In the present paper, we present some important properties of N-transform, which is the Laplace transform for the nabla derivative on the time scale of integers (Bohner and Peterson in Dynamic equations on time scales, Birkhauser, Boston, 2001; Advances in dynamic equations on time scales, Birkhauser, Boston, 2002). We obtain the N-transform of nabla fractional sums and differences and then apply this transform to solve some nabla fractional difference equations with initial value problems. Finally, using N-transforms, we prove that discrete Mittag-Leffler function is the eigen function of Caputo type nabla fractional difference operator $\nabla ^{\alpha }$.

For a large class of integral operators or second-order differential operators, their isospectral (or cospectral) operators are constructed explicitly in terms of $h$-transform (duality). This provides us a simple way to extend the known knowledge on the spectrum (or the estimation of the principal eigenvalue) from a smaller class of operators to a much larger one. In particular, an open problem about the positivity of the principal eigenvalue for birth–death processes is solved in the paper.

We get sharp degree bound for generic smoothness and connectedness of the space of lines and conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on hypersurfaces. As a consequence, we prove that for a Fano complete intersection $X$ with index $\ge 2$, the $1$-Griffiths group generated by algebraic $1$-cycles homologous to $0$ modulo algebraic equivalence is trivial, which is a conjecture for general rationally connected varieties.

We study the forward price dynamics in commodity markets realised as a process with values in a Hilbert space of absolutely continuous functions defined by Filipović (Consistency problems for Heath–Jarrow–Morton interest rate models, 2001). The forward dynamics are defined as the mild solution of a certain stochastic partial differential equation driven by an infinite-dimensional Lévy process. It is shown that the associated spot price dynamics can be expressed as a sum of Ornstein–Uhlenbeck processes, or more generally, as a sum of certain stationary processes. These results link the possibly infinite-dimensional forward dynamics to classical commodity spot models. We continue with a detailed analysis of multiplication and integral operators on the Hilbert spaces and show that Hilbert–Schmidt operators are essentially integral operators. The covariance operator of the Lévy process driving the forward dynamics and the diffusion term can both be specified in terms of such operators, and we analyse in several examples the consequences on model dynamics and their probabilistic properties. Also, we represent the forward price for contracts delivering over a period in terms of an integral operator, a case being relevant for power and gas markets. In several examples, we reduce our general model to existing commodity spot and forward dynamics.

We propose a local model called moving multiple curves/surfaces approximation to separate mixed scanning points received from a thin-wall object, where data from two sides of the object may be mixed due to measurement error. The cases of two curves (including plane curves and space curves) and two surfaces in one model are mainly elaborated, and a lot of examples are tested.