We study (p, q)-inequalities of Ulyanov type for moduli of smoothness of fractional order in the Lp and the Lp(ℝn) setting, p ≥ 1. In particular, we obtain estimates for the modulus of smoothness of a generalized Liouville derivative of a function via the modulus of smoothness of the function itself. We give examples showing the sharpness of these inequalities.

This review paper is concerned with the stability analysis of the continuity equation in the DiPerna–Lions setting in which the advecting velocity field is Sobolev regular. Quantitative estimates for the equation were derived only recently, but optimality was not discussed. We revisit the results from our 2017 paper, compare the new estimates with previously known estimates for Lagrangian flows and demonstrate how these can be applied to produce optimal bounds in applications from physics, engineering and numerical analysis.

We introduce the concept of an elementary operator between two algebras, and thereby extend the study of operators of the form

where ai, bi are fixed elements in an algebra. Elementary operators on some classes of algebras are considered and their general form is described in some special cases.

We describe surjective linear maps preserving commutativity from (symmetric elements of) any algebra (with involution) onto (symmetric elements of) a prime algebra (with involution) not satisfying polynomial identities of low degree. Bijective commutativity preservers on skew elements of centrally closed prime algebras with involution of the first kind are also investigated.

We discuss existence and lowersemicontinuity for clusters of materials minimising an energy given by a collection of norms φij on the interfaces between regions Ri and Rj. Following Ambrosio and Braides, we exhibit a problem for which the triangle inequality holds but existence fails, and we state a new sufficient condition for lowersemicontinuity, which may be necessary.

In this note, we present some Liouville type theorems about the non-negative solutions to some indefinite elliptic equations.

In this paper, inverse spectral problems for Sturm–Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b – 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.

We establish that the maximal operator and the Littlewood–Paley g-function associated with the heat semigroup defined by multidimensional Bessel operators are of weak type (1, 1). We also prove that Riesz transforms in the multidimensional Bessel setting are of strong type (p, p), for every 1 < p < ∞, and of weak type (1, 1).

This paper deals with local and global existence for the solutions of the heat equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied is where Ω ⊂ Rn (n ≥ 1) is a regular and bounded domain, ∂Ω = Γ0 ∪ Γ1, m > 1, 2 ≤ p < r, where r = 2(n − 1)/(n − 2) when n ≥ 3, r = ∞ when n = 1, 2 and u0 ∈ H1(Ω), u0 = 0 on Γ0. We prove local existence of the solutions in H1(Ω) when m > r/(r + 1−p) or n = 1, 2 and global existence when p ≤ m or the initial datum is inside the potential well associated to the stationary problem.

Let S and T be formally symmetric ordinary differential operators defined on a real interval I. It is assumed that the order of S is constant and everywhere strictly higher than the possibly varying order of T. The main result of this paper (Theorem 2.3) gives necessary and sufficient conditions for maximality of the deficiency indices of the differential relation Su = Tv considered in a Hilbert space with a scalar product which is a Dirichlet integral (see section 2) belonging to S. The conditions generalise those given in [5] for less general choices of operators S and T. For certain choices of Dirichlet integral they are explicit integrability conditions on the coefficients of the Dirichlet integral and the operator T.

We prove that the Cauchy problem for the one-dimensional parabolic equations , with initial data in Hs(R), cannot be solved by an iterative scheme based on the Duhamel formula for s < −1 if (k, d) = (2, 0) and s < sc(k, d) = ½ − (2 − d)/(k − 1) otherwise. This exactly completes the positive results on the Cauchy problem in Hs(R) for these equations and shows the particularity of the case (k, d) = (2, 0), for which we prove that the critical space Hsc(R) = H−3/2(R), by standard scaling arguments, cannot be reached. Our results also hold in the periodic setting.

The bivariate series defines a partial theta function. For fixed q (∣q∣ < 1), θ(q, ·) is an entire function. For q ∈ (–1, 0) the function θ(q, ·) has infinitely many negative and infinitely many positive real zeros. There exists a sequence of values of q tending to –1+ such that has a double real zero (the rest of its real zeros being simple). For k odd (respectively, k even) has a local minimum (respectively, maximum) at , and is the rightmost of the real negative zeros of (respectively, for k sufficiently large is the second from the left of the real negative zeros of ). For k sufficiently large one has . One has and .

This paper deals with the logistic Keller–Segel model

\[ \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u\nabla v) + \kappa u - \mu u^2, \\ v_t = \Delta v - v + u \end{cases} \]

$\bar {\Omega }\times (0,\infty )$

Euchœta norvegica has been found to be an important constituent of the food of the American pollack (Willey, 1921), and Bigelow (1926) expresses the opinion that it may form an important part of the food of mackerel and herring in the Gulf of Maine. In Loch Fyne in the Clyde sea-area, where Euchœta occurs in abundance at certain times of the year, it has never been found in the stomachs of herring (Scott, 1907). This is surprising for, as Scott remarks, Euchœta is “rich in oily matter, and apparently is of as much value as Calanus for herring food.” Macdonald (1927), however, has found that adult Meganyctiphanes norvegica feed on Euchœta, so, indirectly at least, it contributes to the food of the herring.

Euchæta norvegica is one of the largest known copepods and has a wide range of distribution in northern waters. Sars (1903, p. 38) records it from the deep waters of the fjords along the Norwegian coast and mentions that it occurred in two places in the polar basin crossed by Nansen. With (1915, p. 162) records its presence in North Atlantic waters in various places visited by the Danish Ingolf Expedition. That it is generally distributed throughout the North Atlantic is shown by Bigelow (1924, p. 230). It occurs also in the deep waters of the Clyde sea-area, where at times it is abundant, particularly in Upper Loch Fyne.

Dust, fogs, and clouds seem to have but little connection with each other, and we might think they could be better treated of under two separate and distinct heads. Yet I think we shall presently see that they are more closely related than might at first sight appear, and that dust is the germ of which fogs and clouds are the developed phenomena.

This was illustrated by an experiment in which steam was mixed with air in two large glass receivers; the one receiver was filled with common air, the other with air which had been carefully passed through a cotton-wool filter and all dust removed from it. In the unfiltered air the steam gave the usual and well-known cloudy form of condensation, while in the filtered air no cloudiness whatever appeared. The air remained supersaturated and perfectly transparent.

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear α: A → B, where A, B are subspaces of V. Then P(V) is a semigroup under composition, which is partially ordered by ⊆ (that is, α ⊆ β if and only if dom α ⊆ dom β and α = β | dom α). We compare this order with the so-called 'natural partial order' ≤ on P(V) and we determine their meet and join. We also describe all elements of P(V) that are minimal (or maximal) with respect to each of these four orders, and we characterize all elements that are 'compatible' with them. In addition, we answer similar questions for the semigroup T(V) consisting of all α ∈ P(V) whose domain equals V. Other orders have been defined by Petrich on any regular semigroup: three of them form a chain below ≤, and we show that two of these are equal on the semigroup P(V) and on the ring T(V). We also consider questions for these orders that are similar to those already mentioned

In this note, we consider the Hardy-Littlewood maximal function on R for arbitrary measures, as was done by Peter Sjögren in a previous paper. We determine the best constant for the weak type inequality.

The characteristic polynomial of a finite multigraph G is expressed in terms of characteristic polynomials oflocal modifications of G. The resulting formula is used to investigate the largest eigenvalues of certain theta graphs.

In this paper, we study the perturbation of zeros of maps of Banach spaces where the maps are invariant under continuous groups of symmetries. In some cases, we allow the perturbed maps partially to break the symmetries. Our results improve earlier results of the author by removing smoothness conditions on the group action. The key new idea is a regularity theorem for the zeros of invariant Fredholm maps.