We consider the modulated harmonic wave in discrete series connected Josephson transmission line (JTL). formulate approach to modulation problems for equations based on calculus. check up by applying it Fermi-Pasta-Ulam-Tsingou type problem. Applying JTL, we obtain equation describing amplitude, which turns out be defocusing nonlinear Schr\"odinger (NLS) equation. compare profile of single soli...

We present here a systematic study of general boundary value problems on weighted networks that includes the variational formulation of such problems. In particular, we obtain the discrete version of the Dirichlet Principle and we apply it to the analysis of the inverse problem of identifying the conductivities of the network in a very general framework. Our approach is based on the development...

The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: in contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than all other formalisms supporting analytical proofs. However, deep applicability of inference rules causes greater nondeterminism than i...

We consider a version of the double integral calculus of variations on time scales, which includes as special cases the classical two-variable calculus of variations and the discrete two-variable calculus of variations. Necessary and sufficient conditions for a local extremum are established, among them an analogue of the Euler–Lagrange equation. AMS (MOS) Subject Classification. 39A0, 49K20.

We introduce a version of the calculus of variations on time scales, which includes as special cases the classical calculus of variations and the discrete calculus of variations. Necessary conditions for weak local minima are established, among them the Euler condition, the Legendre condition, the strengthened Legendre condition, and the Jacobi condition. AMS (MOS) Subject Classification. 39A10.

We introduce stochastic model of chemotaxis by fractional Derivative generalizing the deterministic Keller Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. In this work, we study nonlinear Dirichlet boundary conditions, and disturbed multiplicative noise. The required results prove existence uniqueness mild s...

While difference equations deal with discrete calculus and differential equations with continuous calculus, so-called q-difference equations are considered when studying q-calculus. In this paper, we obtain certain oscillation criteria for second-order q-difference equations, among them a q-calculus version of the famous Kneser theorem. PACS numbers: 02.30.Hq, 02.30.Gp, 02.30.Ks, 02.30.Lt, 02.2...

INTRODUCTION Security of public key cryptographic protocols is based on the apparent difficulty of a mathematical problem. Perhaps the most famous of these problems is that of factoring a composite number into primes, on which the security of the system known as RSA [1] relies. The standard cryptographic protocols like Diffie-Hellman [2] key exchange, ElGamal [3] encryption, and digital signatu...

In this paper we present a new theory of exterior calculus over k-dimensional domains in a smooth n-manifold, unifying the discrete and continuum theories. The discrete theory carries a full set of classical operators, products, and relations of calculus and geometry and is free from the standard problems encountered in other theories of discrete exterior calculus. It is based on the Koszul com...