The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the measure of the angle approaches zero, the difference between the approximation and the original function also approaches 0.
Angle Smalls
The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the measure of the angle approaches zero, the difference between the approximation and the original function also approaches 0.
Angle Smalls
In astronomy, the angular size or angle subtended by the image of a distant object is often only a few arcseconds, so it is well suited to the small angle approximation.[6] The linear size (D) is related to the angular size (X) and the distance from the observer (d) by the simple formula:
When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing simple harmonic motion.
The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to simplify equations, e.g. 'fringe spacing' = 'wavelength' Ã 'distance from slits to screen' Ã 'slit separation'.[7]
The small-angle approximation also appears in structural mechanics, especially in stability and bifurcation analyses (mainly of axially-loaded columns ready to undergo buckling). This leads to significant simplifications, though at a cost in accuracy and insight into the true behavior.
Figure 1. Flowchart depicting the ForceBalance-SAS algorithm. An initial set of parameters is input, followed by MD simulation and calculation of ensemble-averaged small-angle scattering intensities. After the simulation stops, the residual between the simulated and experimental scattering intensities is calculated, along with the gradients and Hessians of the residual. If the desired convergence criteria are met, the algorithm stops, and the new force field parameters are output; if not, optimization is performed, a new set of parameters are obtained, and a simulation with the updated parameters are performed, completing the cycle. The current implementation of the ForceBalance (Wang et al., 2014) approach is demarcated from our approach using dotted lines.
We propose a novel fragment assembly method for low-resolution modeling of RNA and show how it may be used along with small-angle X-ray solution scattering (SAXS) data to model low-resolution structures of particles having as many as 12 independent secondary structure elements. We assessed this model-building procedure by using both artificial data on a previously proposed benchmark and publicly available data. With the artificial data, SAXS-guided models show better similarity to native structures than ROSETTA decoys. The publicly available data showed that SAXS-guided models can be used to reinterpret RNA structures previously deposited in the Protein Data Bank. Our approach allows for fast and efficient building of de novo models of RNA using approximate secondary structures that can be readily obtained from existing bioinformatic approaches. We also offer a rigorous assessment of the resolving power of SAXS in the case of small RNA structures, along with a small multimetric benchmark of the proposed method.
We propose a computational procedure that uses small-angle X-ray solution scattering (SAXS) data to obtain low-resolution approximations of RNA structures. This process can be used as a diagnostic tool to help confirm predicted secondary structures with a higher degree of certainty than chemical footprinting approaches alone [10], [11].
There have also been several attempts to develop conventional sequential fragment assembly, which works by copying local coordinates or angles, similar to the ROSETTA protein modeling method [18], [19].
Attempts have been made to produce an approximate model of flexible RNA molecules using residual dipolar coupling (RDC) data acquired from nuclear magnetic resonance (NMR) experiments to restrain relative angles between helices [25]. This approach has achieved considerable success for molecules with a small number of flexible angles [25].
Each replacement step consists of superimposing boundary atoms onto the boundary atoms of the previous element in the topology of the secondary structures, and then minimizing the 3 last dihedral angles of those boundary elements to assure a contiguous backbone.
Consistency checks between frames were performed automatically by X33 automated processing system [45] using ATSAS software [46]. The expected molecular masses of the solutes were estimated from intensity extrapolated to zero angle, and were found to be consistent with the expected masses for the monomers (see table 1. The maximum diameter of each particle was estimated by indirect Fourier transform using GNOM software [47].
Designed the computational method: MJG. Conceived 7SK experiments: ACDB MJG. Prepared HP4 and HP3 samples and verified their quality: ACDB DMZ EU. Performed SAXS experiments: MJG ACDB DMZ EU. Performed the analysis and validation of small-angle scattering data for HP3 and HP4, built the models: MJG. Contributed implementation of the method: MJG. Wrote first draft of the paper: MJG. Edited the paper: MJG ACDB.
Abstract:Small-angle scattering (SAS; X-rays, neutrons, light) is being increasingly used to better understand the structure of fractal-based materials and to describe their interaction at nano- and micro-scales. To this aim, several minimalist yet specific theoretical models which exploit the fractal symmetry have been developed to extract additional information from SAS data. Although this problem can be solved exactly for many particular fractal structures, due to the intrinsic limitations of the SAS method, the inverse scattering problem, i.e., determination of the fractal structure from the intensity curve, is ill-posed. However, fractals can be divided into various classes, not necessarily disjointed, with the most common being random, deterministic, mass, surface, pore, fat and multifractals. Each class has its own imprint on the scattering intensity, and although one cannot uniquely identify the structure of a fractal based solely on SAS data, one can differentiate between various classes to which they belong. This has important practical applications in correlating their structural properties with physical ones. The article reviews SAS from several fractal models with an emphasis on describing which information can be extracted from each class, and how this can be performed experimentally. To illustrate this procedure and to validate the theoretical models, numerical simulations based on Monte Carlo methods are performed.Keywords: small-angle scattering; fractals; structural properties; Monte Carlo simulations; form factor; structure factor
Basic trigonometric identities are the primary trigonometric ratios of sine, cosine, and tangent that are defined in a right triangle. These ratios have numerous applications, but their most important application is finding unknown sides and angles in a right triangle.
To define the three primary trigonometric ratios, first pick an angle eq\theta /eq in a right triangle, and label the opposite and adjacent sides to this angle and the hypotenuse of the right triangle:
In cases where the angle is relatively small (usually less than or equal to 15 degrees or 0.26 radians), an approximation can be used to find a more simplified formula for each of the three primary ratios. This process is called a small-angle approximation or applying the small-angle approximation theorem. It is important to note that these formulas are approximations and as the angle gets bigger (greater than 15 degrees or 0.26 radians), their accuracy decreases. For angles less than or equal to 15 degrees (0.26 radians) however, these formulas have enough accuracy and are almost equal to the exact value. For the purpose of this lesson, from this point on, the term "small-angle" refers to angles less than or equal to 15 degrees or 0.26 radians.
Small-angle approximation has numerous applications in algebra, calculus, and geometry. In situations where angles are small, these formulas can be used to simplify the calculation. In most cases, small-angle approximation formulas can help to simplify expressions by eliminating the trigonometric ratios and functions.
One of the most important applications of small-angle approximation formulas is in calculus. In calculating limits of functions that involve trigonometric expressions when eqx\to 0 /eq (eqx /eq approaches 0), small-angle approximation formulas can be used. In most cases these formulas help to simplify the limit and more importantly resolve some of the indeterminate value cases.
The basic trigonometric identities or the primary trigonometric ratios of sine, cosine, and tangent are defined for an angle eq\theta /eq in a right triangle. To find the values of these ratios, the sides opposite and adjacent to the angle are needed eq\theta /eq, and the hypotenuse of the triangle:
When the angle is small (usually less than or equal to 15 degrees or 0.26 radians), small-angle approximation can be used to simplify the three primary trigonometric formulas. It must be noted that these formulas are approximations and are not as accurate as the angle gets bigger. However, they can be used for angles less than or equal to 15 degrees (0.26 radians). Each of the three primary trigonometric ratios has its own small-angle approximation formula:
Small-angle approximation and small-angle formulas have numerous applications. These formulas can be used in algebra, calculus, and geometry and the only requirement is that the angle must be small enough. One of the most important applications of small-angle approximation is in calculating limits in calculus. Limits that contain trigonometric expressions and functions can be simplified using small-angle approximation formulas when eqx\to 0 /eq (eqx /eq approaches 0). This is especially useful when the limit initially results in an indeterminate value. Small-angle approximations can help eliminate the factor that causes an indeterminate value such as the eq\frac00 /eq case.
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