A Comparison Estimate for the Heat Equation with an Application to the Heat Content of the S-Adic Von Koch Snowflake

Let D be an open set in Euclidean space R^m with boundary D,and let :D[0, ) be a bounded, measurable function. Let u:DDx[0,)[0, ) be the unique weak solution of the heat equation [formula] with initial condition [formula] and with inhomogeneous Dirichlet boundary condition [formula] Then u(x; t) represents the temperature at a point xD at timet if D has initial temperature 0, while the temperature at apoint xD is kept fixed at (x) for all t>0. We define thetotal heat content (or energy) in D at time t by [formula] In this paper we wish to examine the effect of imposing additionalcooling on some subset C on both u and E_D. 1991 MathematicsSubject Classification 35K05, 60J65, 28A80.     

Sub- and Superadditive Properties of Fejer's sine Polynomial

Let be Fejér's sine polynomial. We prove the following statements.

1. The inequality holds for all x, y (0, ) with x + y < if and only if 0 and + rß 1.
2. The converse of the above inequality is valid for allx, y (0, ) with x + y < if and only if 0 and + rß 1.
3. For all n N and x, y [0, ] we have . Both bounds are best possible.

2000 Mathematics Subject Classification 42A05, 26D05 (primary),39B62 (secondary).     

The Dirichlet problem by variational methods

Let ^N be a bounded open set and C( ). Assume that has an extensionC() such that H^–1().Then by the Riesz representation theorem there exists a unique

We show that u+ coincides with the Perron solutionof the Dirichlet problem

This extends recent results by Hildebrandt [Math. Nachr. 278(2005), 141–144] and Simader [Math. Nachr. 279 (2006),415–430], and also gives a possible answer to Hadamard'sobjection against Dirichlet's principle.     

Normal Families and Shared Values

For f a meromorphic function on the plane domain D and a C,let _f(a) = {z D: f(z) = a}. Let F be a family of meromorphicfunctions on D, all of whose zeros are of multiplicity at leastk. If there exist b 0 and h > 0 such that for every f F,_f(0) = _f^(k)(b) and 0 < |f^(k+1)(z)| h whenever z _f(0), thenF is a normal family on D. The case _f(0) = Ø is a celebratedresult of Gu [5]. 1991 Mathematics Subject Classification 30D45,30D35.     

A variant of the Hales-Jewett theorem

It was shown by Bergelson that any set B with positive uppermultiplicative density contains nicely intertwined arithmeticand geometric progressions: for each k there exist a, b, d such that {b(a+id)^j: i, j {1, 2, ..., k}}B. In particular,one cell of each finite partition of contains such configurations.We prove a Hales–Jewett-type extension of this partitiontheorem.     

On Transitive Permutation Groups with Primitive Subconstituents

Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05.     

Packing, Tiling, Orthogonality and Completeness

Let R^d be an open set of measure 1. An open set DR^d is calleda ‘tight orthogonal packing region’ for if D–Ddoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of R^d. The main contribution of this paperis a new way of proving the following result: D tiles R^d whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L²(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in R^d, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about ‘packing’ R^d withtranslates of a certain non-negative function and, additionally,we have completeness of E in L²() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70.     

An Upper Estimate for a Heat Kernel With Neumann Boundary Condition

Using an upper solution we obtain a bound from above for theheat kernel (x,y,t) for a region which is star-shaped withrespect to one of the points, say y. The estimate is for theNeumann problem and holds for short times. The form of the boundis moreover, for x\Y(y), Here Y(y) is a closed subset of R^Nwith measure zero, d(x,y) is the minimum distance between xand y via the boundary :d(x,y) = inf_Z(|x-z| + |y-z|), and f(.,y)is a positive function, continuous away from Y, and equal tounity on .     

Are All Uniform Algebras Amnm?

A Banach algebra a is AMNM if whenever a linear functional on a and a positive number satisfy |(ab)–(a)(b)|||a||·||b||for all a, b a, there is a multiplicative linear functional on a such that ||–||=o(1) as 0. K. Jarosz [1] asked whetherevery Banach algebra, or every uniform algebra, is AMNM. B.E. Johnson [3] studied the AMNM property and constructed a commutativesemisimple Banach algebra that is not AMNM. In this note weconstruct uniform algebras that are not AMNM. 1991 MathematicsSubject Classification 46J10.     

Trivial centralizers for codimension-one attractors

We show that if is a codimension-one hyperbolic attractor fora C^r diffeomorphism f, where 2 r , and f is not Anosov, thenthere is a neighborhood of f in Diff^r(M) and an open and denseset of such that any g has a trivial centralizer on thebasin of attraction for .