The first theorem related to the denseness of the image of a vertical line Re s = σ0, σ0 > 1 by the Riemann Zeta function has been proved by Harald Bohr in 1911. We argue that this theorem is not really a denseness theorem. Later Bohr and Courant proved similar theorems for the case 1/2 < Re s ≤ 1. Their results have been generalized to classes of Dirichlet functions and are at the origin of a burgeoning field in analytic number theory, namely the universality theory. The tools used in this theory are mainly of an arithmetic nature and do not allow a visualization of the phenomena involved. Our method is based on conformal mapping theory and is supported by computer generated illustrations. We generalize and refine Bohr and Courant results.
Aims: A shape optimization technique is developed, using the boundary element method, for two-dimensional anisotropic structures to study the effects of anisotropy on the displacements and stresses, then minimize weight while satisfying certain constraints upon stresses and geometry.
Study Design: Original Research Paper.
Place and Duration of Study: Jamoum University College, Mathematics Department, between June 2016 and July 2017.
Methodology: The shape design sensitivity analysis of a two-dimensional anisotropic structure using a singular formulation of the boundary element method is investigated to study the effects of anisotropy on the displacements and stresses. An Implicit differentiation technique of the discretized boundary integral equations is performed to produce terms that contain derivatives of the fundamental solutions employed in the analysis. This technique allows the coupling between optimization technique and numerical boundary element method (BEM) to form an optimum shape design algorithm that yields shape design sensitivities of the displacement and stress fields for anisotropic materials with very high accuracy. The fundamental solutions of displacements and tractions in terms of complex variables employed in the analysis. The feasible direction method was developed and implemented for use with the golden-section search algorithm based on BEM as a numerical optimization technique for minimizing weight while satisfying all of the constraints.
Results: The proposed method has been verified by using the two-dimensional plate with an elliptical hole as the numerical example. The numerical results show that the proposed method is suitable and effective tool for the computer implementation of the solution.
Conclusion: From the research that has been performed, it is possible to conclude that the optimal shape of the two-dimensional plate with an elliptical hole is crucial when elastic field is sensitive to boundary shape. Also from this knowledge of the effects of anisotropy on the displacements and stresses, we can design various anisotropic structures to meet specific engineering requirements and utilize within which to place new information can be more effective.
The original triple-protocol is proposed, which is a modification of the well-known Massey - Omura protocol, but greatly improves it from the standpoint of efficiency and enables data to be validated for integrity and authenticity without the use of hash functions. This is ensured by forwarding through open channels to an external ("cloud") information carrier (and also directly to a participant in the information exchange) of some data obtained from mathematical and logical transformations above the original message. The combined use of modulo exponentiation and the XOR logical operation ("exclusive OR") ensures that unauthorized access for an intruder is impossible, bypassing the difficult mathematical problem of discrete logarithm (ECDLP), on which all the algorithms for data transfer on elliptical curves are based.
We will study anew graph, this graph called fuzzy tangle graph, we will study the matrices which represent this graph, and we will discuss the relation between fuzzy tangle graph and dual fuzzy tangle graph. In fuzzy tangle graph the concepts of α-cut tangle graph, strength of edge are developed.
As a random variable, the survival time or Time to Failure (TTF) of a certain component or system can be fully characterized by its probability density function (pdf) fT (t) or its Cumulative Distribution Function (CDF) FT (t). Moreover, it might be also identified by transform functions such as the Moment Generating Function (MGF) and the Characteristic Function (CF). In reliability engineering, additional specific equivalent characterizations are used including the reliability function (survival function) which is the Complementary Cumulative Distribution Function (CCDF), and the failure rate (hazard rate), which is the probability density function normalized w.r.t. reliability. In prognostics, a prominent emerging subfield of reliability engineering, the characterizing functions are still supplemented by other specifically tailored ones. Notable among these is the Mean Residual Life (MRL) (also know as the Remaining Useful Life (RUL)). The purpose of this paper is to compile and interrelate the most prominent among these characterizing functions and explore their important properties. The paper points out that there is currently a significant proliferation of characterizing functions emerged in various fields. It shows that, under mild conditions, the product and quotient of two characterizing functions are also characterizing functions. The choice of one characterizing function in a certain application is a matter of convenience and taste. Our survey is far from being a conclusive one as it is intended to be just a brief tutorial guide for prognostics scholars, especially beginners. We had to arbitrarily leave out many of the less known characterizing functions such as the aging intensity function, log-odds rate, and entropy-related functions.