The nonidealness index of circulant matrices

Ideal matrices are precisely those matrices M where the set covering polyhedron $Q^1(M)$ equals the polyhedron $Q(M)=\{x:Mx\ge 1,x\ge 0\}$. In a previous work (2006) we defined a nonidealness index equivalent to $\max \{t:Q(M)\subset t{Q}^1(M)\}$. Given an arbitrary matrix M the nonideal index is NP-hard to compute and for most matrices it remains unknown.A well known family of minimally nonideal matrices is the one of the incidence matrices of chordless odd cycles. A natural generalization of them is given by circulant matrices. Circulant ideal matrices have been completely identified by Cornuéjols and Novick (1994). In this work we obtain a bound for the nonidealness index of circulant matrices and determine it for some particular cases.     

On the singularity probability of discrete random matrices

Let n be a large integer and Mn be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this paper is to prove a general upper bound for the probability that Mn is singular. For a constant 0<p<1 and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of Mn satisfy this property, then the probability that Mn is singular is at most (p1/r+on(1)). All of the results in this paper hold for any characteristic zero integral domain replacing the complex numbers. In the special case where the entries of Mn are “fair coin flips” (taking the values +1,−1 each with probability 1/2), our general bound implies that the probability that Mn is singular is at most , improving on the previous best upper bound of , proved by Tao and Vu [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In the special case where the entries of Mn are “lazy coin flips” (taking values +1,−1 each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that Mn is singular is at most , which is asymptotically sharp. Our method is a refinement of those from [Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223-240; Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In particular, we make a critical use of the structure theorem from [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628], which was obtained using tools from additive combinatorics.     

A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces

Let s∈R, τ∈[0,∞), p∈(1,∞) and q∈(1,∞]. In this paper, we introduce a new class of function spaces which unify and generalize the Triebel-Lizorkin spaces with both p∈(1,∞) and p=∞ and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the relationship between Triebel-Lizorkin spaces and Q spaces, which answers a question posed by Dafni and Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and Qα(Rn), J. Funct. Anal. 208 (2004) 377-422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces and determine their dual spaces , where s∈R, p,q∈[1,∞), max{p,q}>1, , and t^′ denotes the conjugate index of t∈(1,∞); as an application of this, we further introduce certain Hardy-Hausdorff spaces and prove that the dual space of is just when p,q∈(1,∞).     

On locally compact quantum groups whose algebras are factors

In this paper we are interested in examples of locally compact quantum groups (M,Δ) such that both von Neumann algebras, M and the dual , are factors. There is a lot of known examples such that are respectively of type (I_∞,I_∞) but there is no example with factors of other types. We construct new examples of type (I_∞,II_∞), (II_∞,II_∞) and (IIIλ,IIIλ) for each λ∈[0,1]. We also show that there is no such example with M or a finite factor.     

Modules whose hereditary pretorsion classes are closed under products

A module M is called product closed if every hereditary pretorsion class in σ[M] is closed under products in σ[M]. Every module M which is locally of finite length (every finitely generated submodule of M has finite length) is product closed and every product closed module M is semilocal (M/J(M) is semisimple). Let be product closed and projective in σ[M]. It is shown that (1) M is semiartinian; (2) if M is finitely generated then M satisfies the DCC on fully invariant submodules; (3) M has finite length if M is finitely generated and every hereditary pretorsion class in σ[M] is M-dominated. If the ring R is commutative it is proven that M is product closed if and only if M is locally of finite length.     

Solutions of polynomial Pell's equation

Let D=F²+2G be a monic quartic polynomial in Z[x], where . Then for F/G∈Q[x], a necessary and sufficient condition for the solution of the polynomial Pell's equation X²−DY²=1 in Z[x] has been shown. Also, the polynomial Pell's equation X²−DY²=1 has nontrivial solutions X,Y∈Q[x] if and only if the values of period of the continued fraction of are 2, 4, 6, 8, 10, 14, 18, and 22 has been shown. In this paper, for the period of the continued fraction of is 4, we show that the polynomial Pell's equation has no nontrivial solutions X,Y∈Z[x].     

Forked Temperley-Lieb algebras and intermediate subfactors

We consider noncommuting pairs P, Q of intermediate subfactors of an irreducible, finite-index inclusion N⊂M of II₁ factors such that P and Q are supertransitive with Jones index less than 4 over N. We show that up to isomorphism of the standard invariant, there is a unique such pair corresponding to each even value but none for the odd values . We also classify the angle values which occur between pairs of intermediate subfactors with small index over their intersection: if [P:N],[Q:N]<4, then the unique nontrivial angle value is always .     

Primary decomposition II: Primary components and linear growth

We study the following properties about primary decomposition over a Noetherian ring R: (1) For finitely generated modules N⊆M and a given subset X={P₁,P₂,…,P_r}⊆Ass(M/N), we define an X-primary component of N?M to be an intersection Q₁∩Q₂∩?∩Q_r for some P_i-primary components Q_i of N⊆M and we study the maximal X-primary components of N⊆M; (2) We give a proof of the ‘linear growth’ property of Ext and Tor, which says that for finitely generated modules N and M, any fixed ideals I₁,I₂,…,I_t of R and any fixed integer i∈N, there exists a k∈N such that for any there exists a primary decomposition of 0 in (or 0 in ) such that every P-primary component Q of that primary decomposition contains (or ), where .     

Further analysis of the number of spanning trees in circulant graphs

Let 1?s₁<s₂<?<s_k?⌊n/2⌋ be given integers. An undirected even-valent circulant graph, has n vertices 0,1,2,…, n-1, and for each and j(0?j?n-1) there is an edge between j and . Let stand for the number of spanning trees of . For this special class of graphs, a general and most recent result, which is obtained in [Y.P. Zhang, X. Yong, M. Golin, [The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]], is that where a_n satisfies a linear recurrence relation of order 2^s_k-1. And, most recently, for odd-valent circulant graphs, a nice investigation on the number a_n is [X. Chen, Q. Lin, F. Zhang, The number of spanning trees in odd-valent circulant graphs, Discrete Math. 282 (2004) 69-79].In this paper, we explore further properties of the numbers a_n from their combinatorial structures. Comparing with the previous work, the differences are that (1) in finding the coefficients of recurrence formulas for a_n, we avoid solving a system of linear equations with exponential size, but instead, we give explicit formulas; (2) we find the asymptotic functions and therefore we ‘answer’ the open problem posed in the conclusion of [Y.P. Zhang, X. Yong, M. Golin, The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]. As examples, we describe our technique and the asymptotics of the numbers.     

Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means

Let s∈R. In this paper, the authors first establish the maximal function characterizations of the Besov-type space with and τ∈[0,∞), the Triebel-Lizorkin-type space with p∈(0,∞), q∈(0,∞] and τ∈[0,∞), the Besov-Hausdorff space with p∈(1,∞), q∈[1,∞) and and the Triebel-Lizorkin-Hausdorff space with and , where t^′ denotes the conjugate index of t∈[1,∞]. Using this characterization, the authors further obtain the local mean characterizations of these function spaces via functions satisfying the Tauberian condition and establish a Fourier multiplier theorem on these spaces. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking τ=0 and are also new even for Q spaces and Hardy-Hausdorff spaces.     

Representations for the Drazin inverses of 2 × 2 block matrices

Let M denote a 2 × 2 block complex matrix , where A and D are square matrices, not necessarily with the same orders. In this paper explicit representations for the Drazin inverse of M are presented under the condition that BDⁱC = 0 for i = 0, 1, … , n − 1, where n is the order of D.