Almost resolvable decomposition of Kn1

In this paper, we show that the complete symmetric directed graph with n vertices $\text{K}{}_{\mathrm{n}}{}^1$ admits an almost resolvable decomposition into TT₃ (the transitive tournament on 3 vertices) or C₃ (the directed cycle of length 3) if and only if n ≡ 1(mod 3).     

A Hamiltonian decomposition of K2m1, 2m ≥ 8

It is shown that $\text{K}{}_{\mathrm{2m}}{}^1$, 2m ≥ 8, can be decomposed into Hamiltonian circuits. A direct construction utilizing difference methods is given for 2m ≡ 0 (mod 4). The case 2m ≡ 2 (mod 4) is handled inductively by means of a construction which shows that $\text{K}{}_{\mathrm{4m\; ?\; 2}}{}^1$ admits such a decomposition if $\text{K}{}_{\mathrm{2m}}{}^1$ does.     

Lifting commuting pairs of C1 algebras

Given a commuting pair $\text{A}$₁, $\text{A}$₂ of abelian $\text{C}{}^1$ subalgebras of the Calkin algebra, we look for a commuting pair $\text{B}$₁,$\text{B}$₂ of $\text{C}{}^1$ subalgebras of $\text{B(H)}$ which project onto $\text{A}$₁ and $\text{A}$₂. We do not insist that $\text{B}$_i, be abelian, so $\text{B}$_i, may contain nontrivial compact operators. If X is the joint spectrum σ($\text{A}$₁, $\text{A}$₂), it is shown that the existence of a pair $\text{B}$₁, $\text{B}$₂ depends only on the element τ in Ext(X) determined by $\text{A}$₁, $\text{A}$₂. The set L(X) of those τ in Ext(X) which “lift” in this sense is shown to be a subgroup of Ext(X) when Ext(X) is Hausdorff, and also when $\text{A}$_i are singly generated. In this latter case, L(X) can be explicitly calculated for large classes of joint spectra. These results are applied to lift certain pairs of commuting elements of the Calkin algebra to pairs of commuting operators.     

C1-algebras without idempotents

The simplest statement of the main results are these: Let π be a free group on 2 generators. Let Cπ be the complex ring and L¹π the ring extension to L¹ sums. Then L¹π contains no idempotents. Furthermore, if α ? Cπ, β?L¹π are nonzero, then αβ ≠ 0. The first result is in the direction of proving that a certain simple $\text{C}{}^1\text{-}\text{algebra}$ has no idempotents yielding a counter-example to the suggestion that simple $\text{C}{}^1\text{-}\text{algebras}$ are generated by their projections.     

Nonlinear boundary value problems and operators TT1

We consider nonlinear boundary value problems of the type $\text{L? + N? = 0}$ for the existence of solutions. It is assumed that L is a 2nth-order linear differential operator in the real Hilbert space S = L²[a, b] which admits a decomposition of the form $\text{L =}{}^{\mathrm{TT1}}$ where T is an nth-order linear differential operator and N is a nonlinear operator defined on a subspace of S. The decomposition of L induces a natural decomposition of the generalized inverse of L. Using the method of “alternative problems,” we split the boundary value problem into an equivalent system of two equations. The theory of monotone operators and the theory of nonlinear Hammerstein equations are then utilized to consider the solvability of the equivalent system.     

A Dunford-Pettis theorem for L1H∞⊥

The spaces in the title are associated to a fixed representing measure m for a fixed character on a uniform algebra. It is proved that the set of representing measures for that character which are absolutely continuous with respect to m is weakly relatively compact if and only if each m-negligible closed set in the maximal ideal space of L^∞ is contained in an m-negligible peak set for H^∞. J. Chaumat's characterization of weakly relatively compact subsets in $\text{L}{}^1\text{H}{}^{\mathrm{\infty \perp }}$ therefore remains true, and $\text{L}{}^1\text{H}{}^{\mathrm{\infty \perp }}$ is complete, under the first conditions. In this paper we also give a direct proof. From this we obtain that $\text{L}{}^1\text{H}{}^{\mathrm{\infty \perp }}$ has the Dunford-Pettis property.     

Extensions of factorial states of C1-algebras

Let A be a $\text{C}{}^1$-algebra, B be a $\text{C}{}^1$-subalgebra of A, and φ be a factorial state of B. Sometimes, φ may be extended to a factorial state of A by a tensor product method of Sakai (“$\text{C}{}^1$-algebras and $\text{W}{}^1$-algebras, Springer-Verlag, Berlin/Heidelberg/ New York 1971”). Sometimes, there is a weak expectation of A into $\text{π}{}_{\mathrm{\phi }}\text{(B)}$, and then factorial extensions may be found by a method of Sakai and Tsui (Yokohama Math. J.29 (1981), 157–160). These two methods are shown to have the same effect, and the factorial extensions produced by them are analysed.     

On the expansion of Cϱ1(V + I) as a sum of zonal polynomials

The coefficients a_τ?, sometimes called “generalized binomial coefficients” in the expansion $\text{C}{}_{\mathrm{?}}{}^1\text{(V +I) = Σ}{}_{\mathrm{\tau }}\text{a}{}_{\mathrm{?\tau }}\text{C}{}_{\mathrm{\tau }}{}^1\text{(V)}$, are computed explicitly when t = r + 1, where ? is a partition of r and τ a partition of t. A recursion formula permits the calculation of the general a_τ?. Several properties of a_τ? are proved. A connection between the a_τ? and other coefficients is established. The main tools used are Bingham's identity, results from the theory of invariant differential operators, and a lemma concerning zonal polynomials.     

Decomposition of Km,n(Km,n1) into cycles (circuits) of length 2k

In this paper we give necessary and sufficient conditions in order that K_m,n ($\text{K}{}_{\mathrm{m,n}}{}^1$) admits a decomposition into 2k-cycles (2k-circuits). This answers conjectures of J. C. Bermond (Thesis, Paris XI (Orsay), 1975) and J. C. Bermond and V. Faber (J. Combinatorial Theory Ser. B21 (1976), 146–155).     

Santaló's inequality on Cn by complex interpolationInégalité de Santaló sur Cn par interpolation complexe

A new approach to Santaló's inequality on $\text{C}{}^{\mathrm{n}}$ is obtained by combining complex interpolation and Berndtsson's generalization of Prékopa's inequality. To cite this article: D. Cordero-Erausquin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 767–772.     

Convex polyhedra of doubly stochastic matrices: II. Graph of Ωn

Properties of the graph $\text{G(Ω}{}_{\mathrm{n}}\text{)}$ of the polytope $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices are studied. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ which is not a k-dimensional rectangular parallelotope for k ≥ 2, then G($\text{F}$) is Hamilton connected. Prime factor decompositions of the graphs of faces of $\text{Ω}{}_{\mathrm{n}}$ relative to Cartesian product are investigated. In particular, if $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$, then the number of prime graphs in any prime factor decomposition of G($\text{F}$) equals the number of connected components of the neighborhood of any vertex of G($\text{F}$). Distance properties of the graphs of faces of $\text{Ω}{}_{\mathrm{n}}$ are obtained. Faces $\text{F}$ of $\text{Ω}{}_{\mathrm{n}}$ for which G($\text{F}$) is a clique of $\text{G(Ω}{}_{\mathrm{n}}\text{)}$ are investigated.     

The covering problem for the fields Q(−1)12 and Q(−3)12

Let k be a positive square free integer, $\text{N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ the ring of algebraic integers in $\text{Q(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and S the unit sphere in C_n, complex n-space. If A₁,…, A_n are n linearly independent points of C_n then L = {u₁Au + … + u_nA_n} with $\text{u}{}_{\mathrm{r}}\text{∈ N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is called a k-lattice. The determinant of L is denoted by d(L). If L is a covering lattice for S, then $\text{θ(S, L) =}\text{V(S)}\text{d(L)}$ is the covering density. L is called locally (absolutely) extreme if θ(S, L) is a local (absolute) minimum. In this paper we determine unique classes of extreme lattices for k = 1 and k = 3.     

Convex polyhedra of doubly stochastic matrices III. Affine and combinatorial properties of Ωn

Affine and combinatorial properties of the polytope $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices are investigated. One consequence of this investigation is that if $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ of dimension d > 2, then $\text{F}$ has at most 3(d?1) facets. The special faces of $\text{Ω}{}_{\mathrm{n}}$ which were characterized in Part I of our study of $\text{Ω}{}_{\mathrm{n}}$ in terms of the corresponding (0, 1)- matrices are classified with respect to affine equivalence.     

Embedding EnG in Euclidean space

It is shown that if G is an arbitrary upper semicontinuous decomposition of Eⁿ for which π(N_G embeds in S^m for some m?3, then the decomposition space $\text{E}{}^{\mathrm{n}}\text{G}$ embeds as a closed subset of E^n+m+1. The proof consists of constructing a cell-like upper semicontinuous decomposition G? of E^n+m+1 which intersects Eⁿ to yield precisely G and using Edwards' Cell-Like Approximation Theorem to show that G? is shrinkable. As an immediate corollary, $\text{E}{}^{\mathrm{n}}\text{G}$ embeds in E^n+2k+2 whenever G is an arbitrary k-dimensional upper semicontinuous decomposition of Eⁿ. This is an improvement of (n?1)-dimensions over the corresponding dimension theoretic result and examples due to Daverman show that this result is sharp in case n is odd and off by no more than one dimension in case n is even.     

On some properties of convolution operators in K′1 and S′

Let $\text{H}$′ be either the space $\text{K}$′₁ of distributions of exponential growth or the space $\text{S}$′ of tempered distributions, and let $\text{O}$′_C($\text{H}$′:$\text{H}$′) be the space of convolution operators in $\text{H}$′. In each case $\text{H}$′ is the dual of a space $\text{H}$ of C^∞-functions which are in $\text{O}$′_C($\text{H}$′:$\text{H}$′). We establish necessary and sufficient conditions on the Fourier transform $\text{S}\text{?}$ of ? of S ? $\text{O}$′_C($\text{H}$′:$\text{H}$′) in order that every distribution u? $\text{O}$′_C($\text{H}$′:$\text{H}$′) with S1u?$\text{H}$ be in $\text{H}$. If $\text{H}$′ = $\text{K}$′₁, the condition is equivalent to S×H′¹=H′¹.     

Drazin inverses and canonical forms in Mn(Zh)

The existence and construction of the Drazin inverse of a square matrix over the ring $\text{Z}\text{h}$ is considered. The canonical forms of matrices over this ring are used to facilitate the computation of this type of generalized inverse.