On the Stability of a k-Cubic Functional Equation in Intuitionistic Fuzzy n-Normed Spaces

In this paper, we investigate some stability results concerning the k-cubic functional equation f(kx + y) + f(kx?y) = kf(x + y) + kf(x?y) + 2k(k²?1)f(x) in the intuitionistic fuzzy n-normed spaces.     

Polynomial rings over commutative reduced Hopfian local rings

We prove that if R is a commutative, reduced, local ring, then R is Hopfian if and only if the ring R[x] is Hopfian. This answers a question of Varadarajan [16], in the case when R is a reduced local ring. We provide examples of non-Noetherian Hopfian commutative domains by proving that the finite dimensional domains are Hopfian. Also, we derive some general results related to Hopfian rings.     

On the minimum size of binary codes with length 2<Emphasis Type="Italic">R</Emphasis> + 4 and covering radius <Emphasis Type="Italic">R</Emphasis>

The minimum size of a binary code with length n and covering radius R is denoted by K(n, R). For arbitrary R, the value of K(n, R) is known when n ≤  2R +  3, and the corresponding optimal codes have been classified up to equivalence. By combining combinatorial and computational methods, several results for the first open case, K(2R +  4, R), are here obtained, including a proof that K(10, 3) =  12 with 11481 inequivalent optimal codes and a proof that if K(2R +  4, R) <  12 for some R then this inequality cannot be established by the existence of a corresponding self-complementary code.     

Height reducing property of polynomials and self-affine tiles

A monic polynomial \({f(x)\in {\mathbb Z}[x]}\) is said to have the height reducing property (HRP) if there exists a polynomial \({h(x)\in {\mathbb Z}[x]}\) such that

$f(x)h(x)=a_n x^n+a_{n-1}x^{n-1}+\cdots+a_1x\pm q,$

where q = f(0), |a _i | ≤ (|q| ?1), i = 1, . . . , n and a _n > 0. We show that any expanding monic polynomial f(x) has the height reducing property, improving a previous result in Kirat et al. (Discrete Comput Geom 31: 275–286, 2004) for the irreducible case. The proof relies on some techniques developed in the study of self-affine tiles. It is constructive and we formulate a simple tree structure to check for any monic polynomial f(x) to have the HRP and to find h(x). The property is used to study the connectedness of a class of self-affine tiles.

     

Conditional equations for quadratic functions

We consider quadratic functions f that satisfy the additional equation y² f(x) =  x² f(y) for the pairs \({ (x,y) \in \mathbb{R}^2}\) that fulfill the condition P(x, y) =  0 for some fixed polynomial P of two variables. If P(x, y) =  ax +  by +  c with \({ a , b , c \in \mathbb{R}}\) and \({(a^2 + b^2)c \neq 0}\) or P(x,y) =  x ⁿ ? y with a natural number \({n \geq 2}\), we prove that f(x) =  f(1) x² for all \({x \in \mathbb{R}}\). Some related problems, admitting quadratic functions generated by derivations, are considered as well.     

Strongly lifting modules and strongly dual Rickart modules

The concepts of strongly lifting modules and strongly dual Rickart modules are introduced and their properties are studied and relations between them are given in this paper. It is shown that a strongly lifting module has the strongly summand sum property and the generalized Hopfian property, and a ring R is a strongly regular ring if and only if R_R is a strongly dual Rickart module, if and only if aR is a fully invariant direct summand of R_R for every a ∈ R.     

Derivations cocentralizing multilinear polynomials on left ideals

Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f (X ₁, . . . , X _t ) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that

$d(f(x_{1},\ldots,x_{t}))f(x_{1},\ldots,x_{t})-f(x_{1},\ldots,x_{t})\delta(f(x_{1},\ldots,x_{t}))\in C$

for all \({x_1,\ldots,x_t\in\lambda}\). Then either d = 0 and λ δ(λ) = 0 or λ C = RCe for some idempotent e in the socle of RC and one of the following holds:

1. (1)
   f (X₁, . . . , X _t ) is central-valued on eRCe;
    
2. (2)
   λ(d + δ)(λ) = 0 and f (X₁, . . . , X _t )² is central-valued on eRCe;
    
3. (3)
   char R = 2 and eRCe satisfies st ₄(X ₁, X ₂, X ₃, X ₄), the standard polynomial identity of degree 4.
    

     

Most small $${\varvec{}}$$-grous have an automorhism of order 2

Let f(p, n) be the number of pairwise nonisomorphic p-groups of order \(p^n\), and let g(p, n) be the number of groups of order \(p^n\) whose automorphism group is a p-group. We prove that the limit, as p grows to infinity, of the ratio g(p, n) / f(p, n) equals 1/3 for \(n=6,7\).     

Power-Commuting Generalized Skew Derivations in Prime Rings

Let R be a non-commutative prime ring of characteristic different from 2 with extended centroid C, F ≠ 0 a generalized skew derivation of R, and n ≥ 1 such that [F(x), x] ⁿ  = 0, for all x ∈ R. Then there exists an element λ ∈ C such that F(x) = λx, for all x ∈ R.     

The Schröder-Bernstein property for weakly minimal theories

For a countable, weakly minimal theory T, we show that the Schröder-Bernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to each of the following:

1. 1.
   For any U-rank-1 type q ∈ S(acl ^eq (?)) and any automorphism f of the monster model C, there is some n < ω such that f ⁿ (q) is not almost orthogonal to q ? f(q) ? … ? f ^n?1(q)
    
2. 2.
   T has no infinite collection of models which are pairwise elementarily bi-embeddable but pairwise nonisomorphic.
    

We conclude that for countable, weakly minimal theories, the Schröder-Bernstein property is absolute between transitve models of ZFC.     

Generalized (Jordan) left derivations on rings associated with an element of rings

In this paper, we introduce a new notion of generalized (Jordan) left derivation on rings as follows: let R be a ring, an additive mapping F : R → R is called a generalized (resp. Jordan) left derivation if there exists an element w ∈ R such that F(xy) = xF(y) + yF(x) + yxw (resp. F(x ²) = 2xF(x) + x ² w) for all x, y ∈ R. Then, some related properties and results on generalized (Jordan) left derivation of square closed Lie ideals are obtained.     

On power deformations of univalent functions

For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) ? 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f _c (z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator \({f \mapsto f_c}\) maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.     

Irreducibility criteria of Schur-type and Pólya-type

Let \({f(x)=(x-a_1)\cdots (x-a_m)}\), where a ₁, . . . , a _m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether \({(f(x))^{2^k}+1}\) is irreducible for every k ≥ 1. In 1919 Pólya proved that if \({P(x)\in\mathbb{Z}[x]}\) is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2^?N N! where \({N=\lceil m/2\rceil}\), then P(x) is irreducible. A great number of authors have published results of Schur-type or Pólya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Pólya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.     

Holomorphic injective extensions of functions in <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">K</Emphasis>) and algebra generators

We present necessary and sufficient conditions on planar compacta K and continuous functions f on K in order that f generate the algebras P(K), R(K), A(K) or C(K). We also unveil quite surprisingly simple examples of non-polynomial convex compacta K ? C and f ∈ P(K) with the property that f ∈ P(K) is a homeomorphism of K onto its image, but for which f ^?1 ? P(f(K)). As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull \(\widehat K\). On the other hand, it is shown that the restriction f*|_G of the Gelfand-transform f* of an injective function f ∈ P(K) is injective on every regular, bounded complementary component G of K. A necessary and sufficient condition in terms of the behaviour of f on the outer boundary of K is given in order that f admit a holomorphic injective extension to \(\widehat K\). We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

Cardinality of the Ellis semigroup on compact metric countable spaces

Let E(X, f) be the Ellis semigroup of a dynamical system (X, f) where X is a compact metric space. We analyze the cardinality of E(X, f) for a compact countable metric space X. A characterization when E(X, f) and \(E(X,f)^* = E(X,f) \setminus \{ f^n : n \in \mathbb {N}\}\) are both finite is given. We show that if the collection of all periods of the periodic points of (X, f) is infinite, then E(X, f) has size \(2^{\aleph _0}\). It is also proved that if (X, f) has a point with a dense orbit and all elements of E(X, f) are continuous, then \(|E(X,f)| \le |X|\). For dynamical systems of the form \((\omega ^2 +1,f)\), we show that if there is a point with a dense orbit, then all elements of \(E(\omega ^2+1,f)\) are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where \(E(\omega ^2+1,f)\) and \(\omega ^2+1\) are homeomorphic but not algebraically homeomorphic, where \(\omega ^2+1\) is taken with the usual ordinal addition as semigroup operation.     

Lower bounds on the signed total k-domination number of graphs

Let G be a graph with vertex set V(G). For any integer k ≥ 1, a signed total k-dominating function is a function f: V(G) → {?1, 1} satisfying ∑_x∈N(v)f(x) ≥ k for every v ∈ V(G), where N(v) is the neighborhood of v. The minimum of the values ∑_v∈V(G)f(v), taken over all signed total k-dominating functions f, is called the signed total k-domination number. In this note we present some new sharp lower bounds on the signed total k-domination number of a graph. Some of our results improve known bounds.     

Extendability of Contractible Configurations for Nowhere-Zero Flows and Modulo Orientations

Let H be a connected graph and G be a supergraph of H. It is trivial that for any k-flow (D, f) of G, the restriction of (D, f) on the edge subset E(G / H) is a k-flow of the contracted graph G / H. However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow \((D',f')\) of the contracted graph G / H, whether or not the supergraph G admits a k-flow (D, f) that is consistent with \((D',f')\) in the edge subset E(G / H). In this paper, we will investigate contractible configurations and their extendability for integer flows, group flows, and modulo orientations. We show that no integer flow contractible graphs are extension consistent while some group flow contractible graphs are also extension consistent. We also show that every modulo \((2k+1)\)-orientation contractible configuration is also extension consistent and there are no modulo (2k)-orientation contractible graphs.     

Period of the adelic Ikeda lift for <Emphasis Type="Italic">U</Emphasis>(<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">m</Emphasis>)

We give a period formula for the adelic Ikeda lift of an elliptic modular form f for U(m, m) in terms of special values of the adjoint L-functions of f. This is an adelic version of Ikeda’s conjecture on the period of the classical Ikeda lift for U(m, m).     

Centralizers of generalized skew derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2, let Q be the right Martindale quotient ring of R, and let C be the extended centroid of R. Suppose that G is a nonzero generalized skew derivation of R and f(x ₁,..., x _n ) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r ₁,..., r _n ): r _i ∈ R} be the set of all evaluations of f(x ₁,..., x _n ) in R, while A = {[G (f(r ₁,..., r _n )), f(r ₁,..., r _n )]: r _i ∈ R}, and let C _R (A) be the centralizer of A in R; i.e., C _R (A) = {a ∈ R: [a, x] = 0, ? _x ∈ A }. We prove that if A ≠ (0), then C _R (A) = Z(R).