On the number of partitions into parts not congruent to 0, $$\pm 3 \pmod {12}$$ ± 3 ( mod 12 )

Periodica Mathematica Hungarica - We consider the partitions of n into parts not congruent to 0, $$\pm 3\pmod {12}$$ and provide some new results related to the number of these partitions. In this...     

On the Size of a Cap in PG(n,q) with q Even and n ≥ 3

Let m2(n,q), m2(n,q) be, respectively, the maximum value, the second largest value of k for which there exists a complete k-cap in PG(n,q). In this paper, the known upper bound on m2(3,q), q even, q 8, is improved. This new upper bound on m2(3,q) is then used to improve the upper bounds on m2(n,q), q even, q 8 and n 4.     

On the stability of quasidouble step spline approximations for solutions ofy (n) = f(x,y, ⋯, y (n−1) )

In this paper, we continue the study [4] of the stability question of the quasidouble step spline function approximations,s(x) C ^m–2 , to the initial value problemy ⁽ⁿ⁾ = f(x, y, , y ^(n–1) ). It will be shown that the method is unstable and hence divergent form n + 4.     

On Embeddings of AG(tn,k) into AG(t,K) with t≥2, n≥1

Let k and K be commutative fields with dim_kK=2n, n1 and char(k)2, 3. If k satisfies one of the following conditions: (1) k is a finite field and k contains a 3-th root of unity, or (2) K is a cyclic extension of k where k is not 3-closed, and k contains a 2^a3^b-th root of unity, where 6n=2^a3^bc and c is coprime to 2, 3, then there exists an embedding of the tn-dimensional affine space AG(tn,k) into the t-dimensional affine space AG(t,K), for all t2.     

On the unique finite (SI) decomposition of the Jordan operator ⊕ i=1 n S(θ) for certain inner function θ

In this paper, we have proven that for the Jordan blockS() withS() (SI), _i=1 ⁿ S() =S() ⁽ⁿ⁾ (n 1) has unique finite (SI) decomposition up to a similarity. As result, we obtain that ifV is a Volterra operator onH=L ²([0, 1]), thenV ⁽ⁿ⁾ has unique finite (SI) decomposition.This project was supported by National Natural Science Foundation of China.     

On a greedy algorithm in L 1(0, 1) with regard to subsystems of the Haar system and on ω-quasigreedy bases

All quasigreedy subsystems of the Haar system in L ¹(0, 1) are described. The problem of renormalizing the Haar system so that it becomes a quasigreedy basis is also studied.     

On q-analogs of weight multiplicities for the Lie superalgebras \mathfrak{gl}(n,m) and \mathfrak{spo(}2n,M)

The paper is devoted to the generalization of Lusztig’s q-analog of weight multiplicities to the Lie superalgebras $\mathfrak{gl}(n,m)$ and $\mathfrak{spo(}2n,M).$ We define such q-analogs K _λ,μ (q) for the typical modules and for the irreducible covariant tensor $\mathfrak{gl}(n,m)$ -modules of highest weight λ. For $\mathfrak{gl}(n,m),$ the defined polynomials have nonnegative integer coefficients if the weight μ is dominant. For $\mathfrak{spo(}2n,M)$ , we show that the positivity property holds when μ is dominant and sufficiently far from a specific wall of the fundamental chamber. We also establish that the q-analog associated to an irreducible covariant tensor $\mathfrak{gl}(n,m)$ -module of highest weight λ and a dominant weight μ is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape λ and weight μ. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schützenberger.     

A bijective proof of Amdeberhan’s conjecture on the number of (s,s+2)-core partitions with distinct parts

Amdeberhan conjectured that the number of $(s,s+2)$-core partitions with distinct parts for an odd integer $s$ is $2^{s?1}$. This conjecture was first proved by Yan, Qin, Jin and Zhou, then subsequently by Zaleski and Zeilberger. Since the formula for the number of such core partitions is so simple one can hope for a bijective proof. We give the first direct bijective proof of this fact by establishing a bijection between the set of $(s,s+2)$-core partitions with distinct parts and a set of lattice paths.