Abstract:
The de Broglie-Einstein velocity equation is derived for a relativistic particle by using the energy and momentum relations in terms of wave and matter properties. It is shown that the velocity equation is independent from the relativistic effects and is valid also for the non-relativistic case. The results of this property is discussed.

Abstract:
There are three approaches for the solution of the diffraction problem of plane waves by an impedance half-plane in the literature. The diffracted field expressions, obtained by the related methods, are compared numerically. The examination of the scattered field shows that the most reliable solution is the field representation of Raman and Krishnan. Since the diffracted fields of Senior and Maliuzhinets do not compensate the discontinuities of the geometrical optics waves at the transition regions.

Abstract:
An exact form for the equivalent edge current is derived by using the axioms of the modified theory of physical optics and the canonical problem of half-plane. The edge current is expressed in terms of the parameters of incident and scattered rays. The analogy of the method with the boundary diffraction wave theory is put forward. The edge and corner diffracted waves are derived for the problem of a black half-strip.

Abstract:
The memristor theory of Chua [1] provides a connection with the charge and magnetic flux in an electric circuit. We define a similar relation for the electric and magnetic flux densities in electromagnetism. Such an attempt puts forward interesting results. For example, the magnetic charges do not exist in nature however the electric charges behave as the magnetic monopoles in special media. We support our theory with results of the recent experiments on materials named as spin ice.

Abstract:
Diffraction of scalar plane waves by resistive surfaces are investigated by defining a new boundary condition in terms of the Dirichlet and Neumann conditions. The scattering problems of waves by a resistive half-plane and the interface between resistive and perfectly magnetic conducting half-planes are examined with the developed method. The resulting fields are plotted numerically. The numerical results show that the evaluated field expressions are in harmony with the theory.

Abstract:
Positive Dehn twist products for some elements of finite order in the mapping class group of a 2-dimensional closed, compact, oriented surface $\Sigma_g$, which are rotations of $\Sigma_g$ through $2\pi /p$, are presented. The homeomorphism invariants of the resulting simply connected symplectic 4- manifolds are computed.

Abstract:
In this article we find an upper and lower bound for the slope of genus g hyperelliptic Lefschetz fibrations, which is sharp when g = 2, and demonstrate the strong connection, in general, between the slope of hyperelliptic genus g Lefschetz fibrations and the number of separating vanishing cycles. Specifically, we show that the slope is greater than 4-4/g if and only if the fibration contains separating vanishing cycles. We also improve the existing bound on s/n, the ratio of number of separating vanishing cycles to the number of non-separating vanishing cycles, for hyperelliptic Lefschetz fibrations of genus g>1. In particular we show that s<=n for such fibrations when g>5.

Abstract:
Let n and m be infinite cardinals with n ￠ ‰ ¤m and n be a regular cardinal. We prove certain implications of [n,m]-strongly paracompact, [n,m]-paracompact and [n,m]-metacompact spaces. Let X be [n, ￠ ]-compact and Y be a [n,m]-paracompact (resp. [n, ￠ ]-paracompact), Pn-space (resp. wPn-space). If m= ￠ ‘k Keywords strongly paracompact and metacompact spaces.

Abstract:
A positive Dehn twist product for a $\mathbb{Z}_3$ action with $g+2$ fixed points on the 2-dimensional closed, compact, oriented surface $\Sigma_g$ is presented. The homeomorphism invariants of the resulting symplectic 4-manifolds are computed.

Abstract:
We discuss a connection between the lantern relation in mapping class groups and the rational blowing down process for 4-manifolds. More precisely, if we change a positive relator in Dehn twist generators of the mapping class group by using a lantern relation, the corresponding Lefschetz fibration changes into its rational blowdown along a copy of the configuration C_2. We exhibit examples of such rational blowdowns of Lefschetz fibrations whose blowup is homeomorphic but not diffeomorphic to the original fibration.