A prismatic beam made of a behaviorally nonlinear material is analyzed under aharmonic load moving with a known velocity. The vibration equation of motion is derived usingHamilton principle and Euler-Lagrange Equation. The amplitude of vibration, circular frequency,bending moment, stress and deflection of the beam can be calculated by the presented solution.Considering the response of the beam,...

A function f : Ω→ R is called F -measurable if f−1(B) := {f ∈ B} := {ω ∈ Ω : f(ω) ∈ B} ∈ F for all Borel sets B ⊆ R. Denote by F 0 the collection of sets in F with finite measure, i.e., F 0 := {E ∈ F : μ(E) <∞}. The measure space (Ω,F , μ) is called σ-finite if there exist sets Ei ∈ F 0 such that ⋃∞ i=0Ei = Ω. If needed, these sets may be chosen to additionally satisfy either (a) Ei ⊆ Ei+1 or (...

The Fourier transform can be thought of as a map that decomposes a function into oscillatory functions. In this paper, we will apply this decomposition to help us gain valuable insights into the behavior of our original function. Some particular properties of a function that the Fourier transform will help us examine include smoothness, localization, and its L2 norm. We will conclude with a sec...

Sato’s hyperfunctions are known to be represented as the boundary values of harmonic functions as well as those of holomorphic functions. The author obtains a bijective Poisson mapping P : S∗′(Rn) −→ S∗′(S∗Rn) ∩H(S∗Rn) where H(S∗Rn) is a kind of Hardy subspace of B(S∗Rn). Moreover, the author has an isomorphism between Sobolev spaces P : W (R) −→ W s+(n−1)/4(S∗Rn) ∩H(S∗Rn). There are some simil...

1. Calculus on spheres 2. Spherical Laplacian from Euclidean 3. Eigenvectors for the spherical Laplacian 4. Invariant integrals on spheres 5. L spectral decompositions on spheres 6. Sup-norms of spherical harmonics on Sn−1 7. Pointwise convergence of Fourier-Laplace series 8. Irreducibility of representation spaces for O(n) 9. Hecke’s identity • Appendix: Bernstein’s proof of Weierstraß approxi...

For present purposes, we shall define non-commutative harmonic analysis to mean the decomposition of functions on a locally compact G-space X,1 where G is some (locally compact) group, into functions well-behaved with respect to the action of G. The classical cases are of course Fourier series, when G = X = T, the circle group, and the Fourier transform, when G = X = R, but we will mostly be co...