Numerical solutions of integro-differential equations and application of a population model with an improved Legendre method

In this paper, an improved Legendre collocation method is presented for a class of integro-differential equations which involves a population model. This improvement is made by using the residual function of the operator equation. The error differential equation, gained by residual function, has been solved by the Legendre collocation method (LCM). By summing the approximate solution of the error differential equation with the approximate solution of the problem, a better approximate solution is obtained. We give the illustrative examples to demonstrate the efficiency of the method. Also we compare our results with the results of the known some methods. In addition, an application of the population model is made.     

On the Additive Group Structure of the Nonstandard Models of the Theory of Integers

Let $\hat \mathbb{Z}$ denote the inverse limit of all finite cyclic groups. Let F, G and H be abelian groups with H ≤ G. Let FβH denote the abelian group (F × H, +^β), where +^βis defined by (a, x) +^β (b, y) = (a + b, x + y + β(a) + β(b) — β(a + b)) for a certain β : F → G linear mod H meaning that β(0) = 0 and β(a) + β(b) — β(a + b) ∈ H for all a, b in F. In this paper we show that the following hold: (1) The additive group of any nonstandard model ℤ* of the ring ℤ is isomorphic to (ℤ*₊/H)βH for a certain β : ℤ*₊/H → $\hat \mathbb{Z}$ linear mod H. (2) $\hat \mathbb{Z}$ is isomorphic to (ℤ₊/H )βH for some β : $\hat \mathbb{Z}$/H →ℚ linear mod H, though $\hat \mathbb{Z}$ is not the additive group of any model of Th(ℤ, +, ×) and the exact sequence H → $\hat \mathbb{Z}$ → $\hat \mathbb{Z}$/H is not splitting.     

On Darboux helices in Euclidean 4‐space

The notion of Darboux helix in Euclidean 3‐space was introduced and studied by Yayl? et al. 2012. They show that the class of Darboux helices coincide with the class of slant helices. In a special case, if the curvature functions satisfy the equality κ² + τ² = constant, then these curves are curve of the constant precession. In this paper, we study Darboux helices in Euclidean 4‐space, and we give a characterization for a curve to be a Darboux helix. We also prove that Darboux helices coincide with the general helices. In a special case, if the first and third curvatures of the curve are equal, then Darboux helix, general helix, and V₄‐slant helix are the same concepts.     

Commutativity of first‐order discrete‐time linear time‐varying systems

After introducing the concept of commutativity for continuous‐time linear time‐varying systems, the related literature and the results obtained so far are presented. For a simple introduction of the commutativity of discrete‐time linear time‐varying systems, the problem is formulated for first‐order systems. Finally, explicit necessary and sufficient conditions for the commutativity of first‐order discrete‐time linear time‐varying systems are derived, and their advantageous use in digital system design is illustrated, which are the main objectives of the paper. The results are verified by examples which include an application in amplitude modulation for digital telecommunication.     

Distance of convergence in measure on spaces of measurable functions

Positivity - In this paper we investigate the distance of convergence in measure whenever the measure is not $$sigma $$ -finite and identify the topological coreflection of this approach structure...     

Finite element method for the vibration of cracked beams with varying cross section

Vibration analysis of cracked beams having linearly varying cross-sections both in thickness and width was investigated. A computer program using the finite element method has been written to find the dynamical characteristics (natural frequencies and mode shapes) of the cracked beam. The cracked section in the beam has been modeled by a massless spring whose flexibility depens on the local flexibility induced by the crack. The stiffness of spring has been derived from the linear elastic fracture mechanics theory as the inverse of the compliance matrix calculated using stress intensity factors and strain energy release rate expression. Some examples have been given to explain the proposed method and investigate the effects of the depth and location of cracks on the natural frequencies and mode shapes. The results of current study and those in the literature are compared and good agreements have been found. Consequently it is showed that proposed method is reliable and simple. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical damage assessment of Haghia Sophia bell tower by nonlinear FE modeling

In the present paper, the numerical damage assessment of the masonry bell tower called “Haghia Sophia” in Trabzon, Turkey is performed by nonlinear 3D finite element modeling. The behavior of bell tower is determined under several different conditions: nonlinear static analysis containing dead and wind loads and nonlinear seismic analysis. In addition to, an assessment of the tower’s stability with respect to the tilt of the tower is carried out by means of a nonlinear analysis. In the nonlinear dynamic analysis, the east–west component of 1992 Erzincan earthquake is used. Cracking and crushing of the masonry have been taken into account, as well as the influence of material nonlinearity. The numerical analysis has given a valuable picture of possible damage evolution, providing useful hints for the prosecution of structural monitoring. The displacement and stress fields, as well as the distribution of cracking have been calculated and compared to the actual distribution of fractures in the tower. It is seen from the numerical results that there is a good agreement with present damages of the bell tower.     

SLANT SUBMANIFOLDS OF A RIEMANNIAN PRODUCT MANIFOLD

In this article, the geometry of the slant submanifolds of a Riemannian product manifold is studied. Some necessary and sufficient conditions on slant, bi-slant and semi-slant submanifolds are given. We research fundamental properties of the distributions which are involved in definitions of semi- and bi-slant submanifolds in a Riemannian product manifold.     

Korovkin type approximation theorem for BBH type operators via <Emphasis Type="Italic">J</Emphasis>-convergence

This paper provides a Korovkin type approximation theorem for a class of positive linear operators including Bleimann-Butzer and Hahn operators via J-convergence.