Related papers: Local linear dependence of linear partial differen…

The first part of the paper is a survey of some of the results previously obtained by the authors concerning the $L^p$-dissipativity of scalar and matrix partial differential operators. In the second part we give new necessary and,…

We study the fixed point problem for a system of multivariate operators that are coordinate-wise monotone (i.e., nondecreasing or nonincreasing in each of the variables, independently), in the setting of quasi-ordered sets. We show that…

We give extensions of results on nonnegative matrix semigroups which deduce finiteness or boundedness of such semigroups from the corresponding local properties, e.g., from finiteness or boundedness of values of certain linear functionals…

In this paper, we consider the band functions, Bloch functions and spectrum of the self-adjoint differential operator L with periodic matrix coefficients. Conditions are found for the coefficients under which the number of gaps in the…

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…

We define a Lax operator as a monic pseudodifferential operator $L(\partial)$ of order $N\geq 1$, such that the Lax equations $\dfrac{\partial L(\partial)}{\partial t_k}=[(L^{\frac kN}(\partial))_+,L(\partial)]$ are consistent and non-zero…

We prove continuous dependence on Cauchy data for a backward parabolic operator whose coefficients are Log-Lipschitz continuous in time.

It is well known that while the independence of random variables implies zero correlation, the opposite is not true. Namely, uncorrelated random variables are not necessarily independent. In this note we show that the implication could be…

It is shown that any linear estimator that satisfies the moment conditions up to order $p$ is equivalent to a local polynomial regression of order $p$ with some non-negative weight function if and only if the kernel has at most $p$ sign…

For every associative algebra $A$ and every class $\mathcal{C}$ of representations of $A$ the following question (related to nullstellensatz) makes sense: Characterize all tuples of elements $a_1,\ldots,a_n \in A$ such that vectors…

Let B be an algebra over a field k and let Der(B) be the set of k-derivations from B to B. We define what it means for a subset of Der(B) to be a locally nilpotent set. We prove some basic results about that notion and explore the following…

Let D be a self-adjoint operator on a Hilbert space H and x a bounded operator on H. We say that x is n-times weakly D-differentiable, if for any pair of vectors a, b from H the function < exp(itD)x exp(-itD) a, b> is n-times…

A necessary and sufficient condition for local solvability is presented for the linear partial differential operators $-X^2-Y^2+ia(x)[X,Y]$ in $\bold R^3=\{(x,y,t)\}$, where $X=\partial_x,\; Y=\partial_y+x^k\partial_t$, and $a\in…

We show that (for the weak operator topology) the set of unitary operators on a separable infinite-dimensional Hilbert space is residual in the set of all contractions. The analogous result holds for isometries and the strong operator…

We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…

We prove a differential analogue of Hilbert's irreducibility theorem. Let $\mathcal{L}$ be a linear differential operator with coefficients in $C(\mathbb{X})(x)$ that is irreducible over $\overline{C(\mathbb{X})}(x)$, where $\mathbb{X}$ is…

We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When…

We study boundary regional controllability problems for a class of semilinear fractional systems. Sufficient conditions for regional boundary controllability are proved by assuming that the associated linear system is approximately…

This paper provides a method to study the non-negativity of certain linear operators, from other operators with similar spectral properties. If these new operators are formally self-adjoint and non-negative, we can study the complex powers…

Recent decades have provided a host of examples and applications motivating the study of nonlocal differential operators. We discuss a class of such operators acting on bounded domains, focusing on those with integrable kernels having…