Critical exponents and blow-up rate for a nonlinear diffusion equation with logarithmic boundary flux

In this paper, we establish the critical global existence exponent and the critical Fujita exponent for the nonlinear diffusion equation u_t=(log^σ(1+u)u_x)_x, in R₊×(0,+∞), subject to a logarithmic boundary flux , furthermore give the blow-up rate for the nonglobal solutions.     

Ropelengths of closed braids

For a link K, let L(K) denote the ropelength of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the bound on L(K) in terms of Cr(K). It is well known that there exist positive constants c₁, c₂ such that for any link K, c₁⋅(Cr(K))^3/4?L(K)?c₂⋅(Cr(K))^3/2. In this paper, we show that any closed braid with n crossings can be realized by a unit thickness rope of length at most of the order O(n6/5). Thus, if a link K admits a closed braid representation in which the number of crossings is bounded by a(Cr(K)) for some constant a?1, then we have L(K)?c⋅(Cr(K))^6/5 for some constant c>0 which only depends on a. In particular, this holds for any link that admits a reduced alternating closed braid representation, or any link K that admits a regular projection in which there are at most O(Cr(K)) crossings and Seifert circles.     

Almost complex structure and the quotient four-manifold by an anti-symplectic involution

Suppose that X is a closed, symplectic four-manifold with an anti-symplectic involution σ and its two-dimensional fixed point set. We show that the quotient X/σ admits no almost complex structure if .As a partial converse if X is simply-connected and , then the X/σ admits an almost complex structure.Also we show that the quotient X/σ admits an almost complex structure if X is Kähler and .     

Existence of least area planes in hyperbolic 3-space with co-compact metric

Let r be a metric on the hyperbolic 3-space induced from an arbitrary Riemannian metric on a closed hyperbolic 3-manifold. In this paper, we will show that any smooth simple loop in S_∞² spans a properly embedded r-least area plane in . This solves Gabai's conjecture ((J. Amer. Math. Soc. 10 (1997) 37), Conjecture 3.12), affirmatively.     

Spin geometry on quantum groups via covariant differential calculi

Let be a cosemisimple Hopf ∗-algebra with antipode S and let Γ be a left-covariant first-order differential ∗-calculus over such that Γ is self-dual (see Section 2) and invariant under the Hopf algebra automorphism S². A quantum Clifford algebra Cl(Γ,σ,g) is introduced which acts on Woronowicz’ external algebra Γ^∧. A minimal left ideal of Cl(Γ,σ,g) which is an -bimodule is called a spinor module. Metrics on spinor modules are investigated. The usual notion of a linear left connection on Γ is extended to quantum Clifford algebras and also to spinor modules. The corresponding Dirac operator and connection Laplacian are defined. For the quantum group SL_q(2) and its bicovariant 4D_±-calculi these concepts are studied in detail. A generalization of Bochner's theorem is given. All invariant differential operators over a given spinor module are determined. The eigenvalues of the Dirac operator are computed.     

Filtered ends of infinite covers and groups

Let f:A→B be a covering map. We say that A has e filtered ends with respect to f (or B) if, for some filtration {K_n} of B by compact subsets, A−f⁻¹(K_n) “eventually” has e components. The main theorem states that if Y is a (suitable) free H-space, if K<H has infinite index, and if Y has a positive finite number of filtered ends with respect to H?Y, then Y has one filtered end with respect to K?Y. This implies that if G is a finitely generated group and K<H<G are subgroups each having infinite index in the next, then implies that , where is the number of filtered ends of a pair of groups in the sense of Kropholler and Roller.     

Seifert fibered surgeries with distinct primitive/Seifert positions

We call a pair (K,m) of a knot K in the 3-sphere S³ and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K,m), K can be embedded in a genus 2 Heegaard surface of S³ in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions from a different point of view.     

Orders and actions of branched coverings of hyperbolic links

Let M, M^′ be compact oriented 3-manifolds and L^′ a link in M^′ whose exterior has positive Gromov norm. We prove that the topological types of M and (M^′,L^′) determine the degree of a strongly cyclic covering branched over L^′. Moreover, if M^′ is a homology sphere then these topological types determine also the covering up to conjugacy.     

Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets

Let e and n be positive integers and S={x₁,…,x_n} a set of n distinct positive integers. For x∈S, define . The n×n matrix whose (i,j)-entry is the eth power (x_i,x_j)^e of the greatest common divisor of x_i and x_j is called the eth power GCD matrix on S, denoted by (S^e). Similarly we can define the eth power LCM matrix [S^e]. Bourque and Ligh showed that (S)∣[S] holds in the ring of n×n matrices over the integers if S is factor closed. Hong showed that for any gcd-closed set S with |S|≤3, (S)∣[S]. Meanwhile Hong proved that there is a gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S)?[S]. In this paper, we introduce a new method to study systematically the divisibility for the case max_x∈S{|G_S(x)|}≤2. We give a new proof of Hong’s conjecture and obtain necessary and sufficient conditions on the gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S^e)|[S^e]. This partially solves an open question raised by Hong. Furthermore, we show that such factorization holds if S is a gcd-closed set such that each element is a prime power or the product of two distinct primes, and in particular if S is a gcd-closed set with every element less than 12.     

Acceptance inspection by variables when the measurements are subject to error

The situation considered is that in which measurement of the characteristic of interest is not exact but subject to appreciable error. The error is assumed to be unbiased and independent of the actual value of the characteristic measured. The population and error variances,σ ² andσ _e ² , are assumed to be such thatσ/σ _e has a known lower limit which is greater than zero. The probability distributions involved are assumed to be normal while the actual values and measurement errors each form a random sample. For suitable specified acceptable and unacceptable fractions defective, and forσ _e assumed known and unknown, this paper presents one-sided acceptance inspection criteria which are optimum in a specified sense, and which have the property that the producer’s and consumer’s risks have specified upper bounds.     

Diameters of 3-sphere quotients

Let G⊂O(4) act isometrically on S³. In this article we calculate a lower bound for the diameter of the quotient spaces S³/G. We find it to be , which is exactly the value of the lower bound for diameters of the spherical space forms. In the process, we are also able to find a lower bound for diameters for the spherical Aleksandrov spaces, Sn/G, of cohomogeneities 1 and 2, as well as for cohomogeneity 3 (with some restrictions on the group type). This leads us to conjecture that the diameter of Sn/G is increasing as the cohomogeneity of the group G increases.     

The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A₁,…,A_k∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U^∗AU:Uunitary} always lie in S.