On quantum separation of variables beyond fundamental representations

We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate bases in which their spectral problem is separated, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called “non-fundamental” models we construct two different types of SoV bases. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin’s approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic Y(gl_{2})Y(gl2) Yang-Baxter algebra. Our SoV approach also leads to the construction of a QQ-operator in terms of the fused transfer matrices. Finally, we show that the QQ-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases.

Asymptotically almost periodic solutions of fractional evolution equations

We study the asymptotic behavior of solutions of nonlinear fractional evolution equations in Banach spaces. Asymptotically almost periodic solutions on half line are obtained by establishing a sharp estimate on the resolvent operator family of evolution equations. In particular, when the semigroup generated by A is exponentially stable then the conditions of the existence asymptotically almost periodic solutions is satisfied. An application to a fractional partial differential equation with initial boundary condition is also considered.

Approximate Controllability for a Kind of Fractional Neutral Differential Equations with Damping

This paper gains several meaningful results on the mild solutions and approximate controllability for a kind of fractional neutral differential equations with damping (FNDED) and order belonging to 1,2 in Banach spaces. At first, a new expression for the mild solutions of FNDED via the (p, q)-regularized operator family and the technique of Laplace transform is acquired. Then, we consider the approximate controllability of FNDED by means of the approximate sequence method, and simultaneously, some applicable sufficient conditions are obtained.

Classical Solutions to the Initial-Boundary Value Problems for Nonautonomous Fractional Diffusion Equations

In this paper, we investigate a class of nonautonomous fractional diffusion equations (NFDEs). Firstly, under the condition of weighted Hölder continuity, the existence and two estimates of classical solutions are obtained by virtue of the properties of the probability density function and the evolution operator family. Secondly, it focuses on the continuity and an estimate of classical solutions in the sense of fractional power norm. The results generalize some existing results on classical solutions and provide theoretical support for the application of NFDE.

Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equations

This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrödinger equations; a detailed stability and local error analysis is provided for the relevant special case where the Hamilton operator comprises the Laplacian and a regular space-time-dependent potential. In the context of nonautonomous linear ordinary differential equations, CFQM exponential integrators are composed of exponentials involving linear combinations of certain values of the associated time-dependent matrix; this approach extends to nonautonomous linear evolution equations given by unbounded operators. An inherent advantage of CFQM exponential integrators over other time integration methods such as Runge–Kutta methods or Magnus integrators is that structural properties of the underlying operator family are well preserved; this characteristic is confirmed by a theoretical analysis ensuring unconditional stability in the underlying Hilbert space and the full order of convergence under low regularity requirements on the initial state. Due to the fact that convenient tools for products of matrix exponentials such as the Baker–Campbell–Hausdorff formula involve infinite series and thus cannot be applied in connection with unbounded operators, a certain complexity in the investigation of higher-order CFQM exponential integrators for Schrödinger equations is related to an appropriate treatment of compositions of evolution operators; an effective concept for the derivation of a local error expansion relies on suitable linearisations of the evolution equations for the exact and numerical solutions, representations by the variation-of-constants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data.

Farm safety: A prerequisite for sustainable food production in Newfoundland and Labrador

A sustainable approach to food production must address both environmental sustainability and the wellbeing of food producers. Farming is one of the most dangerous occupations globally with high rates of injury, fatality, and occupational disease. However, occupational hazards and the practices that lead to unsafe working environments are often overlooked in sustainable food system research. Poor management of occupational health and safety (OHS) can potentially threaten the survival of individual agricultural operations through injury and illness of the operator, family members, and employees. Gaps in agricultural safety knowledge, prevention, and compensation have been unevenly addressed in Canada. This paper presents findings from the first study of agricultural OHS in Newfoundland and Labrador (NL). Findings from a 2015-2016 survey of 31 food-producing operators representing 34 large and small operations in three NL regions show: 1) that hazards present within these operations are similar to those found in other contexts; 2) accidents are relatively common and most are not reported to workers’ compensation; 3) some participating operators were unsure whether their farms are subject to the regulations in the NL OHS Act; and, 4) there are gaps in workers’ compensation coverage. Some reliance on local and international volunteers and limited safety training point to other potential vulnerabilities. Study findings highlight the need to incorporate a focused strategy for injury prevention and compensation into efforts to develop a stronger and more sustainable food system in NL. We outline an agenda for future action relevant for NL and other places facing similar gaps and challenges.

On Feller semigroup generated by solution of nonlocal parabolic conjugation problem

The paper deals with the problem of construction of Feller semigroup for one-dimensional inhomogeneous diffusion processes with membrane placed at a point whose position on the real line is determined by a given function that depends on the time variable. It is assumed that in the inner points of the half-lines separated by a membrane the desired process must coincide with the ordinary diffusion processes given there, and its behavior on the common boundary of these regions is determined by the nonlocal conjugation condition of Feller-Wentzell's type. This problem is often called a problem of pasting together two diffusion processes on a line. In order to study the described problem we use analytical methods. Such an approach allows us to determine the desired operator family using the solution of the corresponding problem of conjugation for a linear parabolic equation of the second order (the Kolmogorov backward equation) with discontinuous coefficients. This solution is constructed by the boundary integral equations method under the assumption that the coefficients of the equation satisfy the Holder condition with a nonzero exponent, the initial function is bounded and continuous on the whole real line, and the parameters characterizing the Feller-Wentzell conjugation condition and the curve defining the common boundary of the domains, where the equation is given, satisfies the Holder condition with exponent greater than $\frac{1}{2}.$

Existence of Mild Solutions for Sobolev-Type Hilfer Fractional Nonautonomous Evolution Equations with Delay

This paper treats the existence of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations with delay in Banach spaces. We first characterize the definition of mild solutions for the studied problem which was given based on an operator family generated by the operator pair (A,B) and probability density function. And then via Hilfer fractional derivative and combining the techniques of fractional calculus, measure of noncompactness and Sadovskii fixed-point theorem, we obtain new existence result of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations. Particularly, the existence or compactness of an operator $B^{-1} $ is not necessarily needed in our results. Furthermore, our results obtained improve and extend some related conclusions on this topic. At last, an example is given to illustrate our main results.

Solutions to fractional functional differential equations with nonlocal conditions

In this paper, we discuss the solutions to nonlocal initial value problems of fractional order functional differential equations in a Banach space. In particular, we prove the existence and uniqueness of mild and classical solutions assuming that −A generates a resolvent operator family and nonlinear part is a Lipschitz continuous function. We also investigate the global existence of the solution. At the end, a fractional order partial differential equation is given to illustrate the obtained abstract results.