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.     

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.     

The Real Powers of the Convolution of a Gamma Distribution and a Bernoulli Distribution

In this paper, we essentially compute the set of x,y>0 such that the mapping \(z\longmapsto(1-r+re^{z})^{x}(\frac{\lambda}{\lambda-z})^{y}\) is a Laplace transform. If X and Y are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by μ the distribution of X+Y. The above problem is equivalent to finding the set of x>0 such that μ ^*x exists.     

Analogues of some Tauberian theorems for bounded variation

In this paper definitions for “bounded variation”, “subsequences”, “Pringsheim limit points”, and “stretchings” of a double sequence are presented. Using these definitions and the notion of regularity for four dimensional matrices, the following two questions will be answered. First, if there exists a four dimensional regular matrix A such that Ay = Σ _k,l=1,1 ^∞∞ a _m,n,k,l y _k,l is of bounded variation (BV) for every subsequence y of x, does it necessarily follow that x ∈ BV? Second, if there exists a four dimensional regular matrix A such that Ay ∈ BV for all stretchings y of x, does it necessarily follow that x ∈ BV? Also some natural implications and variations of the two Tauberian questions above will be presented.     

On Certain Identity Related to Herstein Theorem on Jordan Derivations

Let R be a 6-torsion-free prime ring and let \({D : R \rightarrow R}\) be an additive mapping satisfying the relation 2D(x ⁴) = D(x ³)x + x ³ D(x) + D(x)x ³ + xD(x ³) for all \({x \in R}\) . The purpose of this paper is to show that D is a derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion-free prime ring is a derivation.     

Stability of Cubic and Quartic Functional Equations in Non-Archimedean Spaces

We prove generalized Hyers-Ulam–Rassias stability of the cubic functional equation f(kx+y)+f(kx?y)=k[f(x+y)+f(x?y)]+2(k ³?k)f(x) for all \(k\in \Bbb{N}\) and the quartic functional equation f(kx+y)+f(kx?y)=k ²[f(x+y)+f(x?y)]+2k ²(k ²?1)f(x)?2(k ²?1)f(y) for all \(k\in \Bbb{N}\) in non-Archimedean normed spaces.     

Norms on L of Periodic Interpolation Splines with Equidistant Nodes

We consider the set S _r,n of periodic (with period 1) splines of degree r with deficiency 1 whose nodes are at n equidistant points x_i=i / n. For n-tuples y = (y₀, ... , y_n-1), we take splines s _r,n (y, x) from S _r,n solving the interpolation problem

$$s_{r,n} (y,t_i ) = y_i,$$

where t _i = x _i if r is odd and t _i is the middle of the closed interval [x _i , x _i+1 ] if r is even. For the norms L _r,n ^* of the operator y → s _r,n (y, x) treated as an operator from l¹ to L¹ [0, 1] we establish the estimate

$$L_{r,n}^ * = \frac{4}{{\pi ^2 n}}log min(r,n) + O\left( {\frac{1}{n}} \right)$$

with an absolute constant in the remainder. We study the relationship between the norms L _r,n ^* and the norms of similar operators for nonperiodic splines.

     

A note on maximum-type functions

The paper studies the differential properties of functions of the form

$g(x) = \mathop {\max }\limits_{y \in Y} f(x,y),$

where x ∈ X (X is an open convex set from ? ^m ) and y ∈ Y (Y is a compact from ? ⁿ ). Apart from the conventional smoothness conditions imposed on f(x, y), the condition of the concavity of g(x) on X is also imposed.

The differentiability of function g(x) on X is proved.The results of the study facilitate the derivation of the conditions ensuring the sufficiency of Pontryagin’s maximum principle.     

Linked pairs of additive functions

In 1990, Benz asked whether a real additive mapping satisfying \(xf(y)=yf(x)\) for all points (x, y) on the unit circle must be linear. In 2005, Boros and Erdei showed that it must be so. Here we generalize the problem to a pair of additive functions f, g related by the functional equation \(xf(y)=yg(x)\) for all points (x, y) on a specified curve. We find that for many (but not all) types of curves this forces f and g to be equal and linear.     

On nonabelian composition factors of a finite prime spectrum minimal group

Assume that G is a primitive permutation group on a finite set X, x ∈ X, y ∈ X \ {x}, and G _x,y \(\underline \triangleleft \) G _x . P. Cameron raised the question about the validity of the equality G _x,y = 1 in this case. The author proved earlier that, if soc(G) is not a direct power of an exceptional group of Lie type, then G _x,y = 1. In the present paper, we prove that, if soc(G) is a direct power of an exceptional group of Lie type distinct from E ₆(q), ² E ₆(q), E ₇(q), and E ₈(q), then G _x,y = 1.     

On EA-equivalence of certain permutations to power mappings

In this paper we investigate the existence of permutation polynomials of the form x ^d  + L(x) on \({{\mathbb{F}_{2^n}}}\) , where \({{L(x)\in\mathbb{F}_{2^n}[x]}}\) is a linearized polynomial. It is shown that for some special d with gcd(d, 2 ⁿ ?1) > 1, x ^d  + L(x) is nerve a permutation on \({{\mathbb{F}_{2^n}}}\) for any linearized polynomial \({{L(x)\in\mathbb{F}_{2^n}[x]}}\) . For the Gold functions \({{x^{2^i+1}}}\) , it is shown that \({{x^{2^i+1}+L(x)}}\) is a permutation on \({{\mathbb{F}_{2^n}}}\) if and only if n is odd and \({{L(x)=\alpha^{2^i}x+\alpha x^{2^i}}}\) for some \({{\alpha\in\mathbb{F}_{2^n}^{*}}}\) . We also disprove a conjecture in (Macchetti Addendum to on the generalized linear equivalence of functions over finite fields. Cryptology ePrint Archive, Report2004/347, 2004) in a very simple way. At last some interesting results concerning permutation polynomials of the form x ^?1 + L(x) are given.     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator L^Jf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?L^J, D) acts in L²(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel p^J (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?L^J, D) extends in L²(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel p^J(t, x, y), and the function p^J(t, x, y) is uniformly comparable in t, x, y to the function p^J(t, x, y), the heat kernel related to the operator (?L^J,D).     

No decreasing sequence of cardinals

In set theory without the Axiom of Choice (AC), we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f (n + 1) ? f (n), where for sets x and y, x ? y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that AC^LO (AC restricted to linearly ordered sets of non-empty sets, and also equivalent to AC in ZF, the Zermelo–Fraenkel set theory minus AC) ? NDS in ZFA set theory (ZF with the Axiom of Extensionality weakened in order to allow the existence of atoms). The latter result provides a strongly negative answer to the question of whether “every Dedekind-finite set is finite” implies NDS addressed in G. H. Moore “Zermelo’s Axiom of Choice. Its Origins, Development, and Influence” and in P. Howard–J. E. Rubin “Consequences of the Axiom of Choice”. We also prove that AC^WO (AC restricted to well-ordered sets of non-empty sets) ? NDS in ZF (hence, “every Dedekind-finite set is finite” ? NDS in ZF, either) and that “for all infinite cardinals m, m + m = m” ? NDS in ZFA.     

On a class of partial planes related to biplanes

In this note we consider partial planes in which for each element x (point or line) there exists a unique opposite element or antipode x* which cannot be joined to x or has no intersection with x. We also require the existence of a triangle. Such partial planes will be called antipodal planes. We are mainly interested in the subclass of regular antipodal planes satisfying: p I L implies p* I L* for all points p and lines L. We shall provide a free construction of infinite regular antipodal planes. The objects thus constructed are not free objects in the usual sense since between antipodal planes there do not exist proper homomorphisms. On the other hand, regular antipodal planes do have a canonical homomorphic image which is a biplane (cf. Payne, J Comb Theory A 12:268–282, 1972). Regular antipodal planes can be coordinatized by certain algebraic systems in a similar way as projective planes are coordinatized by ternary rings. Again by a free construction, we shall provide examples satisfying a configuration theorem comparable to the Fano condition with fixed line at infinity.     

On the Cauchy problem for an Emden-Fowler equation

We consider the equation y″ = P(x)x ^a y ^σ , σ < 0, and prove the unique solvability of the Cauchy problem y(0) = 0, y′(0) = λ.     

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 the length of a finite group and of its 2-generator subgroups

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h* (G) such that F* _h (G) = G, where F* ₁ (G) = F* (G) is the generalized Fitting subgroup, and F* _i+1(G) is the inverse image of F* (G/F*_i (G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, x^g 〉) ≤ k for allx, g ∈ G such that 〈x, x^g 〉 is soluble, then h* (G) is k-bounded.     

Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency

We study the nonlinear Schrödinger equations: \(-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).\) where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has “critical frequency” \(\inf_{x\in{\bf R}^{N}} V(x)=0\). We also assume that V(x) satisfies \(0 < \liminf_{|x|\to\infty}V(x)\le \sup_{x\in{\bf R}^{N}}V(x) < \infty\) and V(x) has k local or global minima. In critical frequency cases, Byeon-Wang [5,6] showed the existence of single-peak solutions which concentrating around global minimum of V(x). Their limiting profiles—which depend on the local behavior of the potential V(x)—are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u _ε(x) as follows: rescaled function \(w_\epsilon(y)=\left(\frac{g(\epsilon)}{\epsilon}\right)^{\frac{2}{p-1}} u_\epsilon(g(\epsilon)y+x_\epsilon)\) converges to a least energy solution of ?Δw + V ₀(y) w = w ^p , w > 0 in Ω₀, \(w\in H^{1}_0(\Omega_0)\). Here g(ε), V ₀(x) and Ω₀ depend on the local behaviors of V(x).