Homogenization in open sets with holes

Let Q^r be a cylindrical bar with r cylindrical cavities having generators parallel to those of Q^r. Let Ω be the cross-section of the bar, Ω1 the cross-section of the domain occupied by the material and $\text{Ω}{}^{\mathrm{i}}\text{(i = 1,…, r)}$ the cross- sections of the cavities:

$\text{Ω}\text{?}{}^{\mathrm{i}}\text{? Ω}\text{Ω}\text{?}{}^{\mathrm{i}}\text{∩}\text{Ω}\text{?}{}^{\mathrm{k}}\text{= φ, i ≠ k}$

. The study of the elastic torsion of this bar leads to the following problem [see 2., 3., 267–320)]:

$\text{Δ?}{}_{\mathrm{r}}\text{+ 2μα = 0 in Ω1}$

$\text{?}{}_{\mathrm{r¦?\Omega }}\text{= 0}$

(1)

$\text{?}{}_{\mathrm{r}}\text{=}\text{constant on}\text{?Ω}{}^{\mathrm{i}}\text{; i = 1,…, r}$

where μ is the shear modulus of the material, α is the angle of twist and $\text{?}{}_{\mathrm{r}}$ represents the stress function. In this paper the problem (1) with an increasing number of holes which are distributed periodically is considered. One would like to know if $\text{?}{}_{\mathrm{r}}$ has a limit $\text{?}{}_{\mathrm{\infty }}\text{as}\text{r → + ∞}$, and if so, the equation satisfied by this limit. This is an “homogenization” problem — the heterogeneous bar Q^r is replaced by a homogeneous one, the response of which under torsion approximates as closely as possible that of Q^r. A more general problem will be studied and the case of elastic torsion will be obtained as an application. The proof uses the energy method [see Lions (Collège de France, 1975–1977), Tartar (Collège de France, 1977)] and extension theorems. A related problem is the homogenization of a perforated plate [cf. Duvaut (to appear)].     

A new result on the problem of Zarankiewicz

Zarankiewicz (Colloq. Math.2 (1951), 301) raised the following problem: Determine the least positive integer z(m, n, j, k) such that each 0–1-matrix with m rows and n columns containing z(m, n, j, k) ones has a submatrix with j rows and k columns consisting entirely of ones. This paper improves a result of Hylten-Cavallius (Colloq. Math.6 (1958), 59–65) who proved: $\text{[}\text{k}\text{2}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{?}\text{lim}{}_{\mathrm{n\to \infty }}\text{inf}\text{z(n, n, 2, k)n}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}\text{?}\text{lim}{}_{\mathrm{n\to \infty }}\text{sup}\text{z(n, n, 2, k)n}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}\text{? (k ? 1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. We prove that $\text{lim}{}_{\mathrm{n\to \infty }}\text{z(n, n, 2, k)n}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}$ exists and is equal to $\text{(k ? 1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. For the special case where k = 2 resp. k = 3 this result was proved earlier by Kövari, Sos and Turan (Colloq. Math.3 (1954), 50–57) resp. Hylten-Cavallius.     

Finite differences versus finite elements for solving nonlinear integro-differential equations

Consider the nonlinear integro-differential equation $\text{u}{}_{\mathrm{t}}\text{(x, t) = ∝}{}_0{}^{\mathrm{t}}\text{a(t?τ)}\text{?}\text{?x}\text{σ(u}{}_{\mathrm{x}}\text{(x, τ)) dτ + f(x, t)}$, 0 < x <, 0 < t < T, with appropriate initial and boundary conditions. This problem serves as a model for one-dimensional heat flow in materials with memory. The numerical solution via finite elements was discussed in B. Neta [J. Math. Anal. Appl.89 (1982), 598–611]. In this paper we compare the results obtained there with finite difference approximation from the point of view of accuracy and computer storage. It turns out that the finite difference method yields comparable results for the same mesh spacing using less computer storage.     

Positive definite distributions on K\GK

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.     

Critical exponents for the Pucci's extremal operatorsLes exposants critiques pour l'opérateur extrémal de Pucci

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation

(1)$\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}\text{(D}{}^2\text{u)+u}{}^{\mathrm{p}}\text{=0,u?0}\text{in}\text{R}{}^{\mathrm{N}}\text{.}$

Here N?3, p>1 and $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}$ denotes the Pucci's extremal operators with parameters 0<λ?Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents $\text{1<p}{}^{\mathrm{s}}{}_{\mathrm{+}}\text{<p}{}^1{}_{\mathrm{+}}\text{<p}{}^{\mathrm{p}}{}_{\mathrm{+}}$, that satisfy: (i) If $\text{1<p<p}{}^1{}_{\mathrm{+}}$ then there is no nontrivial solution of ($\text{1}$). (ii) If $\text{p=p}{}^1{}_{\mathrm{+}}$ then there is a unique fast decaying solution of ($\text{1}$). (iii) If $\text{p}{}^1\text{<p?p}{}^{\mathrm{p}}{}_{\mathrm{+}}$ then there is a unique pseudo-slow decaying solution to ($\text{1}$). (iv) If p^p₊<p then there is a unique slow decaying solution to ($\text{1}$). Similar results are obtained for the operator $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{?}}$. To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914.     

Restricted plane tree representations of four Motzkin-Catalan equations

Four equations are presented relating the well-known Catalan numbers and the Motzkin numbers m_n, defined by

$\text{m(x) = 1 + xm(x) + x}{}^2\text{m}{}^2\text{(x) =}\text{∑}\text{n=0}\text{m}{}_{\mathrm{n}}\text{x}{}^{\mathrm{n}}\text{,}$

first encountered in Th. Motzkin's paper (Bull. Amer. Math. Soc.54 (1948), 352–360) in a circle chording setting.In this paper five pairs of subsets of the plane trees provide natural settings for the four equations. In these settings the equations are immediate consequences of “natural correspondences” of the plane tree families.     

Some properties of irreducible coverings by cliques of complete multipartite graphs

In this paper some recursion formulas and asymptotic properties are derived for the numbers, denoted by N(p, q), of irreducible coverings by edges of the vertices of complete bipartite (labeled) graphs K_p,q. The problem of determining numbers N(p, q) has been raised by I. Tomescu (dans “Logique, Automatique, Informatique,” pp. 269–423, Ed. Acad. R.S.R., Bucharest, 1971). A result concerning the asymptotic behavior of the number of irreducible coverings by cliques of q-partite complete graphs is obtained and it is proved that , $\text{lim}{}_{\mathrm{n\to \infty }}\text{(}\text{log}\text{M(n))}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{= 3}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$, and $\text{lim}{}_{\mathrm{n\to \infty }}\text{C(n)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{(}\text{n}\text{ln}\text{n)}\text{=}\text{1}\text{e}$, where I(n) and M(n) are the maximal numbers of irreducible coverings, respectively, coverings by cliques of the vertices of an n-vertex graph, and C(n) is the maximal number of minimal colorings of an n-vertex graph. It is also shown that maximal number of irreducible coverings by n ? 2 cliques of the vertices of an n-vertex graph (n ≥ 4) is equal to 2^n?2 ? 2 and this number of coverings is attained only for K_2,n?2 and the value of $\text{lim}{}_{\mathrm{n\to \infty }}\text{I(n, n ? k)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}$ is obtained, where I(n, n ? k) denotes the maximal number of irreducible coverings of an n-vertex graph by n ? k cliques.     

Estimées Ck pour l'équation ∂̄b sur un convexe de type fini de Cn

Let q=1,…,n?1 and D be a bounded convex domain in $\text{C}{}^{\mathrm{n}}$ of finite type m. We construct two integral operators T_q and $\text{T}{}_{\mathrm{q}}$ such that for all $\text{p∈}\text{N}\text{,}\text{T}{}_{\mathrm{q}}\text{,}\text{T}{}_{\mathrm{q}}\text{:C}{}^{\mathrm{p}}{}_{\mathrm{0,q}}\text{(bD)→C}{}^{\mathrm{p+1/m}}{}_{\mathrm{0,q?1}}\text{(bD)}$ are continuous, and for all (0,q)-forms h continuous on bD with $\text{?}{}_{\mathrm{b}}\text{h}$ continuous on bD too, with the additional hypothesis when q=n?1 that ∫_bDh∧φ=0 for all φ∈C^∞_n,0(bD) $\text{?}\text{?}{}_{\mathrm{b}}$-fermée, we show $\text{h=}\text{?}\text{?}{}_{\mathrm{b}}\text{(T}{}_{\mathrm{q}}\text{?}\text{T}{}_{\mathrm{q}}\text{)h+(T}{}_{\mathrm{q+1}}\text{?}\text{T}{}_{\mathrm{q+1}}\text{)}\text{?}\text{?}{}_{\mathrm{b}}\text{h}$. For this construction, we use the Diederich–Fornæss support function of Alexandre (Publ. IRMA Lille 54 (III) (2001)). To prove the continuity of T_q, we integrate by parts and take care of the tangential derivatives. The normal component in z of the kernel of $\text{T}{}_{\mathrm{q}}$ will have a bad behaviour, so, in order to find a good representative of its equivalence class, we isolate the tangential component of the kernel and then integrate by parts again. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Uncertainty principle and Lp–Lq-sufficient pairs on noncompact real symmetric spaces

We consider a real semi-simple Lie group G with finite center and a maximal compact sub-group K of G. Let $\text{G=K}\text{exp}\text{(}\text{a}{}_{\mathrm{+}}\text{)K}$ be a Cartan decomposition of G. For x∈G denote ∥x∥ the norm of the $\text{a}{}_{\mathrm{+}}$-component of x in the Cartan decomposition of G. Let $\text{a>0,}\text{b>0}$ and 1?p,q?∞. In this Note we give necessary and sufficient conditions on $\text{a,}\text{b}$ such that for all K-bi-invariant measurable function f on G, if e^a∥x∥2f∈L^p(G) and then f=0 almost everywhere. To cite this article: S. Ben Farah, K. Mokni, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A short note about pursuit games played on a graph with a given genus

On a finite simple graph G = (X,E), p players pursuers $\text{A}{}_1\text{, ∴, A}{}_{\mathrm{p}}$ and one player evader B who move in turn along the edges of G are considered. The p pursuers $\text{A}{}_1\text{, ∴, A}{}_{\mathrm{p}}$ want to catch B who tries to escape. R. Nowakowski and P. Winkler [Discrete Math.43 (1983), 235–240] and A. Quilliot [“Thèse de 3° cycle,” pp. 131–145, Université de Paris VI, 1978] already characterized the graphs such that one pursuer is sufficient to catch the evader B. Very recently, M. Aigner and M. Fromme [Appl. Discrete Math., in press] proved that if G is a planar graph, three pursuers are sufficient to catch the evader B. This result is extended, showing that if G is a graph with a given genus k, then 3 + 2k pursuers are enough to “arrest” the evader B.     

Arithmetic properties of certain functions in several variables

If T is an n × n matrix with nonnegative integral entries, we define a transformation T: Cⁿ → Cⁿ by w = Tz where

$\text{W}{}_1\text{=}\text{∏}\text{j=1}\text{n}\text{z}{}_{\mathrm{j}}{}^{\mathrm{t}}\text{ij}\text{(1?i?n).}$

We consider functions f(z) of n complex variables which satisfy a functional equation giving f(Tz) as a rational function of ¹f(z) and we obtain conditions under which such a function f(z) takes transcendental values at algebraic points.     

Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

On tensor products of certain group C1-algebras

If A and B are C¹-algebras there is, in general, a multiplicity of C¹-norms on their algebraic tensor product A ⊙ B, including maximal and minimal norms ν and α, respectively. A is said to be nuclear if α and ν coincide, for arbitrary B. The earliest example, due to Takesaki [11], of a nonnuclear C¹-algebra was $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, the C¹-algebra generated by the left regular representation of the free group on two generators F₂. It is shown here that W¹-algebras, with the exception of certain finite type I's, are nonnuclear.If $\text{C}{}^1\text{(F}{}_2\text{)}$ is the group C¹-algebra of F₂, there is a canonical homomorphism λ_l of $\text{C}{}^1\text{(F}{}_2\text{)}$ onto $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$. The principal result of this paper is that there is a norm ζ on $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, distinct from α, relative to which the homomorphism $\text{λ ⊙ λ}{}_{\mathrm{l}}\text{: C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{) → C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$ is bounded ($\text{C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{)}$ being endowed with the norm α). Thus quotients do not, in general, respect the norm α; a consequence of this is that the set of ideals of the α-tensor product of C¹-algebras A and B may properly contain the set of product ideals {$\text{I ? B + A ? J: I ? A, J ? B}$}.Let A and B be C¹-algebras. If A or B is a W¹-algebra there are on A ⊙ B certain C¹-norms, defined recently by Effros and Lance [3], the definitions of which take account of normality. In the final section of the paper it is shown by example that these norms, with α and ν, can be mutually distinct.     

Recurrence relations based on minimization and maximization

Explicit and asymptotic solutions are presented to the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + min_{1 ? t ? n}(αM(t) + βM(n + 1 ? t)) for the cases (1) α + β < 1, $\text{log}{}_2\text{α}\text{log}{}_2\text{β}$ is rational, and g(n) = δ_nI. (2) α + β > 1, min(α, β) > 1, $\text{log}{}_2\text{α}\text{log}{}_2\text{β}$ is rational, and (a) g(n) = δ_n1, (b) g(n) = 1. The general form of this recurrence was studied extensively by Fredman and Knuth [J. Math. Anal. Appl.48 (1974), 534–559], who showed, without actually solving the recurrence, that in the above cases $\text{M(n) = Ω(n}{}^{\mathrm{<mtext>1\; +\; </mtext><mtext>1</mtext><mtext>\gamma </mtext>}}\text{)}$, where γ is defined by α^?γ + β^?γ = 1, and that $\text{lim}{}_{\mathrm{n\; \to \; \infty }}\text{M(n)}\text{n}{}^{\mathrm{1\; +\; \gamma }}$ does not exist. Using similar techniques, the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + max_{1 ? t ? n}(αM(t) + βM(n + 1 ? t)) is also investigated for the special case α = β < 1 and g(n) = 1 if n is odd = 0 if n is even.     

The nonorientable genus of the symmetric quadripartite graph

The nonorientable genus of K_4(n) is shown to satisfy:

$\text{γ}\text{(K}{}_{\mathrm{4(n)}}\text{)=2(n?1)}{}^2\text{for n ? 3}$

,

$\text{γ}\text{(K}{}_{\mathrm{4(2)}}\text{)=3,}\text{γ}\text{(K}{}_4\text{(1))=1}$

.     

K2 and K3 of the circle

Let k be $\text{Z}\text{[}\text{1}\text{2}\text{]}$, $\text{Q}$ or $\text{R}$, and set $\text{A =}\text{k[x,y]}\text{(x}{}^2\text{+ y}{}^2\text{? 1)}$. We compute K₂(A) and K₃(A). Our method is to construct a map $\text{? : K}{}_1\text{(k[i])→K}{}_{\mathrm{1\; +\; 1}}\text{(A)}$ and compare this to a localization sequence.We give three applications. We show that ? accounts for the primitive elements in K₂(A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec(A). Third, we can iterate the construction to compute the K-theory of the torus ring A ?_kA.     

Bounce law at the corners of convex billiards

Let C be a convex subset of $\text{R}{}^{\mathrm{n}}$. Given any elastic shock solution x(·) of the differential inclusion$\text{x}\text{̈}\text{(t)+N}{}_{\mathrm{C}}\text{(x(t))∋0,}\text{t>0,}$the bounce of the trajectory at a regular point of the boundary of C follows the Descartes law. The aim of the paper is to exhibit the bounce law at the corners of the boundary. For that purpose, we define a sequence (C_ε) of regular sets tending to C as ε→0, then we consider the approximate differential inclusion , and finally we pass to the limit when ε→0. For approximate sets defined by $\text{C}{}_{\mathrm{\epsilon }}\text{=C+ε}\text{B}$ (where $\text{B}$ is the unit euclidean ball of $\text{R}{}^{\mathrm{n}}$), we recover the bounce law associated with the Moreau–Yosida regularization.