By Narenda Govil, Ram N. Mohapatra, Zuhair Nashed, A. Sharma, J. Szabados
A suite of over 30 conscientiously chosen papers by way of forty five the world over well-known mathematicians, in honor of A. okay. Varma, reflecting his lifelong ardour for investigating polynomials, inequalities in Lp and uniform metrics.
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Extra resources for Approximation Theory: In Memory of A.K.Varma
Hint. ) 9. Using the determinant differentiation rule, by rows, show that hence, if Pd i=1 ℓij (t) d “X ” d d w(t) ≡ det W (t) = ℓij (t) w(t); dt dt i=1 = ℓ(t), one has w(t) = w(t0 ) e Êt t0 ℓ(τ )dτ . 10. , t → ℓij (t), i, j = 1, . . , d are T −periodic functions. Let x(1) , . . , x(d) be d linearly independent solutions for t > 0. Then there exist d2 constants (i) Aj , i, j = 1, . . , d, such that x(i) (t + T ) = d X (i) Aj x(j) (t), t ≥ 0. j=1 Show that det W (T )/ det W (0) = w(T )/w(0) = det A = 0.
E. Show V (ξ) = |ξ|α , α > 1, and show that the period of the motion with energy 1 1 E is proportional to E α − 2 (see Problem 5). 8. Suppose that 9. Find the limit as E → +∞ the period of the motion with energy potential energy V (ξ) = 12 ξ 2 + 14 ξ 4 10. Same as Problem 9 with V 11. Same as Problem 9 with limξ→∞ V (ξ) = +∞. such that V V (ξ) = V (−ξ), limξ→∞ such that E developing with V (ξ) ξ2 = +∞. 8 Equilibrium: Stability in the Absence of Friction In the proof of Proposition 12, p. , solutions like t → ξ0 = constant, correspond to the stationary points of the potential energy function V .
Solutions like t → ξ0 = constant, correspond to the stationary points of the potential energy function V . In such positions, “equilibrium positions”, the exerted force vanishes. It is also possible to futher distinguish the equilibrium points on the basis of a qualitative property: the stability of their equilibria. Stability is an empirical notion susceptible to assuming different precise meanings, depending on the particular problem where it appears necessary to study the stability of an equilibrium point.
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