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System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…

It is proved that the first-order theory of the structure (N,mod) is undecidable. Here mod denotes the operation of computing the remainder for any division between positive integers; i.e. x mod y is the remainder obtained by the division x…

A uniform gradient for functions u which satisfy a system of N second-order partial differential inequalities is given in this paper. Some structure conditions are given for the coefficients of the matrices of second-order terms and of…

We prove that the existence of a Haantjes structure is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. This structure, expressed in terms of suitable operators whose Haantjes…

General coherence theorems are constructed that yield explicit presentations of categorical and algebraic objects. The categorical structures involved are finitary discrete Lawvere 2-theories, though they are approached within the language…

This paper investigates how global decision problems over arithmetically represented domains acquire reflective structure through class-quantification. Arithmetization forces diagonal fixed points whose verification requires reflection…

We elucidate consistency of the so-called corner equations which are elementary building blocks of Euler-Lagrange equations for two-dimensional pluri-Lagrangian problems. We show that their consistency can be derived from the existence of…

We consider second order elliptic divergence form systems with complex measurable coefficients $A$ that are independent of the transversal coordinate, and prove that the set of $A$ for which the boundary value problem with $L_2$ Dirichlet…

This paper presents the proof of the coherence theorem for Ann-categories whose set of axioms and original basic properties were given in [9]. Let $$\A=(\A,{\Ah},c,(0,g,d),a,(1,l,r),{\Lh},{\Rh})$$ be an Ann-category. The coherence theorem…

The prolongation structure of a two-by-two problem is formulated very generally in terms of exterior differential forms on a standard representation of Pauli matrices. The differential system is general without making reference to any…

We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality…

For a given ring (domain) in $\overline{\mathbb{R}}^n$ we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all $n\ge 3\,,$ the…

Fractionally-quadratic transformations which reduce any two-dimensional quadratic system to the special Lienard equation are introduced. Existence criteria of cycles are obtained.

A multiparameter class of integrable systems is introduced.

In this article, we first investigate the partitions whose parts are congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition classes with modulus $k$ introduced by Andrews. Then, we introduce the…

This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new bifurcation results for quasi-linear…

The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases…

Complementing and extending the Inventiones work of Benson, Grodal, Henke [Group cohomology and control of p-fusion, Invent. Math. 197 (2014), 491--507] we give criteria for a space to have cohomology (strongly) F-isomorphic in the sense of…

In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular we here establish…

We prove a converse theorem for split even special orthogonal groups over finite fields. This is the only case left on converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. In this paper,…