The convergence of\tilde a_s^{(n)} h_0 h_k

It is proved that (i) for survives to E∞; (ii) for survives to E∞. Project supported by the Doctoral Program Foundation of China.     

Actions of $$\mathbf {SAut}\varvec{(F_n)}$$

We study actions of SAut\((F_n)\), the unique subgroup of index two in the automorphism group of a free group of rank n, and obtain rigidity results for its representations. In particular, we show that every smooth action of SAut\((F_n)\) on a low dimensional torus is trivial.     

On the regularity of the locally integrable solutions of the functional equations $$\sum\limits_{i = 1}^k {a_i \left( {x,t} \right)f\left( {x + \varphi _i \left( t \right)} \right) = b\left( {x,t} \right)} $$

Aequationes mathematicae -     

A generalization of Serre’s condition $$\mathrm {(F)}$$ ( F ) with applications to the finiteness of unramified cohomology

Mathematische Zeitschrift - In this paper, we introduce a condition ( $$\mathrm {F}_m'$$ ) on a field K, for a positive integer m, that generalizes Serre’s condition (F) and which still...     

Recursion operators and hierarchies of $$\text{mKdV}$$ equations related to the Kac–Moody algebras $$D_4^{(1)}$$ , $$D_4^{(2)}$$ , and $$D_4^{(3)}$$

Theoretical and Mathematical Physics - We construct three nonequivalent gradings in the algebra $$D_4\simeq so(8)$$ . The first is the standard grading obtained with the Coxeter automorphism...     

On the error term $$\Delta _{(k,l)}(x)$$ Δ ( k,l ) ( x )

The Ramanujan Journal - Let k, l be non-negative integers and $$\zeta ^{(k)}(s)$$ denote the kth derivative of the Riemann zeta function $$\zeta (s).$$ Further let $$d_{(k,l)}(n)$$ be the nth...     

On the existence of a p-adic metaplectic Tate-type $\widetilde {\gamma}$-factor

Let \mathbbF\mathbb{F} be a p-adic field, let χ be a character of \mathbbF^*\mathbb{F}^{*}, let ψ be a character of \mathbbF\mathbb{F} and let g_y^-1\gamma_{\psi}^{-1} be the normalized Weil factor associated with a character of second degree. We prove here that one can define a meromorphic function [(g)\tilde](c,s,y)\widetilde{\gamma}(\chi ,s,\psi) via a similar functional equation to the one used for the definition of the Tate γ-factor replacing the role of the Fourier transform with an integration against y·g_y^-1\psi\cdot\gamma_{\psi}^{-1}. It turns out that γ and [(g)\tilde]\widetilde{\gamma} have similar integral representations. Furthermore, [(g)\tilde]\widetilde{\gamma} has a relation to Shahidi‘s metaplectic local coefficient which is similar to the relation γ has with (the non-metalpectic) Shahidi‘s local coefficient. Up to an exponential factor, [(g)\tilde](c,s,y)\widetilde{\gamma}(\chi,s,\psi) is equal to the ratio \fracg(c²,2s,y)g(c,s+\frac12,y)\frac{\gamma(\chi^{2},2s,\psi)}{\gamma(\chi,s+\frac{1}{2},\psi)}.     

$$\varvec{}$$-Classes in finite groups of conjugate type $$\varvec{(n,1)}$$(n,1)

Two elements in a group G are said to be z-equivalent or to be in the same z-class if their centralizers are conjugate in G. In a recent work, Kulkarni et al. (J. Algebra Appl., 15 (2016) 1650131) proved that a non-abelian p-group G can have at most \(\frac{p^k-1}{p-1} +1\) number of z-classes, where \(|G/Z(G)|=p^k\). Here, we characterize the p-groups of conjugate type (n, 1) attaining this maximal number. As a corollary, we characterize p-groups having prime order commutator subgroup and maximal number of z-classes.     

On the diophantine equation $$\varvec{y^{2} = \prod _{i \le 8}(x + k_i)}$$

This paper improves the result of Tengely (Periodica Math. Hung., 72(1) (2016) 23–28).     

An Operator Theoretical Approach Towards Systems over the Signal Space \ell_{2}\mathbb{(Z)}

In this paper a system is considered as a (possibly unbounded) linear operator from l₂\mathbb(Z)^m \ell_{2}\mathbb{(Z)}^{m} to l₂\mathbb(Z)^p \ell_{2}\mathbb{(Z)}^{p} . Georgiou and Smith [6] noted that there are intrinsic difficulties in using l₂\mathbb(Z) \ell_{2}\mathbb{(Z)} as underlying signal space, since even a simple causal convolution system is not closed and an extended definition of the system is not causal. We discuss these difficulties and we develop necessary and sufficient conditions for notions such as causality, closability and causal closability.     

On the functional equation $$G\left( x,G\left( y,x\right) \right) = G\left( y,G\left( x,y\right) \right) $$Gx,Gy,x=Gy,Gx,y and means

Periodica Mathematica Hungarica - We consider the functional equation $$G\left( x,G\left( y,x\right) \right) =G\left( y,G\left( x,y\right) \right) $$, posed in Jarczyk and Jarczyk (Aequ Math...