This volume contains nine chapters that detail advances in mathematics research. In chapter one, the author presents new spectral sufficient conditions for the Hamiltonian and traceable graphs. In chapter two, the authors examine the Green's function in an acoustic gradient interface model of the monotonic Epstein. In chapter three, asymptotic formulae are presented for solutions and Green's functions of Sturm-Liouville eigenvalue problem having eigen parameter both in differential equation and in one boundary condition. The focus of chapter four is the representation of Sum of Products in Boolean cubes. The author in chapter five explains the value of ontological logic in mathematical research. Chapter six proves that mathematical results using Riemannian geometry deeply question the origin and existence of gravitational waves. In chapter seven the author considers the problem of solving strongly nonlinear oscillators making use of averaging methods. In chapter eight, the authors study the existence and regularity of solutions in the a-norm for some second order partial neutral functional differential equations with infinite delay in Banach spaces. The last chapter probes the complexities of the frequency-amplitude formula, illuminating its integral role across both linear and nonlinear oscillatory contexts.