General Explicit Formula Research Articles

Explicit formulas for the variance of discounted life-cycle cost

In life-cycle costing analyses, optimal design is usually achieved by minimising the expected value of the discounted costs. As well as the expected value, the corresponding variance may be useful for estimating, for example, the uncertainty bounds of the calculated discounted costs. However, general explicit formulas for calculating the variance of the discounted costs over an unbounded time horizon are not yet available. In this paper, explicit formulas for this variance are presented. They can be easily implemented in software to optimise structural design and maintenance management. The use of the mathematical results is illustrated with some examples.

Mean Residence Times in Linear Compartmental Systems. Symbolic Formulae for their Direct Evaluation

A complete analysis has been performed of the mean residence times in linear compartmental systems, closed or open, with or without traps and with zero input. This analysis allows the derivation of explicit and simple general symbolic formulae to obtain the mean residence time in any compartment of any linear compartmental system, closed or open, with or without traps, as well as formulae to evaluate the mean residence time in the entire system like the above situations. The formulae are given as functions of the fractional transfer coefficients between the compartments and, in the case of open systems, they also include the excretion coefficients to the environment from the different compartments. The relationship between the formulae derived and the particular connection properties of the compartments is discussed. Finally, some examples have been solved.

Colloid transport in nonuniform temperature

The aim of this paper is to set up a theoretical framework for obtaining the thermodiffusion (or Soret) coefficient of a colloid in a carrier liquid. It is first argued that the expression of the particle-current density in nonuniform temperature cannot be derived from a theoretical formula valid for an isothermal solution. Then the kinetic theory of Brownian motion is used to derive an expression for the current density properly accounting for thermodiffusion. The cases of free and interacting particles are treated, and the thermodiffusion current pertinent to an ideal solution adds up with a current driven by a temperature- and concentration-dependent potential. Accordingly, a general explicit formula for the thermodiffusion coefficient is obtained. Practical use of the framework is illustrated on simple specific models of a colloid in a solvent. Large Soret coefficients of both signs are calculated for realistic values of the physicochemical parameters, in qualitative agreement with published experimental data.

On the two-loop sbottom corrections to the neutral Higgs boson masses in the MSSM

We compute the O(α bα s) two-loop corrections to the neutral Higgs boson masses in the Minimal Supersymmetric Standard Model, using the effective potential approach. Such corrections can be important in the region of parameter space corresponding to tan β⪢1 and sizeable μ. In spite of the formal analogy with the O(α tα s) corrections, there are important differences, since the dominant effects are controlled by the sbottom-Higgs scalar couplings. We propose a convenient renormalization scheme that avoids unphysically large threshold effects associated with the bottom mass, and absorbs the bulk of the O(α bα s+α bα t) corrections into the one-loop expression. We give general explicit formulae for the O(α bα s) corrections to the neutral Higgs boson mass matrix. We also discuss the importance of the O(α b 2) corrections and derive a formula for their contribution to m h in a simple limiting case.

Lognormal option pricing for arbitrary underlying assets: a synthesis

All European option pricing formulas sharing the assumption of a lognormally distributed terminal price for the underlying asset are formally similar. It is thus natural to seek a single explicit general formula for this class of options. This paper provides such a synthesis. The key insight is recognizing that all option pricing equations depend explicitly on the expected terminal price of the arbitrary underlying asset, which is often obtained through basic financial reasoning. To illustrate the power and pedagogical value of this framework, I obtain several classical option pricing formulas as special cases of the general equation.

Selberg trace formulae for heat and wave kernels of Maass Laplacians on compact forms of the complex hyperbolic space [formula omitted], [formula omitted

We establish explicit forms of Selberg trace formulae for heat and wave kernels of the Maass Laplacians D k on compact forms of the complex hyperbolic space H n( C) , n⩾2. This was possible by establishing first an explicit general Selberg trace formula for automorphic kernels of weight k in H n( C) and secondly by evaluating explicitly some important integrals that involve the product of two hypergeometric functions and which are not of classical type. The formulae given in Theorem 4.1 and Lemma 5.1 are of independent interest and might be applied in some other contexts.

Modelling non-Gaussian random fields using transformations and Hermite polynomials

In the context of modelling residual roughness on nominally flat moderately polished metal surfaces, a method is proposed for solving problems related to sample function properties and/or special points such as maxima, minima, saddle points for random fields having non-Gaussian height distributions by recasting them in terms of the corresponding problems for the much more tractable Gaussian random fields by means of transformations. Special reference is made to the expansion of the transformations in series of Hermite polynomials. While the use of Hermite polynomials in connection with transformations of random fields and the useful results they yield with regard to covariance functions are well known, this paper derives the most general explicit formula for the expectation of any product of several Hermite polynomials in correlated Gaussian arguments thereby allowing their application to the higher moments of the transformed random field, in particular, to the third moment, which may be used to measure skewness.

Formule d'homotopie entre les complexes de Hochschild et de De Rham

Let k be the field ℂ or ℝ let M be the space kn and let A be the algebra of polynomials over M. We know from Hochschild and co-workers that the Hochschild homology H·(A,A) is isomorphic to the de Rham differential forms over M: this means that the complexes (C·(A,A),b) and (Ω·(M), 0) are quasi-isomorphic. In this work, I produce a general explicit homotopy formula between those two complexes. This formula can be generalized when M is an open set in a complex manifold and A is the space of holomorphic functions over M. Then, by taking the dual maps, I find a new homotopy formula for the Hochschild cohomology of the algebra of smooth fonctions over M (when M is either a complex or a real manifold) different from the one given by De Wilde and Lecompte. I will finally show how this formula can be used to construct an homotopy for the cyclic homology.

Bures fidelity for diagonalizable quadratic Hamiltonians in multi-mode systems

Fidelity, as a measure of the distinguishability of states, is an important concept in quantum mechanics, quantum optics and quantum information theory. Recently, the explicit expressions of fidelity for single-mode squeezed states have been given. However, in experimental studies, especially in non-degenerate parametric down-conversion, two photons are generated and one studies two- or more-mode systems. In this paper we study the Bures fidelity for thermal states of a diagonalizable quadratic Hamiltonian in multi-mode Fock space. To the best of our knowledge, no one has yet attempted to give an explicit general formula of fidelity of mixed states in multi-mode systems.

Homotopie explicite en (co)homologie de Hochschild

Let k be the first C and R let M be the space k n and let A be the algebra of polynomials over M. We produce a general explicit homotopy formula between the complex of Hochschild chains of A, ( C •( A), b) and the de Rham complex of M, (Ω •( M), 0). This formula can be generalized when M is a contractible open set in a complex manifold and A is the space of holomorphic functions over M. Then we find a new homotopy formula for the Hochschild cohomology of the algebra of smooth fonctions over M different from the one given in [5]. Finally, we show how this formula can be used to construct a homotopy for the cyclic homology.

An Explicit Formula for the Minimum Free Energy in Linear Viscoelasticity

A general explicit formula for the maximum recoverable work from a given state is derived in the frequency domain for full tensorial isothermal linear viscoelastic constitutive equations. A variational approach, developed for the scalar case, is here generalized by virtue of certain factorizability properties of positive-definite matrices. The resultant formula suggests how to characterize the state in the sense of Noll in the frequency domain. The property that the maximum recoverable work represents the minimum free energy according to both Graffi's and Coleman-Owen's definitions is used to obtain an explicit formula for the minimum free energy. Detailed expressions are presented for particular types of relaxation function.

Centered pyramids

Quadtree-like pyramids have the advantage of re-suiting in a multiresolution representation where each pyramid node has four unambiguous parents. Such a centered topology guarantees a clearly defined up-projection of labels. This concept has been successfully and extensively used in applications of contour detection, object recognition and segmentation. Unfortunately, the quadtree-like type of pyramid has poor approximation powers because of the employed piecewise-constant image model. This paper deals with the construction of improved centered image pyramids in terms of general approximation functions. The advantages of the centered topology such a symmetry, consistent boundary conditions and accurate up-projection of labels are combined with a more faithful image representation at coarser pyramid levels. We start by introducing a general framework for the design of least squares pyramids using the standard filtering and decimation tools. We give the most general explicit formulas for the computation of the filter coefficients by any (well behaving) approximation function in both the continuous (L/sub 2/) and the discrete (l/sub 2/) norm. We then define centered pyramids and provide the filter coefficients for odd spline approximation functions. Finally, we compare the centered pyramid to the ordinary one and highlight some applications.

Théorèmes de réflexion

Let K be a number field containing μ p and supplied with a group of automorphisms G of prime to p order ; for all 𝔽 p -irreducible representation V χ of G, with character χ, and all G-module M, let rg χ (M) be the maximal integer r such that M/M p contains V χ r . We obtain for instance the following explicit general formula : rg χ * (Cℓ T S )-rg χ (Cℓ S T )=ρ χ (T,S), where T and S are finite disjoint sets of places of K such that T∪S contains all places above p∞, where Cℓ T S is the generalized class group corresponding, by class field theory, to the Galois group of the maximal abelian p-extension, T-ramified and S-splitted of K, and ρ χ (T,S) is an elementary algebraic expression and * the involution which acts on characters according to classical Kummer duality. This formula, and those obtained without the hypothesis about the places above p ∞ , give the most general “Spiegelungssatz” of Scholz-Leopoldt-Kuroda (i.e. with conductors), generalizations of a great number of isolated results (especially in the subtil case p=2), and rank formulas for the main arithmetical invariants attached to K.

Approximating the Distribution of the Scan Statistic Using Moments of the Number of Clumps

Let X 1, X 2, …, Xn be randomly distributed points on the unit interval. Let Nx,x+d be the number of these points contained in the interval (x, x + d). The scan statistic Nd is defined as the maximum number of points in a window of length d; that is, Nd = sup x Nx,x+d. This statistic is used to test for the presence of nonrandom clustering. We say that m points form an m: d clump if these points are all contained in some interval of length d. Let Y denote the number of m: d clumps. In this article we show how to compute the lower-order moments of Y, and we use these moments to obtain approximations and bounds for the distribution of the scan statistic Nd. Our approximations are based on using the methods of moments technique to approximate the distribution of Y. We try two basic types of methods of moments approximations: one involving a simple Markov chain model and others using various different compound Poisson approximations. Our results compare favorably with other approximations and bounds in the literature. In particular, our approximations MC2 and CPG2, which use only the first two moments of Y, do quite well and should be generally useful. We calculate the moments of Y using recursions given by Huffer. We give explicit general formulas for the first two moments of Y and show how the computer programs of Lin may be used to calculate the third and fourth moments.

Approximating the Distribution of the Scan Statistic Using Moments of the Number of Clumps

Abstract Let X 1, X 2, …, Xn be randomly distributed points on the unit interval. Let Nx,x+d be the number of these points contained in the interval (x, x + d). The scan statistic Nd is defined as the maximum number of points in a window of length d; that is, Nd = sup x Nx,x+d. This statistic is used to test for the presence of nonrandom clustering. We say that m points form an m: d clump if these points are all contained in some interval of length d. Let Y denote the number of m: d clumps. In this article we show how to compute the lower-order moments of Y, and we use these moments to obtain approximations and bounds for the distribution of the scan statistic Nd. Our approximations are based on using the methods of moments technique to approximate the distribution of Y. We try two basic types of methods of moments approximations: one involving a simple Markov chain model and others using various different compound Poisson approximations. Our results compare favorably with other approximations and bounds in the literature. In particular, our approximations MC2 and CPG2, which use only the first two moments of Y, do quite well and should be generally useful. We calculate the moments of Y using recursions given by Huffer. We give explicit general formulas for the first two moments of Y and show how the computer programs of Lin may be used to calculate the third and fourth moments.

1-(2N-1) Shell-and-tube exchanger effectiveness: explicit equations

This note is aimed at presentation of general explicit formulae for any odd number of equally sized tube passes

On 3+1 decompositions with respect to an observer field via differential forms

3+1 decompositions of differential forms on a Lorentzian manifold (M,g; +−−−) with respect to arbitrary observer field and the decomposition of the standard operations acting on them are studied, making use of the ideas of the theory of connections on principal bundles. Simple explicit general formulas are given as well as their application to the Maxwell equations.

Current-mode OTA-C realisation of arbitrary filtercharacteristics

Two universal current-mode integrator-based OTA-C filter structures are presented. General explicit design formulas are derived. The filter architectures proposed are constructed using a canonical FLF configuration with an input distribution or output summation network. They are comprehensive in function. By setting any particular input distributing or output summing weights (transconductances), we can realise arbitrary approximation characteristics. It is also straightforward to obtain universal filter architectures and the corresponding design formulae of any particular order. The universal second and third order filters and design formulas are discussed in detail.

Tensor reduction of two-loop vacuum diagrams and projectors for expanding three-point functions

Explicit general formulae for the tensor reduction of two-loop massive vacuum diagrams are presented. The problem of calculating the corresponding coefficients is shown to be equivalent to the problem of constructing differential operators (projectors) extracting the coefficients of the momentum expansion of massive scalar three-point functions (with any number of loops), so the solution to the latter problem is also given.

Integral representations and coherent states

A scheme of construction of integral representations on spaces of analytic functions is presented. The scheme gives a general explicit formula for the reproducing kernels. Its construction is based on symmetry groups of systems of dierential equations, which dene the \analytic property of functional spaces. Given examples oer a new view on the classical results on this problem