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.     

The Scholz theorem in function fields

The Scholz theorem in function fields states that the l-rank difference between the class groups of an imaginary quadratic function field and its associated real quadratic function field is either 0 or 1 for some prime l. Furthermore, Leopoldt's Spiegelungssatz (= the Reflection theorem) in function fields yields a comparison between the m-rank of some subgroup of the class group of an imaginary cyclic function field L₁ and the m-rank of some subgroup of the class group of its associated real cyclic function field L₂ for some prime number m; then their m-ranks also equal or differ by 1. In this paper we find an explicit necessary condition for their m-ranks (respectively l-ranks) to be the same in the case of cyclic function fields (respectively quadratic function fields). In particular, in the case of quadratic function fields, if l does not divide the regulator of L₂, then their l-ranks are the same, equivalently if their l-ranks differ by 1, then l divides the regulator of L₂.     

Upper bounds on the upper signed total domination number of graphs

Let G=(V,E) be a graph. A function f:V→{−1,+1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A signed total dominating function f is minimal if there does not exist a signed total dominating function g, f≠g, for which g(v)≤f(v) for every v∈V. The weight of a signed total dominating function is the sum of its function values over all vertices of G. The upper signed total domination number of G is the maximum weight of a minimal signed total dominating function on G. In this paper we present a sharp upper bound on the upper signed total domination number of an arbitrary graph. This result generalizes previous results for regular graphs and nearly regular graphs.     

On some questions related to the Gauss conjecture for function fields

We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.     

Non-monotonic solutions and continuously differentiable solutions of conjugacy equations

In this paper solutions of conjugacy equation φ(f(x))=g(φ(x)) for a strictly decreasing continuous given function f and a continuous given function g (maybe non-monotonic) are constructed by piecewise defining. We determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g. Finally, the recursive algorithm is implemented in MATLAB software and two examples are respectively presented for a non-monotonic solution and a continuously differentiable one.     

On Chebyshev functions and Klee functions

Associated to a lower semicontinuous function, one can define its proximal mapping and farthest mapping. The function is called Chebyshev (Klee) if its proximal mapping (farthest mapping) is single-valued everywhere. We show that the function f is 1/λ-hypoconvex if its proximal mapping Pλf is single-valued. When the function f is bounded below, and Pλf is single-valued for every λ>0, the function must be convex. Similarly, we show that the function f is 1/μ-strongly convex if the farthest mapping Qμf is single-valued. When the function is the indicator function of a set, this recovers the well-known Chebyshev problem and Klee problem in Rn. We also give an example illustrating that a continuous proximal mapping (farthest mapping) needs not be locally Lipschitz, which answers one open question by Hare and Poliquin.     

On φ-Families of Probability Distributions

We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a φ-function, resulting in a φ-family of probability distributions. We show how φ-families are constructed. In a φ-family, the analogue of the cumulant-generating function is a normalizing function. We define the φ-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback–Leibler divergence. A formula for the φ-divergence where the φ-function is the Kaniadakis κ-exponential function is derived.     

Differentiability of the volume of a region enclosed by level sets

The level of a function f on Rn encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the function f is smooth, the volume function is again smooth for regular values of f. For critical values of f the volume function is only finitely differentiable. The initial motivation for this study comes from Radiotherapy, where such volume functions are used in an optimization process. Thus their differentiability properties become important.     

On the Ni(x) integral function and its application to the Airy’s non-homogeneous equation

In this article, we discuss a recently introduced function, Ni(x), to which we will refer as the Nield-Kuznetsov function. This function is attractive in the solution of inhomogeneous Airy’s equation. We derive and document some elementary properties of this function and outline its application to Airy’s equation subject to initial conditions. We introduce another function, Ki(x), that arises in connection with Ni(x) when solving Airy’s equation with a variable forcing function. In Appendix A, we derive a number of properties of both Ni(x) and Ki(x), their integral representation, ascending and asymptotic series representations. We develop iterative formulae for computing all derivatives of these functions, and formulae for computing the values of the derivatives at x = 0. An interesting finding is the type of differential equations Ni(x) satisfies. In particular, it poses itself as a solution to Langer’s comparison equation.     

Sharp inequalities for the dyadic square function in the BMO setting

We introduce a method for the simultaneous study of a BMO function φ and its dyadic square function S(φ) that can yield sharp norm inequalities between the two. One of the applications is the sharp bound for the p-th moment of S(φ), 0<p<∞, which in turn implies the square-exponential integrability of the square function. We also present sharp refinements of these inequalities in the more restrictive case when φ is assumed to be bounded.     

Regularized gap function as penalty term for constrained minimization problems

By using the regularized gap function for variational inequalities, Li and Peng introduced a new penalty function Pα(x) for the problem of minimizing a twice continuously differentiable function in closed convex subset of the n-dimensional space Rn. Under certain assumptions, they proved that the original constrained minimization problem is equivalent to unconstrained minimization of Pα(x). The main purpose of this paper is to give an in-depth study of those properties of the objective function that can be extended from the feasible set to the whole Rn by Pα(x). For example, it is proved that the objective function has bounded level sets (or is strongly coercive) on the feasible set if and only if Pα(x) has bounded level sets (or is strongly coercive) on Rn. However, the convexity of the objective function does not imply the convexity of Pα(x) when the objective function is not quadratic, no matter how small α is. Instead, the convexity of the objective function on the feasible set only implies the invexity of Pα(x) on Rn. Moreover, a characterization for the invexity of Pα(x) is also given.     

Read-Once Functions Revisited and the Readability Number of a Boolean Function

In this paper, we present the first polynomial time algorithm for recognizing and factoring read-once functions. The algorithm is based on algorithms for cograph recognition and a new efficient method for checking normality. Its correctness is based on a classical characterization theorem of Gurvich which states that a positive Boolean function is read-once if and only if it is normal and its co-occurrance graph is P₄-free.We also investigate the problem of factoring certain non-read-once functions. In particular, we show that if the co-occurrence graph of a positive Boolean function f is a tree, then the function is read-twice. We then extend this further proving that if f is normal and its corresponding graph is a partial k-tree, then f is a read 2^k function and a read 2^k formula for F for f can be obtained in polynomial time.     

Induced path transit function, monotone and Peano axioms

The induced path transit function J(u,v) in a graph consists of the set of all vertices lying on any induced path between the vertices u and v. A transit function J satisfies monotone axiom if x,y∈J(u,v) implies J(x,y)⊆J(u,v). A transit function J is said to satisfy the Peano axiom if, for any u,v,w∈V,x∈J(v,w), y∈J(u,x), there is a z∈J(u,v) such that y∈J(w,z). These two axioms are equivalent for the induced path transit function of a graph. Planar graphs for which the induced path transit function satisfies the monotone axiom are characterized by forbidden induced subgraphs.     

Biharmonic extensions on trees without positive potentials

Let T be a tree rooted at e endowed with a nearest-neighbor transition probability that yields a recurrent random walk. We show that there exists a function K biharmonic off e whose Laplacian has potential theoretic importance and, in addition, has the following property: Any function f on T which is biharmonic outside a finite set has a representation, unique up to addition of a harmonic function, of the form f=βK+B+L, where β a constant, B is a biharmonic function on T, and L is a function, subject to certain normalization conditions, whose Laplacian is constant on all sectors sufficiently far from the root. We obtain a characterization of the functions biharmonic outside a finite set whose Laplacian has 0 flux similar to one that holds for a function biharmonic outside a compact set in Rn for n=2,3, and 4 proved by Bajunaid and Anandam. Moreover, we extend the definition of flux and, under certain restrictions on the tree, we characterize the functions biharmonic outside a finite set that have finite flux in this extended sense.     

The Möbius function of separable and decomposable permutations

We give a recursive formula for the Möbius function of an interval [σ,π] in the poset of permutations ordered by pattern containment in the case where π is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,…,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Möbius function in the case where σ and π are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142.We also show that the Möbius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Möbius function of an interval [σ,π] of separable permutations is bounded by the number of occurrences of σ as a pattern in π. Another consequence is that for any separable permutation π the Möbius function of (1,π) is either 0, 1 or −1.     

Approximation with the output of linear control systems

In this paper we give a new proof that for controllable and observable linear systems every L²[0,T] function can be approximated in the L²[0,T] sense with an output function generated by an L²[0,T] input function. We also give a new characterization of how continuous functions on [0,T] are uniformly approximated by an output generated by a continuous input function. The relative degree of the transfer function of the system determines those functions that can be approximated. We further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions.     

Roman Domination Dot-critical Graphs

A Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the value ${f(V(G))=\sum_{u \in V(G)}f(u)}$ . The Roman domination number, γ _R (G), of G is the minimum weight of a Roman dominating function on G. In this paper, we study graphs for which contracting any edge decreases the Roman domination number.     

Admissible majorants for model subspaces, and arguments of inner functions

Let Θ be an inner function in the upper half-plane ?⁺ and let K _Θ denote the model subspace H ² ? Θ H ² of the Hardy space H ² = H ²(?⁺). A nonnegative function w on the real line is said to be an admissible majorant for K _Θ if there exists a nonzero function f ∈ K _Θ such that {f} ? w a.e. on ?. We prove a refined version of the parametrization formula for K _Θ-admissible majorants and simplify the admissibility criterion (in terms of arg Θ) obtained in [8]. We show that, for every inner function Θ, there exist minimal K _Θ-admissible majorants. The relationship between admissibility and some weighted approximation problems is considered.     

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 Fourier Transforms of Radial Functions and Distributions

We find a formula that relates the Fourier transform of a radial function on R ⁿ with the Fourier transform of the same function defined on R ⁿ⁺². This formula enables one to explicitly calculate the Fourier transform of any radial function f(r) in any dimension, provided one knows the Fourier transform of the one-dimensional function t?f(|t|) and the two-dimensional function (x ₁,x ₂)?f(|(x ₁,x ₂)|). We prove analogous results for radial tempered distributions.