This paper develops a theory for characterization of DNA sequences based on their measure representation. The measures are shown to be random cascades generated by an infinitely divisible distribution. This probability distribution is uniquely determined by the exponent function in the multifractal theory of random cascades. Curve fitting to a large number of complete genomes of bacteria indica...

For a complete graph $$K_n$$ of order n, an edge-labeling $$c:E(K_n)\rightarrow \{ -1,1\}$$ satisfying $$c(E(K_n))=0$$ , and spanning forest F we consider the problem to minimize $$|c(E(F'))|$$ over all isomorphic copies $$F'$$ in . In particular, ask under which additional conditions there is zero-sum copy, that is, copy with $$c(E(F'))=0$$ We show always $$|c(E(F'))|\le \Delta (F)+1$$ where $...

Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a “definite totality” but rather is “indefinitely extensible”. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this pictur...

In this paper, we study the risk bounds for samples independently drawn from an infinitely divisible (ID) distribution. In particular, based on a martingale method, we develop two deviation inequalities for a sequence of random variables of an ID distribution with zero Gaussian component. By applying the deviation inequalities, we obtain the risk bounds based on the covering number for the ID d...

We prove that many of beta, beta prime, gamma, inverse gamma, Student tand ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, many of ultraspherical and Student t-distributions have free divisibility indicator 1.

The paper is concerned with algebras whose elements can be used to represent runs of a system from a state to a state. These algebras, called multiplicative transition systems, are categories with respect to a partial binary operation called composition. They can be characterized by axioms such that their elements and operations can be represented by partially ordered multisets of a certain typ...

We prove that the q-Gaussian distribution is freely infinitely divisible for all q ∈ [0, 1].