Necessary and sufficient conditions to characterise weakly $r$-preinvex functions on an invex set are obtained and used to establish generalisations of the Hermite–Hadamard inequality for such functions.

The Laub–Ilani inequality [‘A subtle inequality’, Amer. Math. Monthly97 (1990), 65–67] states that $x^{x}+y^{y}\geqslant x^{y}+y^{x}$ for nonnegative real numbers $x,y$. We introduce and prove new trigonometric and algebraic-trigonometric inequalities of Laub–Ilani type and propose some conjectural algebraic-trigonometric inequalities of similar forms.

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

A necessary and sufficient condition for a continuous function $g$ to be almost periodic on time scales is the existence of an almost periodic function $f$ on $\mathbb{R}$ such that $f$ is an extension of $g$. Our aim is to study this question for pseudo almost periodic functions. We prove the necessity of the condition for pseudo almost periodic functions. An example is given to show that the sufficiency of the condition does not hold for pseudo almost periodic functions. Nevertheless, the sufficiency is valid for uniformly continuous pseudo almost periodic functions. As applications, we give some results on the connection between the pseudo almost periodic (or almost periodic) solutions of dynamic equations on time scales and of the corresponding differential equations.

The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total memory and work, where NT and NS represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel t–1–α on the interval [Δt, T] with a uniform absolute error ε. We give the theoretical analysis to show that the number of exponentials Nexp needed is of order for T≫1 or for TH1 for fixed accuracy ε. The resulting algorithm requires only storage and work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.

An iteration technique for characterizing boundedness of certain types of multilinear operators is presented, reducing the problem to a corresponding linear-operator case. The method gives a simple proof of a characterization of validity of the weighted bilinear Hardy inequality

for all non-negative f, g on (a, b), for 1 < p 1, p 2, q < ∞. More equivalent characterizing conditions are presented.

The same technique is applied to various further problems, in particular those involving multilinear integral operators of Hardy type.

In this paper, we first apply cosine radial basis function neural networks to solve the fractional differential equations with initial value problems or boundary value problems. In the examples, we successfully obtained the numerical solutions for the fractional Riccati equations and fractional Langevin equations. The computer graphics and numerical solutions show that this method is very effective.

The second order weighted and shifted Grünwald difference (WSGD) operators are developed in [Tian, Zhou and Deng, Math. Comput., 84 (2015), pp. 1703–1727] to solve space fractional partial differential equations. Along this direction, we further design a new family of second order WSGD operators; by properly choosing the weighted parameters, they can be effectively used to discretize space (Riemann-Liouville) fractional derivatives. Based on the new second order WSGD operators, we derive a family of difference schemes for the space fractional advection diffusion equation. By von Neumann stability analysis, it is proved that the obtained schemes are unconditionally stable. Finally, extensive numerical experiments are performed to demonstrate the performance of the schemes and confirm the convergence orders.

It is well known that the standard Lipschitz space in Euclidean space, with exponent α ∈ (0, 1), can be characterized by means of the inequality , where is the Poisson integral of the function f. There are two cases: one can either assume that the functions in the space are bounded, or one can not make such an assumption. In the setting of the Ornstein–Uhlenbeck semigroup in ℝn, Gatto and Urbina defined a Lipschitz space by means of a similar inequality for the Ornstein–Uhlenbeck Poisson integral, considering bounded functions. In a preceding paper, the authors characterized that space by means of a Lipschitz-type continuity condition. The present paper defines a Lipschitz space in the same setting in a similar way, but now without the boundedness condition. Our main result says that this space can also be described by a continuity condition. The functions in this space turn out to have at most logarithmic growth at infinity.

Let $E$ be a finite-dimensional normed space and $\unicode[STIX]{x1D6FA}$ a non-empty convex open set in $E$. We show that the Lipschitz-free space of $\unicode[STIX]{x1D6FA}$ is canonically isometric to the quotient of $L^{1}(\unicode[STIX]{x1D6FA},E)$ by the subspace consisting of vector fields with zero divergence in the sense of distributions on $E$.

We provide new quantitative versions of Helly’s theorem. For example, we show that for every family $\{P_{i}:i\in I\}$ of closed half-spaces in $\mathbb{R}^{n}$ such that $P=\bigcap _{i\in I}P_{i}$ has positive volume, there exist $s\leqslant \unicode[STIX]{x1D6FC}n$ and $i_{1},\ldots ,i_{s}\in I$ such that

$$\begin{eqnarray}\operatorname{vol}_{n}(P_{i_{1}}\cap \cdots \cap P_{i_{s}})\leqslant (Cn)^{n}\operatorname{vol}_{n}(P),\end{eqnarray}$$

$\unicode[STIX]{x1D6FC},C>0$

We introduce the open degree of a compact space, and we show that for every natural number $n$, the separable Rosenthal compact spaces of degree $n$ have a finite basis.

In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, in both the centred and uncentred versions. In particular, we solve a question originally posed by Bober et al. [‘On a discrete version of Tanaka’s theorem for maximal functions’, Proc. Amer. Math. Soc.140 (2012), 1669–1680].

In this paper the fractional Euler-Lagrange equation is considered. The fractional equation with the left and right Caputo derivatives of order α ∈ (0,1] is transformed into its corresponding integral form. Next, we present a numerical solution of the integral form of the considered equation. On the basis of numerical results, the convergence of the proposed method is determined. Examples of numerical solutions of this equation are shown in the final part of this paper.

Let $\unicode[STIX]{x1D6FA}$ be a domain in $\mathbb{R}^{m}$ with nonempty boundary. In Ward [‘On essential self-adjointness, confining potentials and the $L_{p}$-Hardy inequality’, PhD Thesis, NZIAS Massey University, New Zealand, 2014] and [‘The essential self-adjointness of Schrödinger operators on domains with non-empty boundary’, Manuscripta Math.150(3) (2016), 357–370] it was shown that the Schrödinger operator $H=-\unicode[STIX]{x1D6E5}+V$, with domain of definition $D(H)=C_{0}^{\infty }(\unicode[STIX]{x1D6FA})$ and $V\in L_{\infty }^{\text{loc}}(\unicode[STIX]{x1D6FA})$, is essentially self-adjoint provided that $V(x)\geq (1-\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA}))/d(x)^{2}$. Here $d(x)$ is the Euclidean distance to the boundary and $\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$ is the nonnegative constant associated to the $L_{2}$-Hardy inequality. The conditions required for a domain to admit an $L_{2}$-Hardy inequality are well known and depend intimately on the Hausdorff or Aikawa/Assouad dimension of the boundary. However, there are only a handful of domains where the value of $\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$ is known explicitly. By obtaining upper and lower bounds on the number of cubes appearing in the $k\text{th}$ generation of the Whitney decomposition of $\unicode[STIX]{x1D6FA}$, we derive an upper bound on $\unicode[STIX]{x1D707}_{p}(\unicode[STIX]{x1D6FA})$, for $p>1$, in terms of the inner Minkowski dimension of the boundary.

In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (nonfractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.

An efficient high order numerical method is presented to solve the mobile-immobile advection-dispersion model with the Coimbra time variable-order fractional derivative, which is used to simulate solute transport in watershed catchments and rivers. On establishing an efficient recursive algorithm based on the properties of Jacobi polynomials to approximate the Coimbra variable-order fractional derivative operator, we use spectral collocation method with both temporal and spatial discretisation to solve the time variable-order fractional mobile-immobile advection-dispersion model. Numerical examples then illustrate the effectiveness and high order convergence of our approach.

We identify a close relation between stable distributions and the limiting homomorphisms central to the theory of regular variation. In so doing some simplifications are achieved in the direct analysis of these laws in Pitman and Pitman (2016); stable distributions are themselves linked to homomorphy.

In this paper, we show that the cost of an optimal train journey on level track over a fixed distance is a strictly decreasing and strictly convex function of journey time. The precise structure of the cost–time curves for individual trains is an important consideration in the design of energy-efficient timetables on complex rail networks. The development of optimal timetables for busy metropolitan lines can be considered as a two-stage process. The first stage seeks to find optimal transit times for each individual journey segment subject to the usual trip-time, dwell-time, headway and connection constraints in such a way that the total energy consumption over all proposed journeys is minimized. The second stage adjusts the arrival and departure times for each journey while preserving the individual segment times and the overall journey times, in order to best synchronize the collective movement of trains through the network and thereby maximize recovery of energy from regenerative braking. The precise nature of the cost–time curve is a critical component in the first stage of the optimization.

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann–Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed. Moreover, we establish the fractional Cauchy–Kovalevskaya extension (FCK extension) theorem for fractional monogenic functions defined on ℝd . Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We study the connection between the FCK extension of functions of the form xPl and the classical Gegenbauer polynomials. Finally, we present an example of an FCK extension.