Dense-choice Counter Machines revisited

This paper clarifies the picture about Dense-choice Counter Machines, which have been less studied than (discrete) Counter Machines. We revisit the definition of "Dense Counter Machines" so that it now extends (discrete) Counter Machines, and we provide new undecidability and decidability results. Using the first-order additive mixed theory of reals and integers, we give a logical characterization of the sets of configurations reachable by reversal-bounded Dense-choice Counter Machines

On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases

This article studies the expressive power of finite automata recognizing sets of real numbers encoded in positional notation. We consider Muller automata as well as the restricted class of weak deterministic automata, used as symbolic set representations in actual applications. In previous work, it has been established that the sets of numbers that are recognizable by weak deterministic automata in two bases that do not share the same set of prime factors are exactly those that are definable in the first order additive theory of real and integer numbers. This result extends Cobham's theorem, which characterizes the sets of integer numbers that are recognizable by finite automata in multiple bases. In this article, we first generalize this result to multiplicatively independent bases, which brings it closer to the original statement of Cobham's theorem. Then, we study the sets of reals recognizable by Muller automata in two bases. We show with a counterexample that, in this setting, Cobham's theorem does not generalize to multiplicatively independent bases. Finally, we prove that the sets of reals that are recognizable by Muller automata in two bases that do not share the same set of prime factors are exactly those definable in the first order additive theory of real and integer numbers. These sets are thus also recognizable by weak deterministic automata. This result leads to a precise characterization of the sets of real numbers that are recognizable in multiple bases, and provides a theoretical justification to the use of weak automata as symbolic representations of sets.Comment: 17 page

G-continuity, impatience and G-cores of exact games

This paper is concerned with real valued set functions defined on the set of Borel sets of a locally compact σ-compact topological space Ω. The first part characterizes the strong and weak impatience in the context of discrete and continuous time flows of income (consumption) valued through a Choquet integral with respect to an (exact) capacity. We show that the impatience of the decision maker translates into continuity properties of the capacity. In the second part, we recall the generalization given by Rébillé [8] of the Yosida-Hewitt decomposition of an additive set function into a continuous part and a pathological part and use it to give a characterization of those convex capacities whose core contains at least one G-continuous measure. We then proceed to characterize the exact capacities whose core contains only G-continuous measures. As a dividend, a simple characterization of countably additive Borel probabilities on locally compact σ-compact metric spaces is obtained.Impatience, exact and convex capacities, G-cores, σcores, Yosida-Hewitt decomposition.

On an Additive Characterization of a Skew Hadamard (n, n−1/ 2 , n−3 4 )-Difference Set in an Abelian Group

We give a combinatorial proof of an additive characterization of a skew Hadamard (n, n−1 2 , n−3 4 )-difference set in an abelian group G. This research was motivated by the p = 4k + 3 case of Theorem 2.2 of Monico and Elia [4] concerning an additive characterization of quadratic residues in Z p. We then use the known classification of skew (n, n−1 2 , n−3 4 )-difference sets in Z n to give a result for integers n = 4k +3 that strengthens and provides an alternative proof of the p = 4k + 3 case of Theorem 2.2 of [4]

A characterization of the support map

AbstractIn this short note we give a characterization of the support map from classical convexity. We show it is the unique additive transformation from the class of closed convex sets in Rn which include 0 to the class of positive 1-homogeneous functions on Rn. This will be a consequence of a theorem about transforms from the class of convex sets to itself which preserve Minkowski addition

Level Sets of Multiparameter Stable Processes

We establish the correct Hausdorff measure function for the level sets of additive strictly stable processes derived from strictly stable processes satisfying Taylor's condition (A). This leads naturally to a characterization of local time in terms of the corresponding Hausdorff measure function of the level se

Aritmetikai függvények, egyértelműségi halmazok, általánosított számrendszerek = Arithmetical functions, sets of uniqueness, generalized number systems

Az alábbi területen sikerült érdekes eredményeket elérni: 1. Additív függvények eloszlása, multiplikatív függvények középértéktétele rövid intervallumokon. Ramachandra Hooley-Huxley kontúrra vonatkozó eredmények alkalmazása. 2. $q$-additív függvények értékeloszlása különböző feltételeket kielégítő részhalmazokon. 3. Azoknak a $q$-multiplikatív függvényeknek a karakterizálása, amelyek a prímszámok halmazán egy adott függvényosztályhoz (\Cal L^\alpha, \Cal L^* tartoznak). 4. Egyértelműségi és $\mod 1$ egyértelműségi halmazok. 5. A Daboussi-féle problémakör. 6. Az Euler-féle $\varphi$ függvény és iteráltjai. 7. Az Erdős-Wintner tétel analogonja az $\{n+1|n\leq x,\ \omega(n)=k\}$ halmazra, ahol $\omega(n)$ az $n$ prímosztóinak a száma. Közlésre elfogadott cikkek: 1. J.-M. De Koninck and I. Kátai, On the local distribution of $\omega(GCD(n,\varphi_k(n)))$, Canadian Math. Bull 2. J.-M. De Koninck and I. Kátai, On an estimate of Kanold, International Journal of Mathematics and Analysis 3. I. Kátai, On $q$-additive and $q$-multiplicative functions, Conference in Allahabad, 2006 December 4. K.-H. Indlekofer and I. Kátai, Some remarks on trigonometric sums, Acta Math. Hung. 5. J.-M. De Koninck, N. Doyon and I. Kátai, Counting the number of twin Niven numbers, Ramanujan Journal 6. I. Kátai and M.V. Subbarao, Distribution of additive and $q$-additive functions under some conditions II., Publ. Math. Debrecen | Interesting and important results have been proved in the following topics: 1. Distribution of additive functions on short intervals. 2. Mean-values of multiplicative functions on short intervals. Application of the method of Ramachandra concerning the Hooley-Huxley contour. 3. Distribution of linear combination of $q$-additive functions on some arithmetically characterized subsets of integers. 4. Characterization of those $q$-multiplicative functions which are defined on the set of primes and belong to some classes of functions (\Cal L^\alpha,\Cal L^*). 5. Sets of uniqueness and sets of uniqueness $\mod 1$. 6. Some analogons of the theorem of Daboussi

Characterization of dual mixed volumes via polymeasures

We prove a characterization of the dual mixed volume in terms of functional properties of the polynomial associated to it. To do this, we use tools from the theory of multilinear operators on spaces of continuos functions. Along the way we reprove, with these same techniques, a recently found characterization of the dual mixed volume