On the Matrix Equation X^{ - 1} X* = A

Solvability conditions are examined for the matrix equation , which cannot be found in the well-known reference books on matrix theory. Methods for constructing solutions to this equation are indicated.     

On \mathfrak{F}-reachable subgroups of finite groups

The structure of $\mathfrak{F}$ -reachable subgroups in Θ-Frattini extensions is established.     

Nilpotent p\mspace{1mu}-local finite groups

We provide characterizations of \(p\mspace {1mu}\) -nilpotency for fusion systems and \(p\mspace {1mu}\) -local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.     

On the {H^1}-stability of the {L_2}-projection onto finite element spaces

We study the stability in the $H^1$ -seminorm of the $L_2$ -projection onto finite element spaces in the case of nonuniform but shape regular meshes in two and three dimensions and prove, in particular, stability for conforming triangular elements up to order twelve and conforming tetrahedral elements up to order seven.     

On finite {s-2, s}-semiaffine linear spaces

In this paper finite {s-2, s}-semiaffine linear spaces of order n are studied. It is proved that if s= 6 or then there is only a finite number of such linear spaces. Received 28 May 1999; revised 28 December 1999.     

On the Recursive Sequence x_{n + 1} = {{ \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {Cx_{n-1} + Dx_{n-2}}}

We investigate the global character of solutions of the equation in the title with positive parameters and positive initial conditions. We obtain results about the global attractivity of the equilibrium, the existence and attractivity of the period-two solution and the semicycles.     

Iterative algorithm for solving a class of general Sylvester-conjugate matrix equation \sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C

This paper is concerned with iterative solution to general Sylvester-conjugate matrix equation of the form $\sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C$ . An iterative algorithm is established to solve this matrix equation. When this matrix equation is consistent, for any initial matrices, the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.     

On $$ \mathfrak{F}_h $$-normal subgroups of finite groups

Let G be a finite group, and let $ \mathfrak{F} $ \mathfrak{F} be a formation of finite groups. We say that a subgroup H of G is $ \mathfrak{F}_h $ \mathfrak{F}_h -normal in G if there exists a normal subgroup T of G such that HT is a normal Hall subgroup of G and (H ∩ T)H _G /H _G is contained in the $ \mathfrak{F} $ \mathfrak{F} -hypercenter $ Z_\infty ^\mathfrak{F} $ Z_\infty ^\mathfrak{F} (G/H _G ) of G/H _G . In this paper, we obtain some results about the $ \mathfrak{F}_h $ \mathfrak{F}_h -normal subgroups and then use them to study the structure of finite groups.     

On finite {1,2,3}-semiaffine planes

In this paper, we investigate {1,2,3}-semiaffine planes. All such planes of order n >51 shall be classified. It turns out that they are embeddable into projective planes of the same order n in the most natural way.Work supported by National Research Project on Strutture Geometriche Combinatoria, loro applicazioni of Italian M.P.I. and G.N.S.A.G.A. of C.N.R.     

On solutions of the simultaneous Pell equations $$ x^{2}-\left( a^{2}-1\right) y^{2}=1$$ and $$y^{2}-pz^{2}=1$$y2-pz2=1

Let \(a\ge 2\) be an integer and p prime number. It is well-known that the solutions of the Pell equation have recurrence relations. For the simultaneous Pell equations

$$\begin{aligned}&x^{2}-\left( a^{2}-1\right) y^{2} =1 \\&y^{2}-pz^{2} =1 \end{aligned}$$

assume that \(x=x_{m}\) and \(y=y_{m}\). In this paper, we show that if \(m\ge 3\) is an odd integer, then there is no positive solution to the system. Moreover, we find the solutions completely for \(5\le a\le 14\) in the cases when \(m\ge 2\) is even integer and \(m=1\).

     

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   

On the determination of solutions of simultaneous Pell equations $$x^{2}-(a^{2}-1)y^{2}=y^{2}-pz^{2}=1$$ x 2 - ( a 2 - 1 ) y 2 = y 2 - p z 2 = 1

Periodica Mathematica Hungarica - In this paper, we consider the simultaneous Pell equations 0.1 $$\begin{aligned} x^{2}-(a^{2}-1)y^{2}= & {} 1, \nonumber \\ y^{2}-pz^{2}= & {} 1,...     

On $\frak {F}$-normal Fitting classes of finite soluble groups

Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G_{\frak X}G_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak S_p\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak S_p\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak S_p\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nⁿ, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak S_p1?\frak S_pr, p_i \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.