Shaping a Concept of Terrorist Acts

Chapter 1 is concerned with bringing some clarity to the widespread conceptual confusion around what terms like “terrorist,” “terrorist act,” and “terrorism” mean. Without being too rigid about definition, it is important to operate with some agreed definitional clarity in the area. The chapter defends the value of such a definitional enterprise and then provides what it calls a tactical definition of a terrorist act that aims to capture a central core involved in talk about terrorism, and opens discussion of terrorist acts to cogent moral assessment. The author’s definition of a terrorist act is: “A political act, ordinarily committed or inspired by an organized group, in which violence is intentionally directed at non-combatants (or ‘innocents’ in a suitable sense) or their significant property, in order to cause them serious harm.” The rest of the chapter discusses advantages of the definition and criticizes a number of objections to it.

The extension problem for fractional Sobolev spaces with a partial vanishing trace condition

We construct whole-space extensions of functions in a fractional Sobolev space of order $$s\in (0,1)$$ s ∈ ( 0 , 1 ) and integrability $$p\in (0,\infty )$$ p ∈ ( 0 , ∞ ) on an open set O which vanish in a suitable sense on a portion D of the boundary $${{\,\mathrm{\partial \!}\,}}O$$ ∂ O of O. The set O is supposed to satisfy the so-called interior thickness condition in$${{\,\mathrm{\partial \!}\,}}O {\setminus } D$$ ∂ O \ D , which is much weaker than the global interior thickness condition. The proof works by means of a reduction to the case $$D=\emptyset $$ D = ∅ using a geometric construction.

Minimality of SRPT Networks With Resource Sharing

A general multi-resource network with users requiring service from a number of shared resources simultaneously is considered. It is demonstrated that the Shortest Remaining Processing Time (SRPT) service protocol minimizes, in a suitable sense, the system resource idleness with respect to customers with residual service times not greater than any threshold value on every network route. Our arguments are pathwise, with no assumptions on the model stochastic primitives and the network topology.

Integer-valued definable functions in ℝan,exp

We give two variations on a result of Wilkie’s [A. J. Wilkie, Complex continuations of [Formula: see text]-definable unary functions with a diophantine application, J. Lond. Math. Soc. (2) 93(3) (2016) 547–566] on unary functions definable in [Formula: see text] that take integer values at positive integers. Provided that the function grows slower (in a suitable sense) than the function [Formula: see text], Wilkie showed that it must be eventually equal to a polynomial. Assuming a stronger growth condition, but only assuming that the function takes values sufficiently close to integers at positive integers, we show that the function must eventually be close to a polynomial. In a different variation we show that it suffices to assume that the function takes integer values on a sufficiently dense subset of the positive integers (for instance the primes), again under a stronger growth bound than that in Wilkie’s result.

Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

We consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\frac{\mu }{2} \hspace{0.07em} \frac{\lVert F \rVert ^{2}}{\det F}+f(\det F)$ W ( F ) = μ 2 ∥ F ∥ 2 det F + f ( det F ) ; such an energy is rank-one convex if and only if the function $f$ f is convex.

On generating functions in additive number theory, II: lower-order terms and applications to PDEs

We obtain asymptotics for sums of the form $$\begin{aligned} \sum _{n=1}^P e\left( {\alpha }_k\,n^k\,+\,{\alpha }_1 n\right) , \end{aligned}$$ ∑ n = 1 P e α k n k + α 1 n , involving lower order main terms. As an application, we show that for almost all $${\alpha }_2 \in [0,1)$$ α 2 ∈ [ 0 , 1 ) one has $$\begin{aligned} \sup _{{\alpha }_{1} \in [0,1)} \Big | \sum _{1 \le n \le P} e\left( {\alpha }_{1}\left( n^{3}+n\right) + {\alpha }_{2} n^{3}\right) \Big | \ll P^{3/4 + \varepsilon }, \end{aligned}$$ sup α 1 ∈ [ 0 , 1 ) | ∑ 1 ≤ n ≤ P e α 1 n 3 + n + α 2 n 3 | ≪ P 3 / 4 + ε , and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations.

Theory of Probabilistic Connectedness

We introduce and study a notion of probabilistic connectedness, which we term $proconnectedness$, defined in terms of partitions of a probability space into two nonempty disjoint independent events. Both proconnectedness and disproconnectedness are shown to be invariants (in a suitable sense) under isomorphic random elements. We show that a substantial part of the fundamental theory of topological connectedness admits a natural counterpart in the present theory of proconnectedness. Some applications and connections regarding limit theorems, cardinality equality of measurability structures, atomic distributions, and singular distributions are discussed.

The large diffusion limit for the heat equation with a dynamical boundary condition

We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition.

Algebraic Tangent Cones of Reflexive Sheaves

We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geometric counterpart of our previous study on singularities of Hermitian–Yang–Mills connections.

The Noether–Lefschetz locus of surfaces in toric threefolds

The Noether–Lefschetz theorem asserts that any curve in a very general surface [Formula: see text] in [Formula: see text] of degree [Formula: see text] is a restriction of a surface in the ambient space, that is, the Picard number of [Formula: see text] is [Formula: see text]. We proved previously that under some conditions, which replace the condition [Formula: see text], a very general surface in a simplicial toric threefold [Formula: see text] (with orbifold singularities) has the same Picard number as [Formula: see text]. Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in [Formula: see text] in a linear system of a Cartier ample divisor with respect to a [Formula: see text]-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether–Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense.