Convergence and Stability of Locally ℝ N -Invariant Solutions of Ricci Flow

Valuable models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally $\mathcal{G}Valuable models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally G\mathcal{G} -invariant solutions on bundles G^N\hookrightarrowM \oversetp? Bⁿ\mathcal{G}^{N}\hookrightarrow\mathcal{M}\,\overset{\pi }{\mathcal{\longrightarrow}}\,\mathcal{B}^{n} , with G\mathcal{G} a nilpotent Lie group. In this paper, we establish convergence and asymptotic stability, modulo smooth finite-dimensional center manifolds, of certain ℝ ^N -invariant model solutions. In case N+n=3, our results are relevant to work of Lott classifying the asymptotic behavior of all 3-dimensional Ricci flow solutions whose sectional curvatures and diameters are respectively O(t^-1)\mathcal{O}(t^{-1}) and O(t^1/2)\mathcal{O}(t^{1/2}) as t→∞.     

Ordering graphs with cut edges by their spectral radii

Let Gnk denote a set of graphs with n vertices and k cut edges. In this paper, we obtain an order of the first four graphs in Gnk in terms of their spectral radii for 6 ≤ k ≤ (n-2)/3.     

The supersingular locus of the Shimura variety of GU(1,<Emphasis Type="Italic">n</Emphasis>−1) II

We complete the study of the supersingular locus M^ss\mathcal{M}^{\mathrm{ss}} in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ in the case that p is inert. This was started by the first author in Can. J. Math. 62, 668–720 (2010) where complete results were obtained for n=2,3. The supersingular locus M^ss\mathcal{M}^{\mathrm{ss}} is uniformized by a formal scheme N\mathcal{N} which is a moduli space of so-called unitary p-divisible groups. It depends on the choice of a unitary isocrystal N. We define a stratification of N\mathcal{N} indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of N. We show that the combinatorial behavior of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the irreducible components of N_red\mathcal{N}_{\mathrm{red}}) are identified with (generalized) Deligne-Lusztig varieties. We show that the Bruhat-Tits stratification is a refinement of the Ekedahl-Oort stratification and also relate the Ekedahl-Oort strata to Deligne-Lusztig varieties. We deduce that M^ss\mathcal{M}^{\mathrm{ss}} is locally a complete intersection, that its irreducible components and each Ekedahl-Oort stratum in every irreducible component is isomorphic to a Deligne-Lusztig variety, and give formulas for the number of irreducible components of every Ekedahl-Oort stratum of M^ss\mathcal{M}^{\mathrm{ss}}.     

On a bipartition problem of Bollobás and Scott

The bipartite density of a graph G is max {|E(H)|/|E(G)|: H is a bipartite subgraph of G}. It is NP-hard to determine the bipartite density of any triangle-free cubic graph. A biased maximum bipartite subgraph of a graph G is a bipartite subgraph of G with the maximum number of edges such that one of its partite sets is independent in G. Let $ \mathcal{H} $ \mathcal{H} denote the collection of all connected cubic graphs which have bipartite density $ \tfrac{4} {5} $ \tfrac{4} {5} and contain biased maximum bipartite subgraphs. Bollobás and Scott asked which cubic graphs belong to $ \mathcal{H} $ \mathcal{H} . This same problem was also proposed by Malle in 1982. We show that any graph in $ \mathcal{H} $ \mathcal{H} can be reduced, through a sequence of three types of operations, to a member of a well characterized class. As a consequence, we give an algorithm that decides whether a given graph G belongs to $ \mathcal{H} $ \mathcal{H} . Our algorithm runs in polynomial time, provided that G has a constant number of triangles that are not blocks of G and do not share edges with any other triangles in G.     

The Cycle Discrepancy of Three-Regular Graphs

Let G = (V, E) be an undirected graph and C(G){{\mathcal C}(G)} denote the set of all cycles in G. We introduce a graph invariant cycle discrepancy, which we define as

Von Neumann Algebras Associated with the Group SL2（R）

Let M\mathcal{M} and N\mathcal{N} be the von Neumann algebras induced by the rational action of the group SL ₂(ℝ) and its subgroup P on the upper half plane \mathbbH\mathbb{H}. We have shown that N\mathcal{N} is spatial isomorphic to the group von Neumann algebra L_P\mathcal{L}_P and characterized M\mathcal{M} and its commutant M￠\mathcal{M}' and gotten a generalization of the Mautner’s lemma. It is also shown that the Berezin operator commutates with the Laplacian operator.     

Bernoulli measure on strings,and Thompson-Higman monoids

The Bernoulli measure on strings is used to define height functions for the dense R\mathcal{ R}- and L\mathcal{ L}-orders of the Thompson-Higman monoids M _k,1. The measure can also be used to characterize the D\mathcal{ D}-relation of certain submonoids of M _k,1. The computational complexity of computing the Bernoulli measure of certain sets, and in particular, of computing the R\mathcal{ R}- and L\mathcal{ L}-height of an element of M _k,1 is investigated.     

Some properties for the largest component of random geometric graphs with applications in sensor networks

In this paper we consider the standard Poisson Boolean model of random geometric graphs G（Hλ,s; 1） in Rd and study the properties of the order of the largest component L1 （G（Hλ,s; 1）） . We prove that ElL1 （G（Hλ,s; 1））] is smooth with respect to A, and is derivable with respect to s. Also, we give the expression of these derivatives. These studies provide some new methods for the theory of the largest component of finite random geometric graphs （not asymptotic graphs as s - co） in the high dimensional space （d 〉 2）. Moreover, we investigate the convergence rate of E[L1（G（Hλ,s; 1））]. These results have significance for theory development of random geometric graphs and its practical application. Using our theories, we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks, which has the significance of theoretical foundation and guidance for the design of network layout.     

Pinning a Line by Balls or Ovaloids in ℝ<Superscript>3</Superscript>

We show that if a line ℓ is an isolated line transversal to a finite family F\mathcal{F} of (possibly intersecting) balls in ℝ³ and no two balls are externally tangent on ℓ, then there is a subfamily G í F\mathcal{G}\subseteq\mathcal{F} of size at most 12 such that ℓ is an isolated line transversal to G\mathcal{G}. We generalize this result to families of semialgebraic ovaloids.     

Parametric estimation with a class of M-estimators

Consider a class of M-estimators indexed by a criterion function ψ. When the function ψ is taken to be in a class of functions $ \mathcal{F} $ \mathcal{F} , a family of processes indexed by the class $ \mathcal{F} $ \mathcal{F} is obtained and called M-processes. Pooling the M-estimators in such class may be used to define new kind of estimators. In order to get the asymptotic properties of these pooled estimators, the convergence in probability of the corresponding M-process is studied uniformly on $ \mathcal{F} $ \mathcal{F} together with their weak convergence towards a Gaussian process. An application to location estimation is presented and discussed.     

Initial Enlargement of Filtrations and Entropy of Poisson Compensators

Let μ be a Poisson random measure, let \mathbbF\mathbb{F} be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let \mathbbG\mathbb{G} be the initial enlargement of \mathbbF\mathbb{F} with the σ-field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the σ-algebra generated by the Poisson random measure μ is equal to the expected relative entropy of the \mathbbG\mathbb{G}-compensator relative to the \mathbbF\mathbb{F}-compensator of the random measure μ. We then use this link to gain some insight into the changes of Doob–Meyer decompositions of stochastic processes when the filtration is enlarged from  \mathbbF\mathbb{F} to  \mathbbG\mathbb{G}. In particular, we show that if the mutual information between G and the σ-algebra generated by the Poisson random measure μ is finite, then every square-integrable \mathbbF\mathbb{F}-martingale is a \mathbbG\mathbb{G}-semimartingale that belongs to the normed space S¹\mathcal{S}^{1} relative to  \mathbbG\mathbb{G}.     

Criteria for the solubility of finite groups by their centralizers

For any group G, let C(G){\mathcal{C}(G)} denote the set of centralizers of G. We say that a group G has n centralizers (G is a C_n{\mathcal{C}_n}-group) if |C(G)| = n{|\mathcal{C}(G)| = n}. In this note, we show that the derived length of a soluble C_n{\mathcal{C}_n}-group (not necessarily finite) is bounded by a function of n.     

Regular Polytopes of Nearly Full Rank

An abstract regular polytope P\mathcal{P} of rank n can only be realized faithfully in Euclidean space \mathbbE^d\mathbb{E}^{d} of dimension d if d≥n when P\mathcal{P} is finite, or d≥n−1 when P\mathcal{P} is infinite (that is, P\mathcal{P} is an apeirotope). In case of equality, the realization P of P\mathcal{P} is said to be of full rank. If there is a faithful realization P of P\mathcal{P} of dimension d=n+1 or d=n (as P\mathcal {P} is finite or not), then P is said to be of nearly full rank. In previous papers, all the at most four-dimensional regular polytopes and apeirotopes of nearly full rank have been classified. This paper classifies the regular polytopes and apeirotopes of nearly full rank in all higher dimensions.     

Circular Favard Length of the Four-Corner Cantor Set

Let C_n\mathcal{C}_{n} be the n-th generation in the construction of the middle-half Cantor set. The Cartesian square K_n\mathcal{K}_{n} of C_n\mathcal{C}_{n} consists of 4 ⁿ squares of side-length 4⁻ⁿ . We drop a circle of radius r on the plane and try to estimate from below the conditional probability of this circle to intersect K_n\mathcal{K}_{n} if it already intersects a disc containing K_n\mathcal{K}_{n}. If the radius is very large  ≈4 ⁿ then clearly this should not differ too much from the usual Buffon needle probability. But it turns out that the best known lower bound (Bateman and Volberg in , 2008) persists even when the radius is much smaller than this—r>Cn ^ε suffices—and the intersection probability is at least \fracC_elognn\frac{C_{\varepsilon}\log n}{n}. This suggests that the method of Bateman and Volberg (, 2008) may be of use in proving a certain estimate for the lacunary circular maximal function from Seeger et al. (Preprint, 2005).     

On equivariant embedding of Hilbert <Emphasis Type="Italic">C</Emphasis>* modules

We prove that an arbitrary (not necessarily countably generated) Hilbert G - module on a G - C ^* algebra admits an equivariant embedding into a trivial G - module, provided G is a compact Lie group and its action on is ergodic.     

On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime

Given a finite family F\mathcal{F} of linear forms with integer coefficients, and a compact abelian group G, an F\mathcal{F}-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F\mathcal{F}. We denote by d_F(G)d_{\mathcal{F}}(G) the supremum of μ(A) over F\mathcal{F}-free sets A⊂G, where μ is the normalized Haar measure on G. Our main result is that, for any such collection F\mathcal{F} of forms in at least three variables, the sequence d_F(\mathbb Z_p)d_{\mathcal{F}}({\mathbb {Z}}_{p}) converges to d_F(\mathbb R/\mathbb Z)d_{\mathcal{F}}({\mathbb {R}}/{\mathbb {Z}}) as p→∞ over primes. This answers an analogue for ℤ _p of a question that Ruzsa raised about sets of integers.     

Arithmetic of semigroups of series in multiplicative systems

We study the arithmetic of a semigroup M_P\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?_{n = 0}^￥ a_nc_n(x)    ( a_n 3 0,?_{n = 0}^￥ a_n = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { c_n }_{n = 0}^￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup M_P\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R ⁿ are true. We describe the class I₀(M_P)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M_P\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.     

Distortio n theorems o n the Lie ball $$ \mathcal{R}_{IV} $$ ( n) i n ℂ n

In this paper, we introduce the subfamilies H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) of holomorphic mappings defined on the Lie ball $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n) which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m = 1 and m → +∞, respectively. Various distortion theorems for holomophic mappings H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are established. The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk. When m = 1 and m → +∞, the distortion theorems reduce to the results obtained by Gong for $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n), respectively. Moreover, our method is different. As an application, the bounds for Bloch constants of H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are given.