In this paper we consider fuzzy functional and fuzzy multivalued dependencies introduced by Sozat and Yazici. We appropriately relate these dependencies to fuzzy formulas. In particular, we relate any subset of the universal set of attributes to fuzzy conjunction of its attributes. Thus, being in the form of implication between such subsets, we naturally relate a fuzzy dependency to fuzzy implication between corresponding fuzzy conjunctions. In this paper we choose standard min, max as well as Yager’s fuzzy implication operator for definitions of fuzzy conjunction, fuzzy disjunction and fuzzy implication, respectively. If any two-element fuzzy relation instance on a given scheme, known to satisfy some set of fuzzy functional and fuzzy multivalued dependencies, satisfies some fuzzy functional or fuzzy multivalued dependency f which is not member of the given set of fuzzy dependencies, then, we prove that satisfiability of the related set of fuzzy formulas yields satisfiability of the fuzzy formula related to f and vice versa. A methodology behind the proofs of our results is mainly based on an application of definitions of the introduced fuzzy logic operators. Our results can be verified for various choices of fuzzy logic operators however.

— Applying a definition of attribute conformance based on a similarity relation, we introduce an interpretation as a function associated to some fuzzy relation instance and defined on the universal set of attributes. As a consequence, the attributes become fuzzy formulas. Conjunctions, disjunctions and implications between the attributes become fuzzy formulas as well in view of the requirement that the interpretation has to agree with the minimum t-norm, the maximum t-conorm and appropriately chosen fuzzy implication. The purpose of this paper is to derive a number of results related to these fuzzy formulas if the fuzzy implication is selected so to be either Reichenbach or some f-generated fuzzy implication.

Applying a suitably derived, Titchmarsh-Landau style approximate formula for the logarithmic derivative of the Ruelle zeta function, we obtain an another proof of the recently improved variant of the prime geodesic theorem for compact, even-dimensional, locally symmetric spaces of real rank one

Prime geodesic theorem gives an asymptotic estimate for the number of prime geodesics over underlying symmetric space counted by their lengths. In any setting, the search for the optimal error term is widely open. Our objective is to derive a weighted, generalized form of the prime geodesic theorem for compact, even-dimensional, locally symmetric Riemannian manifolds of strictly negative sectional curvature. We base our methodology on an application of the integrated, Chebyshev-type counting function of appropriate order. The obtained error term improves the corresponding, and best known one in the case of classical prime geodesic theorem. Our conclusion in the case at hand is that a weighted sense yields a better result.

In this paper we generalize our most recent results that are related to the algorithm that has been developed to automatically derive a fuzzy functional or a fuzzy multivalued dependency from a given set of fuzzy functional and fuzzy multivalued dependencies. Fuzzy dependencies are considered as fuzzy formulas. The first result states that a two-element fuzzy relation instance actively satisfies a fuzzy multivalued dependency if and only if the tuples of the instance are conformant on some known set of attributes with degree of conformance larger than some known constant, and the corresponding fuzzy formula is valid in appropriate interpretations. The second result states that a fuzzy functional or a fuzzy multivalued dependency follows from a set of fuzzy functional and fuzzy multivalued dependencies in two-element fuzzy relation instances if and only if the corresponding fuzzy formula is a logical consequence of the corresponding set of fuzzy formulas. Our earlier research in this direction consisted in an application of some individual fuzzy implication operator, such as Yager, Reichenbach, Kleene-Dienes fuzzy implication operator. The main purpose of this paper is to prove that the aforementioned results remain valid for a wider class of fuzzy implication operators, in particular for the family of f-generated fuzzy implication operators

To prove that a fuzzy dependency follows from a set of fuzzy dependences can be a very demanding task. As far as we know, an algorithm or an application that generally and automatically solves the problem, does not exist. The main goal of this paper is to offer such an algorithm. In order to achieve our goal we consider fuzzy dependences as fuzzy formulas. In particular, we fix fuzzy logic operators: conjunction, disjunction and implication, and allow only these operators to appear within fuzzy formulas. Ultimately, we prove that a fuzzy dependency follows from a set of fuzzy dependences if and only if the corresponding fuzzy formula is a logical consequence of the corresponding set of fuzzy formulas. To prove an implication of the last type, one usually uses the resolution principle, i.e., the steps that can be fully automated. Our methodology assumes the use of soundness and completeness of fuzzy dependences inference rules as well as the extensive use of active fuzzy multivalued dependences fulfillment

We obtain improved asymptotic estimate for the function enumerating prime geodesics over compact locally symmetric space of real rank one.

We use B. Randol’s method to improve the error term in the prime geodesic theorem for a noncompact Riemann surface having at least one cusp. The case considered is a general one, corresponding to a Fuchsian group of the first kind and a multiplier system with a weight on it.

We prove that the meromorphic continuations of the Ruelle and Selberg zeta functions considered by Bunke and Olbrich are of finite order not larger than the dimension of the underlaying compact, odd-dimensional, locally symmetric space.

Abstract We derive approximate formulas for the logarithmic derivative of the Selberg and the Ruelle zeta functions over compact, even-dimensional, locally symmetric spaces of real rank one. The obtained formulas are given in terms of zeta singularities.

For compact, even-dimensional, locally symmetric spaces, we obtain precise estimates on the number of singularities of Selberg's and Ruelle's zeta functions considered by U. Bunke and M. Olbrich.

In this paper we pay our particular attention to the error term in the prime geodesic theorem for compact symmetric spaces represented as double coset spaces of the special linear group of order four over real numbers. It is known that in the case of compact locally symmetric Riemannian manifolds of strictly negative sectional curvature, the corresponding error term depends on classification of Riemannian symmetric spaces of real rank one. In particular, the error term is a function depending on the dimension of the underlying locally symmetric space. In this research we prove that the error term in the case at hand is a function depending on the degree of the polynomial that appears in the functional equation of the corresponding Selberg zeta function.