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

formula here

with initial condition

formula here

and with inhomogeneous Dirichlet boundary condition

formula here

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

formula here

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

The purpose of this note is to obtain a restriction on the fundamental groups of non-elliptic compact complex surfaces of class VII in Kodaira's classification [9]. We recall that these are the compact complex surfaces with first Betti number one and no non-constant meromorphic functions. This seems to be the class of compact complex surfaces whose structure is least understood. The first and simplest examples are the general Hopf surfaces [9, III]. Then there are various classes of examples found by Inoue [5, 6], and which have been studied in more detail in [11]. The only known topological restriction beyond the first Betti number is that intersection form in two-dimensional homology is negative definite. There seems to be little known as to how wide this class of surfaces is. We prove the following theorem.

In this note we give some sharp estimates for norms of polynomials via the products of norms of their linear terms. Different convex norms on the unit disc are considered.

We produce a family of algebraic curves (closed Riemann surfaces) Sλ admitting two cyclic groups H1 and H2 of conformal automorphisms, which are topologically (but not conformally) conjugate and such that S/Hi is the Riemann sphere [Copf ]ˆ. The relevance of this example is that it shows that the subvarieties of moduli space consisting of points parametrizing curves which occur as cyclic coverings (of a fixed topological type) of [Copf ]ˆ need not be normal.

Soient F un corps commutatif localement compact non archimédien et ψ un caractère additif non trivial de F. Soient σ une représentation du groupe de Weil–Deligne de F, et ψσˇ sa contragrédiente. Nous calculons le facteur ε(σ[otimes ]σˇ, ψ, ½). De manière analogue, nous calculons le facteur ε(π×πˇ, ψ, ½) pour toute représentation admissible irréductible π de GLn(F). En conséquence, si F est de caractéristique nulle et si σ et π se correspondent par la correspondance de Langlands construite par M. Harris, ou celle construite par les auteurs, alors les facteurs ε(σ[otimes ]σˇ, ψ, s) et ε(π×πˇ, ψ, s) sont égaux pour tout nombre complexe s.

Let F be a non-Archimedean local field and ψa non-trivial additive character of F. Let σ be a representation of the Weil–Deligne group of F and σˇ its contragredient representation. We compute ε(σ[otimes ]σˇ, ψ, ½). Analogously, we compute ε(π×πˇ, ψ, ½) for all irreducible admissible representations π of GLn(F). Consequently, if F has characteristic zero, and σ, π correspond via the Langlands correspondence established by M. Harris or the correspondence constructed by the authors, then we have ε(σ[otimes ]σˇ, ψ, s) = ε(π×πˇ, ψ, s) for all s∈[Copf ].

In this paper, the minimal positive entropy of pure braids is shown to be greater than $\log(2+\sqrt5)$, independent of the braid index. For pure 3-braids, the minimal entropy is shown to be equal to $\log(3+2\sqrt2)$.

We show that operators on n × n matrices which are representable in the form T(X) = [sum ][lscr ]i=1aiXbi (where ai and bi are n × n matrices) and are k-positive for k = [√[lscr ]] must be completely positive. As a consequence, elementary operators on a C*-algebra with minimal length [lscr ] which are k-positive for k = [√[lscr ]] must be completely positive.

A proof is given that the full symmetric group over any infinite set is the product of finitely many Abelian subgroups. In fact, 289 subgroups suffice. Sharp bounds are also obtained on the minimal number $k$ , such that the finite symmetric group $S_n$ is the product of $k$ Abelian subgroups. Using this, $S_n$ is proved to be the product of $72n^{1/2}(\log n)^{3/2}$ cyclic subgroups.

In this paper, perturbations of the left and right essential spectra of $2\times 2$ upper triangular operator matrix $M_C$ are studied, where $M_C= \left(\begin{smallmatrix} A & C 0 & B \end{smallmatrix}\right)$ is an operator acting on the Hilbert space ${\cal H}\oplus{\cal K }$. For given operators $A$ and $B$, the sets $\bigcap_{C\in B({\cal K},{\cal H})}\sigma_{\rle}(M_C)$ and $\bigcap_{C\in B({\cal K},{\cal H})}\sigma_{\rre}(M_C)$ are determined, where $ \sigma_{\rle}(T)$ and $\sigma_{\rre}(T)$ denote, respectively, the left essential spectrum and the right essential spectrum of an operator $T$.

In this paper we are concerned with the asymptotic description of the sum of digits function of canonical number systems in number fields. We derive an asymptotic formula for its summatory function whose second term contains a periodic fluctuation. In imaginary quadratic fields this function is uniformly distributed with respect to the argument.

G[Lscr ][Uscr ][Cscr ] is the largest semigroup compactification of the locally compact group G. When G is not compact, given q ∈ G* = G[Lscr ][Uscr ][Cscr ] \ G, there is p ∈ G* such that x [map ] qx is discontinuous at p (Theorem 2). If G is σ-compact, there is one p which will serve for all q (Theorem 1).

Let $K$ be the (real closed) field of Puiseux series in $t$ over ${\mathbf R}$ endowed with the natural linear order. Then the elements of the formal power series rings ${\mathbf R}[\![\xi_1,\dots,\xi_n]\!]$ converge $t$-adically on $[-t,t]^n$, and hence define functions $[-t,t]^n\to K$. Let ${\mathcal L}$ be the language of ordered fields, enriched with symbols for these functions. By Corollary 3.15, $K$ is o-minimal in ${\mathcal L}$. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis.

The statement of the numerical values $\kappa$ and $z_0$ on page 167 of [1, Section 3] contained an error. The values that were actually used were (to nine decimal places): $$\begin{eqnarray*} \kappa &=& 2 \omega_1 -\int_1^\infty (4t^3-4t+1)^{- {1}/{2}}\,dt \\[4pt] &=&1.859\,185\,431, \\[6pt] z_0&=&2\omega_3+\int_{-1}^\infty (4t^3-4t+1)^{- {1}/{2}}\,dt \\[4pt] &=&0.204\,680\,500+1.225\,694\,691i, \end{eqnarray*}$$ these being shifted, by the periods $2\omega_1$ and $2\omega_3$ respectively, compared with the values given in [1] (with $\omega_1=1.496\,729\,323$ and $\omega_3=1.225\,694\,691i$). With $\tau_0=\tau_1 =1$ and $\sigma (z)$ denoting the sigma function $\sigma (z;g_2,g_3)$ with invariants $g_2=4$, $g_3=-1$ associated with the elliptic curve given by equation (3.2), these values of $\kappa$ and $z_0$ yield $$\begin{eqnarray*} \sigma (\kappa )&=&1.555\,836\,426, \\[4pt] A=\frac{\tau_0}{\si (z_0 )} &=& 0.112\,724\,016-0.824\,911\,686i, \\[4pt] B=\frac{\si (\ka )\si (z_0 )\,\tau_1} {\si (z_0+\ka )\,\tau_0}&=&0.215\,971\,963+0.616\,028\,193i. \end{eqnarray*}$$ and the latter three values all agree with those stated in the paper (apart from rounding down the last digit in the imaginary part of $A$).

In this paper we obtain the precise structure of right nonsingular semiperfect right CS-rings. It is shown that such rings are direct sums of certain specific block matrix rings.

Let $\idemp$ be the set of idempotents in a finite-dimensional real algebra $A$. Let $p$ and $q$ be idempotents that lie in the same component of $\idemp$. Then, among the continuous paths connecting $p$ and $q$ in $\idemp$, there exist a polynomial path of degree at most $3$ and a polygonal path consisting of at most three segments.

In this paper we compute the subgroup zeta functions of nilpotent groups of the form \[G_n := \langle x_1,\dots,x_{n},y_1,\dots,y_{n-1}|\;[x_i,x_n]=y_i,\;1\leq i \leq n-1\], all other [,] trivial 〉 and deduce local functional equations.

Examples are discussed of natural statements about irrational numbers that are equivalent, provably in ZFC, to strong set-theoretical hypotheses, and of apparently classical statements provable in ZFC of which the only known proofs use strong set-theoretical concepts.

let $e/\mathbb{q}$ be an elliptic curve. for a prime $p$ of good reduction, let $e(\mathbb{f}_p)$ be the set of rational points defined over the finite field $\mathbb{f}_p$. denote by $\omega(\#e(\mathbb{f}_p))$ the number of distinct prime divisors of $\#e(\mathbb{f}_p)$. for an elliptic curve with complex multiplication, the normal order of $\omega(\#e(\mathbb{f}_p))$ is shown to be $\log \log p$. the normal order of the number of distinct prime factors of the exponent of $e(\mathbb{f}_p)$ is also studied.

The special values of multiple polylogarithms, which include multiple zeta values, appear in several fields of mathematics and physics. Many kinds of their linear relations have been investigated, as well as their algebraic relations. In this paper, from the viewpoint of a connection matrix of Fuchsian equations, two kinds of duality of these values are derived.

Let δ be a coaction of a locally compact group G on a C*-algebra A. We show that if I is a δ-invariant ideal in A, then 0 → I×δIG → A×δG → (A/I)×δIG → 0 for full crossed products, as Landstad et al. have done for spatial crossed products by coactions. We prove that for suitable coactions, the crossed products of C0(X)-algebras are again C0(X)-algebras, and the crossed products of continuous C*-bundles by a locally compact group are again continuous C*-bundles.