heat content asymptotics of a riemannian manifold with boundary abstract let m be a compact riemannian manifold with a smooth c boundary m suppose m has initial temperature 1 at time t 0 and suppose that m is kept at temperature 0 for all t 0 we study the asymptotic behavior of the amount of heat in m as t 0 we generalize these results to arbitrary operators of laplace type with dirichlet boundary conditions and to arbitrary smooth initial conditions

asymptotics of the heat exchange abstract let k be a compact subset in euclidean space r m and let e k t denote the total amount of heat in r m k at time t if k is kept at fixed temperature 1 for all t 0 and if r m k has initial temperature 0 for two disjoint compact subsets k 1 and k 2 we define the heat exchange h k 1 k 2 t e k 1 t e k 2 t e k 1 k 2 t we obtain the leading asymptotic behaviour of h k 1 k 2 t as t 0 under mild regularity conditions on k 1 and k 2

heat content asymptotics with singular initial temperature distributions abstract we study the heat content asymptotics with either dirichlet or robin boundary conditions where the initial temperature exhibits radial blowup near the boundary we show that there is a complete small time asymptotic expansion and give explicit geometrical formulas for the first few terms in the expansion

heat flow heat content and the isoparametric property let m be a riemannian manifold and omega a compact domain of m with smooth boundary we study the solution of the heat equation on omega having constant unit initial conditions and dirichlet boundary conditions the purpose of this paper is to study the geometry of domains for which at any fixed value of time the normal derivative of the solution heat flow is a constant function on the boundary we express this fact by saying that such domains have the constant flow property in constant curvature spaces known examples of such domains are given by geodesic balls and more generally by domains whose boundary is connected and isoparametric the question is are they all like that this problem is the analogous for the heat equation of the classical serrin s problem for harmonic domains in this paper we give an affirmative answer to the above question in fact we prove more generally that if a domain in an analytic riemannian manifold has the constant flow property then every component of its boundary is an isoparametric hypersurface for space forms we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the r mean curvatures of the boundary and with the isoparametric property finally we discuss the constant flow property in relation to other well known overdetermined problems involving the laplace operator like the serrin problem or the schiffer problem

on the covering radius of codes the covering radius r of a code is the maximal distance of any vector from the code this work gives a number of new results concerning t n k the minimal covering radius of any binary code of length n and dimension k for example t n 4 and t n 5 are determined exactly and reasonably tight bounds on t n k are obtained for any k when n is large these results are found by using several new constructions for codes with small covering radius one construction the amalgamated direct sum involves a quantity called the norm of a code codes with norm leq 2 r 1 are called normal and may be combined efficiently the paper concludes with a table giving bounds on t n k for n leq 64

new lower bounds for binary covering codes g d chen et al ibid vol it 32 p 680 94 1986 presented two new lower bounds for k n r where k n r denotes the minimum cardinality of a binary code of length n and covering radius r the author shows that a slight modification gives further improvements and some examples are given to confirm the argument codes that have a certain partitioning property are considered

dirichlet spectrum and heat content abstract let m be a complete riemannian manifold and d m a smoothly bounded domain with compact closure we use brownian motion to study the relationship between the dirichlet spectrum of d and the heat content asymptotics of d central to our investigation is a sequence of invariants associated to d defined using exit time moments we prove that our invariants determine that part of the spectrum corresponding to eigenspaces which are not orthogonal to constant functions that our invariants determine the heat content asymptotics associated to the manifold and that when the manifold is a generic domain in euclidean space the invariants determine the dirichlet spectrum

area versus capacity and independence in the crushed ice model we study the heat flow from an infinite union b of disjoint closed balls in euclidean space rm kept at constant temperature 1 while the surrounding medium is at temperature 0 at time 0 we obtain conditions on the geometry of b such that the heat flow for small time is governed by the area of the boundary of b

on the boundedness of solutions of perturbed linear systems y a 0 y w w where x y g h are n vectors a t is a continuous n x n matrix for t 0 g t y is continuous for 12 0 y e r and h t is continuous for t 0 strauss and yorke 8 9 have studied the perturbing uniform asymptotically stable systems and furuno and hara 4 have shown some more detailed results bernfeld 1 and lovelady 6 have studied the perturbing uniformly bounded and uniformly ultimately bounded systems on the other hand coppel 2 3 has studied the boundedness of solutions of pl from the point of view of the dichotomy theory lovelady 5 referred to the connections between the perturbation problem and the dichotomy theory here we shall give some further results on the boundedness of solutions of perturbed linear systems this paper is much influenced by strauss and yorke 9 the purpose of this paper is to prove theorems on the perturbation from l to pl and pl of uniform boundedness theorem 3 1 uniform boundedness and ultimate boundedness theorem 4 1 and uniform boundedness and uniform ultimate boundedness theorem 5 1 let g be the class of functions g t y such that i g t r l 0 and ii y r where j y t dt r 0 and

further results on the covering radius of codes a number of upper and lower bounds are obtained for k n r the minimal number of codewords in any binary code of length n and covering radius r several new constructions are used to derive the upper bounds including an amalgamated direct sum construction for nonlinear codes this construction works best when applied to normal codes and we give some new and stronger conditions which imply that a linear code is normal an upper bound is given for the density of a covering code over any alphabet and it is shown that k n 2 r 1 leq k n r holds for sufficiently large n

heat content asymptotics with singular data we study the asymptotic behaviour of the heat content on a compact riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions assuming the existence of a complete asymptotic series we determine the first three terms in that series in addition to the general setting the interval is studied in detail this article is part of a special issue of journal of physics a mathematical and theoretical in honour of stuart dowker s 75th birthday devoted to applications of zeta functions and other spectral functions in mathematics and physics

improved sphere bounds on the covering radius of codes the sphere bound is a trivial lower bound on k n r the minimal cardinality of any binary code of length n and with covering radius r by simple arguments it is considerably improved to k n 1 or 2 sup n n for n even a table of lower and upper bounds on k n r for n

diffusions on graphs poisson problems and spectral geometry we study diffusions variational principles and associated boundary value problems on directed graphs with natural weightings we associate to certain subgraphs domains a pair of sequences each of which is invariant under the action of the automorphism group of the underlying graph we prove that these invariants differ by an explicit combinatorial factor given by stirling numbers of the first and second kind we prove that for any domain with a natural weighting these invariants determine the eigenvalues of the laplace operator corresponding to eigenvectors with nonzero mean as a specific example we investigate the relationship between our invariants and heat content asymptotics expressing both as special values of an analog of a spectral zeta function

asymptotically short term behavior of solutions to one dimensional diffusion processes consider the problem of solving the diffusion equation u s p x u 0 x p t 0 t x x u 0 t s u p t s 0 t 0 1 u x 0 s u x 0 w x 1 where p x is a smooth positive function on 0 p and u g l 0 p r 0 if we define the operator a d a a l 0 p r via d a w s p x w x dx for all w g d a s w 2 1 0 p r l w 1 1 0 p r 0

heat content and brownian motion for some regions with a fractal boundary letd be an open bounded set in euclidean space m m 2 3 with boundary d supposed has temperature 0 at timet 0 while d is kept at temperature 1 for allt 0 we use brownian motion to obtain estimates for the solution of corresponding heat equation and to obtain results for the asymptotic behaviour ofe d t the amount of heat ind at timet ast 0 for the triadic von koch snowflakek our results imply that c 1 t 1 mathbf mathbf log mathbf 2 log mathbf 3 leqq mathbf e k t leqq ct 1 mathbf mathbf log mathbf 2 log mathbf 3 mathbf 0 leqq t leqq c 1 for some constantc 1

growth of heat trace and heat content asymptotic coefficients abstract we show in the smooth category that the heat trace asymptotics and the heat content asymptotics can be made to grow arbitrarily rapidly in the real analytic context however this is not true and we establish universal bounds on their growth

continuability of solutions of perturbed differential equations on considere les systemes d equations differentielles suivants e x f t x p x f t x g t x per x f t x h t ou x f g h sont des n vecteurs f t x g t x sont continues sur 0 r n et h t est continue sur 0 on demontre des theoremes sur la perturbation de la continuite des solutions de e a p et per

the hybrid spectral problem and robin boundary conditions the hybrid spectral problem where the field satisfies dirichlet conditions d on part of the boundary of the relevant domain and neumann n on the remainder is discussed in simple terms a conjecture for the c1 coefficient is presented and the conformal determinant on a 2 disc where the d and n regions are semi circles is derived comments on higher coefficients are made a separable second order hemisphere hybrid problem is introduced that involves robin boundary conditions and leads to logarithmic terms in the heat kernel expansion which are evaluated explicitly

a retarded gronwall like inequality and its applications we prove a gronwall like inequality and present some of its applications to the qualitative study of retarded differential equations the problems of global continuation of the solutions and the existence of nonoscillatory solutions are considered by means of examples we show the usefulness of our results

the heat content asymptotics for variable geometries let be a time dependent family of riemannian metrics on a manifold m with a smooth boundary let be the initial temperature of m and let be the specific heat of m impose dirichlet or neumann boundary conditions and let be the resulting total heat energy content of m as one can expand in an asymptotic series in half integer powers of the parameter t we determine for in terms of geometric quantities this extends previous results from the autonomous setting where the metric was independent of the parameter t to a dynamic setting where the metric is permitted to be time dependent

on normal and subnormal q ary codes the authors extend to the q ary case the notions of a normal code a subnormal code and the amalgamated direct sum construction in order to investigate problems related to the covering radius of codes for example the authors prove that every nonbinary nontrivial perfect code is absubnormal they also include some linear programming lower bounds on ternary codes with covering radius 2 or 3

a new construction for covering codes a novel method for constructing normal binary nonlinear covering codes is presented the construction improves several upper bounds on k n r the minimum cardinality of a binary code of length n and covering radius r

heat equation on the arithmetic von koch snowflake we investigate the asymptotic behaviour of the heat content as the time t 0 for an s adic von koch snowflake generated by a square we show that the heat content satisfies a functional equation which after appropriate transformations takes the form of an inhomogeneous renewal equation we obtain the structure of the solution of this equation in the arithmetic case up to an exponentially small remainder in t