Let D be an open set in Euclidean space IRm with boundary ∂D, and let ϕ[ratio ]∂D→[0, ∞) be a bounded, measurable function. Let u[ratio ]D∪∂D×[0, ∞)→[0, ∞) be the unique weak solution of the heat equation

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with initial condition

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and with inhomogeneous Dirichlet boundary condition

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Then u(x; t) represents the temperature at a point x∈D at time t if D has initial temperature 0, while the temperature at a point x∈∂D is kept fixed at ϕ(x) for all t>0. We define the total heat content (or energy) in D at time t by

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In this paper we wish to examine the effect of imposing additional cooling on some subset C on both u and ED.

Since the very beginning of the multidimensional theory of quasiregular mappings, it has been widely believed that the class of K-quasiregular mappings in ℝn is closed with respect to uniform convergence, where K stands for the linear dilatation. In this note we give a striking example which refutes this belief. The key element of our construction is that the linear dilatation function fails to be rank-one convex in dimensions higher than 2.

Bernard Scott was born in London on 27 August 1915. At the time of his birth his name was Schultz. He was the only son of a Jewish family of fur and skin merchants who lived in North London; he had two sisters.

Bernard attended the City of London School from 1925 to 1934. From school records it appears that he evinced an early taste for argument and debate. There is also testimony of his enthusiasm for sport, especially rugby. His talents for mathematics and for chess became evident at an early stage. While mathematics became his profession, it was the game of chess that aroused in him an abiding passion and made him a first-class player throughout his life.

Bernard won an Open Scholarship in Mathematics to Magdalene College, Cambridge. He graduated with First Class Honours and a Distinction in Part III of the Mathematical Tripos in 1937. While pursuing his mathematical studies with evident success, he continued to cultivate his talent as a chess player. B. H. Neumann, who was a research student at Cambridge at that time, relates that he and Bernard Schultz (as he then was) went to London to take part in a weekend tournament and returned with all the prize money between them (about £7).

We prove that a proper holomorphic map on the unit disk in the complex plane is uniquely determined up to post-composition with a Möbius transformation by its critical points.

Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side. Let μ(m, n) be the expected number of survivors. Clearly, we have boundary conditions:

μ(m, 0)=m, μ(0, n)=n. (1.1)

We also have the equation

formula here

This is because the probability that the first successful shot is made by the side with m gunmen is m/(m+n). On using the recurrence relation (1.2) together with the boundary condition (1.1), the computer produces Table 1 below, in which

m=8192+k, n=8192−k, d(m, n) =√(m2−n2 =128√(2k).

We observe that the Bargmann transform arises from the polar decomposition of a simple restriction operator. By virtue of the Bargmann transform, the Wigner transform is imbedded in the geometric symmetry group of phase space. From this imbedding, we easily obtain that the spectrum of the Wigner transform consists of {eikπ/12[ratio ]k=0, 1, …, 23}.

In a previous paper, the second author introduced a compact topology τr on the space of closed ideals of a unital Banach algebra A. If A is separable, then τr is either metrizable or else neither Hausdorff nor first countable. Here it is shown that τr is Hausdorff if A is C1[0, 1], but that if A is a uniform algebra, then τr is Hausdorff if and only if A has spectral synthesis. An example is given of a strongly regular, uniform algebra for which every maximal ideal has a bounded approximate identity, but which does not have spectral synthesis.

We introduce a game called Squares where the single player is presented with a pattern of black and white squares and has to reduce the pattern to white by making as few moves as possible. We present a method for solving the game, and show that the following problem is NP-complete.

Problem 1 (Squares-Solvability). Given a pattern X and k∈N, can X be solved in k or less moves?

We demonstrate a reduction to this problem from Not-All-Equal-3SAT. We also present another NP-complete problem that Squares-Solvability can be reduced to.

We prove that the spectrum of a suitably projected Dirac operator — sometimes called the no-pair operator — introduced by Brown and Ravenhall is bounded from below by some positive constant when the nuclear charge is below some critical charge. This bound improves a result obtained recently by Evans et al., and disproves a conjecture made by Hardekopf and Sucher.

Let B=(Bt)t[ges ]0 be standard Brownian motion started at zero. We prove

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for all c>1 and all stopping times τ for B satisfying E(τr)<∞ for some r>1/2. This inequality is sharp, and equality is attained at the stopping time

τ*=inf{t>0[mid ]St [ges ]u*, Xt =1∨αSt},

where u*=1+1/ec(c−1) and α=(c−1)/c for c>1, with Xt=[mid ]Bt[mid ] and St= max0[les ]r[les ]t[mid ]Br[mid ]. Likewise, we prove

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for all c>1 and all stopping times τ for B satisfying E(τr<∞ for some r>1/2. This inequality is sharp, and equality is attained at the stopping time

σ*=inf{t>0[mid ]St [ges ]v*, Xt =αSt},

where v*=c/e(c−1) and α=(c−1)/c for c>1. These results contain and refine the results on the Llog L-inequality of Gilat [6] which are obtained by analytic methods. The method of proof used here is probabilistic and is based upon solving the optimal stopping problem with the payoff

formula here

where F(x) equals either xlog+x or xlog x. This optimal stopping problem has some new interesting features, but in essence is solved by applying the principle of smooth fit and the maximality principle. The results extend to the case when B starts at any given point (as well as to all non-negative submartingales).

The purpose of this note is to establish a new version of the local Steiner formula and to give an application to convex bodies of constant width. This variant of the Steiner formula generalizes results of Hann [3] and Hug [6], who use much less elementary techniques than the methods of this paper. In fact, Hann asked for a simpler proof of these results [4, Problem 2, p. 900]. We remark that our formula can be considered as a Euclidean analogue of a spherical result proved in [2, p. 46], and that our method can also be applied in hyperbolic space.

For some remarks on related formulas in certain two-dimensional Minkowski spaces, see Hann [5, p. 363].

For further information about the notions used below, we refer to Schneider's book [9]. Let [Kscr ]n be the set of all convex bodies in Euclidean space IRn, that is, the set of all compact, convex, non-empty subsets of IRn. Let Sn-1 be the unit sphere. For K∈[Kscr ]n, let NorK be the set of all support elements of K, that is, the pairs (x, u)∈IRn×Sn−1 such that x is a boundary point of K and u is an outer unit normal vector of K at the point x. The support measures (or generalized curvature measures) of K, denoted by Θ0(K, ·), …, Θn−1(K, ·), are the unique Borel measures on IRn×Sn−1 that are concentrated on NorK and satisfy

formula here

for all integrable functions f[ratio ]IRn→IR; here λ denotes the Lebesgue measure on IRn. Equation (1), which is a consequence and a slight generalization of Theorem 4.2.1 in Schneider [9], is called the local Steiner formula. Our main result is the following.

Let A1,…, An be Lipschitz functions on R such that A′1,…, A′n∈VMO. We show that on any bounded interval, the Calderón commutator associated with the kernel (A1(x)−A1(y)) …(An(x) −An(y))/(x−y) n+1 is a compact perturbation of −πiHMA′1… A′n, where H is the Hilbert transform.

The precise indeterminacy of Milnor's concordance μ invariants is determined. It is shown that the (nilpotent quotient) Artin representation of string links is surjective. Orr's computation of the number of linearly independent μ invariants of a fixed length is recovered. A structure theorem for the set of links up to concordance is proven. We define an action of 2l-component string links on l-component string links, which passes to concordance classes. It is shown that the set of links up to concordance is in bijection with the orbit space of the restriction of this action to the stabilizer of the identity. Via the Artin representation, the action passes to a unipotent action, defined purely algebraically and consequently algorithmically computable, on the corresponding automorphism groups.

A series of complexes Qp indexed by all primes p is constructed with catQp=2 and catQp×Sn=2 for either n[ges ]2 or n=1 and p=2. This disproves Ganea's conjecture on Lusternik–Schnirelmann (LS) category.

For all smooth nondecreasing functions f on [0, 1], the local time Lf spent by standard Brownian motion on the graph of f satisfies P(Lf<x) [ges ]P(2L0<x), for x>0. The constant 2 in this inequality may not be replaced with a smaller one.

In [8, 6] it was shown that for each k and n such that 2k>n, there exists a contractible k-dimensional complex Y and a continuous map ϕ[ratio ][ ]n→Y without the antipodal coincidence property, that is, ϕ(x)≠ϕ(−x) for all x∈[ ]n. In this paper it is shown that for each k and n such that 2k>n, and for each fixed-point free homeomorphism f of an n-dimensional paracompact Hausdorff space X onto itself, there is a contractible k-dimensional complex Y and a continuous map ϕ[ratio ]X→Y such that ϕ(x)≠ϕ(f(x)) for all x∈X. Various results along these lines are obtained.

Alan Breach Tayler, CBE, Director of the Oxford Centre for Industrial and Applied Mathematics, died on 28 January 1995, aged 63.

Alan went up to Oxford in 1951 to read Mathematics at Brasenose College. He obtained a first, and after a brief excursion to the Bristol Aircraft Company, he returned to work for a DPhil with George Temple. His thesis, completed in 1959 and entitled ‘Problems in compressible flow’, contained a mixture of analytic, approximate and numerical solutions which foreshadowed the new practical applied mathematics that he embraced later. He became University Lecturer and Tutorial Fellow of St Catherine's Society in 1959.

During the next twenty-five years, Alan Tayler brought a new ethos to applied mathematics. This change came about through his recognition that the status quo in the 1960s, which comprised a delicate balance between theory and practice in the area of applied mechanics, was capable of far-reaching generalisation; indeed, he saw that such a development was essential since the following decades were to be dominated by computers and an ever-increasing need for mathematical modelling. In 1967, with Leslie Fox, he initiated the mathematical Study Groups with Industry, wherein academic and industrial researchers interact in week-long workshops. These were an immediate success: (1) with industry, who found new insights into their problems and new recruiting possibilities; (2) with students, whose enthusiasm to use their theoretical knowledge soon led to the highly popular MSc in Mathematical Modelling and Numerical Analysis; and (3) with faculty, both pure and applied, who found an undreamed of source of fascinating new theoretical problems. For example, one intellectual consequence was the use of industrial case studies to uncover the new field of ‘free boundary problems’, on which several thousand learned articles have appeared since 1970.

Let 0<n1<n2<… <nN be N distinct integers, and let a1, a2, …, aN be complex numbers. We set

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and

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There are two well-known problems concerning the case a1=a2=…=aN=1.