# closed-form Green function

## Top PDF closed-form Green function: ### Point mass dynamics on spherical hypersurfaces

The equations governing point mass dynamics on spherical (hyper-)surfaces S n for n > 2 have been derived. The key result is that the gravitational potential just depends on the form of the Green function, the solution to Poisson’s equation (or the Laplace-Beltrami equation) on S n which incorporates the Gauss condition, namely that the surface integral of the mass density vanishes on any closed surface. This condition cannot be relaxed: it is a mathematical requirement on any closed surface. ### Quasi-Static Complex Image Method for a Current Point Source in Horizontally Stratified Multilayered Earth

In this paper, QSCIM has been fully discussed based on the MP method. First, the procedure for vector or scalar Green’s function of vector dipole or scalar monopole buried in horizontal multilayered earth model is compendiously introduced. Then the process to achieve finite exponential series for the kernel of Green’s function based on MP method is discussed. Finally, numerical results for the kernel of the closed form of Green’s function are carefully studied and discussed. 2. THEORY OF QUASI-STATIC COMPLEX IMAGE METHOD ### A Model System for Dimensional Competition in Nanostructures: A Quantum Wire on a Surface

It is well known in surface science that Green’s functions can also be given in closed form in mixed representations, where momentum coordinates are used along the surface and configuration space coordinates are used for the transverse directions. The same observation applies here. In particular, the Green’s function with one transverse momentum replaced by a transverse coordinate is given by ### Levy Subordinator Model: A Two Parameter Model of Default Dependency

The article is organized as follows. To start with, a brief review of subordinators is pre- sented in section 1. In section 2, a one-factor framework of default dependency is formulated starting with the de Finetti theorem from probability theory. Two classes of models arise naturally that are termed type-I and type-II. These models formulated on an infinitely large homogeneous collection to start with are further extended in section 3 to be applicable to finite heterogeneous collections and general hazard rate curves. Type-I models are well- known that include many of the reduced form models that have a systemic component to their dynamics. Type-II models are formulated as new that include the subordinator models introduced here but it turns out, as discussed in section 4, that the popular copula models can also be recast into this framework. Section 5 discusses type-II models governed by sta- ble subordinators, in particular the L´ evy subordinator model based on the α = 1/2 stable subordinator called the L´ evy subordinator. Section 6 discusses some niceties of the α = 1/2 choice found numerically as appropriate. An attractive feature of the L´ evy subordinator model is the possibility of semi-analytical pricing that is discussed in section 7. Section 8 discusses the large homogeneous pool approximation. Section 9 is a short presentation of an efficient pricing technique based on the Fast Fourier Transform. The model can also be investigated numerically with a Monte Carlo simulation algorithm as discussed in section 10. As is now well appreciated, random recovery is helpful in better pricing of the senior tranches, and section 11 presents a tractable framework. The effect of initial conditions is analyzed in section 12. Section 13 discusses default contagion within the present framework. Some possible extensions such as intensity based modeling are discussed in section 14. Sec- tion 15 concludes with a discussion and a brief summary. Figures 1-10 present the results of a numerical investigation into the model’s implications. ### Analytic Approximations of Projectile Motion with Quadratic Air Resistance

We study projectile motion with air resistance quadratic in speed. We consider three regimes of approximation: low-angle trajectory where the horizontal velocity, u, is assumed to be much larger than the vertical velocity w; high-angle trajectory where w  u ; and split-angle trajectory where w  u . Closed form solutions for the range in the first regime are obtained in terms of the Lambert W function. The approximation is simple and accurate for low angle ballistics problems when compared to measured data. In addition, we find a surprising behavior that the range in this approximation is symmetric about  / 4 , although the trajectories are asymmetric. We also give simple and prac- tical formulas for accurate evaluations of the Lambert W function. ### Some Explicit Formulae for the Hull and White Stochastic Volatility Model

K  K  K  . These prices are observed in the time period that goes from April 2nd, 2012, to July 25th, 2012 Note that the observations are daily observ - tions. Recall that in the study of financial data time series a year is made of about 252 trading days and a month is made of about 21 trading days. Figure 1 shows the value of the USA S&P 500 index as a function of time during the period of interest. Figures 2 and 3 show respectively th ces of the European call and t o n the index with maturity time arch 6th, 2 strike pri ### A Note on the Beta Function And Some Properties of Its Partial Derivatives

t x − 1 (1 − t) y − 1 dt (1) for Rex > 0 and Rey > 0. It is often applied in many fields such as mathematical equations and probability theory. Its definition was extended to complex numbers values of x and y by using the neutrix limit in . Furthermore, the partial derivatives of the Beta function on the complex numbers x and y exist a close relationship with many special functions and special integrals. For example, the following relation was proved in ### On the Impact of Antenna Correlation on the Pilot-Data Balance in Multiple Antenna Systems

uplink channel by the BS. Recall from Section III that the channel estimation process is independent for each MS and we can therefore focus on a single user. The covariance matrix C of the channel h as the function of the antenna spacing, mean angle of arrival and angular spread is modeled as by the well known spatial channel model, which is known to be accurate in non-line-of-sight environment with rich scattering and all antenna elements identically polarized, see . For uniformly distributed angle of arrivals, the (m, n) element of the covariance matrix C are given by ### Optimal Mortgage Refinancing: A Closed Form Solution

The actual behavior of mortgage holders often diﬀers from the predictions of the optimal refinancing model. In the 1980s and 1990s — when mortgage interests rates generally fell — many borrowers failed to refinance despite holding options that were deeply in the money (Giliberto and Thibodeau, 1989, Green and LaCour-Little (1999) and Deng and Quigley, 2006). On the other hand, Chang and Yavas (2006) have noted that over one-third of the borrowers refinanced too early during the period ### Plant Form and Function

Biologists have long suspected that plants are the evolu- tionary descendants of green algae. Now we are sure. The evolutionary history of plants was laid bare at the 1999 In- ternational Botanical Congress held in St. Louis, Mis- souri, by a team of 200 biologists from 12 countries that had been working together for five years with U.S. federal funding. Their project, Deep Green (more formally known as The Green Plant Phylogeny Research Coordi- nation Group), coordinated the efforts of laboratories using molecular, morphological, and anatomical traits to create a new “Tree of Life.” Deep Green confirmed the long-standing claim that green algae were ancestral to plants. More surprising was the finding that just a single species of green algae gave rise to the entire terrestrial plant lineage from moss through the flowering plants (an- giosperms). Exactly what this ancestral alga was is still a mystery, but close relatives are believed to exist in fresh- water lakes today. DNA sequence data is consistent with the claim that a single “Eve” gave rise to the entire king- dom Plantae 450 million years ago. At each subsequent step in evolution, the evidence suggests that only a single family of plants made the transition. The fungi appear to have branched later than the plants and are more closely related to us. ### Risk Management of Policyholder Behavior in Equity-Linked Life Insurance

We start from a standard VA contract combining guaranteed minimum death benefit (GMDB) and guaranteed minimum accumulation benefit (GMAB) riders financed by a fee paid continuously as a constant percentage of the account value. We then derive an explicit closed-form expression for a model-free minimal surrender charge schedule that eliminates the surrender incentive for all account values throughout the whole term of the contract. However, we find that these minimal surrender penalties are generally too high to lead to a marketable VA product. To address this issue, we consider the state-dependent fee structure introduced by Bernard, Hardy, and MacKay (2013), where the fee is paid to the insurer only when the account value is below a certain threshold (see also Delong, 2014; Moenig and Zhu, 2014 and Zhou and Wu, 2015). 5 The motivation is the following: since ### Closed form solution of an exponential kernel integral equation

Having reached a formal solution, there is no general recipe for transforming this into a more amenable form and it is hard even to know if such a form exists. Symbolic algebra software may be helpful in simple cases but for this example did not lead to simplification of the result. However, a literature search led to a relatively obscure result known as the Mehler formula (Srivastava & Manocha, 1984), that contains bilinear sums of Hermite polynomials: ### Graphical Lasso and Thresholding: Equivalence and Closed-form Solutions

the hard-to-obtain graph supp(S opt ), without having to solve the GL. Furthermore, we will show that the GL problem has a simple closed-form solution that can be easily derived merely based on the thresholded sample covariance matrix, provided that its underlying graph has an acyclic structure. This result will then be generalized to obtain an approxi- mate solution for the GL in the case where the thresholded sample covariance matrix has an arbitrary sparsity structure. This closed-form solution converges to the exact solution of the GL as the length of the minimum-length cycle in the support graph of the thresholded sample covariance matrix grows. The derived closed-form solution can be used for two pur- poses: (1) as a surrogate to the exact solution of the computationally heavy GL problem, and (2) as an initial point for common numerical algorithms to numerically solve the GL (see Friedman et al. (2008); Hsieh et al. (2014)). The above results unveil fundamental properties of the GL in terms of sparsification and computational complexity. Although conic optimization problems almost never benefit from an exact or inexact explicit formula for their solutions and should be solved numerically, the formula obtained in this paper suggests that sparse GL and related graph-based conic optimization problems may fall into the category of problems with closed-form solutions (similar to least squares problems). 3. Main Results ### Closed form representations and properties of the generalised Wendland functions

In this paper we investigate the generalisation of Wendland’s com- pactly supported radial basis functions to the case where the smoothness parameter is not assumed to be a positive integer or half-integer and the parameter ℓ, which is chosen to ensure positive deﬁniteness, need not take on the minimal value. We derive suﬃcient and necessary conditions for the generalised Wendland functions to be positive deﬁnite and deduce the native spaces that they generate. We also provide closed form repre- sentations for the generalised Wendland functions in the case when the smoothness parameter is an integer and where the parameter ℓ is any suitable value that ensures positive deﬁniteness, as well as closed form representations for the Fourier transform when the smoothness parame- ter is a positive integer or half-integer. ### Closed form solutions for the generalized extreme value distribution

Order statistics play important roles in theoretical and applied microeconomic anal- ysis, typically in a setting of incomplete information. Conditional moments can serve to impute unobservables, given relevant information. Closed form expressions for them are attractive in theoretical work because they facilitate, or indeed en- able, further analytical analysis, such as comparative statics. And they are useful in empirical work because they avoid costly simulation and render moment based estimation computationally more eﬃcient. In applied demand analysis, the ana- lytical tractability of the Generalized Extreme Value (GEV) distribution has made it a popular modelling choice, giving rise to the family of models involving logit choice probabilities 1 . This paper presents novel analytical results on conditional ### Structure of the Paramyxovirus Parainfluenza Virus 5 Nucleoprotein in Complex with an Amino-Terminal Peptide of the Phosphoprotein

PIV5 N is comprised of an N-terminal core region (N-core) (residues 1 to 402) and a C-terminal disordered tail (N-tail) (residues 403 to 510) (Fig. 1A). The N-core region is ﬂanked by N-arm and C-arm at its amino and carboxyl termini, respectively. We have previously determined the structure of the oligomeric form of the PIV5 N bound to RNA (15). The N-tail of N is highly disordered and was found to be cleaved from the rest of the protein during puriﬁcation. The structure shows the presence of an RNA binding site in the cleft at the hinge region between the N-terminal domain (NTD) and C-terminal domain (CTD). In this structure, N encapsidates nonspeciﬁc Escherichia coli RNA that was resistant to nuclease digestion. PIV5 follows the “rule of six,” according to which the genome length is divisible by six (16). The PIV5 N crystal structure showed the binding of six nucleotides per N monomer (three nucleotides facing inwards and three outwards, consistent with the rule of six. The structure also showed the impor- tance of both N and C arms in oligomerization as they are involved in binding to the neighboring N protomers. After determining the structure of oligomeric PIV5 N, we were interested in determining the structure of the unassembled form of PIV5 N, the result of the chaperone activity of P. The mechanism by which P chaperones N is not known, and there is also a lack of knowledge concerning the conformational changes in N at both the initial stage and during RNA replication. ### Closed form inverse of local multi-trace operators

We are interested here in the inverse of local multi-trace operators. We exhibit a closed form of this inverse for a model problem with three subdo- mains in the special case where the coefficients are homogeneous. An essential ingredient to obtain this closed form inverse are several remarkable identities which were recently discovered, see . We illustrate our findings with a nu- merical experiment that shows that discretizing the closed form inverse gives indeed and approximate inverse of the discretized local multi-trace operator. ### Exploring the application of closed form solutions in geotechnical problems

A way in which one may understand these benefits is to look deeper into the structure of the closed form solution. The closed-form solution involves an under- standing of the definitions of the system parameters and the product/s of their interactions, and the combination of this information to reach a desired conclu- sion. This is an integral process in solving physical processes analytically and an effective way in understanding their true nature as it can unveil the limitations and assumptions made. It is through this fundamental structure that it may be said closed form solutions offer an advantage over numerical solutions, as the analysis of a system and the interactions of its components is one which can yield great insight. One can examine an equation, alter it, incorporate it; as methods to see the interconnections and intricacies of the processes within it.  