Abstract:
Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. The most interesting solutions are probably those inspired from an integral formula that was first introduced in this theory by Herbert Busemann. Besides that, Busemann and his school made a thorough investigation of metrics defined on subsets of projective space for which the projective lines are geodesics and they obtained several results, characterizing several classes of such metrics. We review some of the developments and important results related to Hilbert's problem, especially those that arose from Busemann's work, mentioning recent results and connections with several branches of mathematics, including Riemannian geometry, the foundations of mathematics, the calculus of variations, metric geometry and Finsler geometry.

Abstract:
The matter-antimatter asymmetry problem, corresponding to the virtual nonexistence of antimatter in the universe, is one of the greatest mysteries of cosmology. According to the prevailing cosmological model, the universe was created in the so-called “Big Bang” from pure energy and it is generally considered that the Big Bang and its aftermath produced equal numbers of particles and antiparticles, although the universe today appears to consist almost entirely of matter rather than antimatter. This constitutes the matter-antimatter asymmetry problem: where have all the antiparticles gone? Within the framework of the Generation Model (GM) of particle physics, it is demonstrated that the asymmetry problem may be understood in terms of the composite leptons and quarks of the GM. It is concluded that there is essentially no matter-antimatter asymmetry in the present universe and that the observed hydrogen-antihydrogen asymmetry may be understood in terms of statistical fluctuations associated with the complex many-body processes involved in the formation of either a hydrogen atom or an antihydrogen atom.

Abstract:
We present an existence result for a nonlinear singular differential equation of fourth order with Navier boundary conditions. Under appropriate conditions on the nonlinearity $f(t,x,y)$, we prove that the problem $$displaylines{ L^{2}u=L(Lu) =f(.,u,Lu)quad hbox{a.e. in }(0,1), cr u'(0) =0,quad (Lu) '(0)=0,quad u(1) =0,quad Lu(1) =0. }$$ has a positive solution behaving like $(1-t)$ on $[0,1]$. Here $L$ is a differential operator of second order, $Lu=frac{1}{A}(Au')'$. For $f(t,x,y)=f(t,x)$, we prove a uniqueness result. Our approach is based on estimates for Green functions and on Schauder's fixed point theorem.

Abstract:
A fourth chiral generation, with $m_{t^\prime}$ in the range $\sim (300 - 500)$ GeV and a moderate value of the CP-violating phase can explain the anomalous like-sign dimuon charge asymmetry observed recently by the D0 collaboration. The required parameters are found to be consistent with constraints from other $B$ and $K$ decays. The presence of such quarks, apart from being detectable in the early stages of the LHC, would also have important consequences in the electroweak symmetry breaking sector.

Abstract:
By means of Minimax theory, we study the existence of one nontrivial solution and multiple nontrivial solutions for a fourth-order semilinear elliptic problem with Navier boundary conditions.

Abstract:
In this paper, we use the lower and upper solution method to obtain an existence theorem for the fourth-order boundary-value problem $$displaylines{ u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)),quad 0 Keywords Fourth-order boundary-value problem --- upper and lower solution --- Riemann-Stieltjies integral --- Nagumo-type condition

Abstract:
We prove the existence of a nonzero solution for the fourth order elliptic equation $$Delta^2u= mu u +a(x)g(u)$$ with boundary conditions $u=Delta u=0$. Here, $mu$ is a real parameter, $g$ is superlinear both at zero and infinity and $a(x)$ changes sign in $Omega$. The proof uses a variational argument based on the argument by Bahri-Lions cite{BL}.

Abstract:
We study the existence and multiplicity of positive solutions for a nonlinear fourth-order two-point boundary value problem. The approach is based on critical point theorems in conical shells, Krasnoselskii's compression-expansion theorem, and unilateral Harnack type inequalities.

Abstract:
In this study, we treat for the first time a Lacunary data (0, 1, 4) by constructing spline function of degree six in all subinterval of given partition which interpolates the Lacunary data (0, 1, 4) and the constructed spline function. We applied it to solve the fourth order initial value problem which is defined in section one. The numerical example showed that the presented spline function proved their effectiveness in solving the fourth order initial value problem. Also, we note that, the better error bounds are obtained for a small step size h.