A note on stochastic semilinear equations and their associated Fokker–Planck equations

The main purpose of this paper is to prove existence and uniqueness of (probabilistically weak and strong) solutions to stochastic differential equations (SDE) on Hilbert spaces under a new approximation condition on the drift, recently proposed in [6] to solve Fokker–Planck equations (FPE), extended in this paper to a considerably larger class of drifts. As a consequence we prove existence of martingale solutions to the SDE (whose time marginals then solve the corresponding FPE). Applications include stochastic semilinear partial differential equations with white noise and a non-linear drift part which is the sum of a Burgers-type part and a reaction diffusion part. The main novelty is that the latter is no longer assumed to be of at most linear, but of at most polynomial growth. This case so far had not been covered by the existing literature. We also give a direct and more analytic proof for existence of solutions to the corresponding FPE, extending the technique from [6] to our more general framework, which in turn requires to work on a suitable Gelfand triple rather than just the Hilbert state space.     

A symmetric generalization of Sturm–Liouville problems in q-difference spaces

Classical Sturm–Liouville problems of q-difference variables are extended for symmetric discrete functions such that the corresponding solutions preserve the orthogonality property. Some illustrative examples are given in this sense.     

A note on the Moll–Arias de Reyna integral

The Ramanujan Journal - The Moll–Arias de Reyna integral $$\begin{aligned} \int _0^{\infty }\frac{\mathrm{d}x}{(x^2+1)^{3/2}}\frac{1}{\sqrt{\varphi (x)+\sqrt{\varphi (x)}}} \quad \text {...     

A note on Clebsch–Gordan integral,Fourier–Legendre expansions and closed form for hypergeometric series

The Ramanujan Journal - Let $$p\ge 3$$ be a prime number. In this article, we study the canonical expansion of the primitive $$p^n$$ -th root of unity $$\zeta _{p^n}$$ in p-adic...     

A note on q-difference equations for Ramanujan’s integrals

The Ramanujan Journal - This short paper derives the relationship between solutions of q-difference equations and generating functions for q-orthogonal polynomials. The key of the method is to...     

A note on p-adic q-ζ-functions II

We show that p-adic q-ζ-function constructed by Koblitz [7] (see also D?browski [4]) can be obtained as Γ-transform of some p-adic measure coming from Lubin–Tate formal group.     

A note on exact finite difference schemes for modified Lotka–Volterra differential equations

We construct a class of modified Lotka–Volterra ordinary differential equations (ODE’s) and show that a nonlinear change of the dependent variables transform them into a set of coupled, linear ODE’s. Using the latter equations, we calculate the corresponding exact finite difference schemes using a technique given by Mickens. Next, we show how to reconfigure these relations to obtain the exact finite difference representation of the original modified, nonlinear Lotka–Volterra ODE’s.     

A note on super-hedging for investor–producers

We study the situation of an agent who can trade on a financial market and can also transform some assets into others by means of a production system, in order to price and hedge derivatives on produced goods. This framework is motivated by the case of an electricity producer who wants to hedge a position on the electricity spot price and can trade commodities which are inputs for his system. This extends the essential results of Bouchard and Nguyen (Math Finance, 2011) to continuous time markets. We introduce the generic concept of conditional sure profit along the idea of the no sure profit condition of Rásonyi (Optimality and risk-modern trends in mathematical finance: the Kabanov Fetschrift, 2009). The condition allows one to provide a closedness property for the set of super-hedgeable claims in a very general financial setting. Using standard separation arguments, we then deduce a dual characterization of the latter and provide an application to power futures pricing.     

Study on existence of solutions for some nonlinear functional–integral equations

Using the technique of measures of noncompactness in Banach algebra, we employ abstract fixed point theorems such as Darbo’s theorem to study the existence solution in Banach algebra C[0,a]$C[0,a]$ for some functional–integral equations which include many key integral and functional equations that arise in nonlinear analysis.     

A comment on “Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations”

In a recent paper Alikhani and Bahrami (in press) [1], we found some defects about the exact solutions of given examples. Also, the main result (Theorem 4.5) is not fulfilled. For this purpose, some examples are given.     

A note on regularity criteria for the viscous Camassa–Holm equations in multiplier spaces

In this note, regularity criteria for the viscous Camassa–Holm equations are established in multiplier spaces; these improve on previous results.     

A numerical method based on hybrid orthonormal Bernstein and improved block-pulse functions for solving Volterra–Fredholm integral equations

This paper deals with the numerical solution of the integral equations of linear second kind Volterra–Fredholm. These integral equations are commonly used in engineering and mathematical physics to solve many of the problems. A hybrid of Bernstein and improved block-pulse functions method is introduced and used where the key point is to transform linear second-type Volterra–Fredholm integral equations into an algebraic equation structure that can be solved using classical methods. Numeric examples are given which demonstrate the related features of the process.     

A note on Chudnovsky?s Fuchsian equations

We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these equations and their counterparts on tori. The latters are the Fuchsian equations on elliptic curves and their equivalence is characterized by transcendental transformations which are represented explicitly in terms of elliptic and theta functions.