Level sets of the Takagi function: Hausdorff dimension

The Takagi function τ(x) is a continuous non-differentiable function on the unit interval defined by Takagi in 1903. This paper studies level sets L(y) = {x : τ(x) = y} of the Takagi function and bounds their Minkowski dimensions and Hausdorff dimensions above by 0.668. There exist level sets with Minkowski dimension 1/2. The method of proof involves a multiscale analysis that relies upon the self-similarity of τ(x) up to affine shifts.     

Level sets of the Takagi function: local level sets

The Takagi function ??: [0,1] ?? [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y)?=?{x : ??(x)?=?y} of the Takagi function ??(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a ??generic?? full Lebesgue measure set of ordinates y, the level sets are finite sets. In contrast, here it is shown for a ??generic?? full Lebesgue measure set of abscissas x, the level set L(??(x)) is uncountable. An interesting singular monotone function is constructed associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly ${\frac{3}{2}}$ .     

Level sets of signed Takagi functions

This paper examines level sets of functions of the form $$ f(x)=\sum_{n=0}^\infty \frac{r_n}{2^n}\phi(2^n x), $$ where $\phi(x)=\operatorname{dist}\, (x,\mathbb {Z})$ , the distance from x to the nearest integer, and r _n =±1 for each n. Such functions are referred to as signed Takagi functions. The case when r _n =1 for all n is the classical Takagi function, a well-known example of a continuous but nowhere differentiable function. For f of the above form, the maximum and minimum values of f are expressed in terms of the sequence {r _n }. It is then shown that almost all level sets of f are finite (with respect to Lebesgue measure on the range of f), but the set of ordinates y with an uncountably large level set is residual in the range of f. The concept of a local level set of the Takagi function, due to Lagarias and Maddock, is extended to arbitrary signed Takagi functions. It is shown that the average number of local level sets contained in a level set of f is the reciprocal of the height of the graph of f, and consequently, this average lies between 3/2 and 2. These results generalize recent findings by Buczolich [8], Lagarias and Maddock [14], and Allaart [3].     

General solution of a simple Levi-Civitá functional equation on non-abelian groups

The general solution of the functional equation f(xy) = f(x)h(y) + f(y) on abelian groups is well-known. We present methods for solving this equation on various non-abelian groups. In particular we treat the equation on semidirect products, then extend this treatment to solvable groups. We also find the solution on perfect groups. These results also apply to the more general equation f(xy) = g(x)h(y) + k(y).     

Topological Hypergroups in the Sense of Marty

In this paper, we introduce the concept of topological hypergroups as a generalization of topological groups. A topological hypergroup is a nonempty set endowed with two structures, that of a topological space and that of a hypergroup. Let (H, ○) be a hypergroup and (H, τ) be a topological space such that the mappings (x, y) → x ○ y and (x, y) → x/y from H × H to 𝒫*(H) are continuous. The main tool to obtain basic properties of hypergroups is the fundamental relation β*. So, by considering the quotient topology induced by the fundamental relation on a hypergroup (H, ○) we show that if every open subset of H is a complete part, then the fundamental group of H is a topological group.

It is important to mention that in this paper the topological hypergroups are different from topological hypergroups which was initiated by Dunkl and Jewett.     

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.     

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.     

Sphere coverings and identifying codes

In any connected, undirected graph G = (V, E), the distance d(x, y) between two vertices x and y of G is the minimum number of edges in a path linking x to y in G. A sphere in G is a set of the form S _r (x) = {y ∈ V : d(x, y) = r}, where x is a vertex and r is a nonnegative integer called the radius of the sphere. We first address in this paper the following question: What is the minimum number of spheres with fixed radius r ≥ 0 required to cover all the vertices of a finite, connected, undirected graph G? We then turn our attention to the Hamming Hypercube of dimension n, and we show that the minimum number of spheres with any radii required to cover this graph is either n or n + 1, depending on the parity of n. We also relate the two above problems to other questions in combinatorics, in particular to identifying codes.     

On the Rédei zeta function

Let L be a locally finite lattice. An order function ν on L is a function defined on pairs of elements x, y (with x ≤ y) in L such that ν(x, y) = ν(x, z) ν(z, y). The Rédei zeta function of L is given by ?(s; L) = Σ_x∈Lμ(Ô, x) ν(Ô, x)^?s. It generalizes the following functions: the chromatic polynomial of a graph, the characteristic polynomial of a lattice, the inverse of the Dedekind zeta function of a number field, the inverse of the Weil zeta function for a variety over a finite field, Philip Hall's φ-function for a group and Rédei's zeta function for an abelian group. Moreover, the paradigmatic problem in all these areas can be stated in terms of the location of the zeroes of the Rédei zeta function.     

Projective bichains

An algebra with two binary operations · and +  that are commutative, associative, and idempotent is called a bisemilattice. A bisemilattice that satisfies Birkhoff’s equation x · (x + y) =  x + (x · y) is a Birkhoff system. Each bisemilattice determines, and is determined by, two semilattices, one for the operation +  and one for the operation ·. A bisemilattice for which each of these semilattices is a chain is called a bichain. In this note, we characterize the finite bichains that are weakly projective in the variety of Birkhoff systems as those that do not contain a certain three-element bichain. As subdirectly irreducible weak projectives are splitting, this provides some insight into the fine structure of the lattice of subvarieties of Birkhoff systems.     

Dimensions of some fractals defined via the semigroup generated by 2 and 3

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Σ _m ={0, ...,m?1}^? that are invariant under multiplication by integers. The results apply to the sets {x∈Σ _m :? k, x _k x _2k ... x _nk =0}, where n ≥ 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.     

Hyers–Ulam Stability of General Jensen-Type Mappings in Banach Algebras

Using the fixed point method, we prove the Hyers–Ulam stability of homomorphisms in complex Banach algebras and complex Banach Lie algebras and also of derivations on complex Banach algebras and complex Banach Lie algebras for the general Jensen-type functional equation f(α x + β y) + f(α x ? β y) = 2α f(x) for any \({\alpha, \beta \in \mathbb{R}}\) with \({\alpha, \beta \neq 0}\) . Furthermore, we prove the hyperstability of homomorphisms in complex Banach algebras for the above functional equation with α = β = 1.     

Boundary proximity of SLE

This paper examines how close the chordal SLE _κ curve gets to the real line asymptotically far away from its starting point. In particular, when κ ? (0, 4), it is shown that if β > β _κ  := 1/(8/κ ? 2), then the intersection of the SLE _κ curve with the graph of the function y = x/(log x) ^β , x > e, is a.s. bounded, while it is a.s. unbounded if β = β _κ . The critical SLE₄ curve a.s. intersects the graph of $y=x^{{-({\rm log\,log\,x})}^{\alpha}}, x > e^e$ , x > e ^e , in an unbounded set if α ≤ 1, but not if α > 1. Under a very mild regularity assumption on the function y(x), we give a necessary and sufficient integrability condition for the intersection of the SLE _κ path with the graph of y to be unbounded. When the intersection is bounded a.s., we provide an estimate for the probability that the SLE _κ path hits the graph of y. We also prove that the Hausdorff dimension of the intersection set of the SLE _κ curve and the real axis is 2 ? 8/κ when 4 < κ < 8.     

ABCs of the bomber problem and its relatives

In a classic Markov decision problem of Derman et al. (Oper. Res. 23(6):1120–1130, 1975) an investor has an initial capital x from which to make investments, the opportunities for which occur randomly over time. An investment of size y results in profit P(y), and the aim is maximize the sum of the profits obtained within a given time t. The problem is similar to a groundwater management problem of Burt (Manag. Sci. 11(1):80–93, 1964), the notorious bomber problem of Klinger and Brown (Stochastic Optimization and Control, pp. 173–209, 1968), and types of fighter problems addressed by Weber (Stochastic Dynamic Optimization and Applications in Scheduling and Related Fields, p. 148, 1985), Shepp et al. (Adv. Appl. Probab. 23:624–641, 1991) and Bartroff et al. (Adv. Appl. Probab. 42(3):795–815, 2010a). In all these problems, one is allocating successive portions of a limited resource, optimally allocating y(x,t), as a function of remaining resource x and remaining time t. For their investment problem, Derman et al. (Oper. Res. 23(6):1120–1130, 1975) proved that an optimal policy has three monotonicity properties: (A) y(x,t) is nonincreasing in t, (B) y(x,t) is nondecreasing in x, and (C) x?y(x,t) is nondecreasing in x. Theirs is the only problem of its type for which all three properties are known to be true. In the bomber problem the status of (B) is unresolved. For the general fighter problem the status of (A) is unresolved. We survey what is known about these exceedingly difficult problems. We show that (A) and (C) remain true in the bomber problem, but that (B) is false if we very slightly relax the assumptions of the usual model. We give other new results, counterexamples and conjectures for these problems.     

On the Invariant Fields and Rings of Some Groups of Polynomial Automorphisms

Abstract

We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = ? y + u(x) where u ∈ C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C.     

Extrema of functions of eigenvalues

We consider the question of finding an extreme value for some function of the eigenvalues of the differential equation y″ + λφ(x) y = 0,y(0) = y(1) = 0, as φ(x) varies over a region in a function space. A characterization of the φ(x) at which the function of the eigenvalues achieves its extremum is derived.     

On an internal property of absolute retracts,II

Let X be a (metrizable) space. A mixer for X is, roughly speaking, a map μ:X³→X such that μ(x, x, y) = μ(x, y, x) = μ(y, x, x) = x for all x, y ∈ X. We show that each AR has a mixer and that a finite dimensional path connected space with a mixer is an AR. Our main result is that each separable space with a mixer and having an open cover by sets contractible within the whole space, is LEC.     

An Engel condition with skew derivations

Let R be a prime ring and set [x, y]₁ = [x, y] = xy ? yx for ${x,y\in R}$ and inductively [x, y] _k = [[x, y] _k-1, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [δ(x), x] _k  = 0 for all ${x \in L}$ , where k is a fixed positive integer, then charR = 2 and ${R\subseteq M_{2}(F)}$ for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne.