Peter Roquette gave an arithmetic proof of the Riemann hypothesis for algebraic function fields over a finite constants field, which was proved by André Weil in 1940. The construction of Weil’s scalar product is essential in Roquette’s theory. In this paper a scalar product for algebraic function fields over a number field is constructed which is the analogue of Weil’s scalar product.

We give an alternative construction for a family of partition generating functions due to Kanade and Russell. In our construction, we use ordinary partitions instead jagged partitions. also present new which are evidently positive series To obtain those functions, first construct key infinite product. that combinatorial moves is used decompose arbitrary into base together with some auxiliary bi...

A quantum stabilizer code over GF$(q)$ corresponds to a classical additive GF$(q^2)$ that is self-orthogonal with respect symplectic inner product. We study the decoding of low-density parity-check (LDPC) codes binary finite fields GF$(q=2^l)$ by sum-product algorithm, also known as belief propagation (BP). Conventionally, message in nonbinary BP for GF$(2^l)$ represents probability vector GF$(...

We present phase-plane analysis of cosmologies containing a dilatonic scalar field coupling to a baryotropic fluid with an exponential potential. We show that for baryotropic fluid being radiation the properties of critical points are the same as the conventional scalar field in Ref[11]. For baryotropic fluid being ordinary matter there are coupling between dilatonic scalar field and ordinary m...

We present phase-plane analysis of cosmologies containing a dilatonic scalar field coupling to a baryotropic fluid with an exponential potential. We show that for baryotropic fluid being radiation the properties of critical points are the same as the conventional scalar field in Ref[11]. For baryotropic fluid being ordinary matter there are coupling between dilatonic scalar field and ordinary m...

We apply nonlocal symmetry-like operators to systems of two first and two second-order ordinary differential equations to seek reduction to quadratures. The reduction of order of such systems is carried out with the help of analytic continuation of scalar equations in the complex plane. Examples are taken from the literature. Precisely it is shown how the reduction to quadratures of a system of...