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Constant (mathematics)
About: Constant (mathematics) is a research topic. Over the lifetime, 30206 publications have been published within this topic receiving 452957 citations.
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TL;DR: In this article, the authors studied square integrable coefficients of an irreducible representation of the non-unimodular $ax + b$-group and obtained explicit expressions in the case of a particular analyzing family that plays a role analogous to coherent states (Gabor wavelets) in the usual $L_2 $ -theory.
Abstract: An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant shape, (i.e. obtained by shifts and dilations from any one of them.) The resulting integral transform is isometric and self-reciprocal if the wavelets satisfy an “admissibility condition” given here. Explicit expressions are obtained in the case of a particular analyzing family that plays a role analogous to that of coherent states (Gabor wavelets) in the usual $L_2 $ -theory. They are written in terms of a modified $\Gamma $-function that is introduced and studied. From the point of view of group theory, this paper is concerned with square integrable coefficients of an irreducible representation of the nonunimodular $ax + b$-group.
3,423 citations
2,569 citations
TL;DR: In this paper, the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions), is discussed.
Abstract: Publisher Summary This chapter reviews the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions). It presents a generalization of Phelps' model to include portfolio choice and consumption. The explicit form of the optimal solution is derived for the special case of utility functions having constant relative risk aversion. The optimal portfolio decision is independent of time, wealth, and the consumption decision at each stage. Most analyses of portfolio selection, whether they are of the Markowitz–Tobin mean-variance or of more general type, maximize over one period. The chapter only discusses special and easy cases that suffice to illustrate the general principles involved and presents the lifetime model that reveals that investing for many periods does not itself introduce extra tolerance for riskiness at early or any stages of life.
2,369 citations
TL;DR: In this paper, a good approximation is obtained by relating τ to the slope of the cooling curve at the closure temperature of a geochronological system, which is defined as its temperature at the time corresponding to its apparent age.
Abstract: Closure temperature (Tc) of a geochronological system may be defined as its temperature at the time corresponding to its apparent age. For thermally activated diffusion (D=Doe−E/RT it is given by
$$T_c = R/[E ln (A \tau D_0 /a^2 )]$$
(i) in which R is the gas constant, E the activation energy, τ the time constant with which the diffusion coefficient D diminishes, a is a characteristic diffusion size, and A a numerical constant depending on geometry and decay constant of parent. The time constant τ is related to cooling rate by
$$\tau = R/(Ed T^{ - 1} /dt) = - RT^2 /(Ed T/dt).$$
(ii) Eq. (i) is exact only if T−1 increases linearly with time, but in practice a good approximation is obtained by relating τ to the slope of the cooling curve at Tc.
2,338 citations
TL;DR: A new definition of fractional derivative and fractional integral is given and it is shown that it is the most natural definition, and the most fruitful one.
2,068 citations