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.

11 Read more

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

18 Read more

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

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.

Show more
34 Read more

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**.

Show more
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

20 Read more

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 [1]. 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 [2]

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 [14]. For uniformly distributed angle of arrivals, the (m, n) element of the covariance matrix C are given by

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

46 Read more

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.

Show more
20 Read more

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

Show more
37 Read more

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:

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

Show more
44 Read more

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.

Show more
26 Read more

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

Show more
33 Read more

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.

Show more
14 Read more

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 [3]. 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.

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.

Show more
91 Read more

which are in the physiologically expected range. However, in their study, iterative procedures such as steepest descent algorithm were used to find minimum of the RLS cost **function** and a **closed** **form** solution was not provided. Bias and variance of the estimator were also estimated using some Monte-Carlo simulations.

One presents and discusses an alternative solution writeable in **closed** **form** of the Heston’s PDE, for which the solution is known in literature, up to an inverse Fourier transform. Since the algorithm to compute the inverse Fourier Transform is not able to be applied easily for every payoff, one has elaborated a new methodology based on changing of variables which is independent of payoffs and does not need to use the inverse Fourier transform algorithm or numerical methods as Finite Difference and Monte Carlo simulation. In particular, one will compute the price of Vanilla Options for small maturities in order to validate numerically the Geometrical Transformations technique. The principal achievement is to use an analytical formula to compute the prices of derivatives, in order to manage, balance any portfolio through the Greeks, that by the proposed solution one is able to compute analytically. The above mentioned numerical techniques are computationally expensive in the stochastic volatility market models and for this reason is usually employed the Black Scholes model, that is unsuitable, as it has been widely proven in literature to compute the sensitivities of a portfolio and the price of derivatives. The present article wants to introduce a new approach to solve PDEs complicated, such as, those coming out from the stochastic volatility market models, with the achievement to reduce the computational cost and thus the time machine; besides, the proposed solution is easy to be generalized by adding jump processes as well. The present research work is rather technical and one does wide use of functional analysis. For the conceptual simplicity of the technique, one confides which many applications and studies will follow, extending the applications of the Geometrical Transformations technique to other derivative contracts of different styles and asset classes.

Show more
15 Read more