The purpose of this paper is two-sided. First, we obtain the correct estimate of the error term in the classical prime geodesic theorem for compact symmetric space SL4. As it turns out, the corrected error term depends on the degree of a certain polynomial appearing in the functional equation of the attached zeta function. This is in line with the known result in the case of compact Riemann surface, or more generally, with the corresponding result in the case of compact locally symmetric spaces of real rank one. Second, we derive a weighted form of the theorem. In particular, we prove that the aforementioned error term can be significantly improved when the classicalapproachisreplacedbyitshigherlevelanalogue. Key–Words: Symmetric spaces, counting functions, length spectrum, zeta functions, Riemann zeta Received: August7, 2019. Revised: February 10, 2020. Accepted: February 21, 2020. Published: March 9, 2020.

—In 1949, A. Selberg discovered a real variable (an elementary) proof of the prime number theorem. A number of authors have adapted Selberg’s method to achieve quite a good corresponding error term. The Riemann hypothesis has never been proved or disproved however. Any generalization of the prime number theorem to the more general situations is known in literature as a prime geodesic theorem. In this paper we derive yet another proof of the prime geodesic theorem for compact symmetric spaces formed as quotients of the Lie group SL 4 ( R ) . While the first known proof in this setting applies contour integration over square boundaries, our proof relies on an application of modified circular boundaries. Recently, A. Deitmar and M. Pavey applied such prime geodesic theorem to derive an asymptotic formula for class numbers of orders in totally complex quartic fields with no real quadratic subfields.

This paper represents a natural continuation of our previous study. In our earlier research we proved that the inclusive inference rule and the union inference rule for new vague functional dependencies are sound, and sketched a proof of the fact that the set of the main inference rules is a complete set. In the present paper we rigorously prove that: reflexive, augmentation, transitivity, pseudo-transitivity, and decomposition inference rules are also sound. Some additional insights in completeness of the main inference rules are also provided.

In this paper we complement the most recent results on soundness of inference rules for new vague multivalued dependencies. Motivated by the fact that the inclusive and the augmentation rules are sound, we prove that: complementation, transitivity, replication, coalescence, union, pseudo-transitivity, decomposition, and mixed pseudo-transitivity rules are also sound. Our research relies on definitions of vague functional and vague multivalued dependencies based on appropriately selected similarity measures between vague values, vague sets, and tuples on sets of attributes.

Klir-Yuan fuzzy implication, as fuzzy implication generated from the standard strong fuzzy negation, the probabilistic sum t-conorm, and the product t-norm, represents a classical example of QL-implication, where QL-implications are the short for quantum logic fuzzy implications. In this paper we prove that the recent results on equivalences between fuzzy formulas and fuzzy dependencies remain invariant with respect to QL-implications when considered through Klir-Yuan fuzzy implication.

In the present paper we give a new definition of vague multivalued dependencies in database relations. The definition is based on application of arbitrary similarity measure on vague values, which is known to be reflexive, symmetric, and max-min transitive. The definition is adapted in order to include the imprecise and precise vague multivalued dependencies. The inference rules for new vague multivalued dependencies are listed, and are shown to be sound.

Abstract: We derive a prime geodesic theorem for compact, odd-dimensional, real hyperbolic spaces. The obtained result corresponds to the best known result obtained in the compact, even-dimensional case, as well as to the best known result obtained in the case of non-compact, real hyperbolic manifolds with cusps. The result derived in this paper follows from the fact that the prime geodesic theorem gives a growth asymptotic for the number of closed geodesics counted by their lengths, and the fact that free homotopy classes of closed paths on compact locally symmetric Riemannian manifold with negative sectional curvature are in natural one-to-one correspondence with the set of conjugacy classes of the corresponding discrete, co-compact, torsion-free group. The current article is dedicated to quotients of the real hyperbolic space.

Abstract: In this paper we refine the error term in the prime geodesic theorem for compact, even-dimensional locally symmetric Riemannian manifolds with strictly negative sectional curvature. The ingredients for the starting prime geodesic theorem come from our most recent research of the zeta functions of Selberg and Ruelle associated with locally homogeneous bundles over compact locally symmetric spaces of rank one. In this paper, we shall restrict our investigations to compact, even-dimensional, locally symmetric spaces. For this class of spaces, we prove that there exists a set ∇ of positive real numbers, which is relatively small in the sense that its logarithmic measure is finite, such that the error term of the aforementioned prime geodesic theorem is improved outside ∇. The derived prime geodesic theorem generalizes the corresponding prime number theorem, where it is proved that the error term under Riemann hypothesis assumption can be further reduced except on a set of finite logarithmic measure.

In this paper we introduce a new definition of vague functional dependency based on application of appropriately chosen similarity measures. The definition is adjusted in order to be applicable to both, the imprecise and precise vague functional dependencies. Ultimately, the set of inference rules for new vague functional dependencies is given, and is proven to be sound and complete.

In this paper we pay attention to completeness of the inference rules for vague functional and vague multivalued dependencies in two-element, vague relation instances. Motivated by the fact that the set of the inference rules is a complete set, that is, these exists a vague relation instance on given relation scheme which satisfies all vague functional and vague multivalued dependencies in the closure of the union of some set of vague functional and some set of vague multivalued dependencies, and violates a vague functional, respectively, a vague multivalued dependency outside of the closure, we prove that the vague relation instance may be chosen to contain only two elements.

Abstract: In this paper we investigate the Selberg zeta functions and the Ruelle zeta functions associated with locally homogeneous bundles over compact locally symmetric spaces of rank one. Our basic object will be a compact locally symmetric Riemannian manifold with negative sectional curvature. In particular, our research will be restricted to compact, odd-dimensional, real hyperbolic spaces. For this class of spaces, the Titchmarsh-Landau style approximate formulas for the logarithmic derivative of the aforementioned zeta functions are derived. As expected in this setting, the obtained formulas are given in terms of zeros of the attached Selberg zeta functions. Our results follow from the fact that these zeta functions can be represented as quotients of two entire functions of order not larger than the dimension of the underlying compact, odd-dimensional, locally symmetric space, and the application of suitably chosen Weyl asymptotic law. The obtained formulas can be further applied in the proof of the corresponding prime geodesic theorem.

As it is known, there have been a number of attempts to obtain precise estimates for the number of primes not exceeding x. A lot of them are related to the ones done by Chebyshev. Thus, a good deal is known about them and their limitations. The truth, or otherwise, of the Riemann hypothesis, however, has still not been established. In this paper we derive a prime geodesic theorem for a compact Riemann surface regarded as a quotient of the upper halfplane by a discontinuous group. We assume that the surface at case, considered as a compact Riemannian manifold, is equipped with classical Poincare metric. Our result follows from the standard theory of the zeta functions of Selberg and Ruelle. The closed geodesics in this setting are in one-to-one correspondence with the conjugacy classes of the corresponding group, so analysis conducted here is reminiscent of the relationship between the distribution of rational primes and Riemann zeta function. By analogy with the classical arithmetic case and the fact that the Riemann hypothesis is true in our setting, one would certainly expect to obtain an analogous error term in the prime geodesic theorem. Bearing in mind that the corresponding Selberg zeta funcion has much more zeros than the Riemann zeta, the latter is not satisfied however. Keywords—Compact Riemann surfaces, prime geodesic theorem, upper half-plane, zeta functions, Laplace operator.

In this paper we use the g-generated fuzzy implications to research the concept of automatization in the process of derivation of new fuzzy functional and fuzzy multivalued dependencies from some given set of fuzzy functional and fuzzy multivalued dependencies. The formal definitions of fuzzy functional and fuzzy multivalued dependencies that we apply are based on application of similarity relations and conformance values. In this context, the paper follows similarity based fuzzy relational database approach. In order derive and then apply our results, we identify fuzzy dependencies with fuzzy formulas. The obtained results are verified through the resolution principle.

In this paper we prove that the set of the main inference rules for new vague functional and vague multivalued dependencies is complete set. More precisely, we prove that there exists a vague relation instance on given scheme, which satisfies all vague functional and vague multivalued dependencies from the set of all vague functional and vague multivalued dependencies that can be derived from given ones by repeated applications of the main inference rules, and violates given vague functional resp. vague multivalued dependency which is initially known not to be an element of the aforementioned set of derived vague dependencies. The paper can be considered as a natural continuation of our previous study, where new definitions of vague functional and vague multivalued dependencies are introduced, the corresponding inference rules are listed, and are shown to be sound

A fuzzy formula does not necessarily follow from a set of fuzzy formulas. In the case when fuzzy formulas and fuzzy dependencies are mutually identified, the corresponding equivalent statement has an obvious meaning. An affirmative statement, however, rises the question of automatization. In our earlier research, we offered an efficient algorithm based on application of selected fuzzy logic operators and resolution principle. In this paper we prove that those ingredients of the algorithm that explicitly depend on the choice of fuzzy implication operator, hold also true for the class of g-generated fuzzy implications.