Toeplitz operators on connected domains"

The proof of the index formula of the Toeplitz operator with a continuous symbol on the Hardy space for the unit circle in the complex plane depends on the Hopf theorem. However, the analogue result of the Hopf theorem does not hold on a general connected domain. Hence, the extension of the index formula of the Toeplitz operator on a general domain needs a method which is different from that for the case of the unit circle. In the present paper, the index formula of the Toeplitz operator with a continuous symbol on the finite complex connected domain in the complex plane is obtained, and the cohomology groups of Toeplitz algebras on general domains are discussed. In addition, the Toeplitz operators with symbols in QC are also discussed.     

K-groups of Toeplitz algebras on connected domains

In the present paper, it is proved that the K0-group of a Toeplitz algebra on any connected domain is always isomorphic to the K0-group of the relative continuous function algebra. In addition, the cohomotopy groups of essential boundaries of some connected domains are computed, and the K0-groups of the continuous function algebras on these domains are also computed.     

Edge-decompositions of highly connected graphs into paths

We prove that a graph of edge-connectivity at least has an edge-decomposition into paths of length 4 if and only its size is divisible by 4. We also prove that a graph of girth >m and of edge-connectivity at least 8 ^m has an edge-decomposition into paths of length m provided its size is divisible by m, and m is a power of 2.      

Rationally connected varieties—On a conjecture of Mumford

We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.     

Pluriharmonic mappings and linearly connected domains in ℂ n

In this paper we obtain certain sufficient conditions for the univalence of pluriharmonic mappings defined in the unit ball \(\mathbb{B}^n \) of ? ⁿ . The results are generalizations of conditions of Chuaqui and Hernández that relate the univalence of planar harmonic mappings with linearly connected domains, and show how such domains can play a role in questions regarding injectivity in higher dimensions. In addition, we extend recent work of Hernández and Martín on a shear type construction for planar harmonic mappings, by adapting the concept of stable univalence to pluriharmonic mappings of the unit ball \(\mathbb{B}^n \) into ? ⁿ .     

The Carathéodory topology for multiply connected domains II

We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result which establishes several equivalent conditions for boundedness. This allows us to extend the notions of convergence and equicontinuity to families of functions defined on varying domains.     

Instanton Floer homology for connected sums of Poincaré spheres

In this paper, we apply the Mayer-Vietoris principle to compute the integer graded Floer homology of the connected sum of a Poincaré homology sphere with itself. We then apply the Fintushel-Stern spectral sequence to deduce its regular Floer homology and observe the presence of 3-torsion in the Floer groups.Partially supported by NSF grant number DMS-0245323.     

A mapping connected with the Schur–Szegő composition

Every monic polynomial in one variable of the form $(x+1)S$, $\mathrm{deg}S=n?1$, is presentable in a unique way as a Schur–Szeg? composition of $n?1$ polynomials of the form ${(x+1)}^{n?1}(x+a_i)$. We prove geometric properties of the affine mapping associating to the coefficients of S the $(n?1)$-tuple of values of the elementary symmetric functions of the numbers $a_i$. To cite this article: V.P. Kostov, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A class of hyperrings connected to preordered generalized Γ-rings

In this paper, we define the notion of preordered weak $\Gamma$-rings as a generalization of the concept of $\Gamma$-rings, the basic properties of prime ideals in weak $\Gamma$-rings are developed. At the end we construct a class of hyperrings in the sense of Rota from weak $\Gamma$-rings and also a connection between morphisms of the weak $\Gamma$-rings and the morphism of associated weak $\Gamma$-hyperrings has been investigated.     

Characterizations of connected orthogonality graphs of projections of Rickart ∗-rings

In this paper, we study the orthogonality graphs (see Definition 1.2) of ortholattices. We provide a graph theoretic condition for an ortholattice to be orthomodular. We prove that, the orthogonality graphs of two orthomodular lattices are isomorphic if and only if the lattices are isomorphic. As an application, it is proved that the zero-divisor graph of a Rickart ?-ring is obtained by successively duplicating the vertices of the orthogonality graph of the lattice of projections in the ring. We characterize the finite Rickart ?-rings for which the orthogonality graph of projections is connected.     

Thompson’s conjecture for simple groups with connected prime graph

We deal with finite simple groups G with the property π(G) ⊆ {2, 3, 5, 7, 11, 13, 17}, where π(G) is the set of all prime divisors of the order of a group G. The set of all such groups is denoted by ζ ₁₇. Thompson’s conjecture in [1, Question 12.38] is proved valid for all groups in ζ ₁₇ whose prime graph is connected.     

A theorem connected with results of Steinhaus and Smítal

We will prove that if A and B are subsets of the real line, each having positive outer Lebesgue measure, then A + B, the set of all numbers a + b with a ϵ A and b ϵ B, is “full,” in the sense of outer Lebesgue measure, in some interval K. This result is related to theorems of Steinhaus and Smítal.     

On Hardy’s uncertainty principle for connected nilpotent Lie groups

In (Kaniuth and Kumar in Math. Proc. Camb. Phil. Soc. 131, 487–494, 2001) Hardy’s uncertainty principle for was generalized to connected and simply connected nilpotent Lie groups. In this paper, we extend it further to connected nilpotent Lie groups with non-compact centre. Concerning the converse, we show that Hardy’s theorem fails for a connected nilpotent Lie group G which admits a square integrable irreducible representation and that this condition is necessary if the simply connected covering group of G satisfies the flat orbit condition.     

Proper holomorphic embeddings of finitely connected planar domains into ℂ n

In this paper we consider proper holomorphic embeddings of finitely connected planar domains into ? ⁿ that approximate given proper embeddings on smooth curves. As a side result we obtain a tool for approximating a $\mathcal{C}^{\infty}$ diffeomorphism on a polynomially convex set in ? ⁿ by an automorphism of ? ⁿ that satisfies some additional properties along a real embedded curve.     

Growth of generalized Temperley–Lieb algebras connected with simple graphs

We prove that the generalized Temperley–Lieb algebras associated with simple graphs Γ have linear growth if and only if the graph Γ coincides with one of the extended Dynkin graphs [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , or [(E)\tilde]₇ {\tilde E_7} . An algebra TL_{G, t} T{L_{\Gamma, \tau }} has exponential growth if and only if the graph Γ coincides with none of the graphs A_n {A_n} , D_n {D_n} , E_n {E_n} , [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , and [(E)\tilde]₇ {\tilde E_7} .     

When does the union of random spherical caps become connected?

DropN random caps all of the same angular radius on a unit sphere. LetU denote the part of the surface covered by these caps. We prove that if , then the probability thatU is connected tends to 1 asN→∞, while ifc<1, then the probability thatU is connected tends to 0 asN→∞.