CR-Admissible $$\mathbb{{Z}}_{2}$$-gradations and CR-symmetries

Complementing the results of (Lotta and Nacinovich, Adv. Math. 191(1): 114–146, 2005), we show that the minimal orbit M of a real form G of a semisimple complex Lie group in a flag manifold is CR-symmetric (see (Kaup and Zaitsev Adv. Math. 149(2):145–181, 2000)) if and only if the corresponding CR algebra admits a gradation compatible with the CR structure.      

Orbits and Invariants Associated with a Pair of Spherical Varieties: Some Examples

Let H and K be spherical subgroups of a reductive complex group G. In many cases, detailed knowledge of the double coset space H\G/K is of fundamental importance in group theory and representation theory. If H or K is parabolic, then H\G/K is finite, and we recall the classification of the double cosets in several important cases. If H=K is a symmetric subgroup of G, then the double coset space K\G/K (and the corresponding invariant theoretic quotient) are no longer finite, but several nice properties hold, including an analogue of the Chevalley restriction theorem. These properties were generalized by Helminck and Schwarz (Duke Math. J. 106(2) (2001), pp. 237–279) to the case where H and K are fixed point groups of commuting involutions. We recall Helminck and Schwarz's main results. We also give examples to show the difficulty in extending these results if we allow H=K to be a reductive spherical (nonsymmetric) subgroup or if we have H symmetric and K spherical reductive.     

Counting SL2（F2^s） Representations of Torus Knot Groups

In this paper,we count the number of SL2(F2^s)-representations of torus knot groups up to a conjugacy.For the finite field F2^s with character 2,the counting method is similar to that in out previous work[1].Explicit formulae of the effective counting are given in this paper.Twisted Alexander polynomials related to those reprsentations are discussed.     

2-极大子群的$F_s$-拟正规性

$F$是一个群系. $G$的子群$H$在$G$中称为$F_s$-拟正规的,如果存在$G$的正规子群$T$,使得$HT$在$G$中是$s$-置换的并且$(H\cap T)H_G/H_G$包含在$G/H_G$的$F$超中心$Z^F_\infty(G/H_G)$中.本文利用$F_s$-拟正规子群研究了有限群的结构.获得了某些新的结果.     

Principal Congruence Subgroups of Hecke Groups H（√q）

Using the notion of quadratic reciprocity, we discuss the principal congruence subgroups of the Hecke groups H（√q）,q 〉 5 prime number.     

The estimate for mean values on prime numbers relative to \frac{4} {p} = \frac{1} {{n_1 }} + \frac{1} {{n_2 }} + \frac{1} {{n_3 }}

If n is a positive integer,let f （n） denote the number of positive integer solutions （n 1,n 2,n 3） of the Diophantine equation 4/n=1/n₁ + 1/n₂ + 1/n₃.For the prime number p,f （p） can be split into f 1 （p） + f 2 （p）,where f i （p） （i=1,2） counts those solutions with exactly i of denominators n 1,n 2,n 3 divisible by p.In this paper,we shall study the estimate for mean values ∑ p相似文献   

Unbounded solutions of the max-type difference equation x_{n + 1} = max\left\{ {\frac{{A_n }} {{X_n }},\frac{{B_n }} {{X_{n - 2} }}} \right\}

We investigate the boundedness nature of positive solutions of the difference equation $$ x_{n + 1} = max\left\{ {\frac{{A_n }} {{X_n }},\frac{{B_n }} {{X_{n - 2} }}} \right\},n = 0,1,..., $$ where {A _n } _n=0 ^∞ and {B _n } _n=0 ^∞ are periodic sequences of positive real numbers.     

Remarks on the differential equation\frac{{dy}}{{dx}} = \frac{{y(y + x + 2)}}{{x(2x - y)}}

Zusammenfassung Es wird eine Übersicht über den Verlauf der Integralkurven einer in der Grenzschichttheorie auftretenden Differentialgleichung gegeben. Die Lösungen wurden mittels eines Analogrechners berechnet.     

Neighbor sum distinguishing colorings of graphs with maximum average degree less than $$\tfrac{{37}} {{12}}$$

Let G be a graph and let its maximum degree and maximum average degree be denoted by Δ(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by χ′_∑(G). Flandrin et al. proposed the following conjecture that χ′_∑ (G) ≤ Δ(G) + 2 for any connected graph with at least 3 vertices and G ≠ C₅. In this paper, we prove that the conjecture holds for a normal graph with mad(G) < \(\tfrac{{37}}{{12}}\) and Δ(G) ≥ 7.     

Similarity of $$sgn x( - \tfrac{{d^2 }}{{dx^2 }} + c\delta )$$ Type Operators to Normal and Self-adjoint Operators

Mathematical Notes -     

On the Evaluation of the Integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$

The integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$ where m and n are relatively prime positive integers, is evaluated exactly in terms of elementary functions and the Catalan constant G.     

不动点集为RP(4)∪ P(4,2n-1)的对合

设(M,T)是一个光滑闭流形上的对合,不动点集为F=RP(4)UP(4,2n-1),则它的每一个对合(M,T)必协边(RP(4)×RP(4),twist)和(P(4,2n),T')之一.     

Alternating groups and flag‐transitive 2‐(v,k, 4) symmetric designs

In this article, we study the classification of flag‐transitive, point‐primitive 2‐ (v, k, 4) symmetric designs. We prove that if the socle of the automorphism group G of a flag‐transitive, point‐primitive nontrivial 2‐ (v, k, 4) symmetric design ?? is an alternating group A_n for n≥5, then (v, k) = (15, 8) and ?? is one of the following: (i) The points of ?? are those of the projective space PG(3, 2) and the blocks are the complements of the planes of PG(3, 2), G = A₇ or A₈, and the stabilizer G_x of a point x of ?? is L₃(2) or AGL₃(2), respectively. (ii) The points of ?? are the edges of the complete graph K₆ and the blocks are the complete bipartite subgraphs K_{2, 4} of K₆, G = A₆ or S₆, and G_x = S₄ or S₄ × Z₂, respectively. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:475‐483, 2011     

Isomorphy classes of finite order automorphisms of SL(2,k)

In this paper, we consider the order m k-automorphisms of SL(2,k). We first characterize the forms that order m k-automorphisms of SL(2,k) take and then we find simple conditions on matrices A and B, involving eigenvalues and the field that the entries of A and B lie in, that are equivalent to isomorphy between the order m k-automorphisms Inn_A and Inn_B. We examine the number of isomorphy classes and conclude with examples for selected fields.