نشریه پژوهشهای ریاضی
سال هشتم شماره 3 (پیاپی 22، پاییز 1401)

Throughout this paper R will denote an associative ring with identity, M a unitary right R-module. A functor 𝜏from the category of the right R-modules Mod-R to itself is called a preradicalif it satisfies the following properties: (i) 𝜏(𝑀)is a submodule of M, for every R-module M; (ii) If 𝑓: 𝑀′ → 𝑀is an R-module homomorphism, then 𝑓(𝜏(𝑀′ )) ≤ 𝜏(𝑀) and 𝜏(𝑓) is the restriction of 𝑓to 𝜏(𝑀′ ). For example Rad, Soc, and 𝑍𝑀are preradicals. Note that if K is a summand of M, then 𝐾 ∩ 𝜏(𝑀) = 𝜏(𝐾). For a preradical 𝜏, Al-Takhman, Lomp and Wisbauer defined and studied the concept of 𝜏-lifting and 𝜏-supplemented modules. A module M is called 𝜏- lifting if every submodule N of M has a decomposition 𝑁 = 𝐴 ⊕ 𝐵 such that A is a direct summand of M and 𝐵 ⊆ 𝜏(𝑀).A submodule 𝐾 ⊆ 𝑀 is called 𝜏 −supplement (weak𝜏-supplement) provided there exists some 𝑈 ⊆ 𝑀such that 𝑀 = 𝑈 + 𝐾 and 𝑈 ∩ 𝐾 ⊆ 𝜏(𝐾) (𝑈 ∩ 𝐾 ⊆ 𝜏(𝑀)). M is called 𝜏-supplemented (weakly 𝜏-supplemented) if each of its submodules 𝜏-supplement (weak 𝜏-supplement) in M.Talebi, Moniri Hamzekolaei and Keskin-Tütüncü, defined 𝜏-H-supplemented modules. A module M is called𝜏- H-supplemented if for every 𝑁 ≤ 𝑀 there exists a direct summand D of Msuch that (𝑁 + 𝐷)/𝑁 ⊆ 𝜏(𝑀/𝑁)and(𝑁 + 𝐷)/𝐷 ⊆ 𝜏(𝑀/𝐷). The 𝛽 ∗ relation is introduced and investigated by Birkenmeier, Takil Mutlu, Nebiyev, Sokmez and Tercan. Let X and Y be submodules of M. X and Yare 𝛽 ∗ equivalent, X𝛽 ∗Y, provided 𝑋+𝑌 𝑋 ≪ 𝑀 𝑋 𝑎𝑛𝑑 𝑋+𝑌 𝑌 ≪ 𝑀 𝑌 . Based on definition of 𝛽 ∗ relation they introduced two new classes of modules namely 𝐺𝑜𝑙𝑑𝑖𝑒 ∗ -lifting and 𝐺𝑜𝑙𝑑𝑖𝑒 ∗ −supplemented.They showed that two concept of H-supplemented modules and 𝐺𝑜𝑙𝑑𝑖𝑒 ∗ −lifting modules coincide. In this paper, we introduce Goldie−𝜏 −supplemented and strongly 𝜏-Hsupplemented modules. We introduce the̅𝛽̅̅̅∗ relation. We investigate some properties of this relation and prove that this relation is an equivalence relation. We define Goldie−𝜏 −supplemented and strongly 𝜏-H-supplemented modules. We call a module M, Goldie−𝜏 −supplemented (strongly 𝜏-H-supplemented) if for any submodule N of M, there exists a 𝜏-supplement submodule (a direct summand) D of M such thatN𝛽̅̅̅∗D. Clearly every strongly 𝜏-H-supplemented module is Goldie 𝜏 -supplemented. We will study direct sums of Goldie 𝜏 -Hsupplemented modules. Let 𝑀 = 𝐴 ⊕ 𝐵 be a distributive module. Then M is Goldie 𝜏 -upplemented (strongly 𝜏 -H-supplemented) if and only if A and B are Goldie 𝜏 -supplemented (strongly 𝜏 -H-supplemented). We also define 𝜏 -Hcofinitely supplemented modules and obtain some conditions which under the factor module of a 𝜏 -H-cofinitely supplemented module will be 𝜏 -H-cofinitely supplemented.

In this paper, first we define and investigate the 𝛽𝜏 ∗ relation on submodules of a module. We show that the 𝛽𝜏 ∗ relation is an equivalence relation. We apply this relation to define and investigate the classes of Goldie-𝜏 -supplemented modules and strongly𝜏-H-supplemented modules.

We investigate some properties of this relation and prove that this relation is an equivalence relation. We define Goldie−𝜏 −supplemented and strongly 𝜏-Hsupplemented modules. We call a module M, Goldie−𝜏 −supplemented (strongly 𝜏 -H-supplemented) if for any submodule N of M, there exists a 𝜏- supplement submodule (a direct summand) D of M such that N𝛽̅̅̅∗ D. Clearly every strongly 𝜏 -H-supplemented module is Goldie 𝜏 -supplemented. We will study direct sums of Goldie 𝜏 -H-supplemented modules. Let 𝑀 = 𝐴 ⊕ 𝐵 be a distributive module. Then M is Goldie 𝜏 -upplemented (strongly 𝜏 -Hsupplemented) if and only if A and B are Goldie 𝜏 -supplemented (strongly 𝜏 - H-supplemented). We also define 𝜏 -H-cofinitely supplemented modules and obtain some conditions which under the factor module of a 𝜏 -H-cofinitely supplemented module will be 𝜏 -H-cofinitely supplemented.

The following conclusions were drawn from this research.  Let 𝑀 = 𝑀1 ⊕ 𝑀2 , where 𝑀1 is a fully invariant submodule of M. Assume that 𝜏 is a cohereditary preradical. If M is strongly 𝜏-Hsupplemented, then 𝑀1 𝑎𝑛𝑑 𝑀2 are strongly 𝜏 −H-supplemented.  Let M be an 𝜏-H-cofinitely supplemented module and let 𝑁 ≤ 𝑀 be a submodule. Suppose that for every direct summand K of M, there exists a submodule L of M such that 𝑁 ⊆ 𝐿 ⊆ 𝐾 + 𝑁, L/N is a direct summand of M/N and 𝐾+𝑁 𝑁 𝐿/𝑁 ⊆ 𝜏( 𝑀 𝑁 )+ 𝐿 𝑁 𝐿/𝑁 . Then M/N is 𝜏-Hcofinitelysupplemented.  Let M be a module and let 𝑁 ≤ 𝑀 be a submodule such that for each decomposition 𝑀 = 𝑀1 ⊕ 𝑀2 we have 𝑁 = (𝑁 ∩ 𝑀1 ) ⊕ (𝑁 ∩ 𝑀2). If M is 𝜏-H-cofinitely supplemented, then M/N is 𝜏-H-cofinitely supplemented.

In this paper, the robust vertex centdian  location  problem with uncertain vertex weights on general graphs is studied. The used criterion to solve the problem is the min-max  regret criterion. This problem  is  investigated  with objective function contains $lambda$  and  a polynomial time algorithm for the problem is presented. It is shown that the vertex centdian problem on general graphs is solved in cubic  time.

In this paper, first, the concept of the fuzzy semi maximal filters in BL-algebras has been introduced, then several properties of  fuzzy semi maximal filters have been studied and  its relationship with other fuzzy filters including fuzzy  (positive)implicative  fuzzy filters and fuzzy fantastic  filters are investigated. In the following, using the level subsets  of fuzzy sets in a BL-algebra, fuzzy  semi maximal filters have been studied.  It has also been proved that a fuzzy filter is a fuzzy semi maximal filters  if and only if all level subsets  are  semi maximal filters filter. In addition, it proves that the preimage  of a semi maximal fuzzy filter under the BL-homomorphism  and the image of a fuzzy  semi maximal filter under the BL-isomorphism , is also a fuzzy semi maximal  filter. Also, the Extension  property has been proved  for fuzzy semi maximal filters in BL-algebra. Finally, the relationships between the fuzzy  semi maximal filters and semi simple BL-algebras have been studied.

Let 𝑅 =⊕𝑛𝜖ℕ0 𝑅𝑛 be a standard graded Noetherian ring, i.e. 𝑅0 is a commutative Noetherian ring and 𝑅 is generated (as an 𝑅0 -algebra) by finitely many elements of degree one, 𝑅+ =⊕𝑛𝜖ℕ 𝑅𝑛 be the irrelevant ideal of 𝑅 and 𝒂 stands for a homogeneous ideal of 𝑅. Also, 𝑀 denotes a finitely generated graded 𝑅-module. For 𝑖𝜖ℕ0 and 𝑛𝜖ℤ let 𝐻𝒂 𝑖 (𝑀)𝑛 denotes the n-th component of the i-th graded local cohomology module 𝐻𝒂 𝑖 (𝑀) of 𝑀 with support in 𝒂 (our terminology on local cohomology comes from [3]). It is well-known that 𝐻𝑅+ 𝑖 (𝑀)𝑛 is a finitely generated 𝑅0 -module for all 𝑛𝜖ℤ and it vanishes for all sufficiently large values of n ([3, 15.1.5]). In spite of the case 𝑛 → +∞, the asymptotic behavior of 𝐻𝑅+ 𝑖 (𝑀)𝑛 when 𝑛 → −∞ is so complicated, see for example [4], and it attracts lots of interest, see [6] and [1], for a good survey on this topic. Also, in [5] the authors studied the graded components of 𝐻𝒂 𝑖 (𝑀) in the case where 𝒂 is a homogeneous ideal of 𝑅 containing the irrelevant ideal. In the case where 𝑅0 is an Artinian ring, for all 𝑖𝜖ℕ0 , ℎ𝑀 𝑖 : ℤ → ℤ(𝑛 → 𝑙𝑒𝑛𝑔𝑡ℎ𝑅0 (𝐻𝑅+ 𝑖 (𝑀)𝑛 ), is called the i-th Hilbert cohomological function of 𝑀. In [2], Brodmann et. all. studied the problem of finiteness of the set of Hilbert cohomological functions of some classes of graded modules. In this paper, we also consider this problem. Another reason of this paper, is to study the support of graded local cohomology modules 𝐻𝒂 𝑖 (𝑀) of 𝑀 with support in an arbitrary homogeneous ideal 𝒂 of 𝑅. Let cd𝒂(𝑀) ≔ sup{iϵℤ | 𝐻𝒂 𝑖 (𝑀) ≠ 0}, denotes the cohomological dimension of 𝑀 with respect to 𝒂. In [1, 3.7], it is shown that the set Supp𝑅0 (𝐻𝑅+ cd𝑅+ (𝑀) (𝑀)𝑛) is eventually stable when 𝑛 → −∞, i.e. there exists 𝑋 ⊆ Spec(𝑅0) such that Supp𝑅0 (𝐻𝑅+ 𝑐𝑑𝑅+ (𝑀) (𝑀)𝑛) = 𝑋, for all 𝑛 ≪ 0. In this paper, we study the asymptotic behavior of the set Supp𝑅0 (𝐻𝒂 𝑖 (𝑀)𝑛 ) when 𝑛 → −∞.

First, in a special case we describe 𝐻𝒂 𝑖 (𝑀) in terms of some homologies of the minimal graded free resolution of 𝑀. Then, as a consequence, we find a class of graded modules with a finite set of Hilbert cohomological functions.

We show that, in a special case, the support and dimension of 𝐻𝑅+ 𝑖 (𝑀)𝑡 have upper bound which doesn't depend on i and 𝑀 for sufficiently small values of t. Also, it is shown that, in some cases, the set {Supp𝑅0 (𝐻𝒂 𝑐𝑑𝒂(𝑀) (𝑀)𝑛)} 𝑛𝜖ℤ is eventually increasing, in the sense that Supp𝑅0 (𝐻𝒂 𝑐𝑑𝒂(𝑀) (𝑀)𝑛) ⊆ Supp𝑅0 (𝐻𝒂 𝑐𝑑𝒂(𝑀) (𝑀)𝑛−1) for all 𝑛 ≪ 0.

It is worth to find cases for the existence of a non-zero divisor x on the module M in the ideal a for which cd𝒂 (𝑀/𝑥𝑀) = cd𝒂(𝑀)− 1. It helps us to study the last non-vanishing local cohomology module of M with support in a.

The directed power graph of a semigroup S was defined by Kelarev and Quinn as the digraph ( ) S with vertex set S , in which there is an arc from x to y if and only if m x y  or m y x  for some positive integer m. Motivated by this, Chakrabarty et al. defined the (undirected) power graph ( ) S , in which distinct x and y are joined if one is a power of the other. The concept of power graphs has been studied extensively by many authors. Let  be a graph. We denote V ( )  and E ( )  for vertices and edges of  , respectively. The (open) neighborhood N a( ) of vertex a V ( ) is the set of vertices adjacent to a. Also the closed neighborhood of a , N a N a a [ ] ( ) { }  . Throughout this paper, all groups and graphs are assumed to be finite and the following notation is used: Aut G( ) denotes the group of automorphisms of G ; Z m the cyclic group of order m ; n Z m the direct product of n copies of Z m .

The power graph of a group is the graph whose vertex set is the set of nontrivial elements of the group and two elements are adjacent if one is a power of the other. We introduce some ways to find the automorphism groups of some graphs. Let L be a graph and x y V L N y N x    { ( ) | [ ] [ ]}. We define the weighted graph L as follows: V L x x V L ( ) { | ( )}   , weight x x ( ) | |  , And two vertices x and y are adjacent if x and y are adjacent in L . As an application, we describe the automorphism group of the power graph of a finite group G as: | | ( ( )) ( ( )) ( ( )) . x x V P G Aut P G Aut P G S    Let G be a finite nilpotent group, 1 2 1 2 | | t n n n G p p p  t and G P P P     1 2 t where | | i n P p i i  .We obtain the automorphism group of the power graph of abelian and nilpotent groups by their sylow subgroups which is: 1 2 | | ( ( )) ( ( )) ( ( ( )) ( ( )) ( ( ))) . t x x V P G Aut P G Aut P P Aut P P Aut P P S       Finally, we calculate the automorphism group of the power graph of homocyclic group ( ) m n p G Z  , n  1 as: 1 2 1 1 ( ) ( ( )) (( ( ) ) ) ( ( ) ),

In this paper, subcategories with threshold c are examined from an active general fuzzy automaton and the generator from an active general fuzzy automaton is defined and the connections between them are examined. Then, under the main and maximum automata with threshold c, one of the active general fuzzy automata will be defined, and it will be proved that each active general fuzzy automaton can be written socially from below the distinct primary and maximum automata with threshold c.

Let G = (V, E) be a simple graph with vertex set 𝑉 and let 𝑓: 𝑉 → {0,1,2} be a function of weight 𝜔(𝑓) = ∑ 𝑓(𝑣) 𝑣∈𝑉(𝐺) . A vertex 𝑣 is protected with respect to 𝑓, if 𝑓(𝑣) > 0 or 𝑓(𝑣) = 0 and 𝑣 is adjacent to a vertex 𝑢 such that 𝑓(𝑢) > 0. The function 𝑓 is a co-Roman dominating function, abbreviated CRDF if: (i) every vertex 𝑢 with 𝑓(𝑢) = 0 is adjacent to a vertex 𝑣 for which 𝑓(𝑣) > 0, and (ii) every vertex 𝑣 with 𝑓(𝑣) > 0 has a neighbor 𝑢 for which 𝑓(𝑢) = 0, such that each vertex of 𝐺 is protected with respect to the function 𝑓 ′ : 𝑉(𝐺) → {0,1,2}, defined by 𝑓 ′ (𝑣) = 𝑓(𝑣) −1, 𝑓 ′ (𝑢) = 1 and 𝑓 ′ (𝑥) = 𝑓(𝑥) for 𝑥 ∈ 𝑉(𝐺) − {𝑢, 𝑣}. The co-Roman domination number of a graph G, denoted by 𝛾𝑐𝑟(𝐺), is the minimum weight of a co-Roman dominating function on G. In this paper, we study the co-Roman domination number of grid graphs and we obtain this parameter for 𝑃2 × 𝑃𝑛 and 𝑃3 × 𝑃𝑛.

The fractional calculus deals with the generalization of integration and differentiation of integer order to those ones of any order .The q-fractional differential equation usually describes the physical process imposed on the time scale set Tq. In this paper, we first propose a difference formula for discretizing the fractional q-derivative 𝐷 𝑐 𝑞 𝛼𝑥(𝑡), of Caputo type with order 0 < 𝛼 < 1 and scale index 0 < 𝑞 < 1. We establish a rigorous truncation error boundness and prove that this difference formula is unconditionally stable. Then, we consider the difference method for solving the initial problem of q-fractional differential equation: 𝐷𝑞 𝛼𝑥(𝑡) = 𝑓(𝑡, 𝑥(𝑡)) 𝑐 . We prove the existence and stability of the difference solution and give the convergence analysis. Numerical experiments show the effectiveness and high accuracy of the proposed difference method.

In this scheme, we first present a difference formula (called the 𝐿1,𝑞 formula) to discretize the fractional q-derivative 𝐷 𝑐 𝑞 𝛼𝑥(𝑡), of Caputo type with 0 < 𝛼, 𝑞 < 1. This difference formula is constructed by using the piecewise linear interpolation to approximate the integral function. Then, by using this difference formula, we establish a difference method for solving the initial problem of q-fractional differential equation.

We prove that this difference method is unconditionally stable and give an error estimation of ∆𝑡𝑛 2 -order. Numerical experiments show the high accuracy and effectiveness of this difference formula. To the authors’ best knowledge, it is the first time that an unconditionally stable difference formula is presented and analyzed for the q-fractional problems. Our work provides a numerical approach for solving the q-fractional problems.

The following conclusions were drawn from this research.  We present the difference formula and drive truncation error boundness.  The formula is contributed to the stability analysis.  The difference technique is used to solve the initial value problem of q-fractional differential equation.  The existence of the solution, stability and error estimation are given for the difference formula.

A wide variety of problems in mathematics and particularly in computational mathematics is formulated as second-order partial differential equations (PDEs), which mostly do not admit closed-form solutions. Due to this, constructing numerical solver for such problems is requisite as well as of special importance when the computational domain is not anymore regular. Such problems, which mainly occur in engineering modeling problems, are defined on irregular domain. Classic methods such as finite difference (FD) or spectral solvers are hard to be employed on irregular domains with arbitrary geometries. One remedy is to rely on finite element method or the meshless radial basis function (RBF) method. This is discussed in this paper. Toward this purpose, an RBF-FD method is discussed and its weighting coefficients are constructed theoretically. Numerical reports and comparison will confirm the efficacy and competitiveness of the presented approach on domains with arbitrary geometries.

The meshfree RBF methods are categorized as globalized and localized meshless schemes. A localized version of such schemes is the RBF-FD estimates, which is an attractive choice in contrast to the global meshfree RBF schemes by providing better-conditioned and sparse discretization matrices. This clearly reduces the computational load of applying RBF methods for practical problems in higher dimensions. Unlike the standard procedures, the RBF-FD method can deal with scattered node layouts and irregular domains. In addition, their locality makes them more flexible with respect to local refinement techniques than the globalized RBF methods.

In this paper, we obtain weights of the radial basis function finite difference formula for some differential operators. These weights are used to obtain the local truncation error in function derivative approximating. We discuss how to select the shape parameter for the RBF-FD formulas so as to gain the accuracy as well as stability. We apply these formulas for Poisson equation with irregular domains and show that the proposed method can be used as a fully meshfree method.

The following conclusions were drawn from this research.  Closed-form coefficients for RBF-FD formulas to approximate derivatives of the function can be obtained under some conditions.  It is discussed that how we can choose the shape parameters to have efficient RBF-FD formulations in solving PDE problems on irregular domains.  Several experiments on different types of irregular geometries are discussed and compared and showed how much RBF-FD is useful for practical engineering PDE problems on irregular domains.

A ring R is called a refinement ring if the monoid of finitely generated projective R- modules is refinement.  Let R be a commutative refinement ring and M, N, be two finitely generated projective R-nodules, then M~N  if and only if Mm ~Nm for all maximal ideal m of  R. A rectangular matrix A over R admits diagonal reduction if there exit invertible matrices p and Q such that PAQ is a diagonal matrix. We also prove that for every refinement ring R, every regular matrix over R admits diagonal reduction if and only if every regular matrix over R/J(R)  admits diagonal reduction.

The notion of Leibniz algebras is introduced by Blokh in 1965 as a noncommutative version of Lie algebras and rediscovered by Loday in 1993. A Leibniz algebra is a vector space A over a field F together with a bilinear map [ , ] : A × A A usually called the Leibniz bracket of A, satisfying the Leibniz identity: [x,[y,z]] = [[x,y],z] – [[x,z],y] , x,y,z ∈ A. The classification of nilpotent Leibniz algebras is one of the most important subject in the study of Leibniz algebras. In the present paper, we obtain an upper bound for the nilpotency class of a finitely generated nilpotent Leibniz algebra A in terms of the maximum of the nilpotency classes of maximal subalgebras of A and the minimal number of generators of A.

Throughout the paper, any Leibniz algebra A is considered over a fixed field F, c denotes the maximum of the nilpotency classes of maximal subalgebras of A, d is the minimal number of generators of A and ⌊ ⌋ denotes the integral part. Moreover, we inductively define: A 1=A and A n = [A n-1 ,A], for n ≥ 2. Lemma 1. Let A be a nilpotent Leibniz algebra and M be a maximal subalgebra of A. Then M is a two-sided ideal of A. Lemma 2. Let I be a two-sided ideal of a Leibniz algebra A and {x1,x2, … ,xk} be a subset of A which contains at least n elements of I (n ≤ k). Then [[[x1,x2],x3], … ,xk] ∈ I n . Theorem 3. Let A be a finitely generated nilpotent Leibniz algebra such that d > 1. Then A is nilpotent of class at most ⌊𝑐𝑑/(𝑑 − 1)⌋. Corollary 4. Let A be a finitely generated nilpotent Leibniz algebra and {x1,x2, … ,xd} be a minimal generating set of A with d > 1. Then 𝑐𝑙(𝐴) ≤ 𝑚𝑎𝑥 1≤𝑖≤𝑑 ⌊ 𝑐𝑙(𝑀𝑖) 𝑑 𝑑−1 ⌋, where Mi is the two-sided ideal generated by the set { x1 , …, xi-1 , xi+1 , … ,xd} and cl denotes the nilpotency class. Corollary 5. Let A be a finitely generated nilpotent Leibniz algebra of class k with d > 1. Then c ≤ k ≤ 2c. Corollary 6. Let A be a finitely generated nilpotent Leibniz algebra such that d > c+1. Then A is nilpotent of class c. Corollary 7. Let A be a finitely generated nilpotent Leibniz algebra of class 2c such that d > 1. Then d=2. Proposition 8. There is not any non-Lie Leibniz algebra with at least two generators, whose maximal subalgebras are all abelian.

Data Envelopment Analysis (DEA) combined with Analytic Hierarchy Process (AHP) provides Data Envelopment Analytic Hierarchy Process (DEAHP) methodology for weight derivation and aggregation. A double-frontier analysis approach is proposed in this article to derive priorities in AHP. The proposed method combines DEA variable weight with AHP from both optimistic and pessimistic views to obtain the most desirable and undesirable weights based on a pairwise comparison matrix for decision criteria or alternatives. This approach is able to eliminate DEAHP drawbacks to provide an improved estimate of priorities and better decisions than DEAHP. Two numerical examples including a real AHP application to recruit a research fellow for a research project were presented to demonstrate advantages of the proposed method and its potential applications in multiple-criteria decision analysis.

A Fibonacci string of length $n$ is a binary string $b = b_1b_2ldots b_n$ in which for every $1 leq i < n$, $b_icdot b_{i+1} = 0$. In other words, a Fibonacci string is a binary string without 11 as a substring. Similarly, a Lucas string is a Fibonacci string $b_1b_2ldots b_n$ that $b_1cdot b_n = 0$. For a natural number $ngeq1$, a Fibonacci cube of dimension $n$ is denoted by $Gamma_n$ and is defined as a graph whose vertices are  Fibonacci strings of length $n$ such that two vertices $b_1b_2ldots b_n$ and $b'_1b'_2ldots b'_n$ are adjacent if $b_ineq b'_i$ holds for exactly one $iin{1,ldots, n}$. A Lucas cube of  dimension $n$, $Lambda_n$, is a subgraph of $Gamma_n$ induced by the Lucas strings of length $n$. Let $G=(V,E)$ be a simple undirected graph. A perfect code is a subset $C$ of $V$ in such a way that for every $vin C$, the sets ${uin V | d(u, v) = 1}$ are pairwise disjoint and make a partition for $V$. In other words, each vertex of $G$ is either in $C$ or is adjacent to exactly one of the elements of $C$. It is proved that Fibonacci cube $Gamma_n$, admits a perfect code if and only if $nleq3$. In this paper, we prove the same result for Lucas cubes i.e, $Lambda_n$ admits a perfect code if and only if $nleq3$.

Wavelet analysis is one of the useful techniques in mathematics which is used much in statistics science recently. In this paper, in addition to introduce the wavelet transformation, the wavelet threshold estimation of semiparametric regression model with correlated errors with having Gaussian distribution is determined and the convergence ratio of estimator computed. To evaluate the wavelet threshold estimation, the block function and sinusoidal function are used as objective functions and using the simulation method the average of mean square error and standard deviation of this estimator are compared with the average of mean square error and standard deviation of kernel method. Also, the wavelet semiparametric regression model has been fitted to data on the growth rate of the teeth.

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.