Comparison principle and analogues of the phragmen—lindelöf theorem for solutions of parabolic inequalities

It is well known that a Monotonicity Condition and a Coerciveness Condition principally lie in the basis of most results of the Theory of PDE's. The necessity of these important assumptions for the validity of a comparison principle and analogues of the Phragmen-Lindelöf theorem for solutions of quasilinear parabolic inequalities is discussed in the paper. In the first part of the work we introduce a new concept of monotonicity for nonlinear differential operators-nonlinear monotonicity concept-and on its basis we obtain new phenomena for solutions, subsolutions and supersolutions of the well-known quasilinear differential equations. In the second part we omit the current coerciveness condition and change it by a weaker one. In spite of this we obtain a series of new qualitative properties of solutions for wide classes of quasilinear parabolic inequalities. Most of these properties are also new for solutions of the well-known equations, which we consider in the paper.     

Bivariate delta-evolution equations and convolution polynomials: Computing polynomial expansions of solutions

This paper describes an application of Rota and collaborator’s ideas, about the foundation on combinatorial theory, to the computing of solutions of some linear functional partial differential equations. We give a dynamical interpretation of the convolution families of polynomials. Concretely, we interpret them as entries in the matrix representation of the exponentials of certain contractive linear operators in the ring of formal power series. This is the starting point to get symbolic solutions for some functional-partial differential equations. We introduce the bivariate convolution product of convolution families to obtain symbolic solutions for natural extensions of functional-evolution equations related to delta-operators. We put some examples to show how these symbolic methods allow us to get closed formulas for solutions of genuine partial differential equations. We create an adequate framework to base theoretically some of the performed constructions and to get some existence and uniqueness results.     

Asymptotic Solution of a Differential System Related to the Generalized Hypergeometric Equation

We introduce a first‐order differential system Y′(x) =A(x)Y(x) on [a, ∞) particular cases of which are equivalent to standard forms of the generalized hypergeometric equation. Our purpose is to obtain the asymptotic solution of the system as x → ∞ by defining suitable transformations of the solution vector Y and using ideas from a unified asymptotic theory of differential systems. Thus our methods place the system within the scope of this unified theory, and they are independent of specialized properties of the Meijer G‐function solutions of generalized hypergeometric equations. As such, our methods are also capable of extension to other situations not covered by these special functions.     

Perfect Derived Categories of Positively Graded DG Algebras

We investigate the perfect derived category dgPer(A){{\rm dgPer}}(\mathcal{A}) of a positively graded differential graded (dg) algebra A\mathcal{A} whose degree zero part is a dg subalgebra and semisimple as a ring. We introduce an equivalent subcategory of dgPer(A){{{\rm dgPer}}}(\mathcal{A}) whose objects are easy to describe, define a t-structure on dgPer(A){{{\rm dgPer}}}(\mathcal{A}) and study its heart. We show that dgPer(A){{{\rm dgPer}}}(\mathcal{A}) is a Krull–Remak–Schmidt category. Then we consider the heart in the case that A\mathcal{A} is a Koszul ring with differential zero satisfying some finiteness conditions.     

Some combinatorial properties of the Hurwitz series ring

We study some properties and perspectives of the Hurwitz series ring \(H_R[[t]]\), for an integral domain R, with multiplicative identity and zero characteristic. Specifically, we provide a closed form for the invertible elements by means of the complete ordinary Bell polynomials, we highlight some connections with well–known transforms of sequences, and we see that the Stirling transforms are automorphisms of \(H_R[[t]]\). Moreover, we focus the attention on some special subgroups studying their properties. Finally, we introduce a new transform of sequences that allows to see one of this subgroup as an ultrametric dynamic space.     

Hermite and ps-rings of hurwitz series ∗

Let R be a commutative ring and H R the ring of Hurwitz series over R. In this note, we consider some properties of rings, which are shared by R and HR. In particular, we show that for the rings R and H R, if either ring is (i) a Hermite ring, or (ii) a PF-ring in the sense that every finitely generated projective R-module is free, then so is another. We also show that if R is a PS-ring in the sense that the socle Soc( _RR) is projective and char(R) = 0, then H R is also a PS-ring.     

On the operations of sequences in rings and binomial type sequences

Given a commutative ring with identity R, many different and interesting operations can be defined over the set \(H_R\) of sequences of elements in R. These operations can also give \(H_R\) the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well-known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between \(H_R\) equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.

     

Periodic solutions of second order differential equations in Banach spaces

We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish maximal regularity results in L^p and C^s for strong solutions of a complete second order equation. In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When the operator A involved is the generator of a strongly continuous cosine function, we give characterizations in terms of Fourier multipliers and spectral properties of the cosine function. The results obtained are applied to elliptic partial differential operators. The first author is supported in part by Convenio de Cooperación Internacional (CONICYT) Grant # 7010675 and the second author is partially financed by FONDECYT Grant # 1010675     

Clifford Valued Weighted Variable Exponent Spaces with an Application to Obstacle Problems

In this paper, we introduce the weighted variable exponent spaces in the context of Clifford algebras. After discussing the properties of these spaces, we obtain the existence of weak solutions for obstacle problems for nondegenerate A-Dirac equations with variable growth in the setting of these spaces. Furthermore, we also obtain the existence and uniqueness of weak solutions to the scalar parts of nondegenerate A-Dirac equations in Dirac Sobolev spaces.     

Second-order differentiability with respect to parameters for differential equations with adaptive delays

In this paper, we study the second-order differentiability of solutions with respect to parameters in a class of delay differential equations, where the evolution of the delay is governed explicitly by a differential equation involving the state variable and the parameters. We introduce the notion of locally complete triple-normed linear space and obtain an extension of the well-known uniform contraction principle in such spaces. We then apply this extended principle and obtain the second-order differentiability of solutions with respect to parameters in the W ^1,p -norm (1 ⩽ p < ∞).     

Differential polynomials generated by linear differential equations

This article is devoted to considering value distribution theory of differential polynomials generated by solutions of linear differential equations in the complex plane. In particular, we consider normalized second-order differential equations f″+A(z)f=0, where A(z) is entire. Most of our results are treating the growth of such differential polynomials and the frequency of their fixed points, in the sense of iterated order.     

On Nilpotent and Metabelian Products of Groups

This work is devoted to the study of properties of verbal products and interlacings of groups. More precisely we study approximation properties (for example, the finite approximability, approximability by finite p-groups, and so on) of nilpotent and metabelian products and interlacings. We apply verbal interlacings to the problem of finite approximability with respect to the conjugacy for some nilpotent interlacings and to the study of stable automorphisms of groups of the form $ {F \left/ {V} \right.}\left( \mathcal{N} \right) $ .     

Least squares and bounded variation regularization with nondifferentiable functionals

Let A be an operator from a real Banach space into a real Hilbert space. In this paper we study least squares regularization methods for the ill-posed operator equation A(u) = f using nonlinear nondifferentiable penalty functionals. We introduce a notion of distributional approximation, and use constructs of distributional approximations to establish convergence and stability of approximations of bounded variation solutions of the operator equation. We also show that the results provide a framework for a rigorous analysis of numerical methods based on Euler-Lagrange equations to solve the minimization problem. This justifies many of the numerical implementation schemes of bounded variation minimization that have been recently proposed.     

Invariant tensors and partial differential equations

We consider tensors with coefficients in a commutative differential algebra A. Using the Lie derivative, we introduce the notion of a tensor invariant under a derivation on an ideal of A. Each system of partial differential equations generates an ideal in some differential algebra. This makes it possible to study invariant tensors on such an ideal. As examples we consider the equations of gas dynamics and magnetohydrodynamics.     

Spinors associated with theV 4 3 distribution and the generalized sen-witten equation

On the basis of the methods of nonholonomic differential geometry we introduce the concept of spinors associated with the V ₄ ³ distribution and their spatial covariant derivatives. We obtain the equations of the fundamental spinor fields under a local3+1-stratification of space-time and we study certain properties of their solutions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 133–139.     

Two explicit realizations of a Lie algebra

We introduce a Lie algebra whose some properties are discussed, including its proper ideals, derivations and so on. Then, we again give rise to its two explicit realizations by adopting subalgebra of the Lie algebra A₂ and a column-vector Lie algebra, respectively. Under the frame of zero curvature equations, we may use the realizations to generate the same Lax integrable hierarchies of evolution equations and their Hamiltonian structure.     

Transseries for a class of nonlinear difference equations *

Given a nonlinear analytic difference equation of level 1 with a formal power series solution ? ₀ we associate with it a stable manifold of solutions with asymptotic expansion ? ₀. This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ? ₀ in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in [7]     

Divided powers and composition in integro-differential algebras

An integro-differential algebra of arbitrary characteristic is given the structure of a uniform topological space, such that the ring operations as well as the derivation (= differentiation operator) and Rota–Baxter operator (= integral operator) are uniformly continuous. Using topological techniques and the central notion of divided powers, this allows one to introduce a composition for (topologically) complete integro-differential algebras; this generalizes the series case, viz. meaning formal power series in characteristic zero and Hurwitz series in positive characteristic. The canonical Hausdorff completion for pseudometric spaces is shown to produce complete integro-differential algebras.The setting of complete integro-differential algebras allows us to describe exponential and logarithmic elements in a way that reflects the “integro-differential properties” known from analysis. Finally, we prove also that any complete integro-differential algebra is saturated, in the sense that every (monic) linear differential equation possesses a regular fundamental system of solutions.While the paper focuses on the commutative case, many results are given for the general case of (possibly noncommutative) rings, whenever this does not require substantial modifications.     

On the convergence of solutions of certain inhomogeneous fourth-order differential equations

The main purpose of this paper is to give sufficient conditions for the convergence of solutions of a certain class of fourth-order nonlinear differential equations using Lyapunov’s second method. Nonlinear functions involved are not necessarily differentiable, but a certain incrementary ratio for a function h lies in a closed subinterval of the Routh–Hurwitz interval.     

A criterion forP-stability properties of Runge-Kutta methods

P-stability is an analogous stability property toA-stability with respect to delay differential equations. It is defined by using a scalar test equation similar to the usual test equation ofA-stability. EveryP-stable method isA-stable, but anA-stable method is not necessarilyP-stable. We considerP-stability of Runge-Kutta (RK) methods and its variation which was originally introduced for multistep methods by Bickart, and derive a sufficient condition for an RK method to have the stability properties on the basis of an algebraic characterization ofA-stable RK methods recently obtained by Schere and Müller. By making use of the condition we clarify stability properties of some SIRK and SDIRK methods, which are easier to implement than fully implicit methods, applied to delay differential equations.