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The object of the research are fuzzy functional dependencies on given relation scheme, and the question of their obtaining using the classical and innovated techniques. The attributes of the universal set are associated to the elements of the unit interval, and are turned into fuzzy formulas in this way. We prove that the dependency (which is treated as a fuzzy formula with respect to appropriately chosen valuation) is valid whenever it agrees with the attached two-elements fuzzy relation instance. The opposite direction of the claim is proven to be incorrect in this setting. Generalizing things to sets of attributes, we prove that particular fuzzy functional dependency follows form a set of fuzzy dependencies (in both, the world of two-element and the world of arbitrary fuzzy relation instances) if and only if the dependency is valid with respect to valuation anytime the set of fuzzy formulas agrees with the valuation. The results derived in paper show that the classical techniques in the procedure for generating new fuzzy dependencies may be replaced by the resolution ones, and hence automated. The research is conducted with respect to Willmott fuzzy implication operator

In this paper we consider all possible dependencies that can be built upon similarity-based fuzzy relations, that is, fuzzy functional and fuzzy multivalued dependencies. Motivated by the fact that the classical obtaining of new dependencies via inference rules may be tedious and uncertain, we replace it by the automated one, where the key role is played by the resolution principle techniques and the fuzzy formulas in place of fuzzy dependencies. We prove that some fuzzy multivalued dependency is actively correct with respect to given fuzzy relation instance if and only if the corresponding fuzzy formula is in line with the attached interpretation. Additionally, we require the tuples of the instance to be conformant (up to some extent) on the leading set of attributes. The equivalence as well as the conclusion are generalized to sets of attributes. The research is conducted by representing the attributes and fuzzy dependencies in the form of fuzzy formulas, and the application of fuzzy implication operators derived from carefully selected Frank’s classes of additive generators

In his recent research, the author improved the error term in the prime geodesic theorem for compact, even-dimensional, rank one locally symmetric spaces. It turned out that the obtained estimate $O(x^{2\rho-\frac{\rho}{n}}(\log x)^{-1})$ coincides with the best known results for compact Riemann surfaces, three manifolds, and manifolds with cusps, where $n$ stands for the dimension of the space, and $\rho$ is the half-sum of positive roots. The above bound was then reduced to $O(x^{2\rho-\rho\frac{2\cdot(2n)+1}{2n\cdot(2n)+1}}(\log x)^{\frac{n-1}{2n\cdot(2n)+1}-1}(\log\log x)^{\frac{n-1}{2n\cdot(2n)+1}+\varepsilon})$ in the Gallagherian sense, with $\varepsilon$ $>$ $0$, and the key role played by the counting function $\psi_{2n}(x)$. The purpose of this research is to prove that the latter $O$-term can be further reduced. To do so, we derive new explicit formulas for the functions $\psi_{j}(x)$, $j$ $\geq$ $n$, and conditional formula for $\psi_{n-1}(x)$. Applying the Gallagher-Koyama techniques, we deduce the asymptotics for $\psi_{0}(x)$, and the Gallagherian prime geodesic theorems. The obtained error terms $O(x^{2\rho-\rho\frac{2j+1}{2nj+1}}(\log x)^{\frac{n-1}{2nj+1}-1}(\log\log x)^{\frac{n-1}{2nj+1}+\varepsilon})$, $n-1$ $\leq$ $j$ $<$ $2n$, improve the $O$-term given above, with the optimal unconditional (conditional) size achieved for $j$ $=$ $n$ ($j$ $=$ $n-1$). If $j$ $=$ $n$ $\geq$ $4$, our new bound coincides with the best known estimate in the manifolds with cusps case. If $j$ $=$ $n-1$, the $O$-term fully agrees with the results in the Riemann surface case ($n$ $=$ $2$, $\rho$ $=$ $\frac{1}{2}(n-1)$ $=$ $\frac{1}{2}$), and the three manifolds case ($n$ $=$ $2$, $\rho$ $=$ $1$). Finally, for $j$ $=$ $n-1$, $n$ $\geq$ $4$, our result improves the best known bound in the manifolds with cusps case.

We obtain precise estimates for the number of singularities of Selberg’s and Ruelle’s zeta functions for compact, higher-dimensional, locally symmetric Riemannian manifolds of strictly negative sectional curvature. The methods applied in this research represent a generalization of the methods described in the case of a compact Riemann surface. In particular, this includes an application of the Phragmen-Lindelof theorem, the variation of the argument of certain zeta functions, as well as the use of some classical analytic number theory techniques.

In this paper we apply the h-generated fuzzy implications to prove a number of results which are of fundamental importance to the theory of fuzzy and vague functional and multivalued dependencies defined on given scheme. Our research is motivated by the fact that some analogous results already hold true for the families of f- and g-generated fuzzy implications, and the fact that these three collections of implications share many similar mutual properties. While some of the aforementioned implications are introduced in order to be applied in approximate reasoning, the results derived in this paper represent the main tool in the process of automation and are also used to complement the resolution principle. More precisely, the main result of this research states that the fact that some fuzzy (vague) relation instance r, |r| = 2, satisfies some fuzzy (vague) functional or fuzzy (vague) multivalued dependency c /∈ C (under assumption that r satisfies some set C of fuzzy (vague) functional and fuzzy (vague) multivalued dependencies), yields that the fuzzy formula attached to c is valid whenever all of the fuzzy formulas attached to the elements of C are valid. What is more important is that the opposite claim is also proven. Its importance stems from the fact that the verification by hand, which means purely theoretical verification, that C implies c is not required anymore. Now, in order to prove that some C yields some c, it is enough to make the use of the resolution principle, and automatically verify whether or not the set of the attached fuzzy formulas yields the fuzzy formula attached to c. In the case of affirmative answer, the desired dependency follows. The research conducted in this paper represent a natural generalization of our previous research since it includes and considers both, fuzzy and vague theories.

Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime geodesic theorems that are associated with this class of spaces. First, following classical Randol’s appraoch in the compact Riemann surface case, we improve the error term in the corresponding result. Second, we reduce the exponent in the newly acquired remainder by using the Gallagher–Koyama techniques. In particular, we improve DeGeorge’s bound Oxη, 2ρ − ρn ≤ η < 2ρ up to Ox2ρ−ρnlogx−1, and reduce the exponent 2ρ − ρn replacing it by 2ρ − ρ4n+14n2+1 outside a set of finite logarithmic measure. As usual, n denotes the dimension of the underlying locally symmetric space, and ρ is the half-sum of the positive roots. The obtained prime geodesic theorem coincides with the best known results proved for compact Riemann surfaces, hyperbolic three-manifolds, and real hyperbolic manifolds with cusps.

To tackle a specific class of engineering problems, in this paper, we propose an effectively integrated bat algorithm with simulated annealing for solving constrained optimization problems. Our proposed method (I-BASA) involves simulated annealing, Gaussian distribution, and a new mutation operator into the simple Bat algorithm to accelerate the search performance as well as to additionally improve the diversification of the whole space. The proposed method performs balancing between the grave exploitation of the Bat algorithm and global exploration of the Simulated annealing. The standard engineering benchmark problems from the literature were considered in the competition between our integrated method and the latest swarm intelligence algorithms in the area of design optimization. The simulations results show that I-BASA produces high-quality solutions as well as a low number of function evaluations.

In this paper, we consider the remainder in a weighted form of the length spectrum for compact Riemann surfaces of genus greater than or equal to two. Earlier, we conducted a similar research where we applied the Cauchy residue theorem over two different square boundaries, one of which intersected the corresponding critical line, and some, quite complex estimates for the logarithmic derivative of the associated zeta functions of Selberg and Ruelle. The main goal of this paper is to achieve the same length spectrum with the same remainder as in our previous study, but in a much simpler way.

In this paper we consider a generalized length spectrum in the case of compact symmetric spaces generated as quotients of the special linear group of order four over real numbers. While the classical length spectrum is given as an estimate for a yes function counting prime geodesics of appropriate length, its generalized form is usually represented by a higher order counting function of Chebyshev type. Our goal is to prove that the error term that appears in the classical case in this setting can be significantly improved when derived via analogous, generalized apparatus.

The goal of the paper is to derive some approximate formulas for the logarithmic derivative of several zata functions of Selberg’s type for compact symmetric spaces formed as quotients of the Lie group SL4 (R). Such formulas, known in literature as Tutchmarsh-Landau style approximate formulas, are usually applied in order to obtain prime geodesic theorems in various settings of underlying locally symmetric spaces.

This paper is devoted to some counting functions of level one and level three in the case of quotient space generated by some strictly hyperbolic Fuchsian group and the upper half-plane. Each of the functions is represented as a sum of some explicit part plus the error term. The explicit part is indexed over singularities of the corresponding Selberg zeta function. In particular, the obtained error term is not larger than O ( x 3/4) . The method applied in this paper follows traditional approach for achieving the error terms in the case of locally symmetric spaces of real rank one. In order to establish an analogy with the classical case, we consider the counting functions divided by x and x3, respectively.

Our object of research are certain higher order counting functions of Chebyshev type, associated to the compact symmetric space SL4. In particular, we consider the functionψ1(x)resp.ψ3(x), of order1resp.3.As it is well known, any such function can be represented as a sum of some explicit part, and the corresponding error term. The explicit part is usually indexed over singularities of the attached Selberg zeta functions, while the error term depends on the dimension of the underlying symmetric space. Thus, these functions generalize the classical yes functionπ(x)counting prime geodesics of appropriate length. More precisely, the Chebyshev functions divided by adequate power of x, represent quite natural approximations for the functionπ(x). In this research, we are particularly interested in the error terms ofψ1(x)/xandψ3(x)/x3.

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 classical approach is replaced by its higher level analogue.