On the functional equation $${\varvec{f}}({\varvec{x}})+{\varvec{f}}({\varvec{y}})=\mathbf{max} \{{\varvec{f}}({\varvec{xy}}),{\varvec{f}}({\varvec{xy}}^{-{\varvec{1}}})\}$$ on groups

We analyse the functional equation

$$\begin{aligned} f(x)+f(y)=\max \{f(xy),f(xy^{-1})\} \end{aligned}$$

for a function \(f:G\rightarrow \mathbb R\) where G is a group. Without further assumption it characterises the absolute value of additive functions. In addition \(\{z\in G\mid f(z)=0\}\) is a normal subgroup of G with abelian factor group.

     

On the Number of Solutions of the Equation a_{1}x_{1}^{m_{1}}+\cdots +a_{n}x_{n}^{m_{n}}=bx_{1}\cdots x_{n} in a Finite Field

We obtain an explicit formula for the number of solutions of a special equation in a finite field under a certain restriction on the exponents. Mathematics Subject Classifications (2000) primary: 11G25, secondary: 11T24.     

Characterizations of the Vertex Operator Algebras ${V_{L}^{T}}$ and VLO${V_{L}^{O}}$

In this paper, we give characterizations of the rational vertex operator algebras \({V_{L}^{T}}\) and \({V_{L}^{O}}\), where L is the root lattice of type A ₁, T is the tetrahedral group, and O is the octahedral group. By these two characterizations, the classification of rational VOAs of central charge 1 is reduced to the characterization of \({V_{L}^{I}}\) where I is the icosahedral group.     

On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.     

Bispectral and (\mathfrak{gl}_{N},\mathfrak{gl}_{M}) dualities

Let be a space of quasipolynomials of dimension N=N ₁+⋅⋅⋅+N _n . We define the regularized fundamental operator of V as the polynomial differential operator D=∑ _i=0 ^N A _N−i (x)∂ _x ⁱ annihilating V and such that its leading coefficient A ₀ is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials whose regularized fundamental operator is the differential operator ∑ _i=0 ^N u ⁱ A _N−i (∂ _u ). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical points of the two master functions associated with the -dual Gaudin models and also between the corresponding Bethe vectors. The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460). The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579).     

On the Rational Recursive Sequence x_{n+1}=Ax_{n}+Bx_{n-k}+\frac{\beta x_{n}+\gamma x_{n-k}}{Cx_{n}+Dx_{n-k}}

In this article, we study the global and asymptotic properties of the solutions of the difference equation $$x_{n+1}=Ax_{n}+Bx_{n-k}+(\beta x_{n}+\gamma x_{n-k})/(Cx_{n}+Dx_{n-k}),\quad n=0,1,2,\ldots,$$ where the initial conditions x _?k ,…,x _?1,x ₀ are arbitrary positive real numbers and the coefficients A,B,C,D,β and γ are positive constants, while k is a positive integer number. Some numerical examples will be given to illustrate our results.     

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.     

Blocks with Defect Group \boldsymbol{Q_{2^n}\times C_{2^m}} and \boldsymbol{SD_{2^n}\times C_{2^m}}

We determine the numerical invariants of blocks with defect group $Q_{2^n}\times C_{2^m}$ and $SD_{2^n}\times C_{2^m}$ , where $Q_{2^n}$ denotes a quaternion group of order 2 ⁿ , $C_{2^m}$ denotes a cyclic group of order 2 ^m , and $SD_{2^n}$ denotes a semidihedral group of order 2 ⁿ . This generalizes Olsson’s results for m?=?0. As a consequence, we prove Brauer’s k(B)-Conjecture, Olsson’s Conjecture, Brauer’s Height-Zero Conjecture, the Alperin–McKay Conjecture, Alperin’s Weight Conjecture and Robinson’s Ordinary Weight Conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper follows (and uses) (Sambale, J Pure Appl Algebra 216:119–125, 2012; Proc Amer Math Soc, 2012).     

Frobenius problem for semigroups \mathsf{S}(d_{1},d_{2},d_{3})

We find the matrix representation of the set Δ(d ³), where d ³=(d ₁,d ₂,d ₃), of integers that are unrepresentable by d ₁,d ₂,d ₃ and develop a diagrammatic procedure for calculating the generating function Φ(d ³;z) for the set Δ(d ³). We find the Frobenius number F(d ³), the genus G(d ³), and the Hilbert series H(d ³;z) of a graded subring for nonsymmetric and symmetric semigroups and enhance the lower bounds of F(d ³) for symmetric and nonsymmetric semigroups.      

On the Diophantine equation \dfrac{ax^{n+2l}+c}{abt^{2}x^{n}+c}{\displaystyle\frac{ax^{n+2l}+c}{abt^{2}x^{n}+c}}=by^{2}

We obtain all solutions of the equation $\frac{ax^{n+2l}+c}{abt^{2}x^{n}+c} = by^{2}$ with c??{??1,??2,??4}.     

Isometric embeddings of \mathbb{Z}_{p^k} in the Hamming space \mathbb{F}_{p}^{N} and \mathbb{Z}_{p^k}-linear codes

Isometric embeddings of $\mathbb{Z}_{p^n+1}$ into the Hamming space ( $\mathbb{F}_{p}^{p^n},w$ ) have played a fundamental role in recent constructions of non-linear codes. The codes thus obtained are very good codes, but their rate is limited by the rate of the first-order generalized Reed–Muller code—hence, when n is not very small, these embeddings lead to the construction of low-rate codes. A natural question is whether there are embeddings with higher rates than the known ones. In this paper, we provide a partial answer to this question by establishing a lower bound on the order of a symmetry of ( $\mathbb{F}_{p}^{N},w$ ).     

On the rational difference equation x_{n}=1+\frac{(1-x_{n-k})(1-x_{n-l})(1-x_{n-m})}{x_{n-k}+x_{n-l}+x_{n-m}}

In this note we consider the following higher order rational difference equations $$x_{n}=1+\frac{(1-x_{n-k})(1-x_{n-l})(1-x_{n-m})}{x_{n-k}+x_{n-l}+x_{n-m}},\quad n=0,1,\ldots,$$ where 1≤k<l<m, and the initial values x _?m ,x _?m+1,…,x _?1 are positive numbers. We give some sufficient conditions for the persistence of positive solutions for the above equation, and prove that the positive equilibrium point of this equation is globally asymptotically stable.     

Dynamics of the max-type difference equation x_{n}=\max\{\frac{ 1}{ x_{n-m}} , \frac{A_{n} }{x_{n-r} }\}

In this paper, we study the periodicity, the boundedness and the convergence of the following max-type difference equation $$x_n =\max\biggl\{\frac{ 1}{ x_{n-m}} , \frac{A_n }{x_{n-r} }\biggr \},\quad n =0, 1,2,\ldots,$$ where $\{A_{n}\}^{+\infty}_{n=0}$ is a periodic sequence with period k and A _n ??(0,1) for every n??0, m??{1,2} and r??{2,3,??} with m<r, the initial values x _?r ,??,x _?1??(0,+??). The special case when $m = 1, r = 2, \{A_{n}\}^{+\infty}_{ n=0}$ is a periodic sequence with period k and A _n ??(0,1) for every n??0 has been completely investigated by Y.?Chen. Here we extend his results to the general case.     

On the Dynamics of x_{\bf{n} + \bf{1}} = {{ \mgreek{a} + \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {A + x_{n-2}}}

We investigate the global stability, the periodic character and the boundedness nature of solutions of the equation in the title for all admissible nonnegative values of the parameters and the initial conditions. We show that the solutions exhibit a trichotomy character depending on how the parameter γ compares to the sum of the parameters δ and A.     

关于同伦元素 $\alpha_{1}\beta_{1}\beta_{2}\gamma_{s}$

设 $p\geq 7$ 为任意奇素数. 证明了当 $3\leq s 相似文献   

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 irreducible p,q‐representations of Lie algebras \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {G} (0,1)$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {G} (0,0)$\end{document}

In this paper, the irreducible p, q‐representations of the Lie algebras $\mathcal {G}(0,1)$ and $\mathcal {G}(0,0)$ are discussed. We prove two theorems that classify certain irreducible p, q‐representations of these Lie algebras and construct their one variable models in terms of p, q‐derivative and dilation operators. As an application, we derive a p, q‐special function identity based on one such model.