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Physics - Imperfections Lower the Simulation Cost of Quantum Computers
Lorentz Institute, Leiden University, Leiden, Netherlands
Classical computers can efficiently simulate the behavior of quantum computers if the quantum computer is imperfect enough.
Figure 1: Researchers show that they can simulate the above 1D quantum circuit with a classical one if the quantum computer is imperfect enough.
N
is the number of qubits, and
D
is the number of gates the circuit executes and is equivalent to increasing time in the image. The colored squares represent random one-qubit gates, and the dots connected to a cross-containing box represent two-qubit Controlled-NOT or Controlled-Z gates.Researchers show that they can simulate the above 1D quantum circuit with a classical one if the quantum computer is imperfect enough.
N
D
is the number of gates the circuit executes and is equivalent to increasing time i... Show more
With a few quantum bits, an ideal quantum computer can process vast amounts of information in a coordinated way, making it significantly more powerful than a classical counterpart. This predicted power increase will be great for users but is bad for physicists trying to simulate on a classical computer how an ideal quantum computer will behave. Now, a trio of researchers has shown that they can substantially reduce the resources needed to do these simulations if the quantum computer is “imperfect” [1]. The arXiv version of the trio’s paper is one of the most “Scited” papers of 2020 and the result generated quite a stir when it first appeared back in February—I overheard it being enthusiastically discussed at the Quantum Optics Conference in Obergurgl, Austria, at the end of that month, back when we could still attend conferences in person.
In 2019, Google claimed to have achieved the quantum computing milestone known as “quantum advantage,” publishing results showing that their quantum computer Sycamore had performed a calculation that was essentially impossible for a classical one [2]. More specifically, Google claimed that they had completed a three-minute quantum computation—which involved generating random numbers with Sycamore’s 53 qubits—that would take thousands of years on a state-of-the-art classical supercomputer, such as IBM’s Summit. IBM quickly countered the claim, arguing that more efficient memory storage would reduce the task time on a classical computer to a couple of days [3]. The claims and counterclaims sparked an industry clash and an intense debate among supporters in the two camps.
Resolving the disparity between these estimates is one of the goals of the new work by Yiqing Zhou, of the University of Illinois at Urbana–Champaign, and her two colleagues [1]. In their study, they focused on algorithms for classically replicating “imperfect” quantum computers, which are also known as NISQ (noisy intermediate-scale quantum) devices [4]. Today’s state-of-the-art quantum computers—including Sycamore—are NISQ devices. The algorithms the team used are based on so-called tensor network methods, specifically matrix product states (MPS), which are good for simulating noise and so are naturally suited for studying NISQ devices. MPS methods approximate low-entangled quantum states with simpler structures, so they provide a data-compression-like protocol that can make it less computationally expensive to classically simulate imperfect quantum computers (see Viewpoint: Pushing Tensor Networks to the Limit).
Zhou and colleagues first consider a random 1D quantum circuit made of neighboring, interleaved two-qubit gates and single-qubit random unitary operations. The two-qubit gates are either Controlled-NOT gates or Controlled-Z (CZ) gates, which create entanglement. They ran their algorithm for NISQ circuits containing different numbers of qubits, , and different depths, —a parameter that relates to the number of gates the circuit executes (Fig. 1). They also varied a parameter in the MPS algorithm. is the so-called bond dimension of the MPS and essentially controls how well the MPS capture entanglement between qubits.
The trio demonstrate that they can exactly simulate any imperfect quantum circuit if and are small enough and is set to a value within reach of a classical computer. They can do that because shallow quantum circuits can only create a small amount of entanglement, which is fully captured by a moderate . However, as increases, the team finds that cannot capture all the entanglement. That means that they cannot exactly simulate the system, and errors start to accumulate. The team describes this mismatch between the quantum circuit and their classical simulations using a parameter that they call the two-qubit gate fidelity . They find that the fidelity of their simulations slowly drops, bottoming out at an asymptotic value as increases. This qualitative behavior persists for different values of and . Also, while their algorithm does not explicitly account for all the error and decoherence mechanisms in real quantum computers, they show that it does produce quantum states of the same quality (perfection) as the experimental ones.
In light of Google’s quantum advantage claims, Zhou and colleagues also apply their algorithm to 2D quantum systems—Sycamore is built on a 2D chip. MPS are specifically designed for use in 1D systems, but the team uses well-known techniques to extend their algorithm to small 2D ones. They use their algorithm to simulate an , circuit, roughly matching the parameters of Sycamore (Sycamore has 54 qubits but one is unusable because of a defect). They replace Google’s more entangling “iSWAP” gates with less entangling CZ gates, which allow them to classically simulate the system up to the same fidelity as reported in Ref. [2] with a single laptop. The simulation cost should increase quadratically for iSWAP-gate circuits, and although the team proposes a method for performing such simulations, they have not yet carried them out because of the large computational cost it entails.
How do these results relate to the quantum advantage claims by Google? As they stand, they do not weaken or refute claims—with just a few more qubits, and an increase in or , the next generation of NISQ devices will certainly be much harder to simulate. The results also indicate that the team’s algorithm only works if the quantum computer is sufficiently imperfect—if it is almost perfect, their algorithm provides no speed up advantage. Finally, the results provide numerical insight into the values of , , , and for which random quantum circuits are confined to a tiny corner of the exponentially large Hilbert space. These values give insight into how to quantify the capabilities of a quantum computer to generate entanglement as a function of , for example.
So, what’s next? One natural question is, Can the approach here be transferred to efficiently simulate other aspects of quantum computing, such as quantum error correction? The circuits the trio considered are essentially random, whereas quantum error correction circuits are more ordered by design [5]. That means that updates to the new algorithm are needed to study such systems. Despite this limitation, the future looks promising for the efficient simulation of imperfect quantum devices [6, 7].
Y. Zhou et al., “What limits the simulation of quantum computers?” Phys. Rev. X 10, 041038 (2020).
E. Pednault et al., “Leveraging secondary storage to simulate deep 54-qubit Sycamore circuits,” arXiv:1910.09534.
X. Gao and L. Duan, “Efficient classical simulation of noisy quantum computation,” arXiv:1810.03176.
S. Cheng et al., “Simulating noisy quantum circuits with matrix product density operators,” arXiv:2004.02388.
K. Noh et al., “Efficient classical simulation of noisy random quantum circuits in one dimension,” Quantum 4, 318 (2020).
Jordi Tura is an assistant professor at the Lorentz Institute of the University of Leiden, Netherlands. He also leads the institute’s Applied Quantum Algorithms group. Tura obtained his B.Sc. degrees in mathematics and telecommunications and his M.Sc. in applied mathematics from the Polytechnic University of Catalonia, Spain. His Ph.D. was awarded by the Institute of Photonic Sciences, Spain. During his postdoctoral stay at the Max Planck Institute of Quantum Optics in Germany, Tura started working in the field of quantum information processing for near-term quantum devices.
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EUDML | Schauder decompositions and multiplier theorems EuDML | Schauder decompositions and multiplier theorems
Schauder decompositions and multiplier theorems
P. Clément; B. de Pagter; F. Sukochev; H. Witvliet
We study the interplay between unconditional decompositions and the R-boundedness of collections of operators. In particular, we get several multiplier results of Marcinkiewicz type for
{L}^{p}
-spaces of functions with values in a Banach space X. Furthermore, we show connections between the above-mentioned properties and geometric properties of the Banach space X.
Clément, P., et al. "Schauder decompositions and multiplier theorems." Studia Mathematica 138.2 (2000): 135-163. <http://eudml.org/doc/216695>.
@article{Clément2000,
abstract = {We study the interplay between unconditional decompositions and the R-boundedness of collections of operators. In particular, we get several multiplier results of Marcinkiewicz type for $L^p$-spaces of functions with values in a Banach space X. Furthermore, we show connections between the above-mentioned properties and geometric properties of the Banach space X.},
author = {Clément, P., de Pagter, B., Sukochev, F., Witvliet, H.},
keywords = {Marcinkiewicz-type multiplier theorems; -boundedness; Schauder decomposition},
title = {Schauder decompositions and multiplier theorems},
AU - Clément, P.
AU - Sukochev, F.
AU - Witvliet, H.
TI - Schauder decompositions and multiplier theorems
AB - We study the interplay between unconditional decompositions and the R-boundedness of collections of operators. In particular, we get several multiplier results of Marcinkiewicz type for $L^p$-spaces of functions with values in a Banach space X. Furthermore, we show connections between the above-mentioned properties and geometric properties of the Banach space X.
KW - Marcinkiewicz-type multiplier theorems; -boundedness; Schauder decomposition
[BG94] E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math. 112 (1994), 13-49. Zbl0823.42004
[Bou83] J. Bourgain, Some remarks on Banach spaces in which martingale differences are unconditional, Ark. Mat. 21 (1983), 163-168. Zbl0533.46008
[Bou85] J. Bourgain, Vector-valued singular integrals and the
{H}^{1}
-BMO duality, in: Probability Theory and Harmonic Analysis, Dekker, New York, 1985, 1-19.
[Bur83] D. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in: Proc. Conf. on Harmonic Analysis in Honor of Antoni Zygmund (Chicago, 1981), Wadsworth, Belmont, 1983, 270-286.
[DJT95] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press, 1995. Zbl0855.47016
[DU77] J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, RI, 1977.
[DS97] P. Dodds and F. Sukochev, Non-commutative bounded Vilenkin systems, preprint, 1997.
[EG77] R. E. Edwards and G. I. Gaudry, Littlewood-Paley and Multiplier Theory, Ergeb. Math. Grenzgeb. 90, Springer, Berlin, 1977.
[GK70] I. C. Gohberg and M. G. Kreĭn, Theory and Applications of Volterra Operators in Hilbert Space, Transl. Math. Monogr. 24, Amer. Math. Soc., Providence, RI, 1970. Zbl0194.43804
[KP79] N. J. Kalton and N. T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1-30. Zbl0424.46004
[KP70] S. Kwapień and A. Pełczyński, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43-68. Zbl0189.43505
[LT77] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, Ergeb. Math. Grenzgeb. 92, Springer, Berlin, 1977.
[Mar39] J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math. 8 (1939), 78-91.
[Mau75] B. Maurey, Système de Haar, in: Séminaire Maurey-Schwartz 1974-1975: Espaces
{L}_{p}
, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. I et II, Centre Math., École Polytech., Paris, 1975, p. 26.
[Pal32] R. E. A. C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-279. Zbl0005.24901
[Pis78] G. Pisier, Some results on Banach spaces without local unconditional structure, Composito Math. 37 (1978), 3-19. Zbl0381.46010
[SWS90] F. Schipp, W. R. Wade and P. Simon, Walsh Series, Adam Hilger, Bristol, 1990.
[Ste70] E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Ann. of Math. Stud. 63, Princeton Univ. Press, Princeton, NJ, 1970.
[SF94] A. Sukochev and S. V. Ferleger, Harmonic analysis in symmetric spaces of measurable operators, Dokl. Akad. Nauk 339 (1994), 307-310 (in Russian); English transl.: Russian. Acad. Sci. Dokl. Math. 50 (1995), 432-437.
[SF95] F. A. Sukochev and S. V. Ferleger, Harmonic analysis in (UMD)-spaces: Applications to the theory of bases, Mat. Zametki 58 (1995), 890-905 (in Russian); English transl.: Math. Notes 58 (1995), 1315-1326. Zbl0857.46006
[Sun51] G. I. Sunouchi, On the Walsh-Kaczmarz series, Proc. Amer. Math. Soc. 2 (1951), 5-11. Zbl0044.07103
[Wat58] C. Watari, On generalized Walsh Fourier series, Tôhoku Math. J. (2) 10 (1958), 211-241. Zbl0085.05803
[Wen93] J. Wenzel, Mean convergence of vector-valued Walsh series, Math. Nachr. 162 (1993), 117-124. Zbl0799.42015
Christian Le Merdy, On square functions associated to sectorial operators
Alberto Venni, A Note on Sectorial and R-Sectorial Operators
Peer Christian Kunstmann, Lutz Weis, Perturbation theorems for maximal
{L}_{p}
Bahloul Rachid, Existence and uniqueness of solutions of the fractional integro-differential equations in vector-valued function space
Vitalii Marchenko, Isomorphic Schauder decompositions in certain Banach spaces
Marcinkiewicz-type multiplier theorems,
R
-boundedness, Schauder decomposition
Articles by P. Clément
Articles by B. de Pagter
Articles by F. Sukochev
Articles by H. Witvliet
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On the Solutions of Some Linear Complex Quaternionic Equations
Cennet Bolat, Ahmet İpek, "On the Solutions of Some Linear Complex Quaternionic Equations", The Scientific World Journal, vol. 2014, Article ID 563181, 6 pages, 2014. https://doi.org/10.1155/2014/563181
Cennet Bolat 1 and Ahmet İpek2
1Department of Mathematics, Faculty of Art and Science, Mustafa Kemal University, Tayfur Sökmen Campus, 31100 Hatay, Turkey
2Department of Mathematics, Faculty of Kamil Özdağ Science, Karamanoğlu Mehmetbey University, 70100 Karaman, Turkey
Academic Editor: José Carlos Costa
Some complex quaternionic equations in the type are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.
Mathematics, as with most subjects in science and engineering, has a long and varied history. In this connection one highly significant development which occurred during the nineteenth century was the quaternions, which are the elements of noncommutative algebra. Quaternions have many important applications in many applied fields, such as computer science, quantum physics, statistic, signal, and color image processing, in rigid mechanics, quantum mechanics, control theory, and field theory; see, for example, [1].
In recent years, quaternionic equations have been investigated by many authors. For example, the author of the paper [2] classified solutions of the quaternionic equation . In [3], the linear equations of the forms and in the real Cayley-Dickson algebras (quaternions, octonions, and sedenions) are solved and the form for the roots of such equations is established. In [4], the solutions of the equations of the forms and for some generalizations of quaternions and octonions are investigated. In [5], the linear quaternionic equation with one unknown, , is solved. In [6], Bolat and İpek first defined the quaternion intervals set and the quaternion interval numbers, second, they presented the vector and matrix representations for quaternion interval numbers and then investigated some algebraic properties of these representations, and finally they computed the determinant, norm, inverse, trace, eigenvalues, and eigenvectors of the matrix representation established for a quaternion interval number. In [7], the quaternionic equation is studied. In [8], Bolat and İpek first considered the linear octonionic equation with one unknown of the form , with ; second, they presented a method which allows to reduce any octonionic equation with the left and right coefficients to a real system of eight equations and finally reached the solutions of this linear octonionic equation from this real system. In [9], Flaut and Shpakivskyi investigated the left and right real matrix representations for the complex quaternions. The theory of the quaternion equations and matrix representations of quaternions is considered completely in [2–14].
In this paper, we aim to obtain the solutions of some linear equations with two terms and one unknown by the method of matrix representations of complex quaternions over the complex quaternion field and to investigate the solutions of some complex quaternionic linear equations.
The paper is organized as follows. In Section 2, we start with some basic concepts and results from the theory of the quaternion equations and matrix representations of quaternions which are necessary for the following. In Section 3, we obtain the solutions of some linear equations with two terms and one unknown by the method of matrix representations of complex quaternions over the complex quaternion field. We finish the paper with some conclusions about the study presented.
The following notations, definitions, propositions, lemmas, and theorems will be used to develop the proposed work. We now start the definitions of the quaternion and complex quaternion and their basic properties that will be used in the sequel.
It is well known that a complex number is a number consisting of a real and imaginary part. It can be written in the form , where is the imaginary unit with the defining property . The set of all complex numbers is usually denoted by . From here, it can be easily said that the set of complex numbers is an extension of the set of real numbers, usually denoted by . That is, .
In the literature, firstly, the set of quaternions introduced as with by Irish mathematician Sir William Rowan Hamilton in 1843, is a generalized set of complex numbers. is an algebra over the field , and this algebra is called the real quaternion algebra and the set is a basis in . The elements in take the form , where , which can simply be written as , where and . The conjugate of is defined as , which satisfies for all . The norm of is defined to be . Some simple operation properties on quaternions are listed below:
Theorem 1 (see [10, 15]). Let be quaternions. Then and are similar if and only if and , that is, and .
A complex quaternion is an element of the form , where , and being uniquely determined by and . We denote by the set of the complex quaternions and is an algebra over the field and this algebra is called the complex quaternion algebra. The set is a basis in .
The element , , can be written as where , , and . Therefore, we can write a complex quaternion as the form , where and are in . The conjugate of the complex quaternion is the element , and it satisfies Throughout this note, the algebra is denoted by . For the quaternion , if is defined as then it satisfies the following properties: For the quaternion algebra, , the map where , is an isomorphism between and the algebra of the matrices: We remark that the matrix has as columns the coefficients in of the basis for the elements . The matrix is called the left matrix representation of the element .
Analogously with the left matrix representation, we have for the element the right matrix representation: where .
We remark that the matrix has as columns the coefficients in of the basis for the elements .
Proposition 2 (see [12]). For and , one has(i) , , , , (ii) , , , , (iii) , , where is the inverse of nonzero quaternion.
Proposition 3 (see [12]). For , let be the vector representation of the element . Therefore for all the following relations are fulfilled:(i) ,(ii) ,(iii) ,(iv) ,(v) , where and it is the weak norm of .
For details about the matrix representations of the real quaternions, the reader is referred to [12].
The matrix where is a complex quaternion, with , , and , is called the left real matrix representation for the complex quaternion . The right real matrix representation for the complex quaternion is the matrix: We remark that , ; see [9].
Proposition 4 (see [9]). Let be two quaternions; then, the following relations are true.(i) , where . (ii) , where . (iii) , where . (iv) .(v)For ,
Proposition 5 (see [9]). Let be given. Then
Definition 7 (see [9]). Let , be given. Then is the vector representation of the element , where and , are the vector representations of the quaternions and .
Proposition 8 (see [9]). Let , be given. Then(i) , where is the identity matrix and is the zero matrix;(ii) ;(iii) , where ;(iv) ;(v) ;(vi) ;(vii) , where .
In this section, the complex quaternionic equations in the type are considered. Using the representation matrices and of complex quaternions, the necessary and sufficient conditions for the solvability and the general expression of the solutions are obtained.
According to (ii) and (v) cases in Proposition 8, (18) is equivalent to where , which is a simple system of linear equations over . In order to symbolically solve it, we need to examine some operation properties on the matrix .
Lemma 9. Let be given, and denote . Then(i)the determinant of is where .(ii)if , or , then is nonsingular and its inverse can be expressed as or (iii)if and , then is singular and has a generalized inverse as follows:
Proof. It is a known result that, for all , , there are nonzero such that where and . Now applying Propositions 5, 6, and 8 to both of them we obtain Thus we can derive Consequently, substituting and into it, the proof of th case in Lemma 9 is completed.
The results in Lemma 9 (ii) come from the following two equalities:
Finally, under the conditions that and , it is easily seen that From it and a simple fact that , we can easily deduce the following equality: . So, the proof of th case in Lemma 9 is completed.
Based on Lemma 9, we have the following several results.
Theorem 10. Let be given and . Then the general solution of the equation is where is arbitrary.
Proof. According to (ii) and (v) cases in Proposition 8, (29) is equivalent to and since , (31) has a nonzero solution. In that case, the general solution of (31) can be expressed as where is an arbitrary vector in . Now substituting in Lemma 9 (iii) in it, we get Returning it to complex quaternion form by (ii), (v), and (vii) in Proposition 8, we have (30).
Theorem 11. Let be given. Then(i)the linear equation has a nonzero solution; that is, and are similar, if and only if (ii)in that case, the general solution of (34) is where is arbitrary.
Proof. According to (ii) and (v) cases in Proposition 8, (34) is equivalent to and this equation has a nonzero solution if and only if , which is equivalent, by Lemma 9 (i), to (35). In that case, the general solution of this equation can be expressed as where is an arbitrary vector in . Now substituting in Lemma 9 (iii) in it, we get Returning it to complex quaternion form by (ii), (v), and (vii) in Proposition 8, we have (36).
Theorem 12. Let , , be given with and being not similar; that is, or . Then (18) has a unique solution
Proof. Under the assumption of this theorem, is nonsingular by Lemma 9 (ii). Hence (19) has a unique solution as follows: and the result follows using (ii) and (v) of Proposition 8.
Theorem 13. Let be given. Then the general solution of the equation is where is arbitrary.
Proof. According to (ii) and (v) cases in Proposition 8, (42) is equivalent to and since , (44) has a nonzero solution. In that case, the general solution of (44) can be expressed as where is an arbitrary vector in . Now substituting in Lemma 9 (iii) in it, we get Returning it to complex quaternion form by (ii), (v), and (vii), we have (43).
Theorem 14. Let , , be given with . Then (18) has a solution if and only if in which case the general solution of (18) can be written as where is arbitrary.
Proof. According to (ii) and (v) cases in Proposition 8, (18) is equivalent to This equation is solvable if and only if which is equivalent to Returning it to complex quaternion form by (ii) and (v) cases in Proposition 8, we obtain which is equivalent to , and then (47). In that case the general solution of (49) can be expressed as where is an arbitrary vector in . Returning it to complex quaternion form, we find (48).
Starting from known results and referring to the real matrix representations of the complex quaternions, in this paper we have investigated solutions of some linear equations with two terms and one unknown by the method of matrix representations of complex quaternions over the complex quaternion field.
The methods and results developed in this paper can also extend to complex octonionic equations. We will present them in another paper.
This study is a part of the corresponding author's Ph.D. thesis. The authors would like to thank the anonymous referees for their careful reading and valuable suggestions which improved this work.
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Copyright © 2014 Cennet Bolat and Ahmet İpek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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FalsePosition - Maple Help
Home : Support : Online Help : Education : Student Packages : Numerical Analysis : Visualization : FalsePosition
numerically approximate the real roots of an expression using the method of false position
FalsePosition(f, x=[a, b], opts)
FalsePosition(f, [a, b], opts)
The maximum number of iterations to to perform when numerically approximating a root of f. The default value of maxiterations depends on which type of output is chosen:
output=value : default maxiterations = 100
output=sequence : default maxiterations = 10
output=information : default maxiterations = 10
output=plot : default maxiterations = 5
output=animation : default maxiterations = 10
output = value, sequence, plot, animation, information
output=value returns the final numerical approximation of the root.
output=sequence returns an expression sequence,
{p}_{k}
k
0..n
, where the first
n-1
elements are the subintervals that contain an approximation and the
\mathrm{nth}
element is the final approximate root.
output=plot returns a plot of f with each iterative approximation shown and the relevant information about the numerical approximation displayed in the caption of the plot.
output=animation returns an animation of the numerical approximation on the plot of f.
output=information returns detailed information about the iterative approximations of the root of f.
Whether to display lines that accentuate each approximate iteration when output=plot. By default, this option is set to true. To control the vertical lines, see the showverticallines and verticallineoptions options.
Whether to display the points at each approximate iteration on the plot when output=plot. By default, this option is set to true.
Whether to display the vertical lines at each iterative approximation on the plot when output=plot. By default, this option is set to true.
stoppingcriterion = relative, absolute, function_value
The criterion that the approximation must meet before discontinuing the iterations. The following describes each criterion:
\frac{|{p}_{n}-{p}_{n-1}|}{|{p}_{n}|}
|{p}_{n}-{p}_{n-1}|
|f\left({p}_{n}\right)|
By default, stoppingcriterion=relative.
The tickmarks when output=plot or output=animation. By default, tickmarks are placed at the initial and final approximations with the labels a and b for the two initial approximates and the label
{p}_{n}
for the final approximation, where
n
is the total number of iterations used to reach the final approximation. If the stopping criterion is not met, no tickmark will be placed on the last approximation. See plot/tickmarks for more detail on specifying tickmarks.
\frac{1}{10000}
The FalsePosition command numerically approximates the roots of an algebraic function, f, using a technique similar to the Secant method, but bracketing is incorporated.
Given an expression f and an initial approximate a, the FalsePosition command computes a sequence
{p}_{k}
k
0..n
n
The FalsePosition command is a shortcut for calling the Roots command with the method=falseposition option.
\mathrm{with}\left(\mathrm{Student}[\mathrm{NumericalAnalysis}]\right):
f≔{x}^{3}-7{x}^{2}+14x-6:
\mathrm{FalsePosition}\left(f,x=[2.7,3.2],\mathrm{tolerance}={10}^{-2}\right)
\textcolor[rgb]{0,0,1}{3.014898139}
\mathrm{FalsePosition}\left(f,x=[2.7,3.2],\mathrm{tolerance}={10}^{-2},\mathrm{output}=\mathrm{sequence}\right)
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EUDML | A regularity property of -harmonic functions. EuDML | A regularity property of -harmonic functions.
A regularity property of
p
Greco, Luigi; Verde, Anna
Greco, Luigi, and Verde, Anna. "A regularity property of -harmonic functions.." Annales Academiae Scientiarum Fennicae. Mathematica 25.2 (2000): 317-323. <http://eudml.org/doc/120699>.
author = {Greco, Luigi, Verde, Anna},
keywords = {-harmonic functions; regularity; very weak solutions; Hodge decomposition; -harmonic functions},
title = {A regularity property of -harmonic functions.},
TI - A regularity property of -harmonic functions.
KW - -harmonic functions; regularity; very weak solutions; Hodge decomposition; -harmonic functions
p
-harmonic functions, regularity, very weak solutions, Hodge decomposition,
p
Articles by Greco
Articles by Verde
|
EUDML | On the Cayley transform of positivity classes of matrices. EuDML | On the Cayley transform of positivity classes of matrices.
On the Cayley transform of positivity classes of matrices.
Fallat, Shaun M.; Tsatsomeros, Michael J.
Fallat, Shaun M., and Tsatsomeros, Michael J.. "On the Cayley transform of positivity classes of matrices.." ELA. The Electronic Journal of Linear Algebra [electronic only] 9 (2002): 190-196. <http://eudml.org/doc/122976>.
@article{Fallat2002,
author = {Fallat, Shaun M., Tsatsomeros, Michael J.},
keywords = {inverse -matrix; Cayley transform; -matrices; positive definite matrices; totally nonnegative matrices; stable matrices; matrix factorizations; inverse -matrix; -matrices},
title = {On the Cayley transform of positivity classes of matrices.},
AU - Fallat, Shaun M.
AU - Tsatsomeros, Michael J.
TI - On the Cayley transform of positivity classes of matrices.
KW - inverse -matrix; Cayley transform; -matrices; positive definite matrices; totally nonnegative matrices; stable matrices; matrix factorizations; inverse -matrix; -matrices
M
-matrix, Cayley transform,
P
-matrices, positive definite matrices, totally nonnegative matrices, stable matrices, matrix factorizations, inverse
M
P
Articles by Fallat
Articles by Tsatsomeros
|
A Gene - Open Targets Genetics Documentation
Search by a Gene
Identify loci which functionally implicate a gene
Link out to detailed information on the gene and drugs targeting it
Identify which traits this gene may play a role in, based on variants to which it is assigned
Gene Meta-data
Details of the gene, including link-outs to interrogate the gene in the Open Targets Platform and through other external providers, such as Ensembl, GTEx and Gene Cards. Click through the link to load the Locus View for the gene, with the queried gene pre-selected.
Table summarises published studies and UKB phenotypes with which the queried gene is connected in the Open Targets Genetics pipeline. A gene is connected to a trait in cases where the gene has been functionally assigned to a locus associated with this trait, either via the
V_L
or an assigned proxy
V_T
. Each GWAS Study is linked out via PubMed. UK Biobank traits (Neale et al) do not have a PubMed record. Clicking a Study ID will redirect to the study's page, details of which can be seen here. The table can be sorted by column, filtered by column using the drop-downs, and downloaded in flat and nested formats.
To identify loci and functional variants through which the gene is implicated in a chosen trait, click through to view the Locus Plot. The queried gene, the corresponding
V_L
through which it is associated with the chosen trait, and the
V_T
through which this lead functionally implicates the gene, will be pre-selected when the plot is loaded.
|
EUDML | Common fixed point theorems for commutings -uniformly Lipschitzian mappings in metric spaces. EuDML | Common fixed point theorems for commutings -uniformly Lipschitzian mappings in metric spaces.
Common fixed point theorems for commutings
k
-uniformly Lipschitzian mappings in metric spaces.
Elamrani, M.; Mbarki, A.; Mehdaoui, B.
Elamrani, M., Mbarki, A., and Mehdaoui, B.. "Common fixed point theorems for commutings -uniformly Lipschitzian mappings in metric spaces.." Southwest Journal of Pure and Applied Mathematics [electronic only] 2000.2 (2000): 160-171. <http://eudml.org/doc/232639>.
@article{Elamrani2000,
author = {Elamrani, M., Mbarki, A., Mehdaoui, B.},
keywords = {Kakutani Ryll Nardzewskii fixed point theorem; commuting family of weakly continuous and affine mappings; common fixed point; -uniformly Lipschitzian mappings; normal convexity structure; -uniformly Lipschitzian mappings},
title = {Common fixed point theorems for commutings -uniformly Lipschitzian mappings in metric spaces.},
AU - Elamrani, M.
AU - Mbarki, A.
AU - Mehdaoui, B.
TI - Common fixed point theorems for commutings -uniformly Lipschitzian mappings in metric spaces.
KW - Kakutani Ryll Nardzewskii fixed point theorem; commuting family of weakly continuous and affine mappings; common fixed point; -uniformly Lipschitzian mappings; normal convexity structure; -uniformly Lipschitzian mappings
Kakutani Ryll Nardzewskii fixed point theorem, commuting family of weakly continuous and affine mappings, common fixed point,
k
-uniformly Lipschitzian mappings, normal convexity structure,
k
-uniformly Lipschitzian mappings
A
Articles by Elamrani
Articles by Mbarki
Articles by Mehdaoui
|
In the presence of dynamic obstacles, a local motion planner also requires predictions about the surroundings to assess the validity of planned trajectories. In this example, you represent the surrounding environment using the discrete set of objects approach. For an example using discretized grid, refer to the Motion Planning in Urban Environments Using Dynamic Occupancy Grid Map (Sensor Fusion and Tracking Toolbox) example.
\mathit{k}
{\mathit{x}}_{\mathit{k}}=\left[{\mathit{s}}_{\mathit{k}}\stackrel{˙}{\text{\hspace{0.17em}}{\mathit{s}}_{\mathit{k}\text{\hspace{0.17em}}}\text{\hspace{0.17em}}}{\mathit{d}}_{\mathit{k}\text{\hspace{0.17em}}}\text{\hspace{0.17em}}\stackrel{˙}{{\mathit{d}}_{\mathit{k}\text{\hspace{0.17em}}}}\right]
{\mathit{s}}_{\mathit{k}}
{\mathit{d}}_{\mathit{k}\text{\hspace{0.17em}}}
\left[\begin{array}{c}{\mathit{s}}_{\mathit{k}+1}\\ \stackrel{˙}{{\mathit{s}}_{\mathit{k}+1}}\\ {\mathit{d}}_{\mathit{k}+1}\\ \stackrel{˙}{{\mathit{d}}_{\mathit{k}+1}}\end{array}\right]=\left[\begin{array}{cccc}1& \Delta \mathit{T}& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& \tau \left(1-{\mathit{e}}^{\left(-\frac{\Delta \mathit{T}}{\text{\hspace{0.17em}}\tau }\right)\text{\hspace{0.17em}}}\right)\\ 0\text{\hspace{0.17em}}& 0\text{\hspace{0.17em}}& 0\text{\hspace{0.17em}}& {\mathit{e}}^{\left(-\frac{\Delta \mathit{T}}{\text{\hspace{0.17em}}\tau }\right)\text{\hspace{0.17em}}}\end{array}\right]\left[\begin{array}{c}{\mathit{s}}_{\mathit{k}}\\ \stackrel{˙}{{\mathit{s}}_{\mathit{k}}}\\ {\mathit{d}}_{\mathit{k}}\\ \stackrel{˙}{{\mathit{d}}_{\mathit{k}}}\end{array}\right]+\left[\begin{array}{cc}\frac{\Delta {\mathit{T}}^{2}}{2}& 0\\ \Delta \mathit{T}& 0\\ 0& \frac{\Delta {\mathit{T}}^{2}}{2}\\ 0& \Delta \mathit{T}\end{array}\right]\left[\begin{array}{c}{\mathit{w}}_{\mathit{s}}\\ {\mathit{w}}_{\mathit{d}\text{\hspace{0.17em}}}\end{array}\right]
\Delta \mathit{T}
\mathit{k}
\mathit{k}+1
{\mathit{w}}_{\mathit{s}}
{\mathit{w}}_{\mathit{d}\text{\hspace{0.17em}}}
\tau
The scenario used in this example is created using the Driving Scenario Designer and then exported to a MATLAB® function. The ego vehicle is mounted with 1 forward-looking radar and 5 cameras providing 360-degree coverage. The radar and cameras are simulated using the drivingRadarDataGenerator and visionDetectionGenerator System objects, respectively.
You set up a joint probabilistic data association tracker using the trackerJPDA (Sensor Fusion and Tracking Toolbox) System object. You set the FilterInitializationFcn property of the tracker to helperInitRefPathFilter function. This helper function defines an extended Kalman filter, trackerJPDA (Sensor Fusion and Tracking Toolbox), used to estimate the state of a single object. Local functions inside the helperInitRefPathFilter file define the state transition as well as measurement model for the filter. Further, to predict the tracks at a future time for the motion planner, you use the predictTracksToTime (Sensor Fusion and Tracking Toolbox) function of the tracker.
trackerJPDA (Sensor Fusion and Tracking Toolbox) | referencePathFrenet (Navigation Toolbox) | trajectoryGeneratorFrenet (Navigation Toolbox) | drivingScenario
Optimal Trajectory Generation for Urban Driving (Navigation Toolbox)
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2013 Stabilization Strategies of Supply Networks with Stochastic Switched Topology
Shukai Li, Jianxiong Zhang, Wansheng Tang
In this paper, a dynamical supply networks model with stochastic switched topology is presented, in which the stochastic switched topology is dependent on a continuous time Markov process. The goal is to design the state-feedback control strategies to stabilize the dynamical supply networks. Based on Lyapunov stability theory, sufficient conditions for the existence of state feedback control strategies are given in terms of matrix inequalities, which ensure the robust stability of the supply networks at the stationary states and a prescribed
{H}_{\infty }
disturbance attenuation level with respect to the uncertain demand. A numerical example is given to illustrate the effectiveness of the proposed method.
Shukai Li. Jianxiong Zhang. Wansheng Tang. "Stabilization Strategies of Supply Networks with Stochastic Switched Topology." J. Appl. Math. 2013 1 - 7, 2013. https://doi.org/10.1155/2013/605017
Shukai Li, Jianxiong Zhang, Wansheng Tang "Stabilization Strategies of Supply Networks with Stochastic Switched Topology," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-7, (2013)
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face - Simple English Wiktionary
Face is on the Basic English 850 List.
face is one of the 1000 most common headwords.
enPR: fās
IPA (key): /feɪs/
(countable) A face is the front part of the head.
Synonyms: countenance, features, mug, visage and facade
His face was red with embarrassment.
A look or expression on the face.
Synonyms: expression, look, appearance and air
He made a face at the bitter medicine.
One side of a many-sided shape.
Synonyms: side, surface, aspect, elevation and facet
A dodecahedron has twelve faces.
{\displaystyle x}
{\displaystyle y}
, the front of
{\displaystyle x}
is pointing in the direction of
{\displaystyle y}
Please face me when I speak to you.
The store faces the bank.
It's hard to face the fact that his wife is gone.
You have to face the bully.
(to be opposite) be opposite, be in front of, stand in front of, stand facing, look toward
(admit) admit, accept, be realistic, realize, bite the bullet
(confront) confront, tackle, meet, cope with, challenge, deal with, play, handle, play against, be drawn against
(to be opposite) be adjacent, be beside
(admit) deny, refute, reject, rebuff, contradict, refuse, disallow, decline, renounce, disavow
(confront) avoid, shun, steer clear of, let alone, evade, circumvent, get around, dodge, sidestep, elude, escape, prevent, forestall, preclude, avert
Retrieved from "https://simple.wiktionary.org/w/index.php?title=face&oldid=484109"
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Mathematical_diagram Knowpia
Mathematical diagrams, such as charts and graphs, are mainly designed to convey mathematical relationships—for example, comparisons over time.[1]
Euclid's Elements, ms. from Lüneburg, A.D. 1200
Specific types of mathematical diagramsEdit
Argand diagramEdit
Argand diagram.
A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745–1818).[2] Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.
Butterfly diagramEdit
In the context of fast Fourier transform algorithms, a butterfly is a portion of the computation that combines the results of smaller discrete Fourier transforms (DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into subtransforms). The name "butterfly" comes from the shape of the data-flow diagram in the radix-2 case, as described below. The same structure can also be found in the Viterbi algorithm, used for finding the most likely sequence of hidden states.
The butterfly diagram show a data-flow diagram connecting the inputs x (left) to the outputs y that depend on them (right) for a "butterfly" step of a radix-2 Cooley–Tukey FFT algorithm. This diagram resembles a butterfly as in the morpho butterfly shown for comparison, hence the name.
A commutative diagram depicting the five lemma
Commutative diagramEdit
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition.
Commutative diagrams play the role in category theory that equations play in algebra.
Hasse diagramsEdit
A Hasse diagram is a simple picture of a finite partially ordered set, forming a drawing of the partial order's transitive reduction. Concretely, one represents each element of the set as a vertex on the page and draws a line segment or curve that goes upward from x to y precisely when x < y and there is no z such that x < z < y. In this case, we say y covers x, or y is an immediate successor of x. In a Hasse diagram, it is required that the curves be drawn so that each meets exactly two vertices: its two endpoints. Any such diagram (given that the vertices are labeled) uniquely determines a partial order, and any partial order has a unique transitive reduction, but there are many possible placements of elements in the plane, resulting in different Hasse diagrams for a given order that may have widely varying appearances.
Knot diagramsEdit
In Knot theory a useful way to visualise and manipulate knots is to project the knot onto a plane—;think of the knot casting a shadow on the wall. A small perturbation in the choice of projection will ensure that it is one-to-one except at the double points, called crossings, where the "shadow" of the knot crosses itself once transversely[3]
At each crossing we must indicate which section is "over" and which is "under", so as to be able to recreate the original knot. This is often done by creating a break in the strand going underneath. If by following the diagram the knot alternately crosses itself "over" and "under", then the diagram represents a particularly well-studied class of knot, alternating knots.
Venn diagramEdit
A Venn diagram is a representation of mathematical sets: a mathematical diagram representing sets as circles, with their relationships to each other expressed through their overlapping positions, so that all possible relationships between the sets are shown.[4]
The Venn diagram is constructed with a collection of simple closed curves drawn in the plane. The principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram. That is, the diagram initially leaves room for any possible relation of the classes, and the actual or given relation, can then be specified by indicating that some particular region is null or is not null.[5]
Voronoi centerlines.
Voronoi diagramEdit
A Voronoi diagram is a special kind of decomposition of a metric space determined by distances to a specified discrete set of objects in the space, e.g., by a discrete set of points. This diagram is named after Georgy Voronoi, also called a Voronoi tessellation, a Voronoi decomposition, or a Dirichlet tessellation after Peter Gustav Lejeune Dirichlet.
In the simplest case, we are given a set of points S in the plane, which are the Voronoi sites. Each site s has a Voronoi cell V(s) consisting of all points closer to s than to any other site. The segments of the Voronoi diagram are all the points in the plane that are equidistant to two sites. The Voronoi nodes are the points equidistant to three (or more) sites
Wallpaper group diagram.
Wallpaper group diagramsEdit
A wallpaper group or plane symmetry group or plane crystallographic group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct groups.
Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups, also called space groups. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns which are very different in style, color, scale or orientation may belong to the same group.
Young diagramEdit
A Young diagram or Young tableau, also called Ferrers diagram, is a finite collection of boxes, or cells, arranged in left-justified rows, with the row sizes weakly decreasing (each row has the same or shorter length than its predecessor).
Young diagram.
Listing the number of boxes in each row gives a partition
{\displaystyle \lambda }
of a positive integer n, the total number of boxes of the diagram. The Young diagram is said to be of shape
{\displaystyle \lambda }
, and it carries the same information as that partition. Listing the number of boxes in each column gives another partition, the conjugate or transpose partition of
{\displaystyle \lambda }
; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal.
Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians.
Other mathematical diagramsEdit
^ Working with diagrams at LearningSpace.
^ Wessel's memoir was presented to the Danish Academy in 1797; Argand's paper was published in 1806.
( Whittaker, Edmund Taylor; Watson, G.N. (1927). A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions. Cambridge University Press. p. 9. ISBN 978-0-521-58807-2. )
^ Rolfsen, Dale (1976). Knots and links. Publish or Perish. ISBN 978-0-914098-16-4.
^ "Venn diagram" Archived 2009-11-01 at WebCite, Encarta World English Dictionary, North American Edition 2007. Archived 2009-11-01.
^ Clarence Irving Lewis (1918). A Survey of Symbolic Logic. Republished in part by Dover in 1960. p. 157.
Barker-Plummer, Dave; Bailin, Sidney C. (1997). "The Role of Diagrams in Mathematical Proofs". Machine Graphics and Vision. 6 (1): 25–56. 10.1.1.49.4712. (Special Issue on Diagrammatic Representation and Reasoning).
Barker-Plummer, Dave; Bailin, Sidney C. (2001). "On the practical semantics of mathematical diagrams". In Anderson, M. (ed.). Reasoning with Diagrammatic Representations. Springer Verlag. ISBN 978-1-85233-242-6. CiteSeerX: 10.1.1.30.9246.
Kidman, G. (2002). "The Accuracy of mathematical diagrams in curriculum materials". In Cockburn, A.; Nardi, E. (eds.). Proceedings of the PME 26. Vol. 3. University of East Anglia. pp. 201–8.
Kulpa, Zenon (2004). "On Diagrammatic Representation of Mathematical Knowledge". In Andréa Asperti; Bancerek, Grzegorz; Trybulec, Andrzej (eds.). Mathematical knowledge management: third international conference, MKM 2004, Białowieża, Poland, September 19–21, 2004 : Proceedings. Springer. pp. 191–204. ISBN 978-3-540-23029-8.
Puphaiboon, K.; Woodcock, A.; Scrivener, S. (25 March 2005). "Design method for developing mathematical diagrams". In Bust, Philip D.; McCabe, P.T. (eds.). Contemporary ergonomics 2005 Proceedings of the International Conference on Contemporary Ergonomics (CE2005). Taylor & Francis. ISBN 978-0-415-37448-4.
Wikimedia Commons has media related to Mathematical diagram.
"Diagrams". The Stanford Encyclopedia of Philosophy. Fall 2008.
Kulpa, Zenon. "Diagrammatics: The art of thinking with diagrams". Archived from the original on April 25, 2013.
One of the oldest extant diagrams from Euclid by Otto Neugebauer
Lomas, Dennis (1998). "Diagrams in Mathematical Education: A Philosophical Appraisal". Philosophy of Education Society. Archived from the original on 2011.
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Airy stress function - Wikiversity
1.1 Airy stress function in rectangular Cartesian coordinates
1.2 Airy stress function in polar coordinates
2 Stress equation of compatibility in 2-D
3 Some biharmonic Airy stress functions
4 Displacements in terms of scalar potentials
The Airy stress function (
{\displaystyle \varphi }
Scalar potential function that can be used to find the stress.
Satisfies equilibrium in the absence of body forces.
Only for two-dimensional problems (plane stress/plane strain).
Airy stress function in rectangular Cartesian coordinates[edit | edit source]
If the coordinate basis is rectangular Cartesian
{\displaystyle (\mathbf {e} _{1},~\mathbf {e} _{2})}
with coordinates denoted by
{\displaystyle (x_{1},~x_{2})}
then the Airy stress function
{\displaystyle (\varphi )}
is related to the components of the Cauchy stress tensor
{\displaystyle ({\boldsymbol {\sigma }})}
{\displaystyle {\begin{aligned}\sigma _{11}&=\varphi _{,22}={\cfrac {\partial ^{2}\varphi }{\partial x_{2}^{2}}}\\\sigma _{22}&=\varphi _{,11}={\cfrac {\partial ^{2}\varphi }{\partial x_{1}^{2}}}\\\sigma _{12}&=-\varphi _{,12}=-{\cfrac {\partial ^{2}\varphi }{\partial x_{1}\partial x_{2}}}\end{aligned}}}
Alternatively, if we write the basis as
{\displaystyle (\mathbf {e} _{x},\mathbf {e} _{y})}
and the coordinates as
{\displaystyle (x,y)\,}
, then the Cauchy stress components are related to the Airy stress function by
{\displaystyle {\begin{aligned}\sigma _{xx}&={\cfrac {\partial ^{2}\varphi }{\partial y^{2}}}\\\sigma _{yy}&={\cfrac {\partial ^{2}\varphi }{\partial x^{2}}}\\\sigma _{xy}&=-{\cfrac {\partial ^{2}\varphi }{\partial x\partial y}}\end{aligned}}}
Airy stress function in polar coordinates[edit | edit source]
In polar basis
{\displaystyle (\mathbf {e} _{r},\mathbf {e} _{\theta })}
{\displaystyle (r,\theta )\,}
, the Airy stress function is related to the components of the Cauchy stress via
{\displaystyle {\begin{aligned}\sigma _{rr}&={\cfrac {1}{r}}{\cfrac {\partial \varphi }{\partial r}}+{\cfrac {1}{r^{2}}}{\cfrac {\partial ^{2}\varphi }{\partial \theta ^{2}}}\\\sigma _{\theta \theta }&={\cfrac {\partial ^{2}\varphi }{\partial r^{2}}}\\\sigma _{r\theta }&=-{\cfrac {\partial }{\partial r}}\left({\cfrac {1}{r}}{\cfrac {\partial \varphi }{\partial \theta }}\right)\end{aligned}}}
Do you think the Airy stress function can be extended to three dimensions?
Stress equation of compatibility in 2-D[edit | edit source]
In the absence of body forces,
{\displaystyle \nabla ^{2}{(\sigma _{11}+\sigma _{22})}=0}
{\displaystyle \sigma _{11,11}+\sigma _{11,22}+\sigma _{22,11}+\sigma _{22,22}=0\,}
Note that the stress field is independent of material properties in the absence of body forces (or homogeneous body forces).
Therefore, the plane strain and plane stress solutions are the same if the boundary conditions are expressed as traction BCS.
In terms of the Airy stress function
{\displaystyle \varphi _{,1122}+\varphi _{,2222}+\varphi _{,1111}+\varphi _{,1122}=0\,}
{\displaystyle {\cfrac {\partial ^{4}\varphi }{\partial x_{1}^{4}}}+2{\cfrac {\partial ^{4}\varphi }{\partial x_{1}^{2}\partial x_{2}^{2}}}+{\cfrac {\partial ^{4}\varphi }{\partial x_{2}^{4}}}=0}
{\displaystyle \nabla ^{4}\varphi =0\,}
The stress function
{\displaystyle (\varphi )}
is biharmonic.
Any polynomial in
{\displaystyle x_{1}}
{\displaystyle x_{2}}
of degree less than four is biharmonic.
Stress fields that are derived from an Airy stress function which satisfies the biharmonic equation will satisfy equilibrium and correspond to compatible strain fields.
Some biharmonic Airy stress functions[edit | edit source]
In cylindrical co-ordinates, some biharmonic functions that may be used as Airy stress functions are
{\displaystyle {\begin{aligned}\varphi &=C\theta \\\varphi &=Cr^{2}\theta \\\varphi &=Cr\theta \cos \theta \\\varphi &=Cr\theta \sin \theta \\\varphi &=f_{n}(r)\cos(n\theta )\\\varphi &=f_{n}(r)\sin(n\theta )\\\end{aligned}}}
{\displaystyle {\begin{aligned}f_{0}(r)&=a_{0}r^{2}+b_{0}r^{2}\ln r+c_{0}+d_{0}\ln r\\f_{1}(r)&=a_{1}r^{3}+b_{1}r+c_{1}r\ln r+d_{1}r^{-1}\\f_{n}(r)&=a_{n}r^{n+2}+b_{n}r^{n}+c_{n}r^{-n+2}+d_{n}r^{-n}~,~~n>1\end{aligned}}}
Displacements in terms of scalar potentials[edit | edit source]
If the body force is negligible, then the displacements components in 2-D can be expressed as
{\displaystyle 2\mu u_{1}=-\varphi _{,1}+\alpha \psi _{,2}~,~~2\mu u_{2}=-\varphi _{,2}+\alpha \psi _{,1}}
{\displaystyle \alpha ={\begin{cases}1-\nu &{\text{for plane strain}}\\{\cfrac {1}{1+\nu }}&{\text{for plane stress}}\end{cases}}}
{\displaystyle \psi (x_{1},x_{2})}
is a scalar displacement potential function that satisfies the conditions
{\displaystyle \nabla ^{2}{\psi }=0~,~~\psi _{,12}=\nabla ^{2}{\varphi }}
To prove the above, you have to use the plane strain/stress constitutive relations
{\displaystyle {\begin{matrix}\sigma _{\alpha \beta }&=2\mu \left[\varepsilon _{\alpha \beta }+\left({\cfrac {1-\alpha }{2\alpha -1}}\right)\varepsilon _{\gamma \gamma }\delta _{\alpha \beta }\right]\\\varepsilon _{\alpha \beta }&={\cfrac {1}{2\mu }}\left[\sigma _{\alpha \beta }+\left(1-\alpha \right)\sigma _{\gamma \gamma }\delta _{\alpha \beta }\right]\end{matrix}}}
Note also that the plane stress/strain compatibility equations can be written as
{\displaystyle \nabla ^{2}{\sigma _{\gamma \gamma }}=-{\cfrac {1}{\alpha }}f_{\gamma ,\gamma }}
Retrieved from "https://en.wikiversity.org/w/index.php?title=Airy_stress_function&oldid=2138044"
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less - Simple English Wiktionary
Less is on the Basic English 850 List.
less is one of the 1000 most common headwords.
IPA (key): /lɛs/
SAMPA: /lEs/
(indefinite) (singular) (non-count) A smaller amount or degree.
Synonyms: fewer and reduced
Antonym: more
He does less of the work than she does.
I want less soup.
As we build new roads, these small ones will become less important.
He wore a dark colour that was less likely to be seen.
{\displaystyle x}
{\displaystyle y}
, you do not do
{\displaystyle x}
as much as
{\displaystyle y}
I read less than I used to. = "reading now < reading before"
He works less than she does.
My leg really hurts, but my arm hurts less. = "my arm hurts < my leg hurts"
If you do something less often, easily, well, etc., you do not do it as often, easily, well, etc.
I go out less often than before.
He plays piano less than she does.
If something is less likely, expensive, important, etc., it is not as likely, expensive, important, etc.
This book is less expensive than that book. ="the price of this book < the price of that book"
You should go out. Studying is less important.
(with adjectives) Less identifies the negative comparative form of all comparable adjectives. For example, with the adjective intelligent, the negative comparative is less intelligent. A similar form is the negative superlative.
(mathematics) minus
Antonym: plus
Seven less four is three.
{\displaystyle 7-4=3}
Retrieved from "https://simple.wiktionary.org/w/index.php?title=less&oldid=473779"
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A Variant - Open Targets Genetics Documentation
Search by a Variant
Identify a ranked list of genes which are functionally implicated by the variant
View and dissect the functional data by which genes are assigned to this variant
View PheWAS results for the variant in UK Biobank
View linkage structure around the variant
Whether a variant is a lead or tag cannot be specified whilst searching - simply enter the variant of interest using an rsID or chromosome position e.g. 1_154426264_C_T. When entering a variant in the chromosome position format, options corresponding to the various allele combinations at the locus will be displayed in the search drop-down. There will be slight differences in the returned variant page for lead and tag variants.
All variants in Open Targets Genetics are notated using the format chr_pos_ref_alt, where ref and alt are the reference and alternate alleles as defined by Ensembl. Coordinates are relative to the GRCh38 assembly also known as hg38.
Variant Meta-data
Overall identifying information for the variant, including its rsID if assigned, nearest gene and nearest protein-coding gene according to GRCh38. Clicking through the 'View Locus' link will load the Locus View, with the variant preselected. If the variant is a lead variant in any published study or in UK Biobank sumstats, it will be highlighted on the
V_L
track; otherwise it will be selected as a
V_T
Table summarising the extent of evidence by which the queried variant implicates various genes. The default view summarises the combined evidence for each gene by each functional data source, collapsed across cell types within the data source. Overall G2V is a representation of this combined evidence weighting for each gene. Sort descending on this column to rank the column from most- to least-likely 'causal' gene at this locus based on the current evidence base in Open Targets Genetics.
The presence of a bullet in the table indicates that there is evidence from the given data source linking the gene to the queried variant, in at least one cell type. The radius of each point is proportionate to the relative magnitude of the maximum effect size across all tissues available for the data source, expressed as a quantile. Details of how the V2G score is calculated and weighted are detailed in 'Our Approach'.
To view tissue-specific evidence within a data source, select the data source from the tabs visible at the top of the table widget. An equivalent view segregated by cell type rather than data source will be opened, as above for eQTL evidence in each of 91 cell types. If evidence from the data source being examined can be interpreted directionally, bullets will be coloured according to the direction of effect: blue notating a positive beta, and red a negative. Again, the radius of the bullet is proportionate to the beta's magnitude. Hovering over a bullet will display the underlying beta and p-value represented, as reported by the original data source.
UK Biobank PheWAS
PheWAS results for the selected variant across all UK Biobank phenotypes released by Neale and colleagues are displayed as a PheWAS plot segregated by high-level phenotype grouping, and detailed in an underlying table. The red line denotes the significance level after Bonferroni correction for the number of phenotypes testing, conservatively considering each as independent. The direction of the plot character arrow corresponds to the beta direction of effect, and points are coloured corresponding to their broad phenotype. Details of the association of each phenotype with the trait of interest are displayed in a table beneath the plot, which can be sorted, filtered on column, and downloaded. A direct locus view link is also provided for each phenotype, which will load the locus view with the variant and UK Biobank trait pre-selected.
Finally, we display two tables dedicated to the genetic architecture of the locus to which the variant of interest belongs - 'GWAS Lead Variants' and 'Tag Variants'. The former displays all GWAS lead variants (from either GWAS Catalog or UK Biobank) to which the queried variant has been assigned as a proxy (tag) based on LD or fine-mapping. If the queried variant is itself a GWAS lead variant, the second table displays all variants which have been assigned as a proxy to it. This table will not be displayed if the variant queried is not a lead variant. Note that most of the details shown in the 'Tag Variants' table (for example to PMID and other study meta-data) refer to the lead variant. Fields referring to the tag variant are clearly denoted. LD refers to the LD between the tag and the lead. The posterior probability corresponds to the likelihood that the tag variant is the causal variant for the trait shown at this disease locus. This data will only be available for UK Biobank traits at present, as its calculation requires full summary statistics.
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A Poisson hierarchical modelling approach to detecting copy number variation in sequence coverage data | SpringerLink
\dots \phantom{\rule{0.3em}{0ex}}
f\left(\left\{{n}_{g,i}\right\}|{N}_{g};\eta \right)={N}_{g}!\prod _{i}\frac{{p}_{g}{\left(i\right)}^{{n}_{g,i}}}{{n}_{g,i}!},
{N}_{g}={\sum }_{i}{n}_{g,i}
{p}_{g}\left(i\right)=\frac{\Gamma \left(\alpha +i\right)}{\Gamma \left(i\right)\Gamma \left(\alpha +1\right)}{\left(\frac{\beta }{\beta +1}\right)}^{\alpha }{\left(\frac{1}{\beta +1}\right)}^{i},
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A word u defined over an alphabet
𝒜
is c-balanced (c ∈
ℕ
) if for all pairs of factors v, w of u of the same length and for all letters a ∈
𝒜
, the difference between the number of letters a in v and w is less or equal to c. In this paper we consider a ternary alphabet
𝒜
= {L, S, M} and a class of substitutions
{\phi }_{p}
{\phi }_{p}
(L) = LpS,
{\phi }_{p}
(S) = M,
{\phi }_{p}
(M) = Lp-1S where p > 1. We prove that the fixed point of
{\phi }_{p}
, formally written as
{\phi }_{p}^{\infty }
(L), is 3-balanced and that its abelian complexity is bounded above by the value 7, regardless of the value of p. We also show that both these bounds are optimal, i.e. they cannot be improved.
Mots clés : balance property, abelian complexity, substitution, ternary word
author = {Turek, Ond\v{r}ej},
title = {Balances and abelian complexity of a certain class of infinite ternary words},
AU - Turek, Ondřej
TI - Balances and abelian complexity of a certain class of infinite ternary words
Turek, Ondřej. Balances and abelian complexity of a certain class of infinite ternary words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 3, pp. 313-337. doi : 10.1051/ita/2010017. http://www.numdam.org/articles/10.1051/ita/2010017/
[2] B. Adamczewski, Balances for fixed points of primitive substitutions. Theoret. Comput. Sci. 307 (2003) 47-75. | Zbl 1059.68083
[3] Ľ. Balková, E. Pelantová and Š. Starosta, Sturmian Jungle (or Garden?) on Multiliteral Alphabets. RAIRO: Theoret. Informatics Appl (to appear).
[4] Ľ. Balková, E. Pelantová and O. Turek, Combinatorial and Arithmetical Properties of Infinite Words Associated with Quadratic Non-simple Parry Numbers. RAIRO: Theoret. Informatics Appl. 41 3 (2007) 307-328. | Zbl 1144.11009
[5] V. Berthé and R. Tijdeman, Balance properties of multi-dimensional words. Theoret. Comput. Sci. 273 (2002) 197-224. | Zbl 0997.68091
[6] J. Cassaigne, Recurrence in infinite words, in Proc. STACS, LNCS Dresden (Allemagne) 2010, Springer (2001) 1-11. | Zbl 0976.68524
[7] J. Cassaigne, S. Ferenczi and L.Q. Zamboni, Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier 50 (2000) 1265-1276. | Zbl 1004.37008
[8] E.M. Coven and G.A. Hedlund, Sequences with minimal block growth. Math. Syst. Th. 7 (1973) 138-153. | Zbl 0256.54028
[9] J. Currie and N. Rampersad, Recurrent words with constant Abelian complexity. Adv. Appl. Math. doi:10.1016/j.aam.2010.05.001 (2010).
[10] S. Fabre, Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219-236. | Zbl 0872.11017
[11] C. Frougny, J.P. Gazeau and J. Krejcar, Additive and multiplicative properties of point-sets based on beta-integers. Theor. Comp. Sci. 303 (2003) 491-516. | Zbl 1036.11034
[12] K. Klouda and E. Pelantová, Factor complexity of infinite words associated with non-simple Parry numbers. Integers - Electronic Journal of Combinatorial Number Theory (2009) 281-310. | Zbl 1193.68201
[13] M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002).
[14] M. Morse and G.A. Hedlund, Symbolic dynamics. Am. J. Math. 60 (1938) 815-866. | JFM 64.0798.04
[15] M. Morse and G.A. Hedlund, Symbolic dynamics II. Sturmian Trajectories. Am. J. Math. 62 (1940) 1-42. | JFM 66.0188.03
[16] G. Richomme, K. Saari, L. Q. Zamboni, Balance and Abelian Complexity of the Tribonacci word. Adv. Appl. Math. 45 (2010) 212-231. | Zbl 1203.68131
[17] G. Richomme, K. Saari, L. Q. Zamboni, Abelian Complexity in Minimal Subshifts. J. London Math. Soc. (to appear).
[18] W. Thurston, Groups, tilings and finite state automata. AMS Colloquium Lecture Notes (1989).
[19] O. Turek, Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers. RAIRO: Theoret. Informatics Appl. 41 2 (2007) 123-135. | Zbl 1146.68410
[20] L. Vuillon, Balanced words. Bull. Belg. Math. Soc. Simon Stevin 10 (2003) 787-805. | Zbl 1070.68129
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Maths and Juggling – Math is in the Air
February 21, 2018 3 Comments Written by Alessandro Budroni
A mathematical language for juggling
We have all seen, at least once in our life, a juggler tossing balls in the air. Why is that so impressing at our eyes?
Despite having just two hands, any respectable juggler can juggle three balls at the same time. Considering for simplicity that one can handle one ball for each hand, how is that possible?
Let's try to analyze Animation 1. We can see that each ball is tossed by one hand to the other: the right hand tosses the balls to the left hand and vice versa. Just as the floating ball floating is about to fall down, the juggler tosses another ball up to free his hand and catch the falling one. Juggling three or more balls is possible only by iterating this principle.
Animation 1: cascade
The pattern represented in Animation 1 is known as three-ball cascade. Let's analyze now Animation 2 and compare it with Animation 1.
In this case we immediately note that the number of balls is still 3, but the pattern is different. Indeed, by observing it carefully, we see that the juggler tosses the three balls at three different heights.
As you can easily imagine, there is a wide variety of patterns and, if we were to assign a name to each pattern (as in the case of the cascade), we would have to make a prohibitive effort of memory.
For this reason Paul Klimek and Don Hatch, at the beginning of the 80s, independently invented a notation system to describe and name juggling tricks nowadays called siteswap. Afterwards, this system has been developed and extended by other jugglers, like Bruce Tiemann, Jack Boyce and Ben Beever.
Siteswap is able to describe (and name) all juggling patterns with any number of jugglers and balls, covering both the case of synchronous and asynchronous throws. In the following two animations we can see the same pattern done in both the asynchronous and synchronous versions.
(NOTE: some patterns can be only asynchronous while others can be only synchronous).
For simplicity, we will describe the so-called Vanilla siteswap. This siteswap notation allows us to describe all the patterns where the balls are tossed asynchronously by a single juggler using both hands.
A limitation of the siteswap notation
Before going through the description of this notation method, we must underline that siteswap has a limitation. Let's observe the following two animations.
We have seen already the left-side animation: the three-ball cascade. The right-side pattern, known as three-ball Mill's Mess, is still a cascade but it's done by crossing and switching the hands' position alternatively. Even though the two patterns look very different, they have the same siteswap notation, i.e. they are identical. Indeed, if we focus on the trajectories of the balls with respect to the positions of the hands, we see that Mill's Mess is identical to the normal cascade.
Therefore, siteswap is able to describe juggling patterns by considering the height and the direction in which the balls are tossed (a ball can be tossed to the same or to the other hand) but without considering "how" the pattern is executed.
A number for each toss
After this quick introduction, we will now describe how siteswap works. The basic idea is very simple: we assign a positive integer number to each throw that corresponds to the number of beats (soon we will deepen this concept) that the ball takes to complete his trajectory. We use odd numbers (1, 3, 5, ...) for throws from one hand to the other hand and even numbers (2, 4, 6, ...) for throws from one hand to itself. The number zero (0) is used to indicate when one hand is not holding balls during a beat.
A 0 means a beat when the hand is empty.
A 1 means a direct throw from one hand to the other, during which there is no time to catch or throw other balls, i.e. it is executed in one beat.
A 2 means a very small throw (almost imperceptible) of a ball to the same hand. While the ball is completing its trajectory, the hand who tossed it has no time to do anything else while the other one has a beat to catch and throw another ball.
A 3 means a throw from one hand to the other during which both have a beat to juggle a ball each (so there is time to juggle two other balls).
A 4 means a throw from one hand to the same hand during which the tossing hand can juggle another ball while the other hand can juggle two balls (so there is time to juggle other 3 balls).
Therefore, the numbers indicate the height at which the balls are tossed relatively to the execution speed of the throws. Indeed, it is possible to toss a 5 with top height under our head if we juggle quickly, or over 3 meters if we juggle slowly. What really matters are the beats left to juggle other balls during the trajectory of the toss. This depends, of course, by the speed of the juggler.
Furthermore, as it is easy to guess from the animations above, the patterns are repeated cyclically. In other words, there is a period after which the pattern is repeated (identically or symmetrically). With the siteswap notation we only write the throws that identify the period of the pattern. For example, the period of 531531531 is 531. We refer to it as 531 by removing the redundant part and without loosing any information.
Once the concepts detailed above are clear, we can try to recognize some patterns:
Not every sequence of numbers is a pattern!
Once we are familiar with the concept of siteswap we can go through a little bit of theory. Let's try to imagine the pattern 432. First, say with the right hand, we toss a 4, i.e. the ball will falls in the same had. Then we toss a 3 with the left hand, i.e. the ball will fall in to the right hand. While the two balls are still completing their trajectory, the right hand executes a 2, in other words it performs a small toss to itself. What will happen is that the right hand will find itself with three balls falling on it at the same time. In siteswap jargon this event is called collision, and the pattern is impossible to repeat. Indeed, the sequence 432 is not executable.
How can we distinguish an executable sequence from a non executable one? Fortunately maths comes to the rescue! Indeed, there is a theorem that characterizes siteswaps and gives us a condition such that there are no collisions.
Characterization theorem of siteswaps
A finite sequence of non-negative numbers
s_1s_2...s_n
n
is the number of digits) is executable if
i \neq j
Here, the operator
a\mod b
returns the remainder of the division
\frac{a}{b}
Let's come back to the previous example and verify, using the theorem, that the sequence 432 is not valid:
4 + 1\mod 3 = 2
3 + 2\mod 3 = 2
2 + 3\mod 3 = 2
In this case we get 2 for every digit of the sequence and, according to the theorem, this is not a valid siteswap. We now try to apply the theorem to a valid siteswap that can be obtained by switching the last two digits of the sequence above: 423
4 + 1 \mod 3 = 2
2 + 2 \mod 3 = 1
3 + 3 \mod 3 = 0
It is clear that this siteswap respects the condition imposed by the theorem (and you can actually find it in one of the animations above).
Suppose now to have a valid siteswap, for example 534, How many balls do we need in order to execute it? Again we have another nice and helpful theorem used by the jugglers from all over the world.
Theorem on the number of balls
s_1s_2...s_n
is a valid siteswap (where
n
is the number of digits), then we have that
Let's try how many balls we need for the 534 pattern:
The answer is 4 balls!
Do you think that those patterns are science-fiction? Try to watch the following video by Ofek Snir, a great juggler that executes (among other stuff) some very hard siteswap with 7 balls.
As already specified above, this article only talks about the Vanilla siteswap. Actually there are also siteswap notations to represent other categories of patterns such as synchronous, the patterns where one hand can hold and toss more than one ball at time (in jargon multiplex) and the ones executed by more than one juggler (in jargon passing). Here are some examples
Synchronous: (6x,4)(4,6x)
Maths and JugglinMaths
Creative Mathematics: An Application of Category Theory
Juggling has a Mathematical Language | Delightful & Distinctive COLRS on July 4, 2019 at 5:07 am
Juggling | DiScoro on January 21, 2021 at 8:53 pm
Sjonglere | DiScoroNorge on April 19, 2021 at 8:22 pm
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How to Calculate Equivalent Annual Cost (EAC)
Equivalent annual cost (EAC) is the cost per year for owning or maintaining an asset over its lifetime. Calculating EAC is useful in budgeting decision-making by converting the price of an asset to an equivalent annual amount. EAC helps to compare the cost effectiveness of two or more assets with different lifespans. The formula for EAC is:
{\displaystyle {\text{Asset Price}}*{\frac {\text{Discount Rate}}{1-(1+{\text{Discount Rate}})^{\text{-Periods}}}}+{\text{Annual Maintenance Costs}}}
Let's see how this equation is applied.
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Determine the price of the asset.[1] X Research source For example, suppose you are comparing two analyzers, A and B, costing $100,000 and $130,000, respectively. These are the Asset Prices.
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Determine the expected lifespan for each.[2] X Research source Suppose Analyzer A is expected to last 5 years, while Analyzer B is expected to last 7 years. These are the number of Periods.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/7\/7b\/Calculate-Equivalent-Annual-Cost-%28EAC%29-Step-03.jpg\/v4-460px-Calculate-Equivalent-Annual-Cost-%28EAC%29-Step-03.jpg","bigUrl":"\/images\/thumb\/7\/7b\/Calculate-Equivalent-Annual-Cost-%28EAC%29-Step-03.jpg\/aid7836170-v4-728px-Calculate-Equivalent-Annual-Cost-%28EAC%29-Step-03.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>\u00a9 2022 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is <b>not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc.<br>\n<\/p><p><br \/>\n<\/p><\/div>"}
Determine your discount rate.[3] X Research source Discount rate is the cost of capital, or how much return your capital is required to generate each year. Say your organization uses a Discount Rate of 10%.
Determine the annual maintenance costs for the asset.[4] X Research source Suppose Analyzer A has an annual maintenance expense of $11,000, while Analyzer B has annual maintenance expense of $8,000.
Plug the numbers into the equation Asset Price x Discount rate / (1-(1+Discount Rate)^-Periods) + Annual Maintenance Costs.[5] X Research source It should be apparent that Analyzer B is the more cost effective option, with a net savings of $2,677.03 a year, compared to Analyzer A.
For Analyzer A,
{\displaystyle {\text{EAC}}=\$100,000*{\frac {0.10}{(1-(1+0.10)^{-5})}}+\$11,000=\$37,379.75}
For Analyzer B,
{\displaystyle {\text{EAC}}=\$130,000*{\frac {0.10}{(1-(1+0.10)^{-7})}}+\$8,000=\$34,702.72}
A precision lathe costs $10,000 and will cost $20,000 a year to operate and maintain. If the discount rate is 10% and the lathe will last for 5 years, what is the equivalent annual cost of the tool?
Using the formula above: EAC = $10,000 * (10% / (1 - (1 + 10%)^-5) + $20,000 = $22,637.97
This part of the formula,
{\displaystyle {\frac {\text{Discount Rate}}{1-(1+{\text{Discount Rate}})^{\text{-Periods}}}}}
is the inverse of the annuity factor (AF).
{\displaystyle {\text{AF}}={\frac {1-(1+{\text{Discount Rate}})^{\text{-Periods}}}{\text{Discount Rate}}}}
, is used to calculate present values of annuities.
It is often abbreviated as
{\displaystyle AF(n,r)}
, which can be readily computed by financial calculators plugging in values for n (periods) and r (discount rate). AF can also be looked up from an annuity factor table. The EAC formula may be simplified as
{\displaystyle {\frac {\text{Asset Price}}{\text{Annuity Factor}}}+{\text{Annual Maintenance Cost}}}
From an annuity factor table, AF(5,10%) = 3.7908 for Analyzer A and AF(7,10%) = 4.8684 for Analyzer B, so EAC = $100,000/3.7908 + $11,000 = $37,379.65 for Analyzer A and $130,000/4.8684 + $8,000 = $34,702.81 for Analyzer B.
Note that these figures are very close to the actual figures as calculated above. The minuscule differences arise from rounding errors attributed to AF having only 5 significant figures from the AF table.
File a UCC Financing Statement
↑ https://www.accaglobal.com/us/en/student/exam-support-resources/fundamentals-exams-study-resources/f9/technical-articles/eac.html
↑ https://bizfluent.com/how-6404927-calculate-equivalent-annual-cost.html
↑ https://xplaind.com/143298/equivalent-annual-cost
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40 CFR § 264.1312 - User fee calculation methodology. | CFR | US Law | LII / Legal Information Institute
40 CFR § 264.1312 - User fee calculation methodology.
\begin{array}{c}{\mathrm{Fee}}_{i}=\left(\frac{\text{System Setup Cost}}{\mathrm{Years}×{N}_{t}}\right)+\left({\text{Marginal Cost}}_{i}+\frac{O&M\phantom{\rule{0ex}{0ex}}\text{Cost}}{{N}_{t}}\right)×\left(1+\text{Indirect Cost Factor}\right)\\ \phantom{\rule{0ex}{0ex}}\\ \text{System Setup Cost}=\text{Procurement Cost}+\text{EPA Program Cost}\\ \phantom{\rule{0ex}{0ex}}\\ O&M\phantom{\rule{0ex}{0ex}}\mathrm{Cost}=\\ \text{Electronic System}\phantom{\rule{0ex}{0ex}}O&M\text{}\phantom{\rule{0ex}{0ex}}\text{Cost}+\text{Paper Center}\phantom{\rule{0ex}{0ex}}O&M\text{}\phantom{\rule{0ex}{0ex}}\text{Cost}+\text{Help Desk Cost +}\\ \text{EPA Program Cost}+\text{CROMERR Cost}+\\ \text{LifeCycle Cost to Modify or Upgrade eManfiest System Related Services}\end{array}
(1) If after four years of system operations, electronic manifest usage does not equal or exceed 75% of total manifest usage, EPA may transition to the following formula or methodology to determine per manifest fees:
\begin{array}{c}{\mathrm{Fee}}_{i}=\left(\frac{\text{System Setup Cost}}{\mathrm{Years}×{N}_{t}}\right)+\left({\text{Marginal Cost}}_{i}+\frac{O&{M}_{i}\phantom{\rule{0ex}{0ex}}\text{Cost}}{{N}_{i}}\right)×\left(1+\text{Indirect Cost Factor}\right)\\ \phantom{\rule{0ex}{0ex}}\\ \text{System Setup Cost}=\text{Procurement Cost}+\text{EPA Program Cost}\\ \phantom{\rule{0ex}{0ex}}\\ O&{M}_{\text{fully electronic}}\phantom{\rule{0ex}{0ex}}\mathrm{Cost}=\\ \text{Electronic System}\phantom{\rule{0ex}{0ex}}O&M\text{}\phantom{\rule{0ex}{0ex}}\text{Cost}+\text{Help Desk Cost}+\text{EPA Program Cost}+\text{CROMERR Cost}\\ \text{+ LifeCycle Cost to Modify or Upgrade eManfiest System Related Services}\\ \phantom{\rule{0ex}{0ex}}\\ O&{M}_{\text{all other}}\phantom{\rule{0ex}{0ex}}\mathrm{Cost}=Electronic System\phantom{\rule{0ex}{0ex}}O&M\text{}\phantom{\rule{0ex}{0ex}}\text{Cost}+\text{Paper Center}\phantom{\rule{0ex}{0ex}}O&M\text{}\phantom{\rule{0ex}{0ex}}\mathrm{Cost}+\\ \text{Help Desk Cost}+\text{EPA Program Cost}+\text{CROMERR Cost +}\\ \text{LifeCycle Cost to Modify or Upgrade eManfiest System Related Services}\end{array}
Where Ni refers to the total number of one of the four manifest submission types “i” completed in a year and O&MiCost refers to the differential O&M Cost for each manifest submission type “i.”
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Secondary Novikov-Shubin invariants of groups and quasi-isometry
January, 2007 Secondary Novikov-Shubin invariants of groups and quasi-isometry
We define new
{L}^{2}
-invariants which we call secondary Novikov-Shubin invariants.We calculate the first secondary Novikov-Shubin invariants of finitely generated groups by using random walk on Cayley graphs and see in particular that these are invariant under quasi-isometry.
Shin-ichi OGUNI. "Secondary Novikov-Shubin invariants of groups and quasi-isometry." J. Math. Soc. Japan 59 (1) 223 - 237, January, 2007. https://doi.org/10.2969/jmsj/1180135508
Keywords: Cayley graphs , L^{2}-invariants , Novikov-Shubin invariants , quasi-isometry , Random walk
Shin-ichi OGUNI "Secondary Novikov-Shubin invariants of groups and quasi-isometry," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 59(1), 223-237, (January, 2007)
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Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : FastArithmeticTools Subpackage : Overview
Overview of the RegularChains[FastArithmeticTools] Subpackage of RegularChains
List of RegularChains[FastArithmeticTools] Subpackage Commands
RegularChains[FastArithmeticTools][command](arguments)
The RegularChains[FastArithmeticTools] subpackage contains a collection of commands for computing with regular chains in prime characteristic using asymptotically fast algorithms.
Most of the commands of this package implements core operations on regular chains such as regularity test and polynomial GCD modulo a regular chain. However, these commands have several constraints. On top of the characteristic issue (mentioned above) the current regular chain must have dimension zero or one. There is only one exception: the command RegularGcdBySpecializationCube which makes no assumption on dimension.
In order to call one of the commands of this subpackage the characteristic of the polynomial ring must be a prime number p satisfying the following properties. First, it should not be greater than 962592769. Secondly, the number p-1 should be divisible by a sufficiently large power of 2. 2^20 is often sufficient. The best prime number p under these constraints is
469762049
for which p-1 writes 2^26 * 7. If this power of 2 is not large enough, then a clean error is returned.
The commands IteratedResultantDim0 and IteratedResultantDim1 compute the iterated resultant of a polynomial w.r.t. a regular chain of dimension 0 and 1, respectively.
The commands NormalFormDim0 and ReduceCoefficientsDim0 compute the normal form of a polynomial w.r.t. a zero-dimensional regular chain.
The commands NormalizePolynomialDim0 and NormalizeRegularChainDim0 normalize a polynomial (w.r.t. a zero-dimensional regular chain) and a regular chain (w.r.t. itself).
The commands RandomRegularChainDim0 and RandomRegularChainDim1 compute random regular chains of given degrees.
The command RegularizeDim0 tests whether a polynomial is invertible modulo a zero-dimensional regular chain.
The commands RegularGcdBySpecializationCube, ResultantBySpecializationCube and SubresultantChainSpecializationCube compute resultants and polynomial GCDs modulo a regular chain using fast evaluation and interpolation.
The following is the list of the available commands.
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{FastArithmeticTools}\right):
\mathrm{with}\left(\mathrm{ChainTools}\right):
p≔962592769;
\mathrm{vars}≔[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}]:
R≔\mathrm{PolynomialRing}\left(\mathrm{vars},p\right):
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{962592769}
Randomly generating (dense) regular chain and polynomial
N≔\mathrm{nops}\left(\mathrm{vars}\right):
\mathrm{dg}≔3:
\mathrm{degs}≔[\mathrm{seq}\left(4,i=1..N\right)]:
\mathrm{pol}≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p:
\mathrm{tc}≔\mathrm{RandomRegularChainDim0}\left(\mathrm{vars},\mathrm{degs},p\right):
\mathrm{Equations}\left(\mathrm{tc},R\right):
Compute the iterated resultant of pol w.r.t. tc
\mathrm{r1}≔\mathrm{IteratedResultantDim0}\left(\mathrm{pol},\mathrm{tc},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{r1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{446889812}
\mathrm{r2}≔\mathrm{IteratedResultant}\left(\mathrm{pol},\mathrm{tc},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{r2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{446889812}
The results computed IteratedResultantDim0 and IteratedResultant are equivalent.
\mathrm{Expand}\left(\mathrm{r1}-\mathrm{r2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p
\textcolor[rgb]{0,0,1}{0}
Define another ring of polynomials.
p≔962592769:
\mathrm{vars}≔[x,y,z]:
N≔\mathrm{nops}\left(\mathrm{vars}\right):
R≔\mathrm{PolynomialRing}\left(\mathrm{vars},p\right):
Generate a regular chain and a random polynomial.
\mathrm{c1}≔{z}^{2}+286748183z+134705551:
\mathrm{c2}≔{y}^{2}+914706789yz+686506773y+823875308z+417988453:
\mathrm{c3}≔{x}^{2}+224047618xyz+197329999xy+838274835xz+792563861x+600038529yz+434770098y+251400283z+918968185:
\mathrm{lf}≔[\mathrm{c1},\mathrm{c2},\mathrm{c3}]:
\mathrm{tc}≔\mathrm{Chain}\left(\mathrm{lf},\mathrm{Empty}\left(R\right),R\right):
\mathrm{lf}≔[x+\mathrm{c1}\mathrm{c2}\mathrm{c3},\left(\mathrm{c1}-\mathrm{c2}\right)\left(\mathrm{c1}-\mathrm{c3}\right),1+\mathrm{c1}\left(\mathrm{c2}-\mathrm{c3}\right)]:
\mathrm{dg}≔10:
\mathrm{q1}≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p:
\mathrm{q2}≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p:
\mathrm{q3}≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p:
r≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p:
f≔\mathrm{q1}\mathrm{c1}+\mathrm{q2}\mathrm{c2}+\mathrm{q3}\mathrm{c3}+r:
Compute the normal form of f.
\mathrm{nf1}≔\mathrm{NormalFormDim0}\left(f,\mathrm{tc},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{nf1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{328601399}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{201896638}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{156935587}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{566967624}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{648149447}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{308862802}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{921920449}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{743421169}
Compare with the generic algorithm (non-fast and non-modular algorithm) of the command NormalForm.
\mathrm{nf2}≔\mathrm{NormalForm}\left(f,\mathrm{tc},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{nf2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{328601399}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{201896638}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{156935587}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{566967624}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{648149447}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{308862802}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{921920449}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{743421169}
The results computed by NormalFormDim0 and NormalForm are equivalent.
\mathrm{nf1}-\mathrm{nf2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p
\textcolor[rgb]{0,0,1}{0}
X. Li and M. Moreno Maza. "Efficient implementation of polynomial arithmetic in a multilevel programming environment." Proc. International Congress of Mathematical Software, p.12-23, LNCS Vol.4151, Springer, 2006.
X. Li, M. Moreno Maza and E. Schost. "Fast arithmetic for triangular sets: from theory to practice." ISSAC'2007, ACM Press. p.269-276, 2007.
X. Li and M. Moreno Maza. "Multithreaded parallel implementation of arithmetic operations modulo a triangular set." PASCO'2007, ACM Press, p.53-59, 2007.
A. Filatei, X. Li, M. Moreno Maza and E. Schost. "Implementation techniques for fast polynomial arithmetic in a high-level programming environment." ISSAC'2006, p.93-100, ACM Press, 2006.
X. Li, M. Moreno Maza, R. Rasheed and E. Schost. "High-Performance Symbolic Computation in a Hybrid Compiled-Interpreted Programming Environment." Proc. 2008 International Conference on Computational Science and Applications, IEEE Computer Society, pp 331--341 2008.
X. Li, M. Moreno Maza, R. Rasheed and E. Schost. "The Modpn library: Bringing fast polynomial arithmetic into Maple." Proc. 2008 Milestones in Computer Algebra, Stonehaven Bay, Trinidad and Tobago, 2008.
X. Li, M. Moreno Maza and W. Pan. "Computations modulo regular chains." Technical Report, University of Western Ontario, 2009.
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Recursively thinking about... recursion - DEV Community
This time we'll be looking at one of those foundational concepts in Computer Science which you already know which one it is by the title.
So let's get started with it, shall we?
When we think about recursion, what are the first things that come to mind?
They might be thoughts about recursion which means that we're thinking about recursion and then... shucks here we go again.
We're now inside the recursion loop lol.
Ahhh... recursion, that concept you can define by referencing it on its own definition and still be able to get away with it.
I remember that when I was taught about it in college it was quite hard to grasp and it took me a long while to understand it.
That was mostly because of the previous course I took called "Basic Imperative Programming".
It made me get too used to think in terms of how a program should behave instead of thinking about what it has to do.
That mindset shift of "just tell the program what it has to do and it'll surely find a better way of doing it than you" was liberating and mind-expanding.
And you'll probably be like "you haven't said what Recursion actually is"
That's right my friend, I was hoping the previous image was enough hahaha.
All jokes aside, Recursion in simple terms is. A way to solve problems in which you specify the initial "base case(s)" and then what to do when those cases don't match.
So it'll be like solving a complex problem by breaking it down into smaller problems.
Then you solve those ones so that a solution can be made from all the other little ones.
Let's see it in a more practical way.
But before we do that, this post needs the obligatory XKCD image of recursion to be complete.
Alright, we may now proceed.
Two of the most well-known and classic examples of recursion are the Factorial and the Fibonacci Sequence.
You may recall that the factorial of a number is written with an exclamation mark after the number like 5! and the result can be expressed like.
5! = 5 x 4 x 3 x 2 x 1
So what would be the factorial of 8? Simply as 8 times the factorial of 7 or
8! = 8 x 7!
And we could go on and on until we're looking for the factorial of 1 which is none other than the 1 itself.
So there we have it, the factorial of a number is that number times the factorial of the previous number.
We could express that like.
factorial(n) = n * factorial(n - 1)
And we now can take that mathematical function and use it to make a program that can compute the factorial of any number we want.
The function takes an integer number and returns the value of the factorial of that number (another integer number)
So for the original example we mentioned, the factorial of 8 is 40320 which is 8 times the factorial of 7.
This is another great example which if you recall from your math classes, it's a series of numbers where each number is the sum of the previous two numbers.
The first 10 numbers in the sequence are...
fib(10) = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
Now you can see that the first two numbers are actually a 1 and then they sum up to form the 2.
So if the Fibonacci of 10 is 55, we could write it as the sum of 34 plus 21 which, if you guessed it, is the Fibonacci minus 1 for the first number and minus 2 for the second number. Like so
fib(n) = fib(n - 1) + fib(n - 2)
Notice that we are using the function two times to define the result of the function which will give us another 2 functions to evaluate the result of those functions.
Have I used the word function enough times already? in case not then well... a function equals function plus function function.
How we would go about creating a program that can calculate this sequence for us?
We already know our base cases and what to do when those don't apply.
There we go, now with that we can compute the values of the sequence for any number that we pass to the program.
Now, you probably were familiar with those examples already and want to see something more advanced where recursion gets more interesting.
In that case, let's look at another example that is more fun than those previous ones.
This one indeed is a more interesting example. It is a game that has existed for a long time already.
It's kind of frustrating when you first encounter it and you don't have any notions of programming or maths.
Nevertheless, it's a fun game when you can actually play it with a wooden structure.
Although you can still play the game online if you want.
The objective of this game is to move all the disks from the start to the destination while using the middle rod as a temporary place.
When moving the disks you have the respect two rules.
You cannot place a larger disk on top of a smaller one (or else the weight of the bigger will crush the smaller and that's no bueno)
Only having 3 disks the solution to the problem is pretty straightforward.
We can start moving the little one to the destination then the middle one to the temporary and now we have the disks in 3 separated sticks.
From there, move the little one to the temporary place then move the large one to the destination.
Now, move the little one back to the start and the middle one to the destination.
Finally the little one there too and badabim badabom you're already done.
If you count, it only takes 7 moves to win the game with 3 disks but with 4 disks it takes 15, and for 5 it takes 31.
Now, we can extrapolate that process to solve the problem for much more disks like 10 or 25.
Using recursion, we could say that the process required to move 3 disks from point A to point C is the same process to move n-1 disks.
And the idea is that we move a smaller amount of disks solving a similar problem in every step until we get to our final solution.
We now repeat that process of moving disks until we reach the base case which is moving 1 disk. Easy right? we just pick it and move it to where it has to go.
So if you've been following along 'till this point and are thinking "yeah I get it that's easy" then it's time for action.
Let's turn our strategy of moving disks and our base case into a recursive algorithm that can solve the problem for any number of disks that we want to use.
Remember that as the number of disks increase, so does the number of steps and we'll need more time (and patience) to move that recursive load.
Now speaking of recursive load... I remembered about this comic
And to not make this post longer than it should, I'll leave it to you to implement the solution to this problem as a program that could take 3 inputs:
the number of disks (e.g. 'n')
the label of the starting point (e.g. 'A')
the label of the destination point (e.g. 'C')
The program should output the sequence of steps required to move the n disks from the A point to the C point.
Well, that's it for this post. Thanks for reading so far.
If you want, post your solution in the comments or reach out and let me know how would you implement it.
Thanks again and hope to see you on the next one!
I worked for a company whose tech lead did not allow recursion of any type! He must have had a nightmare once that permanently scared him. Talking about PTSD.
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Assigning Variants to Disease (V2D) - Open Targets Genetics Documentation
Lead Variant Annotation
Open Targets Genetics is based on the human reference genome assembly GRCh38. Lead variants,
V_L
, associated with a phenotype by hypothesis-free approaches (GWAS) are initially annotated with their associated trait(s) as described below:
Traits in Open Targets Genetics are assigned in the format Reported Trait (Author, Year). Multiple studies assessing the same trait are not collapsed into a single annotation. This preserves the integrity of the published record and will allow users to select study-specific analyses within the portal.
Published Associations
Reported variant-phenotype associations in literature were identified via the NHGRI-EBI GWAS Catalog, a manually-curated database of published variants meeting certain inclusion criteria, which will be familiar to most geneticists. On an ongoing basis, GWAS Catalog extracts and records detailed variant and study-level data for variants reported to be associated with any phenotype at a significance level of
p≤1e−5
, and fit the inclusion criteria detailed here.
In Open Targets Genetics we include GWAS Catalog curated associations with
p≤5e−8
. A subset of studies (N=162) then undergo distance based clustering (±500kb) to remove redundant associations that are an artefact of the curation process, as opposed to true independent signals. For associations that have a reported risk allele, we harmonised the effects so that all are with respect to the alternative allele.
GWAS Catalog summary statistics repository
Data from the GWAS Catalog summary statistics repository has been included in the portal as of June 2019. The initial release has been restricted to datasets derived from samples of predominantly European ancestry (N=201) due to the lack of suitable linkage-disequilibrium reference panels for conditional analysis. We are working to include all datasets in a future release of the portal.
UK Biobank Phenotypes
Recent efforts to rapidly and systematically apply established GWAS methods to all available data fields in UK Biobank have made available large repositories of summary statistics. To leverage these data disease locus discovery, we used full summary statistics from:
The Neale lab Round 2 (N=2139). These analyses applied GWAS (implemented in Hail) to all data fields using imputed genotypes from HRC as released by UK Biobank in May 2017, consisting of 337,199 individuals post-QC. Full details of the Neale lab GWAS implementation are available here. We have remove all ICD-10 related traits from the Neale data to reduce overlap with the SAIGE results.
The University of Michigan SAIGE analysis (N=1281). The SAIGE analysis uses PheCode derived phenotypes and applies a new method that "provides accurate P values even when case-control ratios are extremely unbalanced". See Zhou et al. (2018) for further details.
The fine-mapping section below explains how associated-loci were defined using the UK Biobank summary statistics.
Lead Variant to Tag Variant expansion
Two methods are used to expand lead disease-associated variants into a more complete set of possibly causal tag variants. Linkage-disequilibrium expansion using a reference population is applied to all studies in Open Targets Genetics, and expansion by fine-mapping (credible set analysis) is used where full summary statistics are available (currently UK Biobank traits and those included in the GWAS Catalog summary statistics repository).
Head to our FAQs page for the explanation on the differences between lead and tag variants.
Linkage Disequilibrium Expansion
Linkage disequilibrium (LD) information is calculated using the 1000 Genomes Phase 3 (1KG) haplotype panel as a reference. LD is calculated in the 1KG super-population that most closely matches GWAS study ancestry information curated by the GWAS Catalog. If the study is conducted in a mixture of populations, a weighted-average (of Fisher Z-transformed correlation coefficients) across super-populations is used. If ancestry information is unknown, European ancestry is assumed. See here for full methods.
Fine-mapping Expansion
Overview of the fine-mapping pipeline
Summary Statistics Preprocessing
Summary statistics were harmonised to ensure that the ALT allele is always the effect allele, and were pre-filtered to remove variants with low minor allele counts which would lead to inaccurate effect estimation. Variants located in the MHC region (6:28,510,120–33,480,577 GRCh38) are excluded from the fine-mapping pipeline. See here for harmonisation scripts and here for "ingestion" scripts and detailed inclusion criteria.
Top loci detection
Independently associated top loci are detected with GCTA stepwise selection procedure (cojo-slct) using unrelated European ancestry UK Biobank genotypes down-sampled to 10K individuals as an LD reference. Lead variants (the most associated variant at each locus) are kept if both the conditional and nominal p-values have
p≤5e^{-8}
Per locus conditional analysis
Where multiple index SNPs are found at the same locus (within 2Mb of each other), we perform GCTA single-variant association analysis conditional on other index SNPs at the locus. This produces a set of conditional summary statistics for each independently associated locus.
Credible set analysis
Credible set analysis is conducted for each associated locus using the above conditional summary statistics. We calculate an approximate Bayes factors (ABF) for all variants in a defined region around the index variant (±500kb). ABFs are computed using the approx.bf.p method re-implemented from the coloc package. Variants are ordered by their posterior probabilities (PP) and sequentially added to the credible set until the cumulative sum is >0.95 (95% credible set).
The implementation of our fine-mapping pipeline can be found here.
Summary statistics and fine-mapping results from FinnGen are handled differently, described here.
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Constrained optimization - Wikipedia
Optimizing objective functions that have constrained variables
1 Relation to constraint-satisfaction problems
3.2.3 Quadratic programming
3.2.4 KKT conditions
3.2.6 First-choice bounding functions
3.2.6.1 Russian doll search
3.2.7 Bucket elimination
{\displaystyle {\begin{array}{rcll}\min &~&f(\mathbf {x} )&\\\mathrm {subject~to} &~&g_{i}(\mathbf {x} )=c_{i}&{\text{for }}i=1,\ldots ,n\quad {\text{Equality constraints}}\\&~&h_{j}(\mathbf {x} )\geqq d_{j}&{\text{for }}j=1,\ldots ,m\quad {\text{Inequality constraints}}\end{array}}}
{\displaystyle g_{i}(\mathbf {x} )=c_{i}~\mathrm {for~} i=1,\ldots ,n}
{\displaystyle h_{j}(\mathbf {x} )\geq d_{j}~\mathrm {for~} j=1,\ldots ,m}
are constraints that are required to be satisfied (these are called hard constraints), and
{\displaystyle f(\mathbf {x} )}
is the objective function that needs to be optimized subject to the constraints.
For very simple problems, say a function of two variables subject to a single equality constraint, it is most practical to apply the method of substitution.[4] The idea is to substitute the constraint into the objective function to create a composite function that incorporates the effect of the constraint. For example, assume the objective is to maximize
{\displaystyle f(x,y)=x\cdot y}
{\displaystyle x+y=10}
. The constraint implies
{\displaystyle y=10-x}
, which can be substituted into the objective function to create
{\displaystyle p(x)=x(10-x)=10x-x^{2}}
. The first-order necessary condition gives
{\displaystyle {\frac {\partial p}{\partial x}}=10-2x=0}
, which can be solved for
{\displaystyle x=5}
{\displaystyle y=10-5=5}
If the constrained problem has only equality constraints, the method of Lagrange multipliers can be used to convert it into an unconstrained problem whose number of variables is the original number of variables minus the original number of equality constraints. Alternatively, if the constraints are all equality constraints and are all linear, they can be solved for some of the variables in terms of the others, and the former can be substituted out of the objective function, leaving an unconstrained problem in a smaller number of variables.
One way for evaluating this upper bound for a partial solution is to consider each soft constraint separately. For each soft constraint, the maximal possible value for any assignment to the unassigned variables is assumed. The sum of these values is an upper bound because the soft constraints cannot assume a higher value. It is exact because the maximal values of soft constraints may derive from different evaluations: a soft constraint may be maximal for
{\displaystyle x=a}
while another constraint is maximal for
{\displaystyle x=b}
This method[6] runs a branch-and-bound algorithm o{\displaystyle n}
problems, where
{\displaystyle n}
is the number of variables. Each such problem is the subproblem obtained by dropping a sequence of variables
{\displaystyle x_{1},\ldots ,x_{i}}
from the original problem, along with the constraints containing them. After the problem on variables
{\displaystyle x_{i+1},\ldots ,x_{n}}
is solved, its optimal cost can be used as an upper bound while solving the other problems,
In particular, the cost estimate of a solution having
{\displaystyle x_{i+1},\ldots ,x_{n}}
as unassigned variables is added to the cost that derives from the evaluated variables. Virtually, this corresponds on ignoring the evaluated variables and solving the problem on the unassigned ones, except that the latter problem has already been solved. More precisely, the cost of soft constraints containing both assigned and unassigned variables is estimated as above (or using an arbitrary other method); the cost of soft constraints containing only unassigned variables is instead estimated using the optimal solution of the corresponding problem, which is already known at this point.
The bucket elimination algorithm can be adapted for constraint optimization. A given variable can be indeed removed from the problem by replacing all soft constraints containing it with a new soft constraint. The cost of this new constraint is computed assuming a maximal value for every value of the removed variable. Formally, if
{\displaystyle x}
is the variable to be removed,
{\displaystyle C_{1},\ldots ,C_{n}}
are the soft constraints containing it, and
{\displaystyle y_{1},\ldots ,y_{m}}
are their variables except
{\displaystyle x}
, the new soft constraint is defined by:
{\displaystyle C(y_{1}=a_{1},\ldots ,y_{n}=a_{n})=\max _{a}\sum _{i}C_{i}(x=a,y_{1}=a_{1},\ldots ,y_{n}=a_{n}).}
^ Rossi, Francesca; van Beek, Peter; Walsh, Toby (2006-01-01), Rossi, Francesca; van Beek, Peter; Walsh, Toby (eds.), "Chapter 1 – Introduction", Foundations of Artificial Intelligence, Handbook of Constraint Programming, Elsevier, vol. 2, pp. 3–12, doi:10.1016/s1574-6526(06)80005-2, retrieved 2019-10-04
^ Martins, J. R. R. A.; Ning, A. (2021). Engineering Design Optimization. Cambridge University Press. ISBN 978-1108833417.
^ Wenyu Sun; Ya-Xiang Yuan (2010). Optimization Theory and Methods: Nonlinear Programming, Springer, ISBN 978-1441937650. p. 541
^ Prosser, Mike (1993). "Constrained Optimization by Substitution". Basic Mathematics for Economists. New York: Routledge. pp. 338–346. ISBN 0-415-08424-5.
^ Leader, Jeffery J. (2004). Numerical Analysis and Scientific Computation. Addison Wesley. ISBN 0-201-73499-0.
^ Verfaillie, Gérard, Michel Lemaître, and Thomas Schiex. "Russian doll search for solving constraint optimization problems." AAAI/IAAI, Vol. 1. 1996.
Bertsekas, Dimitri P. (1982). Constrained Optimization and Lagrange Multiplier Methods. New York: Academic Press. ISBN 0-12-093480-9.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Constrained_optimization&oldid=1084508378"
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Random symmetric matrices are almost surely nonsingular
1 November 2006 Random symmetric matrices are almost surely nonsingular
Kevin P. Costello, Terence Tao, Van Vu
Kevin P. Costello,1 Terence Tao,2 Van Vu1
1Department of Mathematics, Rutgers University
{Q}_{n}
denote a random symmetric (
n×n\right)
-matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values
0
1
1/2
). We prove that
{Q}_{n}
is nonsingular with probability
1-O\left({n}^{-1/8+\delta }\right)
\delta >0
. The proof uses a quadratic version of Littlewood-Offord-type results concerning the concentration functions of random variables and can be extended for more general models of random matrices
Kevin P. Costello. Terence Tao. Van Vu. "Random symmetric matrices are almost surely nonsingular." Duke Math. J. 135 (2) 395 - 413, 1 November 2006. https://doi.org/10.1215/S0012-7094-06-13527-5
Kevin P. Costello, Terence Tao, Van Vu "Random symmetric matrices are almost surely nonsingular," Duke Mathematical Journal, Duke Math. J. 135(2), 395-413, (1 November 2006)
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Thermodynamics Part-2 - Live session and RevisionContact Number: 9667591930 / 8527521718
A 10 g piece of iron (C = 0.45 J/g°C) at 100 °C is dropped into 25 g of water (C = 4.2 J/g°C) at 27°C. Find the temperature of the iron and water system at thermal equilibrium.
2 mole of zinc is dissolved at HCI at
25°\mathrm{C}
The work done in open vessel is:
1. -2.477 KJ
3. 0.0489 KJ
A sample of an ideal gas is expanded 1
{\mathrm{m}}^{3}
{\mathrm{m}}^{3}
in a reversible process for which P = K
{\mathrm{V}}^{2}
, with K = 6 bar/m. Work done by the gas is
1. 5200 kJ
2. 15600 kJ
An ideal gas is taken around the cycle ABCA as shown in P-V diagram. The network done during the cycle is equal to :
{\mathrm{P}}_{1}{\mathrm{V}}_{1}
{\mathrm{P}}_{1}{\mathrm{V}}_{1}
{\mathrm{P}}_{1}{\mathrm{V}}_{1}
{\mathrm{P}}_{1}{\mathrm{V}}_{1}
A gas expands against a variable pressure given by P = 20/V (where P in atm and V in L). During expansion from the volume of 1 liter to 10 liters, the gas undergoes a change in internal energy of 400 J. How much heat is absorbed by the gas during expansion?
3. 5065.8 J
5 mole of an ideal gas expand isothermally and irreversibly from a pressure of 10 atm to 1 atm against a constant external pressure of 1 atm,
{\mathrm{W}}_{\mathrm{irr}}
at 300 K is:
1. -15.921 KJ
3. -110.83 KL
One mole of a non-ideal gas undergoes a change of state from (1.0 atm, 3.0 L, 200 K) to (4.0 atm, 5.0 L, 250 K) with a change in internal energy (
∆
U) = 40 L-atm. The change in enthalpy of the process in L-atm:
Consider the reaction at 300 K
{\mathrm{C}}_{6}{\mathrm{H}}_{6}\left(\mathrm{l}\right) + \frac{15}{2}{\mathrm{O}}_{2}\left(\mathrm{g}\right) \to 6{\mathrm{CO}}_{2}\left(\mathrm{g}\right) +3{\mathrm{H}}_{2}\mathrm{O}\left(\mathrm{I}\right); \left(∆\mathrm{H}\right) = -3271 \mathrm{kJ}
. What is AU for the combustion of 1.5 moles of benzene at 27°C
1. -3267.25 kJ
3. -4906.5 kJ
When two moles of an ideal gas (
{\mathrm{C}}_{\mathrm{P}, \mathrm{m}}
\frac{5}{2}
R) heated from 300 K to 600 K at constant pressure. The change in entropy of gas
∆
\frac{3}{2} \mathrm{R} \mathrm{In} 2
2. -
\frac{3}{2} \mathrm{R} \mathrm{In} 2
5 \mathrm{R} \mathrm{In} 2
\frac{5}{2} \mathrm{R} \mathrm{In} 2
When one mole of an ideal gas is compressed to half of its initial volume and simultaneously heated to twice its initial temperature, the change in entropy of gas (
∆
S) is :
{\mathrm{C}}_{\mathrm{P}, \mathrm{m}}
{\mathrm{C}}_{\mathrm{v}, \mathrm{m}}
3. R In 2
4. (
{\mathrm{C}}_{\mathrm{v}, \mathrm{m}}
= R) In 2
What is the change in entropy when 2.5 moles of water is heated from 27 °C to 87°C? Assume that the capacity is constant (
{\mathrm{C}}_{\mathrm{P},\mathrm{m}} \left({\mathrm{H}}_{2}\mathrm{O}\right) = 4.2 \mathrm{J}/\mathrm{g}-\mathrm{k} \mathrm{in} \left(1.2\right) = 0.18
1. 16.6 J/K
2. 9 J/K
3. 34.02 J/K
4. 1.89 J/K
At 25 °C,
∆\mathrm{G}°
for the process
{\mathrm{H}}_{2}\mathrm{O}
⇌
{\mathrm{H}}_{2}\mathrm{O}
(g) is 8.6 kJ. The vapor pressure of water at this temperature is near:
2. 285 torr
3. 32.17 torr
The standard enthalpy of formation of gaseous
{\mathrm{H}}_{2}\mathrm{O}
at 298 K is -241.82 kJ/mol. Calculate
∆\mathrm{H}°
at 373 K given the following values of the molar heat capacities at constant pressure:
{\mathrm{H}}_{2}\mathrm{O}\left(\mathrm{g}\right) = 33.58 {\mathrm{JK}}^{-1} {\mathrm{mol}}^{-4}
{\mathrm{H}}_{2}\left(\mathrm{g}\right) = 29.84 {\mathrm{JK}}^{-1} {\mathrm{mol}}^{-1}
{\mathrm{O}}_{2}\left(\mathrm{g}\right) = 29.37 {\mathrm{JK}}^{-1} {\mathrm{mol}}^{-1}
Assume that the heat capacities are independent of temperature :
1. -242.6 kJ/mol
Gasoline has an enthalpy of combustion 24000 kJ/gallon. When gasoline burns in an automobile engine, approximately 30% of the energy released is used to produce mechanical work. The remainder is lost as heat transfer to the engine's cooling system. As a start on estimating how much heat transfer is required, calculate what mass of water could be heated from 25 °C to 75 °C by the combustion of 1.0 gallons of gasoline in an automobile? (Given : C(
{\mathrm{H}}_{2}\mathrm{O}
) = 4.18 J/g°C)
A 0.05 L sample of 0.2 M aqueous hydrochloric acid is added to 0.05 L of 0.2 M aqueous ammonia in a calorimeter. The heat capacity of the entire calorimeter system is 480 J/K. The temperature increase is 1.09 K. Calculate
∆\mathrm{rH}°
in kJ/mol for the following reaction:
\mathrm{HCl}\left(\mathrm{aq}.\right)+{\mathrm{NH}}_{3}\left(\mathrm{aq}\right) \to {\mathrm{NH}}_{4}\mathrm{Cl}\left(\mathrm{aq}.\right)
1. -52.32
Boron can undergo the following reactions with the given enthalpy changes: 2B(s)
2\mathrm{B}\left(\mathrm{s}\right) + \frac{3}{2}{\mathrm{O}}_{2} \to {\mathrm{B}}_{2}{\mathrm{O}}_{2}\left(\mathrm{s}\right); ∆\mathrm{H} = -1260 \mathrm{kJ}\phantom{\rule{0ex}{0ex}}2\mathrm{B}\left(\mathrm{s}\right) + 3{\mathrm{H}}_{2} \to {\mathrm{B}}_{2}{\mathrm{H}}_{6}\left(\mathrm{s}\right); ∆\mathrm{H} = 30 \mathrm{kJ}
Assume no other reactions are occurring. If in a container (operating at constant pressure) which is isolated from the surrounding, a mixture of H2(gas) and
{\mathrm{O}}_{2}
(gas) is passed over excess of B(s), then calculate the molar ratio (
{\mathrm{O}}_{2};{\mathrm{H}}_{2}
) so that the temperature of the container does not change :
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Pre Video Test - Haloalkanes and Haloarenes
Pre Video Test - Haloalkanes and HaloarenesContact Number: 9667591930 / 8527521718
The absolute configuration of the following compound
1. 2 S, 3R
2. 2 S, 3 S
3. 2 R, 3S
4. 2 R, 3R
Which of the following pairs of compounds are
enantiomers?
The correct order of solvolysis in the following alkyl
halide is
{\mathrm{CH}}_{2}\mathrm{Cl}>{\mathrm{CH}}_{2}=\mathrm{CH}-{\mathrm{CH}}_{2}\mathrm{Cl}>{\mathrm{CH}}_{3}{\mathrm{CH}}_{2}{\mathrm{CH}}_{2}\mathrm{Cl}
{\mathrm{CH}}_{2}=\mathrm{CH}-{\mathrm{CH}}_{2}\mathrm{Cl}>
{\mathrm{CH}}_{3}{\mathrm{CH}}_{2}{\mathrm{CH}}_{2}\mathrm{Cl}
{\mathrm{CH}}_{3}{\mathrm{CH}}_{2}{\mathrm{CH}}_{2}\mathrm{Cl}>{\mathrm{CH}}_{3}={\mathrm{CHCH}}_{2}\mathrm{Cl}>
{\mathrm{CH}}_{2}\mathrm{Cl}
{\mathrm{CH}}_{2}=\mathrm{CH}-{\mathrm{CH}}_{2}\mathrm{Cl}>{\mathrm{CH}}_{3}{\mathrm{CH}}_{2}{\mathrm{CH}}_{2}\mathrm{Cl}>
{\mathrm{CH}}_{2}\mathrm{Cl}
{\mathrm{CH}}_{3}{\mathrm{CH}}_{2}{\mathrm{CH}}_{2}\mathrm{Cl}\stackrel{\mathrm{alc}. \mathrm{KOH}}{\to }\mathrm{A}\stackrel{\mathrm{HBr}}{\to }\mathrm{C}\stackrel{\mathrm{Na}/\mathrm{ether}}{\to }\mathrm{D}
2. 2, 3-dimethylbutane
4. Allyl bromide
Which one of the following is most reactive towards
nucleophilic substitution reaction?
1. CH2= CH – Cl
3. CH3CH = CH – Cl
4. ClCH2– CH = CH2
Bottles containing C6H5I and C6H5CH2I lost their
original labels. They were labelled A and B for testing.
A and B were separtely taken in test tube and boiled
with NaOH solution. The end solution in each tube
was made acidic with dilute HNO3 and some AgNO3
solution added. Solution B gave a yellow ppt. which
one of the following statements is true for the
1. Addition of HNO3 was unnecessary
2. A was C6H5I
3. A was C6H5CH2I
4. B was C6H5I
On monochlorination of 2-methylbutane, the total
number of chiral compounds is
Replacement of ‘Cl’ of chlorobenzene to give phenol
requires drastic conditions but chlorine of 2,4-
dinitrochloro benzene is readily replaced because
1. NO2 makes the ring electrons rich at ortho and
2. NO2 withdraws electrons from meta position
3. NO2 donates electrons at m-position
4. NO2 withdraws electrons from ortho and para
Which of the following derivatives of benzene would
undergo hydrolysis most readily with aq. KOH?
The major product obtained when Br2/Fe is reacted with
Among the following bromides given below, the order
of their reactivity in the SN1 reaction is
1. III > II > I
2. II > III > I
3. III > I > II
4. II > I > III
Which haloarenes is most reactive towards
electrophilic substitution reaction ?
The ease of aromatic nucleophilic substitution among
these compounds will be in the order as:
1. IV > I > II > III
2. IV > III > II > I
3. III > II > IV > I
4. I >II > III > IV
Arrange the following in the order of C-Br bond
strength in a polar solvent
1. I < II < III < IV
2. III < IV < I < II
3. IV < III < II < I
4. II < I < III < IV
Which is not correct about (I) ?
1. (I) is more water soluble than bromocyclopropane
2. (I) gives pale yellow precipitate on addition of AgNO3
3. (I) is having lower dipole moment than bromocyclopropane
4. (I) have more ionic character than
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LMIs in Control/Matrix and LMI Properties and Tools/Strict Projection Lemma - Wikibooks, open books for an open world
LMIs in Control/Matrix and LMI Properties and Tools/Strict Projection Lemma
2 Strict Projection Lemma
Projection Lemma is also known as Matrix Elimination Lemma. Strict Projection Lemma is one of the characteristics of the Projection Lemma.
Strict Projection LemmaEdit
{\displaystyle \Psi \in \mathbb {S} ^{n}}
{\displaystyle G\in \mathbb {R} ^{n\times m}}
{\displaystyle \Lambda \in \mathbb {R} ^{m\times p}}
{\displaystyle H\in \mathbb {R} ^{n\times p}.}
{\displaystyle \Lambda }
{\displaystyle {\begin{aligned}\ \Psi +G\Lambda H^{T}+H\Lambda ^{T}G^{T}<0,\end{aligned}}}
{\displaystyle {\begin{aligned}\ N_{G}^{T}\Psi N_{G}<0\end{aligned}}}
{\displaystyle {\begin{aligned}\ N_{H}^{T}\Psi N_{H}<0\end{aligned}}}
{\displaystyle {\mathcal {R}}(N_{G})={\mathcal {N}}(G^{T})}
{\displaystyle {\mathcal {R}}(N_{H})={\mathcal {N}}(H^{T})}
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Matrix_and_LMI_Properties_and_Tools/Strict_Projection_Lemma&oldid=4009617"
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Space-filling tree - Wikipedia
An assembly of curves that branch out from a root to reach every point in a space
Space-filling trees are geometric constructions that are analogous to space-filling curves,[1] but have a branching, tree-like structure and are rooted. A space-filling tree is defined by an incremental process that results in a tree for which every point in the space has a finite-length path that converges to it. In contrast to space-filling curves, individual paths in the tree are short, allowing any part of the space to be quickly reached from the root. [2][3] The simplest examples of space-filling trees have a regular, self-similar, fractal structure, but can be generalized to non-regular and even randomized/Monte-Carlo variants (see Rapidly exploring random tree). Space-filling trees have interesting parallels in nature, including fluid distribution systems, vascular networks, and fractal plant growth, and many interesting connections to L-systems in computer science.
A space-filling tree is defined by an iterative process whereby a single point in a continuous space is connected via a continuous path to every other point in the space by a path of finite length, and for every point in the space, there is at least one path that converges to it.
The concept of a "space-filling tree" in this sense was described in Chapter 15 of Mandelbrot's influential book The Fractal Geometry of Nature (1982).[4] The concept was made more rigorous and given the name "space-filling tree" in a 2009 tech report [5] that defines "space-filling" and "tree" differently than their traditional definitions in mathematics. As explained in the space-filling curve article, in 1890, Peano found the first space-filling curve, and by Jordan's 1887 definition, which is now standard, a curve is a single function, not a sequence of functions. The curve is "space filling" because it is "a curve whose range contains the entire 2-dimensional unit square" (as explained in the first sentence of space-filling curve).
In contrast, a space-filling tree, as defined in the tech report, is not a single tree. It is only a sequence of trees. The paper says "A space-filling tree is actually defined as an infinite sequence of trees". It defines
{\displaystyle T_{\text{square}}}
as a "sequence of trees", then states "
{\displaystyle T_{\text{square}}}
is a space-filling tree". It is not space-filling in the standard sense of including the entire 2-dimensional unit square. Instead, the paper defines it as having trees in the sequence coming arbitrarily close to every point. It states "A tree sequence T is called 'space filling' in a space X if for every x ∈ X, there exists a path in the tree that starts at the root and converges to x.". The standard term for this concept is that it includes a set of points that is dense everywhere in the unit square.
The simplest example of a space-filling tree is one that fills a square planar region. The images illustrate the construction for the planar region
{\displaystyle [0,1]^{2}\subset \mathbb {R} ^{2}}
. At each iteration, additional branches are added to the existing trees.
Square space-filling tree (Iteration 1)
Space-filling trees can also be defined for a variety of other shapes and volumes. Below is the subdivision scheme used to define a space-filling for a triangular region. At each iteration, additional branches are added to the existing trees connecting the center of each triangle to the centers of the four subtriangles.
Subdivision scheme for the first three iterations of the triangle space-filling tree
The first six iterations of the triangle space-filling tree are illustrated below:
Triangle space-filling tree (Iteration 1)
Space-filling trees can also be constructed in higher dimensions. The simplest examples are cubes in
{\displaystyle \mathbb {R} ^{3}}
and hypercubes in
{\displaystyle \mathbb {R} ^{n}}
. A similar sequence of iterations used for the square space-filling tree can be used for hypercubes. The third iteration of such a space-filling tree in
{\displaystyle \mathbb {R} ^{3}}
is illustrated below:
Cube space-filling tree (Iteration 3)
Rapidly exploring random tree (RRTs)
^ Sagan, H. and J. Holbrook: "Space-filling curves", Springer-Verlag, New York, 1994
^ Kuffner, J. J. and S. M. LaValle: Space-filling Trees, The Robotics Institute, Carnegie Mellon University, CMU-RI-TR-09-47, 2009.
^ Kuffner, J. J.; LaValle, S.M.; “Space-filling trees: A new perspective on incremental search for motion planning,” Intelligent Robots and Systems (IROS), 2011 IEEE/RSJ International Conference on , vol., no., pp.2199-2206, 25-30 Sept. 2011
Retrieved from "https://en.wikipedia.org/w/index.php?title=Space-filling_tree&oldid=1077768984"
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Funadamental operations in fractions: Multiplication — lesson. Mathematics State Board, Class 6.
Multiplication of a fractional number with another fractional number can be done in two steps:
\(\text{Product of two fraction}\) \(=\) \(\frac{\text{Products of numerators of the fractions}}{\text{Products of denominators of the fractions}}\)
Multiplication of a fraction using the operator 'of ':
Ravi has \(20\) apples, and Ram has one-fifth of it. How many apples does Ram have?
The number of Apple Ram has \(=\)
\frac{1}{5}
of \(20\)
Therefore, \(= 4\) Apples.
Multiplication of a fraction by a whole number:
Let us understand how to multiply a fraction and a whole number.
Mithra went to a supermarket, where she bought three \(1/4\) kgs of rice packet. What is the total \(kg\) of rice she bought totally?
Let, \(1/4\) \(kg\) of rice packet \(=\)
Total \(kg\) of rice she bought can be illustrated as following.
Multiplication of mixed fractions:
To multiply two mixed fractions first, convert them to improper fractions and then multiply the fractions.
Value of the products:
When two proper fractions are multiplied, the value of the product is less than each proper fraction.
When two improper fractions are multiplied, the value of the product is greater than each of the improper fraction.
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Donadelli, Jair ; Haxell, Penny E. ; Kohayakawa, Yoshiharu
{T}_{s}H
be the graph obtained from a given graph
H
by subdividing each edge
s
times. Motivated by a problem raised by Igor Pak [Mixing time and long paths in graphs, in Proc. of the 13th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002) 321-328], we prove that, for any graph
H
, there exist graphs
G
O\left(s\right)
edges that are Ramsey with respect to
{T}_{s}H
Classification : 05C55, 05D40
Mots clés : The Size-Ramsey number, Ramsey theory, expanders, Ramanujan graphs, explicit constructions
author = {Donadelli, Jair and Haxell, Penny E. and Kohayakawa, Yoshiharu},
title = {A note on the {Size-Ramsey} number of long subdivisions of graphs},
AU - Donadelli, Jair
AU - Haxell, Penny E.
AU - Kohayakawa, Yoshiharu
TI - A note on the Size-Ramsey number of long subdivisions of graphs
Donadelli, Jair; Haxell, Penny E.; Kohayakawa, Yoshiharu. A note on the Size-Ramsey number of long subdivisions of graphs. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 191-206. doi : 10.1051/ita:2005019. http://www.numdam.org/articles/10.1051/ita:2005019/
[1] N. Alon and F.R.K. Chung, Explicit construction of linear sized tolerant networks. Discrete Math. 72 (1988) 15-19. | Zbl 0657.05068
[2] N. Alon, Subdivided graphs have linear Ramsey numbers. J. Graph Theory 18 (1994) 343-347. | Zbl 0811.05046
[3] N. Alon and J.H. Spencer, The probabilistic method, 2nd edition, Ser. Discrete Math.Optim., Wiley-Interscience, John Wiley & Sons, New York, 2000. (With an appendix on the life and work of Paul Erdős.) | MR 1885388 | Zbl 0767.05001
[4] J. Beck, On size Ramsey number of paths, trees, and circuits. I. J. Graph Theory 7 (1983) 115-129. | Zbl 0508.05047
[5] J. Beck, On size Ramsey number of paths, trees and circuits. II. Mathematics of Ramsey theory, Springer, Berlin, Algorithms Combin. 5 (1990) 34-45. | Zbl 0735.05056
[6] V. Chvátal, V. Rödl, E. Szemerédi and W.T. Trotter Jr., The Ramsey number of a graph with bounded maximum degree. J. Combin. Theory Ser. B 34 (1983) 239-243. | Zbl 0547.05044
[7] R. Diestel, Graph theory. Springer-Verlag, New York (1997). Translated from the 1996 German original. | MR 1411445 | Zbl 0873.05001
[8] P. Erdős, R.J. Faudree, C.C. Rousseau and R.H. Schelp, The size Ramsey number. Periodica Mathematica Hungarica 9 (1978) 145-161. | Zbl 0331.05122
[9] P. Erdős and R.L. Graham, On partition theorems for finite graphs, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I. North-Holland, Amsterdam, Colloq. Math. Soc. János Bolyai 10 (1975) 515-527. | Zbl 0324.05124
[10] R.J. Faudree and R.H. Schelp, A survey of results on the size Ramsey number, Paul Erdős and his mathematics, II (Budapest, 1999). Bolyai Soc. Math. Stud., János Bolyai Math. Soc., Budapest 11 (2002) 291-309. | Zbl 1027.05069
[11] J. Friedman and N. Pippenger, Expanding graphs contain all small trees. Combinatorica 7 (1987) 71-76. | Zbl 0624.05028
[12] P.E. Haxell, Partitioning complete bipartite graphs by monochromatic cycles. J. Combin. Theory Ser. B 69 (1997) 210-218. | Zbl 0867.05022
[13] P.E. Haxell and Y. Kohayakawa, The size-Ramsey number of trees. Israel J. Math. 89 (1995) 261-274. | Zbl 0822.05049
[14] P.E. Haxell, Y. Kohayakawa and T. Łuczak, The induced size-Ramsey number of cycles. Combin. Probab. Comput. 4 (1995) 217-239. | Zbl 0839.05073
[15] P.E. Haxell and T. Łuczak, Embedding trees into graphs of large girth. Discrete Math. 216 (2000) 273-278. | Zbl 0958.05030
[16] P.E. Haxell, T. Łuczak and P.W. Tingley, Ramsey numbers for trees of small maximum degree. Combinatorica 22 (2002) 287-320. Special issue: Paul Erdős and his mathematics. | Zbl 0997.05065
[17] T. Jiang, On a conjecture about trees in graphs with large girth. J. Combin. Theory Ser. B 83 (2001) 221-232. | Zbl 1023.05035
[18] Xin Ke, The size Ramsey number of trees with bounded degree. Random Structures Algorithms 4 (1993) 85-97. | Zbl 0778.05060
[19] Y. Kohayakawa, Szemerédi's regularity lemma for sparse graphs, Foundations of computational mathematics (Rio de Janeiro, 1997). Springer, Berlin (1997) 216-230. | Zbl 0868.05042
[20] Y. Kohayakawa and V. Rödl, Regular pairs in sparse random graphs. I. Random Structures Algorithms 22 (2003) 359-434. | Zbl 1022.05076
[21] Y. Kohayakawa and V. Rödl, Szemerédi's regularity lemma and quasi-randomness, in Recent advances in algorithms and combinatorics. CMS Books Math./Ouvrages Math. SMC, Springer, New York 11 (2003) 289-351. | Zbl 1023.05108
[22] A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs. Combinatorica 8 (1988) 261-277. | Zbl 0661.05035
[23] G.A. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii 24 (1988) 51-60. | Zbl 0708.05030
[24] I. Pak, Mixing time and long paths in graphs, manuscript available at http://www-math.mit.edu/~pak/research.html#r (June 2001).
[25] I. Pak, Mixing time and long paths in graphs, in Proceedings of the 13th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002) 321-328. | Zbl 1056.05135
[26] O. Pikhurko, Asymptotic size Ramsey results for bipartite graphs. SIAM J. Discrete Math. 16 (2002) 99-113 (electronic). | Zbl 1029.05103
[27] O. Pikhurko, Size Ramsey numbers of stars versus 4-chromatic graphs. J. Graph Theory 42 (2003) 220-233. | Zbl 1013.05052
[28] L. Pósa, Hamiltonian circuits in random graphs. Discrete Math. 14 (1976) 359-364. | Zbl 0322.05127
[29] D. Reimer, The Ramsey size number of dipaths. Discrete Math. 257 (2002) 173-175. | Zbl 1012.05116
[30] V. Rödl and E. Szemerédi, On size Ramsey numbers of graphs with bounded degree. Combinatorica 20 (2000) 257-262. | Zbl 0959.05076
[31] E. Szemerédi, Regular partitions of graphs, in Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976). CNRS, Paris (1978) 399-401. | Zbl 0413.05055
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Interpret Numeric Tuning Results - MATLAB & Simulink - MathWorks Nordic
Tuning-Goal Scalar Values
Tuning Results at the Command Line
Tuning Results in Control System Tuner
When you tune a control system with systune or Control System Tuner, the software provides reports that give you an overview of how well the tuned control system meets your design requirements. Interpreting these reports requires understanding how the tuning algorithm optimizes the system to satisfy your tuning goals. (The software also provides visualizations of the tuning goals and system responses to help you see where and by how much your requirements are not satisfied. For information about using these plots, see Visualize Tuning Goals.)
The tuning software converts each tuning goal into a normalized scalar value which it then constrains (hard goals) or minimizes (soft goals). Let fi(x) and gj(x) denote the scalar values of the soft and hard goals, respectively. Here, x is the vector of tunable parameters in the control system to tune. The tuning algorithm solves the minimization problem:
\underset{i}{\mathrm{max}}{f}_{i}\left(x\right)
\underset{j}{\mathrm{max}}{g}_{j}\left(x\right)<1
{x}_{\mathrm{min}}<x<{x}_{\mathrm{max}}
xmin and xmax are the minimum and maximum values of the free parameters of the control system. (For information about the specific functions used to evaluate each type of requirement, see the reference pages for each tuning goal.)
\underset{x}{\mathrm{min}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{max}\left(\alpha f\left(x\right),g\left(x\right)\right).
The tuning software reports the final scalar values for each tuning goal. When the final value of fi(x) or gj(x) is less than 1, the corresponding tuning goal is satisfied. Values greater than 1 indicate that the tuning goal is not satisfied for at least some conditions. For instance, a tuning goal that describes a frequency-domain constraint might be satisfied at some frequencies and not at others. The closer the value is to 1, the closer the tuning goal is to being satisfied. Thus these values give you an overview of how successfully the tuned system meets your requirements.
The form in which the software presents the optimized tuning-goal values depends on whether you are tuning with Control System Tuner or at the command line.
The systune command returns the control system model or slTuner interface with the tuned parameter values. systune also returns the best achieved values of each fi(x) and gj(x) as the vector-valued output arguments fSoft and gHard, respectively. See the systune reference page for more information. (To obtain the final tuning goal values on their own, use evalGoal.)
By default, systune displays the best achieved final values of the tuning goals in the command window. For instance, in the example PID Tuning for Setpoint Tracking vs. Disturbance Rejection, systune is called with one soft requirement, R1, and two hard requirements R2 and R3.
This display indicates that the largest optimized value of the hard tuning goals is less than 1, so both hard goals are satisfied. The soft goal value is slightly greater than one, indicating that the soft goal is nearly satisfied. You can use tuning-goal plots to see in what regimes and by how much the tuning goals are violated. (See Visualize Tuning Goals.)
You can obtain additional information about the optimization progress and values using the info output of systune. To make systune display additional information during tuning, use systuneOptions.
In Control System Tuner, when you click , the app compiles a Tuning Report summarizing the best achieved values of fi(x) and gj(x). To view the tuning report immediately after tuning a control system, click Tuning Report at the bottom-right corner of Control System Tuner.
The tuning report displays the final fi(x) and gj(x) values obtained by the algorithm.
The Hard Goals area shows the minimized gi(x) values and indicates which are satisfied. The Soft Goals area highlights the largest of the minimized fi(x) values as Worst Value, and lists the values for all the requirements. In this example, the hard goal is satisfied, while the soft goals are nearly satisfied. As in the command-line case, you can use tuning-goal plots to see where and by how much tuning goals are violated. (See Visualize Tuning Goals.)
You can view a report from the most recent tuning run at any time. In the Tuning tab, click Tune , and select Tuning Report.
systune | systune (for slTuner) (Simulink Control Design) | viewGoal | evalGoal
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EUDML | An analogue of the Krein-Milman theorem for star-shaped sets. EuDML | An analogue of the Krein-Milman theorem for star-shaped sets.
An analogue of the Krein-Milman theorem for star-shaped sets.
Martini, Horst, and Wenzel, Walter. "An analogue of the Krein-Milman theorem for star-shaped sets.." Beiträge zur Algebra und Geometrie 44.2 (2003): 441-449. <http://eudml.org/doc/123722>.
@article{Martini2003,
author = {Martini, Horst, Wenzel, Walter},
keywords = {convex sets; star-shaped sets; closure operators; Krein-Milman theorem; visibility problems; illumination problems; watchman route problem},
title = {An analogue of the Krein-Milman theorem for star-shaped sets.},
AU - Martini, Horst
AU - Wenzel, Walter
TI - An analogue of the Krein-Milman theorem for star-shaped sets.
KW - convex sets; star-shaped sets; closure operators; Krein-Milman theorem; visibility problems; illumination problems; watchman route problem
convex sets, star-shaped sets, closure operators, Krein-Milman theorem, visibility problems, illumination problems, watchman route problem
Convex sets in
Variants of convex sets (star-shaped, (
m,n
Galois correspondences, closure operators
Articles by Wenzel
|
EUDML | Weak asymptotic method for the study of propagation and interaction of infinitely narrow -solitons. EuDML | Weak asymptotic method for the study of propagation and interaction of infinitely narrow -solitons.
Weak asymptotic method for the study of propagation and interaction of infinitely narrow
\delta
-solitons.
Danilov, Vladimir G.; Omel'yanov, Georgii A.
Danilov, Vladimir G., and Omel'yanov, Georgii A.. "Weak asymptotic method for the study of propagation and interaction of infinitely narrow -solitons.." Electronic Journal of Differential Equations (EJDE) [electronic only] 2003 (2003): Paper No. 90, 27 p., electronic only-Paper No. 90, 27 p., electronic only. <http://eudml.org/doc/124118>.
@article{Danilov2003,
author = {Danilov, Vladimir G., Omel'yanov, Georgii A.},
keywords = {KdV type equations; soliton interaction; zero limit dispersion problem},
title = {Weak asymptotic method for the study of propagation and interaction of infinitely narrow -solitons.},
AU - Danilov, Vladimir G.
AU - Omel'yanov, Georgii A.
TI - Weak asymptotic method for the study of propagation and interaction of infinitely narrow -solitons.
KW - KdV type equations; soliton interaction; zero limit dispersion problem
KdV type equations, soliton interaction, zero limit dispersion problem
Articles by Danilov
Articles by Omel'yanov
|
EUDML | Asymptotics of cross sections for convex bodies. EuDML | Asymptotics of cross sections for convex bodies.
Asymptotics of cross sections for convex bodies.
Brehm, Ulrich; Voigt, Jürgen
Brehm, Ulrich, and Voigt, Jürgen. "Asymptotics of cross sections for convex bodies.." Beiträge zur Algebra und Geometrie 41.2 (2000): 437-454. <http://eudml.org/doc/223019>.
@article{Brehm2000,
author = {Brehm, Ulrich, Voigt, Jürgen},
keywords = {convex body; isotropic; cross section; central limit theorem; marginal distribution},
title = {Asymptotics of cross sections for convex bodies.},
AU - Voigt, Jürgen
TI - Asymptotics of cross sections for convex bodies.
KW - convex body; isotropic; cross section; central limit theorem; marginal distribution
Assaf Naor, Dan Romik, Projecting the surface measure of the sphere of
{\ell }_{p}^{n}
convex body, isotropic, cross section, central limit theorem, marginal distribution
{L}^{p}
Articles by Brehm
Articles by Voigt
|
Numeric to Symbolic Conversion - MATLAB & Simulink - MathWorks Italia
Conversion to Rational Symbolic Form
Conversion by Using Floating-Point Expansion
Conversion to Rational Symbolic Form with Error Term
Conversion to Decimal Form
This topic shows how Symbolic Math Toolbox™ converts numbers into symbolic form. For an overview of symbolic and numeric arithmetic, see Choose Numeric or Symbolic Arithmetic.
To convert numeric input to symbolic form, use the sym command. By default, sym returns a rational approximation of a numeric expression.
sym(t)
sym determines that the double-precision value 0.1 approximates the exact symbolic value 1/10. In general, sym tries to correct the round-off error in floating-point inputs to return the exact symbolic form. Specifically, sym corrects round-off error in numeric inputs that match the forms p/q, pπ/q, (p/q)1/2, 2q, and 10q, where p and q are modest-sized integers.
For these forms, demonstrate that sym converts floating-point inputs to the exact symbolic form. First, numerically approximate 1/7, pi, and
1/\sqrt{2}
N2 = pi
N3 = 1/sqrt(2)
Convert the numeric approximations to exact symbolic form. sym corrects the round-off error.
S1 = sym(N1)
To return the error between the input and the estimated exact form, use the syntax sym(num,'e'). See Conversion to Rational Symbolic Form with Error Term.
You can force sym to accept the input as is by placing the input in quotes. Demonstrate this behavior on the previous input 0.142857142857143. The sym function does not convert the input to 1/7.
sym('0.142857142857143')
When you convert large numbers, use quotes to exactly represent them. Demonstrate this behavior by comparing sym(133333333333333333333) with sym('133333333333333333333').
sym(1333333333333333333)
sym('1333333333333333333')
You can specify the technique used by sym to convert floating-point numbers using the optional second argument, which can be 'f', 'r', 'e', or 'd'. The default flag is 'r', for rational form.
Convert input to exact rational form by calling sym with the 'r' flag. This is the default behavior when you call sym without flags.
sym(t, 'r')
If you call sym with the flag 'f', sym converts double-precision, floating-point numbers to their numeric value by using N*2^e, where N and e are the exponent and mantissa respectively.
Convert t by using a floating-point expansion.
sym(t, 'f')
If you call sym with the flag 'e', sym returns the rational form of t plus the error between the estimated, exact value for t and its floating-point representation. This error is expressed in terms of eps (the floating-point relative precision).
Convert t to symbolic form. Return the error between its estimated symbolic form and its floating-point value.
sym(t, 'e')
eps/40 + 1/10
The error term eps/40 is the difference between sym('0.1') and sym(0.1).
If you call sym with the flag 'd', sym returns the decimal expansion of the input. The digits function specifies the number of significant digits used. The default value of digits is 32.
sym(t,'d')
Change the number of significant digits by using digits.
digitsOld = digits(7);
For further calculations, restore the old value of digits.
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EUDML | Eisenstein series for Siegel modular groups. EuDML | Eisenstein series for Siegel modular groups.
Eisenstein series for Siegel modular groups.
Shin-ichiro Mizumoto
Mizumoto, Shin-ichiro. "Eisenstein series for Siegel modular groups.." Mathematische Annalen 297.4 (1993): 581-626. <http://eudml.org/doc/165149>.
@article{Mizumoto1993,
author = {Mizumoto, Shin-ichiro},
keywords = {slowly increasing partial derivative; automorphic -functions; Epstein zeta-functions; growth of derivatives of confluent hypergeometric functions; Siegel-type non-holomorphic Eisenstein series; analytic continuation; functional equation; Fourier expansion; Rankin-Selberg convolution},
title = {Eisenstein series for Siegel modular groups.},
AU - Mizumoto, Shin-ichiro
TI - Eisenstein series for Siegel modular groups.
KW - slowly increasing partial derivative; automorphic -functions; Epstein zeta-functions; growth of derivatives of confluent hypergeometric functions; Siegel-type non-holomorphic Eisenstein series; analytic continuation; functional equation; Fourier expansion; Rankin-Selberg convolution
Shin-ichiro Mizumoto, Certain L-functions at s = 1/2
Siegfried Böcherer, Francesco Ludovico Chiera, On Dirichlet Series and Petersson Products for Siegel Modular Forms
slowly increasing partial derivative, automorphic
L
-functions, Epstein zeta-functions, growth of derivatives of confluent hypergeometric functions, Siegel-type non-holomorphic Eisenstein series, analytic continuation, functional equation, Fourier expansion, Rankin-Selberg convolution
Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
L
Articles by Shin-ichiro Mizumoto
|
EUDML | A Socratic methodological proposal for the study of the equality $0.999\ldots = 1$ EuDML | A Socratic methodological proposal for the study of the equality $0.999\ldots = 1$
A Socratic methodological proposal for the study of the equality
0.999...=1
Maria Angeles Navarro; Pedro Pérez Carreras
Maria Angeles Navarro, and Pedro Pérez Carreras. "A Socratic methodological proposal for the study of the equality $0.999\ldots = 1$." The Teaching of Mathematics (2010): 17-34. <http://eudml.org/doc/256812>.
@article{MariaAngelesNavarro2010,
author = {Maria Angeles Navarro, Pedro Pérez Carreras},
title = {A Socratic methodological proposal for the study of the equality $0.999\ldots = 1$},
AU - Maria Angeles Navarro
AU - Pedro Pérez Carreras
TI - A Socratic methodological proposal for the study of the equality $0.999\ldots = 1$
Articles by Maria Angeles Navarro
Articles by Pedro Pérez Carreras
|
Home : Support : Online Help : Mathematics : Basic Mathematics : Exponential, Trig, and Hyperbolic Functions : Working in Degrees : Degrees package : Simplify
simplify an expression that has degree-form trig functions
Simplify( expr )
The Simplify(expr) command converts the underlying trig functions in the given expression to radians-form, applies simplify, and converts back to degrees form. Some additional care is taken to produce good results for expressions containing Int, Diff, and RootOf.
This function is part of the Degrees package, so it can be used in the short form Simplify(..) only after executing the command with(Degrees). However, it can always be accessed through the long form of the command by using Degrees:-Simplify(..).
\mathrm{with}\left(\mathrm{Degrees}\right):
\mathrm{expr}≔{\mathrm{sind}\left(x\right)}^{2}+{\mathrm{cosd}\left(x\right)}^{2}
\textcolor[rgb]{0,0,1}{\mathrm{expr}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathrm{sind}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{cosd}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}
\mathrm{Simplify}\left(\mathrm{expr}\right)
\textcolor[rgb]{0,0,1}{1}
The Degrees[Simplify] command was introduced in Maple 2021.
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Stress‐Drop Variations of Induced Earthquakes in OklahomaStress‐Drop Variations of Induced Earthquakes in Oklahoma | Bulletin of the Seismological Society of America | GeoScienceWorld
Stress‐Drop Variations of Induced Earthquakes in Oklahoma
Qimin Wu;
Department of Geosciences, Virginia Tech, 4044 Derring Hall, Blacksburg, Virginia 24061, wqimin86@vt.edu, mcc@vt.edu
Now at ConocoPhillips School of Geology and Geophysics, University of Oklahoma, Norman, Oklahoma 73019.
Martin Chapman;
ConocoPhillips School of Geology and Geophysics, University of Oklahoma, 100 East Boyd Street, Norman, Oklahoma 73019, xiaowei.chen@ou.edu
Qimin Wu, Martin Chapman, Xiaowei Chen; Stress‐Drop Variations of Induced Earthquakes in Oklahoma. Bulletin of the Seismological Society of America 2018;; 108 (3A): 1107–1123. doi: https://doi.org/10.1785/0120170335
We calculate corner frequencies and stress drops for 201 earthquakes in four earthquake sequences that are potentially induced by wastewater injection in Oklahoma. Specifically, we determine stress drops for 35 events in the 6 November 2011
Mw
5.6 Prague sequence, 40 events in the 13 February 2016
Mw
5.1 Fairview sequence, 73 events in the 3 September 2016
Mw
5.8 Pawnee sequence, and 53 events in the 7 November 2016
Mw
5.0 Cushing sequence. Although the stress‐drop estimates show large scatter for individual sequences, we find high stress drops for three of the four
Mw
5+ mainshocks (17–34 MPa, Brune stress drop) and lower stress drops for most of the foreshocks/aftershocks in each individual sequence. The exception is the 2011 Prague sequence, which has stress drops ranging between 0.03 and 76 MPa, and the mainshock has a low stress drop of 3.25 MPa. Compared with the other three sequences, the 2016 Fairview sequence exhibits more constant stress‐drop estimates, with a two to three times higher median stress drop. We find significant scatter in the stress‐drop estimates of small earthquakes (
Mw<4
) and note that the earthquakes greater than
Mw
4 have systematically larger stress drops than the average stress drop observed for earthquakes below
Mw
4. We observe no clear evidence for depth dependence or temporal patterns of stress‐drop estimates. The large spatial variability of stress drops reflects strong fault heterogeneity in this area, which is likely influenced by the injection of fluids into the subsurface. Given the large variations of stress drop, our results do not support the suggestion of using low stress drops in ground‐motion prediction models for seismic hazard assessment of induced earthquakes in the central and eastern United States (CEUS).
Increases in Life-Safety Risks to Building Occupants from Induced Earthquakes in the Central United States
Magnitude Recurrence Relations for Colorado Earthquakes
A High‐Resolution Seismic Catalog for the Initial 2019 Ridgecrest Earthquake Sequence: Foreshocks, Aftershocks, and Faulting Complexity
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What Is Unlevered Beta ?
Beta is a measure of market risk. Unlevered beta (or asset beta) measures the market risk of the company without the impact of debt.
'Unlevering' a beta removes the financial effects of leverage thus isolating the risk due solely to company assets. In other words, how much did the company's equity contribute to its risk profile.
Levered beta (commonly referred to as just beta or equity beta) is a measure of market risk. Debt and equity are factored in when assessing a company's risk profile.
Unlevered beta strips off the debt component to isolate the risk due solely to company assets.
High debt-to-equity ratio usually translates to an increase in the risk associated with a company's stock.
A beta of 1 means that the stock is as risky as the market while betas greater or less than 1 reflect risk thresholds higher or lower than the market, respectively.
Understanding Unlevered Beta
Beta is the slope of the coefficient for a stock regressed against a benchmark market index like the Standard & Poor's (S&P) 500 Index. A key determinant of beta is leverage, which measures the level of a company’s debt to its equity. Levered beta measures the risk of a firm with debt and equity in its capital structure to the volatility of the market. The other type of beta is known as unlevered beta.
'Unlevering' the beta removes any beneficial or detrimental effects gained by adding debt to the firm's capital structure. Comparing companies' unlevered betas give an investor clarity on the composition of risk being assumed when purchasing the stock.
\text{Unlevered beta (asset beta)} =\frac{ \text{Levered beta (equity beta)} } {\left( 1 + \frac{\left( 1-\text{tax rate} \right )*\text{Debt}}{\text{Equity}} \right )}
Unlevered beta (asset beta)=(1+Equity(1−tax rate)∗Debt)Levered beta (equity beta)
Take a company that is increasing its debt thus raising its debt-to-equity ratio. This will lead to a larger percentage of earnings being used to service that debt which will amplify investor uncertainty about future earnings stream. Consequently, the company's stock is deemed to be getting riskier but that risk is not due to market risk.
Isolating and removing the debt component of overall risk results in unlevered beta.
The level of debt that a company has can affect its performance, making it more sensitive to changes in its stock price. Note that the company being analyzed has debt in its financial statements, but unlevered beta treats it like it has no debt by stripping any debt off the calculation. Since companies have different capital structures and levels of debt, an analyst can calculate the unlevered beta to effectively compare them against each other or against the market. This way, only the sensitivity of a firm’s assets (equity) to the market will be factored in.
To 'unlever' the beta, the levered beta for the company has to be known in addition to the company’s debt-equity ratio and corporate tax rate.
Systematic risk is the type of risk that is caused by factors beyond a company's control. This type of risk cannot be diversified away. Examples of systematic risk include natural disasters, political elections, inflation, and wars. Beta is used to measure the level of systematic risk, or volatility, of a stock or portfolio.
Beta is a statistical measure that compares the volatility of the price of a stock against the volatility of the broader market. If the volatility of the stock, as measured by beta, is higher, the stock is considered risky. If the volatility of the stock is lower, the stock is said to have less risk.
A beta of one is equivalent to the risk of the broader market. That is, a company with a beta of one has the same systematic risk as the broader market. A beta of two means the company is twice as volatile as the overall market, but a beta of less than one means the company is less volatile and presents less risk than the broader market.
Example of Unlevered Beta
BU = BL ÷ [1 + ((1 - Tax Rate) x D/E)]
For example, calculating the unlevered beta for Tesla, Inc. (as of November 2017):
beta (BL) is 0.73
Debt to Equity (D/E) ratio is 2.2
Tesla BU = 0.73 ÷ [1 +((1 – 0.35) * 2.2)]= 0.30
Unlevered beta is almost always equal to or lower than levered beta given that debt will most often be zero or positive. (In the rare occasions where a company's debt component is negative, say a company is hoarding cash, then unlevered beta can potentially be higher than levered beta.)
If the unlevered beta is positive, investors will invest in the company's stock when prices are expected to rise. A negative unlevered beta will prompt investors to invest in the stock when prices are expected to decline.
How Can Unlevered Beta Help an Investor?
Unlevered beta removes any beneficial or detrimental effects gained by adding debt to the firm's capital structure. Comparing companies' unlevered betas give an investor clarity on the composition of risk being assumed when purchasing the stock. Since companies have different capital structures and levels of debt, an investor can calculate the unlevered beta to effectively compare them against each other or against the market. This way, only the sensitivity of a firm’s assets (equity) to the market will be factored in.
Simply put, beta (ß) is a measure of market risk. More precisely, it is a measure of the volatility—or systematic risk—of a security or portfolio compared to the market as a whole. In statistical terms, it is the slope of the coefficient for a security (stock) regressed against a benchmark market index (S&P 500). Each of these data points represents an individual stock's returns against those of the market as a whole. So, beta effectively describes the activity of a security's returns as it responds to swings in the market.
Levered beta measures the risk of a firm with debt and equity in its capital structure to the volatility of the market. A key determinant of beta is leverage, which measures the level of a company’s debt to its equity. So, a publicly traded security's levered beta measures the sensitivity of that security's tendency to perform in relation to the overall market. A levered beta greater than positive 1 or less than negative 1 means that it has greater volatility than the market. A levered beta between negative 1 and positive 1 has less volatility than the market.
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Japanese community key - zxc.wiki
Japanese community key
The Japanese municipality key ( Japanese 全国地方 公共 団 体 コ ー ド , zenkoku chihō kōkyō dantai kōdo , German code of the local authorities of the whole country ) is a five- or, with a check digit , six-digit sequence to identify regional authorities in Japan . Its function is comparable to the official municipality key in Germany and Austria.
The Japanese community key was introduced in 1968 by the then Japanese Ministry of Self-Government (now the Ministry of Internal Affairs and Communications ), adopted by the Administrative Oversight Authority in 1970, and has been the basis for censuses and official statistics in Japan ever since.
The structure of the community key is abcde-f . The first two digits ab designate the Japanese prefecture beginning with Hokkaidō in the north (01) to Okinawa in the south (47) and correspond to the respective code in ISO 3166-2: JP . If the key only refers to the prefecture, the following digits are cde zeros.
The third digit c is 1 for the cities and districts of Tokyo determined by government decree, 2 for the other large cities ( 市 , shi ) and 3 for the small towns ( 町 , machi or chō ) and villages ( 村 , mura or son ) to 7, and for special purpose associations 8. The state land surveying office ( Kokudo Chiriin ) also uses regional authority keys for former districts ( gun ) , these also have 3 to 7 as the third digit and are grouped with the districts belonging to municipalities.
The calculation of the check digit f is carried out according to the calculation rule .
{\ displaystyle f = 11- \ left (6a + 5b + 4c + 3d + 2e \ right) \ mod 11}
List of parish keys in Japan with the corresponding postal addresses (in Japanese)
Sōmushō : zenkoku chihō kōkyō dantai code (Japanese, with regularly updated lists as pdf and excel, also contains the codes of special-purpose associations ( ichibu jimu kumiai & Kōiki Rengō ))
This page is based on the copyrighted Wikipedia article "Japanischer_Gemeindeschl%C3%BCssel" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Augment - Maple Help
Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : LHPDE : Augment
append some DEs to a LHPDE object
Augment( obj, des, rifReduce = r)
a list or set of DEs
The Augment method appends some DEs des to the LHPDE object obj.
The method returns a new LHPDE object with new conditions des appended.
The returned LHPDE object can be in rif-reduced form (see DEtools[rifsimp]) by specifying rifReduce = true.
\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):
\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):
\mathrm{Typesetting}:-\mathrm{Suppress}\left({\mathrm{\eta },\mathrm{\xi }}\left(x,y\right)\right)
S≔\mathrm{LHPDE}\left([\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0],\mathrm{indep}=[x,y],\mathrm{dep}=[\mathrm{\xi },\mathrm{\eta }]\right)
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}[{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{indep}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{dep}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}]
\mathrm{Augment}\left(S,[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)=0]\right)
[{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{indep}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{dep}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}]
\mathrm{Augment}\left(S,[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)=0],\mathrm{rifReduce}=\mathrm{true}\right)
[{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{indep}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{dep}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}]
The Augment command was introduced in Maple 2020.
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Counting and Probability - Course Hero
College Algebra/Counting and Probability
Counting and probability are used to determine how likely something is to happen. There are a variety of ways to count the number of ways something can happen, including diagrams and formulas. Counting is needed to determine probabilities. The probabilities of events are figured differently depending on the situation. The theoretical probability is what is expected to happen, but it doesn't always match the experimental probability, or what actually happens.
A factorial of a natural number represents a product of all the natural numbers less than or equal to the number.
A permutation is a selection of elements of a set such that order matters; the number of permutations of
n
objects chosen
r
at a time can be determined by a formula.
A combination is a selection of elements of a set such that order does not matter. The number of combinations of
n
r
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Neyman-Pearson Hypothesis Testing - MATLAB & Simulink - MathWorks France
Support for Neyman-Pearson Hypothesis Testing
Threshold for Real-Valued Signal in White Gaussian Noise
Threshold for Two Pulses of Real-Valued Signal in White Gaussian Noise
Threshold for Complex-Valued Signals in Complex White Gaussian Noise
In phased-array applications, you sometimes need to decide between two competing hypotheses to determine the reality underlying the data the array receives. For example, suppose one hypothesis, called the null hypothesis, states that the observed data consists of noise only. Suppose another hypothesis, called the alternative hypothesis, states that the observed data consists of a deterministic signal plus noise. To decide, you must formulate a decision rule that uses specified criteria to choose between the two hypotheses.
When you use Phased Array System Toolbox™ software for applications such as radar and sonar, you typically use the Neyman-Pearson (NP) optimality criterion to formulate your hypothesis test.
When you choose the NP criterion, you can use npwgnthresh to determine the threshold for the detection of deterministic signals in white Gaussian noise. The optimal decision rule derives from a likelihood ratio test (LRT). An LRT chooses between the null and alternative hypotheses based on a ratio of conditional probabilities.
npwgnthresh enables you to specify the maximum false-alarm probability as a constraint. A false alarm means determining that the data consists of a signal plus noise, when only noise is present.
For details about the statistical assumptions the npwgnthresh function makes, see the reference page for that function.
This example shows how to compute empirically the probability of false alarm for a real-valued signal in white Gaussian noise.
Determine the required signal-to-noise (SNR) in decibels for the NP detector when the maximum tolerable false-alarm probability is 10^-3.
T = npwgnthresh(Pfa,1,'real');
Determine the actual detection threshold corresponding to the desired false-alarm probability, assuming the variance is 1.
threshold = sqrt(variance * db2pow(T));
Verify empirically that the detection threshold results in the desired false-alarm probability under the null hypothesis. To do so, generate 1 million samples of a Gaussian random variable, and determine the proportion of samples that exceed the threshold.
x = sqrt(variance) * randn(N,1);
falsealarmrate = sum(x > threshold)/N
falsealarmrate = 9.9500e-04
Plot the first 10,000 samples. The red horizontal line shows the detection threshold.
x1 = x(1:1e4);
line([1 length(x1)],[threshold threshold],'Color','red')
You can see that few sample values exceed the threshold. This result is expected because of the small false-alarm probability.
This example shows how to empirically verify the probability of false alarm in a system that integrates two real-valued pulses. In this scenario, each integrated sample is the sum of two samples, one from each pulse.
Determine the required SNR for the NP detector when the maximum tolerable false-alarm probability is
1{0}^{-3}
Generate two sets of one million samples of a Gaussian random variable.
pulse1 = sqrt(variance)*randn(N,1);
intpuls = pulse1 + pulse2;
Compute the proportion of samples that exceed the threshold.
threshold = sqrt(variance*db2pow(T));
falsealarmrate = sum(intpuls > threshold)/N
The empirical false alarm rate is very close to .001
This example shows how to empirically verify the probability of false alarm in a system that uses coherent detection of complex-valued signals. Coherent detection means that the system utilizes information about the phase of the complex-valued signals.
Determine the required SNR for the NP detector in a coherent detection scheme with one sample. Use a maximum tolerable false-alarm probability of
1{0}^{-3}
T = npwgnthresh(pfa,1,'coherent');
Test that this threshold empirically results in the correct false-alarm rate The sufficient statistic in the complex-valued case is the real part of the received sample.
x = sqrt(variance/2)*(randn(N,1)+1j*randn(N,1));
falsealarmrate = sum(real(x)>threshold)/length(x)
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Overview of VaR Backtesting - MATLAB & Simulink - MathWorks í•œêµ
Kupiec’s POF and TUFF Tests
Christoffersen’s Interval Forecast Tests
Haas’s Time Between Failures or Mixed Kupiec’s Test
Suppose that you have VaR limits and corresponding returns or profits and losses for days t = 1,…,N. Use VaRt to denote the VaR estimate for day t (determined on day t − 1). Use Rt to denote the actual return or profit and loss observed on day t. Profits and losses are expressed in monetary units and represent value changes in a portfolio. The corresponding VaR limits are also given in monetary units. Returns represent the change in portfolio value as a proportion (or percentage) of its value on the previous day. The corresponding VaR limits are also given as a proportion (or percentage). The VaR limits must be produced from existing VaR models. Then, to perform a VaR backtesting analysis, provide these limits and their corresponding returns as data inputs to the VaR backtesting tools in Risk Management Toolbox™.
Kupiec’s tests
Christoffersen’s tests
Haas’s tests
The most straightforward test is to compare the observed number of exceptions, x, to the expected number of exceptions. From the properties of a binomial distribution, you can build a confidence interval for the expected number of exceptions. Using exact probabilities from the binomial distribution or a normal approximation, the bin function uses a normal approximation. By computing the probability of observing x exceptions, you can compute the probability of wrongly rejecting a good model when x exceptions occur. This is the p-value for the observed number of exceptions x. For a given test confidence level, a straightforward accept-or-reject result in this case is to fail the VaR model whenever x is outside the test confidence interval for the expected number of exceptions. “Outside the confidence interval†can mean too many exceptions, or too few exceptions. Too few exceptions might be a sign that the VaR model is too conservative.
{Z}_{bin}=\frac{xâNp}{\sqrt{Np\left(1âp\right)}}
The “red†zone starts at the number of exceptions where this probability equals or exceeds 99.99%. It is unlikely that too many exceptions come from a correct VaR model.
The “yellow†zone covers the number of exceptions where the probability equals or exceeds 95% but is smaller than 99.99%. Even though there is a high number of violations, the violation count is not exceedingly high.
L{R}_{POF}=â2\mathrm{log}\left(\frac{{\left(1âp\right)}^{Nâx}{p}^{x}}{{\left(1â\frac{x}{N}\right)}^{Nâx}{\left(\frac{x}{N}\right)}^{x}}\right)
L{R}_{TUFF}=â2\mathrm{log}\left(\frac{p{\left(1âp\right)}^{nâ1}}{\left(\frac{1}{n}\right){\left(1â\frac{1}{n}\right)}^{nâ1}}\right)
Christoffersen (1998) proposed a test to measure whether the probability of observing an exception on a particular day depends on whether an exception occurred. Unlike the unconditional probability of observing an exception, Christoffersen's test measures the dependency between consecutive days only. The test statistic for independence in Christoffersen’s interval forecast (IF) approach is given by
L{R}_{CCI}=â2\mathrm{log}\left(\frac{{\left(1â\mathrm{Ï}\right)}^{n00+n10}{\mathrm{Ï}}^{n01+n11}}{{\left(1â{\mathrm{Ï}}_{0}\right)}^{n00}{\mathrm{Ï}}_{0}^{n01}{\left(1â{\mathrm{Ï}}_{1}\right)}^{n10}{\mathrm{Ï}}_{1}^{n11}}\right)
Ï€0 — Probability of having a failure on period t, given that no failure occurred on period t − 1 = n01 / (n00 + n01)
Ï€1 — Probability of having a failure on period t, given that a failure occurred on period t − 1 = n11 / (n10 + n11)
Ï€ — Probability of having a failure on period t = (n01 + n11 / (n00 + n01 + n10 + n11)
Haas (2001) extended Kupiec’s TUFF test to incorporate the time information between all the exceptions in the sample. Haas’s test applies the TUFF test to each exception in the sample and aggregates the time between failures (TBF) test statistic.
L{R}_{TBFI}=â2{â}_{i=1}^{x}\mathrm{log}\left(\frac{p{\left(1âp\right)}^{{n}_{i}â1}}{\left(\frac{1}{{n}_{i}}\right){\left(1â\frac{1}{{n}_{i}}\right)}^{{n}_{i}â1}}\right)
Like Christoffersen’s test, you can combine this test with the frequency POF test to get a TBF mixed test, sometimes called Haas’ mixed Kupiec’s test:
L{R}_{TBF}=L{R}_{POF}+L{R}_{TBFI}
[1] Basel Committee on Banking Supervision, Supervisory framework for the use of “backtesting†in conjunction with the internal models approach to market risk capital requirements. January 1996, https://www.bis.org/publ/bcbs22.htm.
[3] Cogneau, P. “Backtesting Value-at-Risk: how good is the model?" Intelligent Risk, PRMIA, July, 2015.
[8] Nieppola, O. “Backtesting Value-at-Risk Models.†Helsinki School of Economics, 2009.
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A New Pressure Regularity Criterion of the Three-Dimensional Micropolar Fluid Equations
2013 A New Pressure Regularity Criterion of the Three-Dimensional Micropolar Fluid Equations
Junbai Ren
This paper concerns the regularity criterion of the weak solutions to the three-dimensional (3D) micropolar fluid equations in terms of the pressure. It is proved that if one of the partial derivatives of pressure satisfies
{\partial }_{3}\pi \in {L}^{p}\left(0,T;{L}^{q}\left({\mathbf{\text{R}}}^{3}\right)\right)
2/p+3/q\le 2,3<q<\infty ,1<p<\infty
, then the weak solution of the micropolar fluid equations becomes regular on
\left(\mathrm{0},T\right]
Junbai Ren. "A New Pressure Regularity Criterion of the Three-Dimensional Micropolar Fluid Equations." J. Appl. Math. 2013 1 - 5, 2013. https://doi.org/10.1155/2013/262131
Junbai Ren "A New Pressure Regularity Criterion of the Three-Dimensional Micropolar Fluid Equations," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-5, (2013)
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Angular Symbols for Standard Solar Relations | EME 810: Solar Resource Assessment and Economics
Angular Symbols for Standard Solar Relations
Summarized below are angle notations that will be used in readings and problems throughout the course. In the literature, you may encounter some variation of symbols and their interpretation, so in the attempt of better consistency, we will try to stick this table as a reference guide.
Range and Sign Convention
\alpha
(alpha) 0o to + 90o; horizontal is zero
\gamma
(gamma) 0o to + 360o; clockwise from North origin
azimuth (alternate)
\gamma
(gamma) 0o to ±180o; zero (origin) faces the equator, East is + ive, West is - ive
Earth-Sun Angles
\varphi
(phi) 0o to ± 90o; Northern hemisphere is +ive
\lambda
(lambda) 0o to ± 180o; Prime Meridian is zero, West is -ive
\delta
(delta) 0o to ± 23.45o; Northern hemisphere is +ive
\omega
(omega) 0o to ± 180o; solar noon is zero, afternoon is +ive, morning is -ive
Sun-Observer Angles
solar altitude angle (complement)
{\alpha }_{s}=\text{ }1\text{ }-\text{ }{\theta }_{z}
{\alpha }_{s}=1-{\theta }_{z}
(alphas is the complement of thetaz) 0o to + 90o
{\gamma }_{s}
(gammas) 0o to + 360o; clockwise from North origin
{\theta }_{z}
(thetaz) 0o to + 90o; vertical is zero
Collector-Sun Angles
surface altitude angle
\alpha
(alpha) 0o to + 90o
slope or tilt (of collector surface)
\beta
(beta) 0o to ±90o; facing equator is +ive
\gamma
\theta
(theta) 0o to + 90o
glancing angle (complement)
\alpha =1-\theta
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Class 10 - Arihant - All in one (Math) - Circles
Class 10 - Arihant - All in one (Math) - CirclesContact Number: 9667591930 / 8527521718
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the center O at a point Q such that PQ = 12 cm. Find the length of OQ.
Find the radius of a circle, if the length of the tangent from a point at a distance of 25 cm from the center of the circle, is 24 cm.
If PQ is a tangent to a circle with center O and radius 6cm such that
\angle \mathrm{PQO} = 60°
, then find the length of a tangent PQ and a line OQ.
If AB is a tangent drawn from a point A to a circle with center O and BOC is a diameter of the circle such that
\angle \mathrm{AOC} = 120°
\angle \mathrm{OAB}
(i) In the given figure, find the value of
\mathrm{X}°
(ii) from the given figure, find the value of
\mathrm{x}° + \mathrm{y}°.
Find the length of tangent to a circle from a point at a distance of 5 cm from the centre of the circle of radius 3 cm.
IF O is the center of a circle. PQ is chord and the tangent PR at P makes an angle of
50°
with PQ, then find
\angle \mathrm{POQ}.
\mathrm{In} \mathrm{the} \mathrm{given}, \mathrm{FG} \mathrm{is} \mathrm{a} \mathrm{tangent} \mathrm{to} \mathrm{the} \mathrm{circle} \mathrm{with} \mathrm{center} \mathrm{A}. \mathrm{If} \angle \mathrm{DCB} = 15° \mathrm{and} \mathrm{CE} = \mathrm{DE}, \mathrm{then} \mathrm{find} \angle \mathrm{GCE} \mathrm{and} \angle \mathrm{BCE}.
In the given figure, XP and XQ are tangents from X to the circle with O. R is a point on the circle.
Prove that XA + AR = XB + BR.
O is the center of a circle of radius 5 cm. T is a point such that OT = 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, then find the length of AB.
Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
\mathrm{ABC} \mathrm{is} \mathrm{a} \mathrm{right}-\mathrm{angled} \mathrm{triangle} \mathrm{with} \angle \mathrm{B} = 90°, \mathrm{BC} = 3 \mathrm{cm} \mathrm{and} \mathrm{AB} = 4 \mathrm{cm}. \mathrm{A} \mathrm{circle} \mathrm{with} \mathrm{center} \mathrm{O} \mathrm{and} \mathrm{radius} \mathrm{r} \mathrm{cm} \phantom{\rule{0ex}{0ex}}\mathrm{has} \mathrm{been} \mathrm{inscribed} \mathrm{in} ∆\mathrm{ABC}. \mathrm{Find} \mathrm{the} \mathrm{radius} \mathrm{of} \mathrm{the} \mathrm{incircle}.
If the radius of a circle is 5 cm. then find the distance between two parallel tangents.
\mathrm{If} \mathrm{AB} \mathrm{and} \mathrm{BC} \mathrm{are} \mathrm{two} \mathrm{tangents} \mathrm{to} \mathrm{a} \mathrm{circle} \mathrm{with} \mathrm{center} \mathrm{O} \mathrm{such} \mathrm{that} \angle \mathrm{AOC} = 120°, \mathrm{then} \mathrm{find} \angle \mathrm{ABC}.
In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose three sides are AB = 6 cm, BC = 7 cm, and CD = 4cm. Find the length of AD.
In the given figure, O is the centre of two concentric circles of radii 4 cm and 6 cm, respectively. PA and PB are tangents to the outer and inner circles, respectively. If PA = 10 cm, then find the length of PB up to one place Of decimal.
ABCD is a quadrilateral such that
\mathrm{ZD} = 90°.
A circle C (O, r) touches the sides AB, BC, CD, and DA at P, Q, R, and S, respectively. If BC = 38 cm, CD = 25 cm and BP = 27 cm. then find the value of r.
In the given figure, PA and PB are tangents to the given circle such that PA = 5 cm and
\angle \mathrm{APB} = 60°.
Find the length of chord AB.
(i) A tangent to a circle intersects it in ____________ point(s).
(ii) A line intersecting a circle in two points is called a ______________.
(iii) A circle can have ____________ parallel tangenst at the most.
(iv) The common point of a tangent to a circle and the circle is called ______________.
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length of PQ is
\sqrt{119}
Draw a circle and two lines parallel to a given line such that one is a tangent and the other a. secant to the circle.
In the given figure, if TP and TQ are the two tangents to a circle with center O, so that
\angle \mathrm{POQ} = 110°, \mathrm{then} \angle \mathrm{PTQ} \mathrm{is} \mathrm{equal} \mathrm{to}
60°
70°
80°
90°
7 Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
In the given figure, XY and X'Y' are two parallel tangents to a circle with center O and another tangent AB with the point of contact C intersecting XY at A and X'Y' at B. Prove that
\angle \mathrm{AOB} = 90°
∆
ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm, respectively (see figure). Find the sides AB and AC.
How many tangents can be drawn to a circle from a point Plies outside the circle?
At which point a tangent is perpendicular to the radius?
What term will you use for a line which intersects a circle at two distinct points?
Write the name of the common point of the tangent to a circle and the circle.
What do you say about the line which is perpendicular to the radius of the circle through the point of contact?
Write the number of tangents to a circle that is parallel to a secant.
Find the distance between two parallel tangents of a circle of radius 3 cm.
If two tangents inclined at an angle of
60°
are drawn to a circle of radius 3 cm, then find the length of each tangent.
In the given figure, AB, AC, and PQ are tangents. If AB = 5 cm, then find the perimeter of
∆\mathrm{APQ}.
\mathrm{In} \mathrm{the} \mathrm{following} \mathrm{figure}, \mathrm{PQ} \mathrm{is} \mathrm{a} \mathrm{tangent} \mathrm{at} \mathrm{a} \mathrm{point} \mathrm{C} \mathrm{to} \mathrm{circle} \mathrm{with} \mathrm{center} \mathrm{O}. \mathrm{If} \mathrm{AB} \mathrm{is} \mathrm{a} \mathrm{diameter} \mathrm{and} \angle \mathrm{CAB} = 30°, \phantom{\rule{0ex}{0ex}}\mathrm{then} \mathrm{find} \angle \mathrm{PCA}.
Two concentric circles with centre O are of radii 5 cm and 3 cm. From an external point P, two tangents PA and PB are drawn to these circles, respectively. If PA = 12 cm, then find the length of PB.
In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length.
If PA and PB are two tangents drawn from a point P to a circle with centre O touching it at A and B, prove that OP is the perpendicular bisector of AB.
In the adjoining figure, AD = 8 cm, AC = 6 cm, and TB is the tangent at B to the circle with center O. Find OT, if BTis 4 cm.
The tangents drawn at the endpoints of two perpendicular diameters Of a circle are parallel to each other, which form of a square and whose length of the side is 2 cm. Find the radius of the circle.
\mathrm{PA} \mathrm{is} \mathrm{a} \mathrm{tangent} \mathrm{to} \mathrm{the} \mathrm{circle} \mathrm{with} \mathrm{center} \mathrm{O}. \mathrm{If} \mathrm{BC} = 3 \mathrm{cm}, \mathrm{AC} = 4 \mathrm{cm} \mathrm{and} ∆\mathrm{ACB} ~ ∆\mathrm{PAO}, \mathrm{then} \mathrm{find} \mathrm{OA} \mathrm{and} \frac{\mathrm{OP}}{\mathrm{AP}}.
In the given figure, common tangents AB and CD to two circles intersect at E. Prove that AB = CD.
In the adjoining figure, from an external point P, two tangents PT and PS are drawn to a circle with center O and radius r. If OP = 2r, then show that
\angle \mathrm{OTS} = \angle \mathrm{OST} = 30°
At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. Find the length Of the chord CD parallel to XY and at a distance 8 cm from A.
\mathrm{PC} \mathrm{is} \mathrm{tangent} \mathrm{to} \mathrm{the} \mathrm{circle} \mathrm{at} \mathrm{C}. \mathrm{AOB} \mathrm{is} \mathrm{the} \mathrm{diameter} \mathrm{which} \mathrm{when} \mathrm{extended} \mathrm{meets} \mathrm{the} \mathrm{tangent} \mathrm{at} \mathrm{P}. \phantom{\rule{0ex}{0ex}}\mathrm{Find} \angle \mathrm{CBA}, \angle \mathrm{AOC}, \mathrm{and} \angle \mathrm{BCO}, \mathrm{if} \angle \mathrm{PCA} = 110°.
\mathrm{A} \mathrm{circle} \mathrm{touches} \mathrm{the} \mathrm{side} \mathrm{BC} \mathrm{of} \mathrm{a} ∆\mathrm{ABC} \mathrm{at} \mathrm{P} \mathrm{and} \mathrm{AB} \mathrm{and} \mathrm{AC} \mathrm{when} \mathrm{produced} \mathrm{at} \mathrm{Q} \mathrm{and} \mathrm{R} \mathrm{respectively} \mathrm{as} \mathrm{shown} \mathrm{in} \mathrm{the} \phantom{\rule{0ex}{0ex}}\mathrm{figure}. \mathrm{Show} \mathrm{that} \mathrm{AQ} = \frac{1}{2}\left(\mathrm{perimeter} \mathrm{of} ∆\mathrm{ABC}\right) \mathrm{or} \mathrm{show} \mathrm{that} \mathrm{AQ} = \frac{1}{2}\left(\mathrm{BC} + \mathrm{CA} + \mathrm{AB}\right)
\mathrm{In} \mathrm{the} \mathrm{given} \mathrm{figure}, \mathrm{O} \mathrm{is} \mathrm{the} \mathrm{center} \mathrm{of} \mathrm{the} \mathrm{circle}. \mathrm{Determine} \angle \mathrm{AQB} \mathrm{and} \angle \mathrm{AMB}, \mathrm{if} \mathrm{PA} \mathrm{and} \mathrm{PB} \mathrm{are} \mathrm{tangents} \mathrm{and} \angle \mathrm{APB} = 75°
\mathrm{Tangents} \mathrm{AP} \mathrm{and} \mathrm{AQ} \mathrm{are} \mathrm{drawn} \mathrm{to} \mathrm{circle} \mathrm{with} \mathrm{center} \mathrm{O} \mathrm{from} \mathrm{an} \mathrm{external} \mathrm{point} \mathrm{A}. \mathrm{Prove} \mathrm{that} \angle \mathrm{PAQ} = 2\angle \mathrm{OPQ}.
\mathrm{In} \mathrm{the} \mathrm{given} \mathrm{figure}, \angle \mathrm{ADC} = 90°, \mathrm{BC} = 38 \mathrm{ac}, \mathrm{CD} =28 \mathrm{cm}, \mathrm{and} \mathrm{BP} = 25 \mathrm{cm}, \mathrm{then} \mathrm{find} \mathrm{the} \mathrm{radius} \mathrm{of} \mathrm{the} \mathrm{circle}.
The radii of two concentric circles are 13 cm and 8 cm. ABis a diameter of the bigger circle. BD is a tangent to the smaller circle touching it at D. Find the length of AD.
A circle is inscribed in a
∆
ABC having sides aB = 8 cm, BC = 10 cm, and CA = 12 cm as shown in figure. Find AD, BE, and CF.
If a hexagon ABCDEF circumscribe a circle, prove that AB + CD + EF = BC+DE + FA
If a, b, c are the sides of a right-angled triangle, where c is hypotenuse, then prove that the radius r of the circle which touches the sides of the triangle is given by
\mathrm{r} = \frac{\mathrm{a} + \mathrm{b} - \mathrm{c}}{2}
\mathrm{AC} \mathrm{and} \mathrm{AD} \mathrm{are} \mathrm{tangents} \mathrm{at} \mathrm{C} \mathrm{and} \mathrm{D}, \mathrm{respectively}. \mathrm{If} \angle \mathrm{BCD} = 44°, \mathrm{then} \mathrm{find} \angle \mathrm{CAD}, \angle \mathrm{ADC}, \angle \mathrm{CBD}, \mathrm{and} \angle \mathrm{ACD}.
In the given figure, AD is a diameter of a circle with center O and AB is a tangent at A. C is a point on the circle such that DC produced intersects the tangent at B and
\angle \mathrm{ABD} = 50°. \mathrm{Find} \angle \mathrm{COA}.
\mathrm{Tangents} \mathrm{PQ} \mathrm{and} \mathrm{PR} \mathrm{are} \mathrm{drawn} \mathrm{to} \mathrm{a} \mathrm{circle} \mathrm{such} \mathrm{that} \angle \mathrm{RPQ} = 30°. \mathrm{A} \mathrm{chord} \mathrm{RS} \mathrm{is} \mathrm{drawn} \mathrm{parallel} \mathrm{to} \mathrm{the} \mathrm{tangent} \mathrm{PQ}.\phantom{\rule{0ex}{0ex}}\mathrm{Find} \angle \mathrm{RQS}.
PA and PB are the tangents to a circle which circumscribes an equilateral
∆
ABQ.
\mathrm{If} \angle \mathrm{PAB} = 60°, \mathrm{as} \mathrm{shown} \mathrm{in} \mathrm{the} \mathrm{figure} \mathrm{prove} \mathrm{that} \mathrm{QP} \mathrm{bisects} \mathrm{AB} \mathrm{at} \mathrm{right} \mathrm{angle}.
In the given figure, from an external point P, a tangent PT and a line segment PAB drawn to a circle with center O. ON is perpendicular to the chord AB. Prove that
\left(\mathrm{i}\right) \mathrm{PA} · \mathrm{PB} = {\mathrm{PN}}^{2} - {\mathrm{AN}}^{2}\phantom{\rule{0ex}{0ex}}\left(\mathrm{ii}\right) {\mathrm{PN}}^{2} - {\mathrm{AN}}^{2} = {\mathrm{OP}}^{2} - {\mathrm{OT}}^{2}\phantom{\rule{0ex}{0ex}}\left(\mathrm{iii}\right) \mathrm{PA} · \mathrm{PB} = {\mathrm{PT}}^{2}
As a part of a campaign, a huge balloon with a message of "AWARENESS OF CANCER" was displayed from the terrace of a tall building. It was held by strings of length 8 m each, which inclined at an angle of
60°
at the point, where it was tied as shown in the figure.
(i) the length of AB?
(ii) If the perpendicular distance from the center of the circle to the chord AB is 3 m, then find the radiUS of the circle.
(iii) Which method should be applied to find the radius of the circle?
(iv) What do you think of such a campaign?
|
Specific latent heat — lesson. Science State Board, Class 9.
Specific latent heat is defined as latent heat expressed per unit mass of a substance. It is denoted by the letter L.
If \(Q\) is the amount of heat energy absorbed or released by the ‘m' mass of a substance during its phase change at a constant temperature, then the specific latent heat is given by
L=\frac{Q}{m}
As a result, specific latent heat is the amount of heat energy absorbed or liberated by a unit mass without causing a temperature change during a state change. J/kg is the SI unit for specific latent heat.
Let us look at the example problem.
1. How much heat energy is required to melt 4 kg of ice? (Specific latent heat of ice = 336 J/g)
Mass (\(m\)) \(=\) \(4 kg\)
Specific latent heat of ice (\(L\)) \(=\) \(336 J/g\)
To find: Heat energy (\(Q\))
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EUDML | Skein-theoretical derivation of some formulas of Habiro. EuDML | Skein-theoretical derivation of some formulas of Habiro.
Skein-theoretical derivation of some formulas of Habiro.
Masbaum, Gregor. "Skein-theoretical derivation of some formulas of Habiro.." Algebraic & Geometric Topology 3 (2003): 537-556. <http://eudml.org/doc/123697>.
@article{Masbaum2003,
author = {Masbaum, Gregor},
keywords = {Kauffman bracket skein module; colored Jones polynomial; twist knots},
title = {Skein-theoretical derivation of some formulas of Habiro.},
AU - Masbaum, Gregor
TI - Skein-theoretical derivation of some formulas of Habiro.
KW - Kauffman bracket skein module; colored Jones polynomial; twist knots
Kauffman bracket skein module, colored Jones polynomial, twist knots
{S}^{3}
Articles by Masbaum
|
Examine the tile pattern shown at right.
On graph paper, draw Figure 4.
How many tiles will Figure 10 have? How do you know?
To find the number of tiles in Figure 10 you can make a table and extend it to Figure 10, write an equation and let
x = 10
10
would look like. The two tiles in the middle row are always there. Notice how many pairs of tiles are in each arm of the figure. Sketch figure 10 and determine the number of tiles.
Use the eTool below to examine and draw the tile pattern.
Click the link at right for a full version of the eTool: Int1 1-15 HW eTool
|
Magician's Offer! | Toph
By Ariful_Efath · Limits 2s, 512 MB
Rosi Vidmun is a greedy landlord who likes to grab as much land as he can. Currently he has
P square km land and he wants more! Nobody likes Rosi, but he doesn’t bother about it. Yesterday, a Magician came to him and gave him an offer of giving a larger land. He will get a land of
P^X
PX square km, but only if he is able to solve a puzzle. Magician has a land having Infinite length and width. Rosi is free to select any length and width from the top left corner of the land. He needs to answer how many possible ways are there to make
P^X
PX square km land where the length and width must be a positive integer. If he takes the offer and fails to answer, the magician will take all of his land. Rosi was confused to take the offer. So, the magician told him, he will count the answer as correct if Rosi could tell the last 7 digits of the actual answer. As 7 is a lucky number, Rosi took the offer and thought that it will bring luck to him. I know you might also have started hating Rosi! But, help him to find the answer he needs. He might give you 10% of the land!
Rosi is assuming that the magician might change the value of
P and
X before asking him the answer. So, you have to solve it for different values of
P and
X.
Input starts with a positive integer
T (
≤ 1000
≤1000) denoting the number of test cases.
Each case starts with a line containing two integers
P and
X where
1 ≤ P ≤ 10^{12}
1≤P≤1012 and
1 ≤ X ≤ 10^5
1≤X≤105.
Output should consist of exactly
T lines, one for each test case. Each line should be of the format "Case X: Y" (without the quotes), where
X is the test case and
Y is the last 7 digits of the answer that Rosi needs. If the answer has less than 7 digits, print it with leading zeros.
Case 1: 0000004
P=6 and
X=1, Rosi has the following ways to select his land.
Counting, NumberTheory uDebug
forthright48Earliest, Feb '16
kfoozminusLightest, 1.3 MB
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Play Sound - Monogatari Documentation
'play sound <sound_id> [with [properties]]'
The play sound action let's you, as it name says, play sound effects in your game. You can play as many sound effects as you want simultaneously.
To stop the sound, check out the Stop Sound documentation.
Action ID: Sound
The name of the sound you want to play. These assets must be declared beforehand.
The fade property let's you add a fade in effect to the sound, it accepts a time in seconds, representing how much time you want it to take until the sound reaches it's maximum volume.
The volume property let's you define how high the sound will be played.
Make the sound loop. This property does not require any value.
To play a sound, you must first add the file to your assets/sound/ directory and then declare it. To do so, Monogatari has an has a function that will let you declare all kinds of assets for your game.
'<sound_id>': 'soundFileName'
'play sound riverFlow'
'riverFlow': 'river_water_flowing.mp3'
The following will play the sound, and once the sound ends, it will start over on an infinite loop until it is stopped using the Stop Sound Action.
'play sound riverFlow with loop'
The following will play the sound, and will use a fade in effect.
'play sound riverFlow with fade 3'
The following will set the volume of this sound to 73%.
'play sound riverFlow with volume 73'
Please note however, that the user's preferences regarding volumes are always respected, which means that this percentage is taken from the current player preferences, meaning that if the player has set the volume to 50%, the actual volume value for the sound will be the result of:
50 * 0.73 = 36.5%
'play sound riverFlow with volume 100 loop fade 20'
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LMIs in Control/Matrix and LMI Properties and Tools/Dualization Lemma - Wikibooks, open books for an open world
LMIs in Control/Matrix and LMI Properties and Tools/Dualization Lemma
Dualization LemmaEdit
{\displaystyle P_{i}\in {\text{S}}^{n}}
and the subspaces
{\displaystyle U,V}
{\displaystyle P}
{\displaystyle U+V={\text{R}}^{n}}
. The following are equivalent.
{\displaystyle X^{T}PX<0}
{\displaystyle X\in {\text{U}}}
{\displaystyle \left\{0\right\}}
{\displaystyle X^{T}PX\geq 0}
{\displaystyle X\in {\text{V}}}
{\displaystyle X^{T}P^{-1}X>0}
{\displaystyle X\in {\text{U}}^{\bot }}
{\displaystyle \left\{0\right\}}
{\displaystyle X^{T}P^{-1}X\leq 0}
{\displaystyle X\in {\text{V}}^{\bot }}
{\displaystyle Q\in {\text{S}}^{n},S\in {\text{R}}^{n\times m},R\in {\text{S}}^{m},M\in {\text{R}}^{m\times n}}
{\displaystyle R\geq 0,}
which define the quadratic matrix inequality
{\displaystyle {\begin{aligned}\qquad {\begin{bmatrix}1&M\\\end{bmatrix}}{\begin{bmatrix}Q&S\\S^{T}&R\\\end{bmatrix}}{\begin{bmatrix}1\\M\\\end{bmatrix}}<0.\qquad (1)\end{aligned}}}
{\displaystyle {\begin{aligned}P={\begin{bmatrix}Q&S\\S^{T}&R\\\end{bmatrix}},U=R({\begin{bmatrix}0\\1\\\end{bmatrix}})\end{aligned}}}
{\displaystyle U+V=R^{n+m}}
{\displaystyle (1)}
{\displaystyle X^{T}PX<0}
{\displaystyle X\in {\text{U}}}
{\displaystyle \left\{0\right\}}
.Additionally,
{\displaystyle X^{T}PX<0\geq }
{\displaystyle X\in {\text{V}}}
is euaivalent to
{\displaystyle {\begin{aligned}\qquad {\begin{bmatrix}0&1\\\end{bmatrix}}{\begin{bmatrix}Q&S\\S^{T}&R\\\end{bmatrix}}{\begin{bmatrix}0\\1\\\end{bmatrix}}=R\geq 0,\end{aligned}}}
which is satisfied based on the definition of
{\displaystyle R}
. By the dualization lemma,
{\displaystyle (1)}
is satisfied with
{\displaystyle R\geq 0}
{\displaystyle {\begin{aligned}\qquad {\begin{bmatrix}-M^{T}&1\\\end{bmatrix}}{\begin{bmatrix}{\tilde {Q}}&{\tilde {S}}\\{\tilde {S}}^{T}&{\tilde {R}}\\\end{bmatrix}}{\begin{bmatrix}-M^{T}\\1\\\end{bmatrix}}>0,\qquad {\tilde {Q}}\leq 0,\end{aligned}}}
{\displaystyle {\begin{aligned}\qquad {\begin{bmatrix}{\tilde {Q}}&{\tilde {S}}\\{\tilde {S}}^{T}&{\tilde {R}}\\\end{bmatrix}}={\begin{bmatrix}Q&S\\S^{T}&R\\\end{bmatrix}}^{-1},U^{\bot }=N([1\quad M^{T}])=R({\begin{bmatrix}-M^{T}\\1\\\end{bmatrix}})\end{aligned}}}
{\displaystyle {\begin{aligned}V^{\bot }=N([0\quad 1])=R({\begin{bmatrix}1\\0\\\end{bmatrix}})\end{aligned}}}
Norm-Preserving Dilations - Norm-preserving dilations and their applications to optimal error bounds (Davis, Kahan, Weinberger).
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Matrix_and_LMI_Properties_and_Tools/Dualization_Lemma&oldid=4011013"
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Repeating things - Nice R Code
If there is one piece of advice I would recommend keeping in your head when working with R, it is this:
Many of the headaches that people end up having stem from repeating themselves. This basically mortgages your future time. It is common to see cases where people have copy-and-pasted some piece of code a bunch of times (changing one thing each time), then copy and pasted that a bunch of times, changing it a bit more. In the end, the script is very long, hard to read, really hard to understand, and full of bugs.
Believe me on the last case — you will write code that has bugs. The more code you write, the more bugs you will have. The good thing is that unlike Excel bugs, you can see and find them. The other goood thing is that the less code you write, the more easily you will spot the bugs.
There are many ways of ensuring that you don’t repeat yourself. Mastering the different approaches can take many years. However, try to notice when you feel like you are battling against the language, and you’ll get a good hint for when you should be doing something differently.
Applying a function over and over and over and over
We’ll be using the data set again, with factors sorted alphabetically as discussed in the last section.
data <- read.csv("data/seed_root_herbivores.csv", stringsAsFactors = FALSE)
data$Plot <- factor(data$Plot, levels = unique(data$Plot))
There are a few data columns in the data set that we’ve been using.
## Plot Seed.herbivore Root.herbivore No.stems Height Weight Seed.heads
## 1 plot-2 TRUE TRUE 1 31 4.16 83
## 2 plot-2 TRUE TRUE 3 41 5.82 175
## 4 plot-2 TRUE FALSE 1 64 7.16 125
## Seeds.in.25.heads
No.stems, Height, Weight, Seed.heads, Seeds.in.25.heads.
What if we wanted to get the mean of each column?
mean.no.stems <- mean(data$No.stems)
mean.Height <- mean(data$Height)
mean.Weight <- mean(data$Weight)
mean.Seed.heads <- mean(data$Seed.heads)
mean.Seeds.in.25.heads <- mean(data$Seeds.in.25.heads)
What if we wanted to get the variance of all these? Copy and paste and change the function name? Very repetitive, hard to see intent, easy to make mistakes, and boring.
Notice the pattern though — we are taking each column in turn and applying the same function to it. Sombody else noticed this pattern too.
The function that we will use is sapply. It takes as its first argument a list, and applies a function to each element in turn.
As a simple example, here is a little list with toy data:
obj <- list(a = 1:5, b = c(1, 5, 6), c = c(pi, exp(1)))
This computes the length of each element in obj
sapply(obj, length)
This computes their sum
sapply(obj, sum)
## 15.00 12.00 5.86
This computes the variance
sapply(obj, var)
This works with any function you want to use as the second argument.
response.variables <- c("No.stems", "Height", "Weight", "Seed.heads", "Seeds.in.25.heads")
Remember when I said that a data.frame is like a list; this is one case where we take advantage of that.
sapply(data[response.variables], mean)
## No.stems Height Weight Seed.heads
## 1.982 55.544 11.198 225.799
## Seeds.in.25.heads
This does all of the repetitive hard work that we did before, but in one line. Read it as:
“Apply to each element in data (subset by my response variables) the function mean”
The coefficient of variation is defined as the standard deviation (square root of the variance) divided by the mean:
\frac{\mathrm{x}}{\bar x}
. Compute the coefficient of variation of these variables, and tell me which variable has the largest CV.
coef.variation <- function(x) {
sqrt(var(x))/mean(x)
Did you just do
sapply(data[response.variables], coef.variation)
## 0.8624 0.2850 0.8841 0.8195
and look for the largest variable?
cvs <- sapply(data[response.variables], coef.variation)
which(cvs == max(cvs))
which.max(cvs)
Similarly, for minimum
which.min(cvs)
There is a function lapply works well when you are not expecting the same length output for the result of applying your function to each element in the list. It gives you the result back in a list, for example
lapply(obj, sum)
This is has its uses. For example, suppose we wanted to double all the elements in obj. We can’t just multiply this list by 2:
obj * 2 # causes error
However, we can do this with lapply, as long as we have a function that will double things:
Apply the double function to each element in our list:
lapply(obj, double)
We have to use lapply rather than sapply because the result of the function on different elements is different lengths.
Note also that sapply actually does the same thing here. But if the elements of obj happened to have the same length they would give different answers.
sapply(obj, double)
obj2 <- list(a = 1:3, b = c(1, 5, 6), c = c(pi, exp(1), log(10)))
lapply(obj2, double)
Returns a matrix:
sapply(obj2, double)
## a b c
## [1,] 2 2 6.283
## [2,] 4 10 5.437
This is probably more than you need to know right now, but it may stick around in your head until needed.
The split–apply–combine pattern
Fairly often, you’ll want to do something like compute means for each level of a treatment. To compute the mean height given the root herbivore treatment here we could do:
height.with <- mean(data$Height[data$Root.herbivore])
height.without <- mean(data$Height[!data$Root.herbivore])
Remember read ! as “not”.
(notice that plant height is taller when herbivores are absent).
However, suppose that we want to get mean height by plot:
height.2 <- mean(data$Height[data$Plot == "plot-2"])
and so on until we go out of our minds.
There is a function tapply that automates this.
It’s arguments are
The first argument, X is our data variable; the thing that we want the means of.
The second argument, INDEX is the grouping variable; the thing that we want means at each distinct value/level of.
The third argument, FUN is the function that you want to apply; in our case mean.
tapply(data$Height, data$Plot, mean)
## plot-2 plot-4 plot-6 plot-8 plot-10 plot-12 plot-14 plot-16 plot-18
## 46.17 57.00 33.33 47.40 42.00 44.60 50.00 70.57 58.50
## plot-20 plot-22 plot-24 plot-26 plot-28 plot-30 plot-32 plot-34 plot-36
## plot-56 plot-58 plot-60
For the first example (present/absent) we have:
tapply(data$Height, data$Root.herbivore, mean)
Notice that there was no more work going from 2 levels to 30 levels!
Also notice that it’s really easy to change variables (as above) or change functions:
tapply(data$Height, data$Root.herbivore, var)
## 218.3 215.5
This approach has been called the “split–apply-combine” pattern. There is a package plyr that a lot of people like that can make this easier.
Two factors at once
The experiment here is a 2x2 factorial design (though imbalanced as all ecological data end up being after they’ve been left in the field for a while).
You can do that with tapply above, but it gets quite hard. The aggregate function can do this nicely though.
There are two interfaces to this (see the help page ?aggregate). In the first, you supply the response variable as the first argument, the grouping variables as the second, and the function as the third (just like tapply).
aggregate(data$Height, data[c("Root.herbivore", "Seed.herbivore")], mean)
## Root.herbivore Seed.herbivore x
## 1 FALSE FALSE 68.27
## 2 TRUE FALSE 52.20
## 3 FALSE TRUE 58.93
## 4 TRUE TRUE 50.14
or similarly, take a 1 column data frame:
aggregate(data["Height"], data[c("Root.herbivore", "Seed.herbivore")], mean)
## Root.herbivore Seed.herbivore Height
## 1 FALSE FALSE 68.27
## 2 TRUE FALSE 52.20
## 3 FALSE TRUE 58.93
## 4 TRUE TRUE 50.14
The other interface uses R’s formula interface, which can be a lot shorter and easier to read.
aggregate(Height ~ Root.herbivore + Seed.herbivore, data = data, mean)
This is why I think it is important to get used to writing functions:
Because your own functions work just as well with these tools:
aggregate(Height ~ Root.herbivore + Seed.herbivore, data = data, standard.error)
## 1 FALSE FALSE 2.408
## 2 TRUE FALSE 1.761
## 3 FALSE TRUE 2.848
## 4 TRUE TRUE 2.224
(now you have everything for an interaction plot to start seeing how the different herbivores interact to effect plant growth).
In contrast with most programming languages, we have not covered a basic loop yet. You may be familiar with these if you’ve written basic, python, perl, etc.
The idea is the same as the first section in this page. You want to apply a function (or do some calculation) over a series of elements in a list.
For example, the print function prints a representation of an object to the screen. Suppose we wanted to print all the elements of our list obj.
sapply(obj, print)
This is because we want to just print the result to the screen and we don’t actually want to do anything with the result (it turns out that print silently returns what is passed into it.
for (el in obj) {
The element i is created at the beginning of the loop, and set to the first element in obj. Each time “around” the loop it is set to the next element (so on the second iteration it will be the second element, and so on).
Note that in contrast with functions, the variables created by loops do actually replace things in the global environment:
Strictly, in the enclosing environment, so a loop within a function leaves the global environment unaffected.
If you want to do something with the results of the loop, you need more scaffolding. Let’s do the mean example again and get the mean of each element in the list.
First, we need somewhere to put the output. The function numeric creates a numeric vector of the requested length.
n <- length(obj)
You could use rep here if you’d rather.
Then, rather than loop over the elements directly as above, we loop over their indices:
out[i] <- mean(obj[[i]])
Read the line within the loop as “into the i’th element of output (out) save the result of running the function on the i’th element if the input (obj)”. The double square brackets are needed (Why? Because out[i] would be a list with one element while out[[i]] is the ith element itself.).
This does not have names like the sapply version did. If we want names, add them back with names.
names(out) <- names(obj)
There are times where loops are absolutely the best (or the only way) of doing something.
For example, if you want a cumulative sum over list elements (so that out[[i]] contains the sum of obj[[1]], obj[[2]], …, obj[[i]]), you might do that with a loop:
total <- 0 # running total
total <- total + sum(obj[[i]])
out[[i]] <- total
Although we can actually do that with the built-in function cumsum and sapplying over the list to get the sums.
cumsum(sapply(obj, sum))
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Discrete Laplacian - MATLAB del2 - MathWorks India
Second Derivative of Vector
Second Derivative of Cosine Vector
Laplacian of Multivariate Function
Laplacian of Natural Logarithm Function
hx,hy,...,hN
Laplace’s differential operator
L = del2(U)
L = del2(U,h)
L = del2(U,hx,hy,...,hN)
L = del2(U) returns a discrete approximation of Laplace’s differential operator applied to U using the default spacing, h = 1, between all points.
L = del2(U,h) specifies a uniform, scalar spacing, h, between points in all dimensions of U.
L = del2(U,hx,hy,...,hN) specifies the spacing hx,hy,...,hN between points in each dimension of U. Specify each spacing input as a scalar or a vector of coordinates. The number of spacing inputs must equal the number of dimensions in U.
The first spacing value hx specifies the x-spacing (as a scalar) or x-coordinates (as a vector) of the points. If it is a vector, its length must be equal to size(U,2).
The second spacing value hy specifies the y-spacing (as a scalar) or y-coordinates (as a vector) of the points. If it is a vector, its length must be equal to size(U,1).
All other spacing values specify the spacing (as scalars) or coordinates (as vectors) of the points in the corresponding dimension in U. If, for n > 2, the nth spacing input is a vector, then its length must be equal to size(U,n).
Calculate the acceleration of an object from a vector of position data.
Create a vector of position data.
p = [1 3 6 10 16 18 29];
To find the acceleration of the object, use del2 to calculate the second numerical derivative of p. Use the default spacing h = 1 between data points.
L = 4*del2(p)
1 1 1 2 -4 9 22
Each value of L is an approximation of the instantaneous acceleration at that point.
Calculate the discrete 1-D Laplacian of a cosine vector.
Define the domain of the function.
This produces 100 evenly spaced points in the range
-2\pi \le x\le 2\pi
Create a vector of cosine values in this domain.
U = cos(x);
Calculate the Laplacian of U using del2. Use the domain vector x to define the 1-D coordinate of each point in U.
L = 4*del2(U,x);
Analytically, the Laplacian of this function is equal to
\Delta U=-\mathrm{cos}\left(x\right)
plot(x,U,x,L)
legend('U(x)','L(x)','Location','Best')
The graph of U and L agrees with the analytic result for the Laplacian.
Calculate and plot the discrete Laplacian of a multivariate function.
Define the x and y domain of the function.
[x,y] = meshgrid(-5:0.25:5,-5:0.25:5);
U\left(x,y\right)=\frac{1}{3}\left({x}^{4}+{y}^{4}\right)
over this domain.
U = 1/3.*(x.^4+y.^4);
Calculate the Laplacian of this function using del2. The spacing between the points in U is equal in all directions, so you can specify a single spacing input, h.
L = 4*del2(U,h);
\Delta U\left(x,y\right)=4{x}^{2}+4{y}^{2}
Plot the discrete Laplacian, L.
surf(x,y,L)
title('Plot of $\Delta U(x,y) = 4x^2+4y^2$','Interpreter','latex')
The graph of L agrees with the analytic result for the Laplacian.
Calculate the discrete Laplacian of a natural logarithm function.
Define the x and y domain of the function on a grid of real numbers.
[x,y] = meshgrid(-5:5,-5:0.5:5);
U\left(x,y\right)=\frac{1}{2}\mathrm{log}\left({x}^{2}y\right)
U = 0.5*log(x.^2.*y);
The logarithm is complex-valued when the argument y is negative.
Use del2 to calculate the discrete Laplacian of this function. Specify the spacing between grid points in each direction.
L = 4*del2(U,hx,hy);
Analytically, the Laplacian is equal to
\Delta U\left(x,y\right)=-\left(1/{x}^{2}+1/2{y}^{2}\right)
. This function is not defined on the lines
x=0
y=0
Plot the real parts of U and L on the same graph.
surf(x,y,real(L))
title('Plot of U(x,y) and $\Delta$ U(x,y)','Interpreter','latex')
The top surface is U and the bottom surface is L.
h — Spacing in all dimensions
Spacing in all dimensions, specified as 1 (default), or a scalar.
hx,hy,...,hN — Spacing in each dimension (as separate arguments)
Spacing in each dimension, specified as separate arguments of scalars (for uniform spacing) or vectors (for nonuniform spacing). The number of spacing inputs must be equal to the number of dimensions in U. Each spacing input defines the spacing between points in one dimension of U:
L — Discrete Laplacian approximation
Discrete Laplacian approximation, returned as a vector, matrix, or multidimensional array. L is the same size as the input, U.
The definition of the Laplace operator used by del2 in MATLAB® depends on the dimensionality of the data in U.
If U is a vector representing a function U(x) that is evaluated on the points of a line, then del2(U) is a finite difference approximation of
L=\frac{\Delta U}{4}=\frac{1}{4}\frac{{\partial }^{2}U}{\partial {x}^{2}}.
If U is a matrix representing a function U(x,y) that is evaluated at the points of a square grid, then del2(U) is a finite difference approximation of
L=\frac{\Delta U}{4}=\frac{1}{4}\left(\frac{{\partial }^{2}U}{\partial {x}^{2}}+\frac{{\partial }^{2}U}{\partial {y}^{2}}\right).
For functions of three or more variables, U(x,y,z,...), the discrete Laplacian del2(U) calculates second-derivatives in each dimension,
L=\frac{\Delta U}{2N}=\frac{1}{2N}\left(\frac{{\partial }^{2}U}{\partial {x}^{2}}+\frac{{\partial }^{2}U}{\partial {y}^{2}}+\frac{{\partial }^{2}U}{\partial {z}^{2}}+...\right),
where N is the number of dimensions in U and
N\ge 2
If the input U is a matrix, the interior points of L are found by taking the difference between a point in U and the average of its four neighbors:
{L}_{ij}=\left[\frac{\left({u}_{i+1,j}+{u}_{i-1,j}+{u}_{i,j+1}+{u}_{i,j-1}\right)}{4}-{u}_{i,j}\right]\text{\hspace{0.17em}}.
Then, del2 calculates the values on the edges of L by linearly extrapolating the second differences from the interior. This formula is extended for multidimensional U.
diff | gradient
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Air changes per hour - Wikipedia
Measure of the air volume added to or removed from a space
Air changes per hour, abbreviated ACPH or ACH, or air change rate is the number of times that the total air volume in a room or space is completely removed and replaced in an hour. If the air in the space is either uniform or perfectly mixed, air changes per hour is a measure of how many times the air within a defined space is replaced each hour.
In many air distribution arrangements, air is neither uniform nor perfectly mixed. The actual percentage of an enclosure's air which is exchanged in a period depends on the airflow efficiency of the enclosure and the methods used to ventilate it. The actual amount of air changed in a well mixed ventilation scenario will be 63.2% after 1 hour and 1 ACH.[1] In order to achieve equilibrium pressure, the amount of air leaving the space and entering the space must be the same.
1.2 Ventilation rates
2 Air change rate recommendations
3 Measure of airtightness
4 Effects of ACH due to forced ventilation in a dwelling
Air changes per hour[edit]
In Imperial units:
{\displaystyle \quad ACPH={\frac {60Q}{Vol}}}
ACPH = number of air changes per hour; higher values correspond to better ventilation
In metric units
{\displaystyle \quad ACPH={\frac {3.6Q}{Vol}}}
Q = Volumetric flow rate of air in liters per second (L/s)
Vol = Space volume L × W × H, in cubic meter
Ventilation rates[edit]
Ventilation rates are often expressed as a volume rate per person (CFM per person, L/s per person). The conversion between air changes per hour and ventilation rate per person is as follows:
{\displaystyle \quad Rp={\frac {ACPH*D*h}{60}}}
Rp = ventilation rate per person (cubic feet per minute (CFM) per person or cubic meters per minute per person)
D = Occupant density (square feet per occupant or square meters per occupant)
h = Ceiling height (feet or meters)
One cubic meter per minute = 16.67 liter/second
Air change rate recommendations[edit]
Find sources: "Air changes per hour" – news · newspapers · books · scholar · JSTOR (October 2013) (Learn how and when to remove this template message)
Air change rates are often used as rules of thumb in ventilation design. However, they are seldom used as the actual basis of design or calculation. For example, laboratory ventilation standards indicate recommended ranges for air change rates,[2] as a guideline for the actual design. Residential ventilation rates are calculated based on area of the residence and number of occupants.[3] Non-residential ventilation rates are based on floor area and number of occupants, or a calculated dilution of known contaminants.[4] Hospital design standards use air changes per hour,[5] although this has been criticized.[6]
The American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) recommends "0.35 air changes per hour but not less than 15 cfm (7.5 L/s) per person" in living areas.[7]
Basement Parking 15–30
Residential Basement 3–4
Bedroom 2-4[8]
Residential Bathroom 6-7
Residential Living Rooms 6-8
Residential Kitchen 7-8
Residential Laundry 8-9
Business Lunch Break Rooms 7-8
Business Conference Rooms 8-12
Business Copy Rooms 10-12
Restaurant Food Staging Area 10-12
Restaurant Bar 15-20
Public Hallway 6-8
Public Retail Store 6-10
Public Foyer 8-10
Church 8-12
Public Auditorium 12-14
Commercial kitchens & Restrooms 15–30
Laboratories 6–12[2]
Warehousing 3-10
Measure of airtightness[edit]
Many if not most uses of ACH are actually referring to results of a standard blower door test in which 50 pascals of pressure are applied (ACH50), rather than the volume of air changed under normal conditions. The Passive House standard requires airtightness so that there will be less than 0.6 ACH with a pressure difference between inside and outside of 50 Pa.[9]
Effects of ACH due to forced ventilation in a dwelling[edit]
Forced ventilation to increase ACH becomes a necessity to maintain acceptable air quality as occupants become reluctant to open windows due to behavioural changes such as keeping windows closed for security.[10]
Air changes are often cited as a means of preventing condensation in houses with forced ventilation systems often rated from 3 - 5 ACH though without referencing the size of the house. However, where ACH is already greater than 0.75 a forced ventilation system is unlikely to be of use at controlling condensation and instead insulation or heating are better remedies.[10] Seven out of eight houses studied in NZ in 2010 had an ACH (corrected for ventilation factors) of 0.75 or greater.[10] The presence of forced ventilation systems has been shown in some cases to actually increase the humidity rather than lower it.[10] By displacing air inside a dwelling with infiltrated air (air brought in from outside the dwelling), positive pressure ventilation systems can increase heating (in winter) or cooling (in summer) requirements in a house.[10][11] For example, to maintain a 15 °C temperature in a certain dwelling about 3.0 kW of heating are required at 0 ACH (no heat loss due to warmed air leaving the dwelling, instead heat is lost due to conduction or radiation), 3.8 kW at 1 ACH and 4.5 kW are required at 2 ACH.[10] The use of roof space for heating or cooling was seen as ineffectual with the maximum heating benefits occurring in winter in more southerly regions (being close to the South Pole in these southern hemisphere reports) but being equivalent only to about 0.5 kW or the heating provided by about five 100 W incandescent light bulbs; cooling effects in summer were similarly small and were more pronounced for more northerly homes (being closer to the equator); in all cases the values assumed that the ventilation system automatically disengaged when the infiltrating air was warmer or cooler (as appropriate) than the air already in the dwelling as it would otherwise exacerbate the undesirable conditions in the house.[11]
^ Bearg, David W. (1993). Indoor Air Quality and HVAC Systems. CRC Press. p. 64. ISBN 0-87371-574-8.
^ a b "Lab Ventilation ACH Rates Standards and Guidelines" (PDF). Retrieved 9 June 2014.
^ "ANSI/ASHRAE Standard 62.2-2013: Ventilation and Acceptable Indoor Air Quality in Low-Rise Residential Buildings". Atlanta, GA: American Society of Heating, Refrigerating and Air-Conditioning Engineers. 2013. {{cite journal}}: Cite journal requires |journal= (help)
^ "ANSI/ASHRAE Standard 62.1-2013: Ventilation for Acceptable Indoor Air Quality". Atlanta, GA: American Society of Heating, Refrigerating and Air-Conditioning Engineers. 2013. {{cite journal}}: Cite journal requires |journal= (help)
^ "ANSI/ASHE/ASHRAE Standard 170: Ventilation for Healthcare Facilities". Atlanta, GA: American Society of Heating, Refrigerating and Air-Conditioning Engineers. 2013. {{cite journal}}: Cite journal requires |journal= (help)
^ "Engineers' Perspectives on Hospital Ventilation". Retrieved 9 June 2014.
^ (PDF) https://www.ashrae.org/File%20Library/Technical%20Resources/Standards%20and%20Guidelines/Standards%20Addenda/62-2001/62-2001_Addendum-n.pdf. {{cite web}}: Missing or empty |title= (help)
^ "Air Changes Per Hour Calculator, Formula, Recommendations". Air Purifier Reviews. Retrieved 2021-11-11. {{cite web}}: CS1 maint: url-status (link)
^ "International Passive House Association - Guidelines". Retrieved 23 March 2013.
^ a b c d e f Pollard, AR and McNeil, S, Forced Air Ventilation Systems, June 2010, Report IEQ7570/3 for Beacon Pathway Limited
^ a b Warren Fitzgerald, Dr Inga Smith and Muthasim Fahmy, Heating and cooling potential of roof space air: implications for ventilation systems, May 2011, Prepared for the Energy Efficiency and Conservation Authority (EECA)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Air_changes_per_hour&oldid=1085293290"
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(Redirected from Lambda expressions)
By the way, why did Church choose the notation “λ”? In [an unpublished 1964 letter to Harald Dickson] he stated clearly that it came from the notation “
{\displaystyle {\hat {x}}}
” used for class-abstraction by Whitehead and Russell, by first modifying “
{\displaystyle {\hat {x}}}
{\displaystyle \land x}
” to distinguish function-abstraction from class-abstraction, and then changing “
{\displaystyle \land }
” to “λ” for ease of printing.
{\displaystyle \operatorname {square\_sum} (x,y)=x^{2}+y^{2}}
{\displaystyle (x,y)\mapsto x^{2}+y^{2}}
(which is read as "a tuple of x and y is mapped to
{\textstyle x^{2}+y^{2}}
"). Similarly, the function
{\displaystyle \operatorname {id} (x)=x}
{\displaystyle x\mapsto x}
The second simplification is that the lambda calculus only uses functions of a single input. An ordinary function that requires two inputs, for instance the
{\textstyle \operatorname {square\_sum} }
function, can be reworked into an equivalent function that accepts a single input, and as output returns another function, that in turn accepts a single input. For example,
{\displaystyle (x,y)\mapsto x^{2}+y^{2}}
{\displaystyle x\mapsto (y\mapsto x^{2}+y^{2})}
Function application of the
{\textstyle \operatorname {square\_sum} }
function to the arguments (5, 2), yields at once
{\textstyle ((x,y)\mapsto x^{2}+y^{2})(5,2)}
{\textstyle =5^{2}+2^{2}}
{\textstyle =29}
{\textstyle {\Bigl (}{\bigl (}x\mapsto (y\mapsto x^{2}+y^{2}){\bigr )}(5){\Bigr )}(2)}
{\textstyle =(y\mapsto 5^{2}+y^{2})(2)}
// the definition of
{\displaystyle x}
has been used with
{\displaystyle 5}
in the inner expression. This is like β-reduction.
{\textstyle =5^{2}+2^{2}}
{\displaystyle y}
{\displaystyle 2}
. Again, similar to β-reduction.
{\textstyle =29}
if t is a lambda term, and x is a variable, then
{\displaystyle (\lambda x.t)}
(sometimes written in ASCII as
{\displaystyle Lx.t}
) is a lambda term (called an abstraction);
if t and s are lambda terms, then
{\displaystyle (ts)}
is a lambda term (called an application).
An abstraction
{\displaystyle \lambda x.t}
is a definition of an anonymous function that is capable of taking a single input x and substituting it into the expression t. It thus defines an anonymous function that takes x and returns t. For example,
{\displaystyle \lambda x.x^{2}+2}
is an abstraction for the function
{\displaystyle f(x)=x^{2}+2}
using the term
{\displaystyle x^{2}+2}
for t. The definition of a function with an abstraction merely "sets up" the function but does not invoke it. The abstraction binds the variable x in the term t.
An application ts represents the application of a function t to an input s, that is, it represents the act of calling function t on input s to produce
{\displaystyle t(s)}
There is no concept in lambda calculus of variable declaration. In a definition such as
{\displaystyle \lambda x.x+y}
{\displaystyle f(x)=x+y}
), the lambda calculus treats y as a variable that is not yet defined. The abstraction
{\displaystyle \lambda x.x+y}
is syntactically valid, and represents a function that adds its input to the yet-unknown y.
Bracketing may be used and may be needed to disambiguate terms. For example,
{\displaystyle \lambda x.((\lambda x.x)x)}
{\displaystyle (\lambda x.(\lambda x.x))x}
denote different terms (although they coincidentally reduce to the same value). Here, the first example defines a function whose lambda term is the result of applying x to the child function, while the second example is the application of the outermost function to the input x, which returns the child function. Therefore, both examples evaluate to the identity function
{\displaystyle \lambda x.x}
{\displaystyle \lambda x.x}
represents the identity function,
{\displaystyle x\mapsto x}
{\displaystyle (\lambda x.x)y}
represents the identity function applied to
{\displaystyle y}
{\displaystyle (\lambda x.y)}
represents the constant function
{\displaystyle x\mapsto y}
, the function that always returns
{\displaystyle y}
, no matter the input. In lambda calculus, function application is regarded as left-associative, so that
{\displaystyle stx}
{\displaystyle (st)x}
A basic form of equivalence, definable on lambda terms, is alpha equivalence. It captures the intuition that the particular choice of a bound variable, in an abstraction, does not (usually) matter. For instance,
{\displaystyle \lambda x.x}
{\displaystyle \lambda y.y}
are alpha-equivalent lambda terms, and they both represent the same function (the identity function). The terms
{\displaystyle x}
{\displaystyle y}
are not alpha-equivalent, because they are not bound in an abstraction. In many presentations, it is usual to identify alpha-equivalent lambda terms.
{\displaystyle x}
are just
{\displaystyle x}
The set of free variables of
{\displaystyle \lambda x.t}
is the set of free variables of
{\displaystyle t}
{\displaystyle x}
{\displaystyle ts}
is the union of the set of free variables of
{\displaystyle t}
and the set of free variables of
{\displaystyle s}
For example, the lambda term representing the identity
{\displaystyle \lambda x.x}
has no free variables, but the function
{\displaystyle \lambda x.yx}
has a single free variable,
{\displaystyle y}
{\displaystyle t}
{\displaystyle s}
{\displaystyle r}
are lambda terms and
{\displaystyle x}
{\displaystyle y}
are variables. The notation
{\displaystyle t[x:=r]}
indicates substitution of
{\displaystyle r}
{\displaystyle x}
{\displaystyle t}
in a capture-avoiding manner. This is defined so that:
{\displaystyle x[x:=r]=r}
{\displaystyle x}
substituted fo{\displaystyle r}
{\displaystyle r}
{\displaystyle y[x:=r]=y}
{\displaystyle x\neq y}
{\displaystyle x}
substituted fo{\displaystyle r}
{\displaystyle y}
{\displaystyle y}
{\displaystyle (ts)[x:=r]=(t[x:=r])(s[x:=r])}
; substitution distributes to further application of the variable
{\displaystyle (\lambda x.t)[x:=r]=\lambda x.t}
; although
{\displaystyle x}
has been mapped to
{\displaystyle r}
, subsequently mapping all
{\displaystyle x}
{\displaystyle t}
will not change the lambda function
{\displaystyle (\lambda x.t)}
{\displaystyle (\lambda y.t)[x:=r]=\lambda y.(t[x:=r])}
{\displaystyle x\neq y}
{\displaystyle y}
is not in the free variables of
{\displaystyle r}
{\displaystyle y}
is said to be "fresh" fo{\displaystyle r}
{\displaystyle (\lambda x.x)[y:=y]=\lambda x.(x[y:=y])=\lambda x.x}
{\displaystyle ((\lambda x.y)x)[x:=y]=((\lambda x.y)[x:=y])(x[x:=y])=(\lambda x.y)y}
The freshness condition (requiring that
{\displaystyle y}
{\displaystyle r}
) is crucial in order to ensure that substitution does not change the meaning of functions. For example, a substitution is made that ignores the freshness condition can lead to errors:
{\displaystyle (\lambda x.y)[y:=x]=\lambda x.(y[y:=x])=\lambda x.x}
. This substitution turns the constant function
{\displaystyle \lambda x.y}
{\displaystyle \lambda x.x}
by substitution.
In general, failure to meet the freshness condition can be remedied by alpha-renaming with a suitable fresh variable. For example, switching back to our correct notion of substitution, in
{\displaystyle (\lambda x.y)[y:=x]}
the abstraction can be renamed with a fresh variable
{\displaystyle z}
{\displaystyle (\lambda z.y)[y:=x]=\lambda z.(y[y:=x])=\lambda z.x}
, and the meaning of the function is preserved by substitution.
The β-reduction rule states that an application of the form
{\displaystyle (\lambda x.t)s}
reduces to the term
{\displaystyle t[x:=s]}
{\displaystyle (\lambda x.t)s\to t[x:=s]}
is used to indicate that
{\displaystyle (\lambda x.t)s}
β-reduces to
{\displaystyle t[x:=s]}
. For example, for every
{\displaystyle s}
{\displaystyle (\lambda x.x)s\to x[x:=s]=s}
. This demonstrates that
{\displaystyle \lambda x.x}
really is the identity. Similarly,
{\displaystyle (\lambda x.y)s\to y[x:=s]=y}
, which demonstrates that
{\displaystyle \lambda x.y}
The lambda calculus may be seen as an idealized version of a functional programming language, like Haskell or Standard ML. Under this view, β-reduction corresponds to a computational step. This step can be repeated by additional β-reductions until there are no more applications left to reduce. In the untyped lambda calculus, as presented here, this reduction process may not terminate. For instance, consider the term
{\displaystyle \Omega =(\lambda x.xx)(\lambda x.xx)}
{\displaystyle (\lambda x.xx)(\lambda x.xx)\to (xx)[x:=\lambda x.xx]=(x[x:=\lambda x.xx])(x[x:=\lambda x.xx])=(\lambda x.xx)(\lambda x.xx)}
. That is, the term reduces to itself in a single β-reduction, and therefore the reduction process will never terminate.
A typed lambda calculus is a typed formalism that uses the lambda-symbol (
{\displaystyle \lambda }
) to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see Kinds of typed lambda calculi). From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus but from another point of view, they can also be considered the more fundamental theory and untyped lambda calculus a special case with only one type.[27]
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(Redirected from Concentrating solar power)
{\displaystyle \eta }
{\displaystyle \eta _{Receiver}}
{\displaystyle \eta _{mechanical}}
{\displaystyle \eta =\eta _{\mathrm {optics} }\cdot \eta _{\mathrm {receiver} }\cdot \eta _{\mathrm {mechanical} }\cdot \eta _{\mathrm {generator} }}
{\displaystyle \eta _{\mathrm {optics} }}
{\displaystyle \eta _{\mathrm {receiver} }}
{\displaystyle \eta _{\mathrm {mechanical} }}
{\displaystyle \eta _{\mathrm {generator} }}
{\displaystyle \eta _{\mathrm {receiver} }}
{\displaystyle \eta _{\mathrm {receiver} }={\frac {Q_{\mathrm {absorbed} }-Q_{\mathrm {lost} }}{Q_{\mathrm {incident} }}}}
{\displaystyle Q_{\mathrm {incident} }}
{\displaystyle Q_{\mathrm {absorbed} }}
{\displaystyle Q_{\mathrm {lost} }}
{\displaystyle \eta _{\mathrm {mechanical} }}
{\displaystyle T_{H}}
{\displaystyle T^{0}}
{\displaystyle \eta _{\mathrm {Carnot} }=1-{\frac {T^{0}}{T_{H}}}}
{\displaystyle I}
{\displaystyle I=1000\,\mathrm {W/m^{2}} }
{\displaystyle C}
{\displaystyle \eta _{Optics}}
{\displaystyle A}
{\displaystyle \alpha }
{\displaystyle Q_{\mathrm {solar} }=ICA}
{\displaystyle Q_{\mathrm {absorbed} }=\eta _{\mathrm {optics} }\alpha Q_{\mathrm {solar} }}
{\displaystyle \epsilon }
{\displaystyle Q_{\mathrm {lost} }=A\epsilon \sigma T_{H}^{4}}
{\displaystyle \eta _{\mathrm {Optics} }}
{\displaystyle \alpha }
{\displaystyle \epsilon }
{\displaystyle \eta =\left(1-{\frac {\sigma T_{H}^{4}}{IC}}\right)\cdot \left(1-{\frac {T^{0}}{T_{H}}}\right)}
{\displaystyle T_{\mathrm {max} }=\left({\frac {IC}{\sigma }}\right)^{0.25}}
{\displaystyle {\frac {d\eta }{dT_{H}}}(T_{\mathrm {opt} })=0}
{\displaystyle T_{opt}^{5}-(0.75T^{0})T_{\mathrm {opt} }^{4}-{\frac {T^{0}IC}{4\sigma }}=0}
{\displaystyle C}
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Key Equations | EME 810: Solar Resource Assessment and Economics
Mathematical Relationships for Light
This is another reference book page, which may become useful throughout the course, especialy when we talk about properties of light and light energy conversion. This content is for your fiuture reference (you will not be tested on it during orientation).
Relation between wave length, frequency, and speed of light
An electromagnetic wave can be characterized by its wavelength (
\lambda
) and frequency (
\nu
). Mathematically, a simple relationship between these properties shows that
\lambda =\frac{c}{\nu }
where c= speed of light in a vacuum (
\approx 3×{10}^{17}\frac{nm}{s}
Dependence of radiation energy on frequency
All electromagnetic radiation is quantized and occurs in photons. The energy (E) depends on the frequency(f) of the electromagnetic radiation. This relationship is elegantly described by Planck's equation
{E}_{p}=h\cdot \nu \frac{hc}{\lambda }
where h=Planck's constant (
\approx 4.1357×{10}^{-15}eV\cdot s
But we want a simple relation to convert between wavelength and energy (as eV). If we multiply Planck's constant times the speed of light (all in units of nm and eV), we get a simple equation like this:
\[ {E}_{p}=\frac{1239.8\text{ }eV\cdot nm}{\lambda } \]
\[ \lambda =\frac{1239.8\quad eV\cdot nm}{Ep} \]
Even easier, just count to 5! For our approximation purposes (plenty good enough for the real spectrum), we can use these simpler equations as long as we remember our units and the decimal point:
\[ {E}_{p}=\frac{1234.5\text{ }eV\cdot nm}{\lambda } \]
\[ \lambda =\frac{1234.5\text{ }eV\cdot nm}{{E}_{p}} \]
Self Check using 1234.5/(argument)
Click on the question to reveal the correct answer.
1. The band gap of Silicon is 1.1 eV. What is that value in nanometers?
1234.5eV\cdot nm/1.1eV
is about 1100 nm. Putting 1100 back into the denominator yields 1.1 eV
2. The band gap of CdTe is about 1.5 eV. What is that in nanometers?
1234.5eV\cdot nm/1.5eV
is about 800 nm. Putting 800 back into the denominator still yields 1.5 eV
3. The visible spectrum is from 380-780 nm. What would that be in terms of eV?
380\text{}nm:\text{}3.2\text{}eV\text{}--\text{}780\text{}nm:\text{}1.6\text{}eV
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A new enrichment space for the treatment of discontinuous pressures in multi‐fluid flows - Samper 2011a - Scipedia
A new enrichment space for the treatment of discontinuous pressures in multi‐fluid flows
R. Ausas, G. Buscaglia, S. Idelsohn
In this work, a new enrichment space to accommodate jumps in the pressure field at immersed interfaces in finite element formulations, is proposed. The new enrichment adds two degrees of freedom per element that can be eliminated by means of static condensation. The new space is tested and compared with the classical P1 space and to the space proposed by Ausas et al (Comp. Meth. Appl. Mech. Eng., Vol. 199, 1019–1031, 2010) in several problems involving jumps in the viscosity and/or the presence of singular forces at interfaces not conforming with the element edges. The combination of this enrichment space with another enrichment that accommodates discontinuities in the pressure gradient has also been explored, exhibiting excellent results in problems involving jumps in the density or the volume forces.
discontinuous pressures
R. Ausas, G. Buscaglia and S. Idelsohn, A new enrichment space for the treatment of discontinuous pressures in multi‐fluid flows, Int. J. Numer. Meth. Fluids (2011). Vol. 70 (7), pp. 829-850 URL https://www.scipedia.com/public/Samper_2011a
In this work, a new enrichment space to accommodate jumps in the pressure field at immersed interfaces in finite element formulations, is proposed. The new enrichment adds two degrees of freedom per element that can be eliminated by means of static condensation. The new space is tested and compared with the classical
{\displaystyle P_{1}}
space and to the space proposed by Ausas et al (Comp. Meth. Appl. Mech. Eng., Vol. 199, 1019–1031, 2010) in several problems involving jumps in the viscosity and/or the presence of singular forces at interfaces not conforming with the element edges. The combination of this enrichment space with another enrichment that accommodates discontinuities in the pressure gradient has also been explored, exhibiting excellent results in problems involving jumps in the density or the volume forces.
H. Ji, Q. Zhang. A simple finite element method for Stokes flows with surface tension using unfitted meshes. Int. J. Numer. Meth. Fluids 81(2) (2015) DOI 10.1002/fld.4176
S. Idelsohn, J. Gimenez, N. Nigro. Multifluid flows with weak and strong discontinuous interfaces using an elemental enriched space. Int J Numer Meth Fluids 86(12) (2017) DOI 10.1002/fld.4477
L. Cattaneo, L. Formaggia, G. Iori, A. Scotti, P. Zunino. Stabilized extended finite elements for the approximation of saddle point problems with unfitted interfaces. Calcolo 52(2) (2014) DOI 10.1007/s10092-014-0109-9
B. Schott, U. Rasthofer, V. Gravemeier, W. Wall. A face-oriented stabilized Nitsche-type extended variational multiscale method for incompressible two-phase flow. Int. J. Numer. Meth. Engng 104(7) (2014) DOI 10.1002/nme.4789
C. Schöler, A. Haeusler, V. Karyofylli, M. Behr, W. Schulz, A. Gillner, M. Niessen. Hybrid simulation of laser deep penetration welding. Mat.-wiss. u. Werkstofftech. 48(12) (2018) DOI 10.1002/mawe.201700164
J. Barrett, H. Garcke, R. Nürnberg. A Stable Parametric Finite Element Discretization of Two-Phase Navier–Stokes Flow. J Sci Comput 63(1) (2014) DOI 10.1007/s10915-014-9885-2
M. Thielmann, D. May, B. Kaus. Discretization Errors in the Hybrid Finite Element Particle-in-cell Method. Pure Appl. Geophys. 171(9) (2014) DOI 10.1007/s00024-014-0808-9
K. Andriamananjara, L. Chevalier, N. Moulin, J. Bruchon, P. Liotier, S. Drapier. Numerical approach for modelling across scales infusion-based processing of aircraft primary structures. DOI 10.1063/1.5007998
M. Cruchaga, L. Battaglia, M. Storti, J. D’Elía. Numerical Modeling and Experimental Validation of Free Surface Flow Problems. Arch Computat Methods Eng 23(1) (2014) DOI 10.1007/s11831-014-9138-4
P. Becker, S. Idelsohn, E. Oñate. A unified monolithic approach for multi-fluid flows and fluid–structure interaction using the Particle Finite Element Method with fixed mesh. Comput Mech 55(6) (2014) DOI 10.1007/s00466-014-1107-0
Multi-fluids • Two-phase flows • Embedded interfaces • Finite element method • Surface tension • Discontinuous pressures • Kinks
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Required SNR using Albersheim’s equation - MATLAB albersheim - MathWorks France
Compute Required SNR for Probability of Detection
Compute Required SNR for Probability of Detection of 10 Pulses
Albersheim's Equation
Required SNR using Albersheim’s equation
SNR = albersheim(prob_Detection,prob_FalseAlarm)
SNR = albersheim(prob_Detection,prob_FalseAlarm,N)
SNR = albersheim(prob_Detection,prob_FalseAlarm) returns the signal-to-noise ratio in decibels. This value indicates the ratio required to achieve the given probabilities of detection prob_Detection and false alarm prob_FalseAlarm for a single sample.
SNR = albersheim(prob_Detection,prob_FalseAlarm,N) determines the required SNR for the noncoherent integration of N samples.
Compute the required SNR of a single pulse to achieve a detection probability of 0.9 as a function of the false alarm probability.
Set the probability of detection to 0.9 and the probabilities of false alarm from .0001 to .01.
Pd=0.9;
Pfa=0.0001:0.0001:.01;
Loop the Albersheim equation over all Pfa's.
snr = zeros(1,length(Pfa));
snr(j) = albersheim(Pd,Pfa(j));
Plot SNR versus Pfa.
semilogx(Pfa,snr,'k','linewidth',1)
xlabel('Probability of False Alarm')
ylabel('Required SNR (dB)')
title('Required SNR for P_D = 0.9 (N = 1)')
Compute the required SNR of 10 non-coherently integrated pulse to achieve a detection probability of 0.9 as a function of the false alarm probability.
Loop over the Albersheim equation over all Pfa's.
snr(j) = albersheim(Pd,Pfa(j),Npulses);
title('Required SNR for P_D = 0.9 (N = 10)')
Albersheim's equation uses a closed-form approximation to calculate the SNR. This SNR value is required to achieve the specified detection and false-alarm probabilities for a nonfluctuating target in independent and identically distributed Gaussian noise. The approximation is valid for a linear detector and is extensible to the noncoherent integration of N samples.
A=\mathrm{ln}\frac{0.62}{{P}_{FA}}
B=\mathrm{ln}\frac{{P}_{D}}{1â{P}_{D}}
{P}_{FA}
{P}_{D}
are the false-alarm and detection probabilities.
Albersheim's equation for the required SNR in decibels is:
\text{SNR}=â5{\mathrm{log}}_{10}N+\left[6.2+4.54/\sqrt{N+0.44}\right]{\mathrm{log}}_{10}\left(A+0.12AB+1.7B\right)
where N is the number of noncoherently integrated samples.
[2] Skolnik, M. Introduction to Radar Systems, 3rd Ed. New York: McGraw-Hill, 2001, p. 49.
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Field of force due to point mass - MATLAB - MathWorks Benelux
Field of force due to point mass
This block represents the gravitational field of a point mass. This field applies a gravitational force at the center of mass of each rigid body. The force magnitude decays with the square distance from the field origin, coincident with the base port frame origin. The force on a rigid body follows from Newton’s universal gravitation law:
{F}_{g}=-\text{\hspace{0.17em}}G\frac{Mm}{{R}_{BF}{}^{2}},
Fg is the force that the gravitational field exerts on a given rigid body.
G is the universal gravitational constant, 6.67384 × 10-11 m3kg-1s-2.
M is the total mass generating the gravitational field.
m is the total mass of the rigid body the gravitational force acts upon.
RBF is the distance between the source mass position and the rigid body center of mass.
The figure shows these variables. The plot shows the inverse square dependence between the gravitational force and distance.
The source mass can be positive or negative. Combine multiple instances of this block to model the gravitational effects that positive and negative mass disturbances impose on a stronger gravitational field, such as a reduction in the gravitational pull of a planet due to a concentration of low-density material along a portion of its surface.
This block excludes the gravitational forces that other rigid bodies exert on the field source mass. To include these forces, you can connect Gravitational Field blocks to other rigid bodies in the model. Alternatively, you can use the Inverse Square Law Force block to model the gravitational forces between a single pair of rigid bodies.
The gravitational field is time invariant. To specify a time-varying, spatially uniform field, use the Mechanism Configuration block.
Total mass generating the gravitational field. The resulting gravitational forces are directly proportional to this mass. This mass adds no inertia to the model. The default value for the mass parameter is 1.0 kg.
Frame port B represents a frame with origin at the point mass responsible for the gravitational field.
Inverse Square Law Force | Mechanism Configuration
Model Gravity in a Planetary System
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Arithmetic_geometry Knowpia
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory.[1] Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.[2][3]
The hyperelliptic curve defined by
{\displaystyle y^{2}=x(x+1)(x-3)(x+2)(x-2)}
has only finitely many rational points (such as the points
{\displaystyle (-2,0)}
{\displaystyle (-1,0)}
) by Faltings's theorem.
In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers.[4]
The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity.[5]
The structure of algebraic varieties defined over non-algebraically-closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fields, étale cohomology provides topological invariants associated to algebraic varieties.[6] p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields.[7]
19th century: early arithmetic geometryEdit
In the early 19th century, Carl Friedrich Gauss observed that non-zero integer solutions to homogeneous polynomial equations with rational coefficients exist if non-zero rational solutions exist.[8]
In the 1850s, Leopold Kronecker formulated the Kronecker–Weber theorem, introduced the theory of divisors, and made numerous other connections between number theory and algebra. He then conjectured his "liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his twelfth problem, which outlines a goal to have number theory operate only with rings that are quotients of polynomial rings over the integers.[9]
Early-to-mid 20th century: algebraic developments and the Weil conjecturesEdit
In the late 1920s, André Weil demonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to the Mordell–Weil theorem which demonstrates that the set of rational points of an abelian variety is a finitely generated abelian group.[10]
Modern foundations of algebraic geometry were developed based on contemporary commutative algebra, including valuation theory and the theory of ideals by Oscar Zariski and others in the 1930s and 1940s.[11]
In 1949, André Weil posed the landmark Weil conjectures about the local zeta-functions of algebraic varieties over finite fields.[12] These conjectures offered a framework between algebraic geometry and number theory that propelled Alexander Grothendieck to recast the foundations making use of sheaf theory (together with Jean-Pierre Serre), and later scheme theory, in the 1950s and 1960s.[13] Bernard Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960.[14] Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together with Michael Artin and Jean-Louis Verdier) by 1965.[6][15] The last of the Weil conjectures (an analogue of the Riemann hypothesis) would be finally proven in 1974 by Pierre Deligne.[16]
Mid-to-late 20th century: developments in modularity, p-adic methods, and beyondEdit
Between 1956 and 1957, Yutaka Taniyama and Goro Shimura posed the Taniyama–Shimura conjecture (now known as the modularity theorem) relating elliptic curves to modular forms.[17][18] This connection would ultimately lead to the first proof of Fermat's Last Theorem in number theory through algebraic geometry techniques of modularity lifting developed by Andrew Wiles in 1995.[19]
In the 1960s, Goro Shimura introduced Shimura varieties as generalizations of modular curves.[20] Since the 1979, Shimura varieties have played a crucial role in the Langlands program as a natural realm of examples for testing conjectures.[21]
In papers in 1977 and 1978, Barry Mazur proved the torsion conjecture giving a complete list of the possible torsion subgroups of elliptic curves over the rational numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves.[22][23] In 1996, the proof of the torsion conjecture was extended to all number fields by Loïc Merel.[24]
In 1983, Gerd Faltings proved the Mordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstrates finite generation of the set of rational points as opposed to finiteness).[25][26]
In 2001, the proof of the local Langlands conjectures for GLn was based on the geometry of certain Shimura varieties.[27]
In the 2010s, Peter Scholze developed perfectoid spaces and new cohomology theories in arithmetic geometry over p-adic fields with application to Galois representations and certain cases of the weight-monodromy conjecture.[28][29]
Arithmetic of abelian varieties
Siegel modular variety
^ Sutherland, Andrew V. (September 5, 2013). "Introduction to Arithmetic Geometry" (PDF). Retrieved 22 March 2019.
^ Klarreich, Erica (June 28, 2016). "Peter Scholze and the Future of Arithmetic Geometry". Retrieved March 22, 2019.
^ Poonen, Bjorn (2009). "Introduction to Arithmetic Geometry" (PDF). Retrieved March 22, 2019.
^ Arithmetic geometry in nLab
^ Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. pp. 43–67. ISBN 3-540-61223-8. Zbl 0869.11051.
^ a b Grothendieck, Alexander (1960). "The cohomology theory of abstract algebraic varieties". Proc. Internat. Congress Math. (Edinburgh, 1958). Cambridge University Press. pp. 103–118. MR 0130879.
^ Serre, Jean-Pierre (1967). "Résumé des cours, 1965–66". Annuaire du Collège de France. Paris: 49–58.
^ Mordell, Louis J. (1969). Diophantine Equations. Academic Press. p. 1. ISBN 978-0125062503.
^ A. Weil, L'arithmétique sur les courbes algébriques, Acta Math 52, (1929) p. 281-315, reprinted in vol 1 of his collected papers ISBN 0-387-90330-5.
^ Zariski, Oscar (2004) [1935]. Abhyankar, Shreeram S.; Lipman, Joseph; Mumford, David (eds.). Algebraic surfaces. Classics in mathematics (second supplemented ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-58658-6. MR 0469915.
^ Weil, André (1949). "Numbers of solutions of equations in finite fields". Bulletin of the American Mathematical Society. 55 (5): 497–508. doi:10.1090/S0002-9904-1949-09219-4. ISSN 0002-9904. MR 0029393. Reprinted in Oeuvres Scientifiques/Collected Papers by André Weil ISBN 0-387-90330-5
^ Serre, Jean-Pierre (1955). "Faisceaux Algebriques Coherents". The Annals of Mathematics. 61 (2): 197–278. doi:10.2307/1969915. JSTOR 1969915.
^ Dwork, Bernard (1960). "On the rationality of the zeta function of an algebraic variety". American Journal of Mathematics. American Journal of Mathematics, Vol. 82, No. 3. 82 (3): 631–648. doi:10.2307/2372974. ISSN 0002-9327. JSTOR 2372974. MR 0140494.
^ Grothendieck, Alexander (1995) [1965]. "Formule de Lefschetz et rationalité des fonctions L". Séminaire Bourbaki. Vol. 9. Paris: Société Mathématique de France. pp. 41–55. MR 1608788.
^ Deligne, Pierre (1974). "La conjecture de Weil. I". Publications Mathématiques de l'IHÉS. 43 (1): 273–307. doi:10.1007/BF02684373. ISSN 1618-1913. MR 0340258.
^ Taniyama, Yutaka (1956). "Problem 12". Sugaku (in Japanese). 7: 269.
^ Shimura, Goro (1989). "Yutaka Taniyama and his time. Very personal recollections". The Bulletin of the London Mathematical Society. 21 (2): 186–196. doi:10.1112/blms/21.2.186. ISSN 0024-6093. MR 0976064.
^ Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255.
^ Shimura, Goro (2003). The Collected Works of Goro Shimura. Springer Nature. ISBN 978-0387954158.
^ Langlands, Robert (1979). "Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen" (PDF). In Borel, Armand; Casselman, William (eds.). Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics. Vol. XXXIII Part 1. Chelsea Publishing Company. pp. 205–246.
^ Mazur, Barry (1977). "Modular curves and the Eisenstein ideal". Publications Mathématiques de l'IHÉS. 47 (1): 33–186. doi:10.1007/BF02684339. MR 0488287.
^ Mazur, Barry (1978). with appendix by Dorian Goldfeld. "Rational isogenies of prime degree". Inventiones Mathematicae. 44 (2): 129–162. Bibcode:1978InMat..44..129M. doi:10.1007/BF01390348. MR 0482230.
^ Merel, Loïc (1996). "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields]. Inventiones Mathematicae (in French). 124 (1): 437–449. Bibcode:1996InMat.124..437M. doi:10.1007/s002220050059. MR 1369424.
^ Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields]. Inventiones Mathematicae (in German). 73 (3): 349–366. Bibcode:1983InMat..73..349F. doi:10.1007/BF01388432. MR 0718935.
^ Faltings, Gerd (1984). "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae (in German). 75 (2): 381. doi:10.1007/BF01388572. MR 0732554.
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IsLeftInvolutary - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Magma : IsLeftInvolutary
test whether a finite magma is left involutary
IsLeftInvolutary( m )
The IsLeftInvolutary command returns true if the given magma satisfies the law X(XY) = Y. It returns false otherwise.
\mathrm{with}\left(\mathrm{Magma}\right):
m≔〈〈〈1|2|3〉,〈1|2|3〉,〈1|3|2〉〉〉
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}\end{array}]
\mathrm{IsLeftInvolutary}\left(m\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
m≔〈〈〈1|1|3〉,〈1|1|1〉,〈1|3|1〉〉〉
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{IsLeftInvolutary}\left(m\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The Magma[IsLeftInvolutary] command was introduced in Maple 15.
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Rotational Flows, Thermodynamics, Angle Vibration and Action of Whirly Flows on Fishes
Department of Solid Mechanics, KTH Royal Institute of Technology, Stockholm, Sweden.
DOI: 10.4236/ojmip.2017.74005 PDF HTML XML 747 Downloads 1,176 Views Citations
A continuum thermodynamic model for how whirls can transform into thermal energy-forms determined by a functional relation for temperature is derived. This is used to describe how fishes maintain circulation in the vascular system, at very low temperatures.
Entropy, Temperature, Thermodynamics, Angular Velocity, Vorticity, Eddy, Harmonic Oscillator, Angle Vibration, Temperature Distribution, Spatial Cylindrical Coordinates, φ(t )
Strömberg, L. (2017) Rotational Flows, Thermodynamics, Angle Vibration and Action of Whirly Flows on Fishes. Open Journal of Molecular and Integrative Physiology, 7, 53-56. doi: 10.4236/ojmip.2017.74005.
Theories for interactions of different types of energy, e.g. kinetic, mechanical and thermal are of importance in many fields. It invokes also transformations between them, dissipation and creation of sublevels and super levels. Here, we will scrutinize the ramifications of thermodynamics and continuum mechanics for a viscous incompressible fluid.
2. Balance Equation for Entropy
For a fluid with viscosity, continuum mechanics with thermodynamics and additional assumptions of heat capacity and heat flux gives an equation for the rate of entropy, reading
\rho \theta {d}_{t}\eta =s-\kappa \Delta \theta +\lambda {\left(trL\right)}^{2}+2\mu tr{L}^{2}
where, with usual notations;
\rho
\eta
, s,
\theta
, trL,
\left(\lambda {\left(trL\right)}^{2}+2\mu tr{L}^{2}\right)
is the density, entropy, heat radiation, temperature, trace of velocity gradient and internal friction in the viscous motion, and dt denotes entire time derivative. Here, we consider a flow consisting of whirls such that L = skew(L) = wx, where x is the cross product. From thermodynamics, entropy is energy conjugated to temperature, and there are different types of energies, c.f. [1] . The energies are potential functions in so called Legendre transformations, to change between (independent) variables. The conjugated variables for mechanical energy are p/ρ and ρ, but these will not be used in the present analysis.
Entropic Forces in Elasticity and Continuum Mechanics
An example of an entropic system in material science is a study of biological tissues with some elasticity, and motion also governed by a statistic random distribution dependent on temperature, c.f. [2] .
It appears that in harmonisation between continuum mechanical formulation and classical statistical mechanics, the molecule mass enters in the statistics caloric equation of state. An exact evaluation gives that
p=\left(k/{m}_{w}\right)\rho \theta
, where k is Boltzman’s constant and mw is the molecule-weight. If altering the example in [2] into continuum mechanics, then k should be replaced with k/mw. Since these systems often are used with self-calibration, the exact expressions do not show, but instead changes and ratios. First, we shall adopt the framework in [2] to rotations instead of displacements. Hereby, interpreting w2 as weighted from a statistical temperature distribution, we obtain that w2 is proportional to
\theta k/{m}_{w}
. Then, with notations of the balance Equation (1), we identify left side with this such that
\rho {d}_{t}\eta =k/{m}_{w}
3. Solutions in Terms of Independent Variables on a Manifold
To connect w2 with kinematics and geometry, alternate frames could be chosen:
With discrete memory as implied from Tti in anco, the acceleration may be from a previous state. This gives change of direction, such that the angular acceleration is the square of angular velocity.
No heat or terms with temp balanced:
\rho \theta {d}_{t}\eta =2\mu {w}^{2}
{d}_{t}\eta
proportional to temperature gives a “symmetry” such that vorticity w is proportional to temperature.
For systems with many (rotational) d.o.f, often harmonic oscillators are assumed as the point of departure. Therefore, only the inertia parts are exported to a multi-dimensional Hamiltonian and the individual potentials, as well as potential energy is invoked in some manner into entire energy, and the kinematic interactions are neglected. This is the foundation of the Schrödinger equation in QM.
Here we will proceed with one d.o.f on the meso-scale provided by continuum mechanics. Formulations in terms of energies depending on
\theta
\eta
w
and energy conjugate to w, could be done comparison with notations of a manifold, c.f. [1] . Then, either a measure of whirl and temperature as forms without connection to coordinates, or the whirl and temperature expressed in R3-coordinates, could be used as the independent variable in analysis.
Next, the latter will be considered to see how the fields matter in real space.
Solutions in Terms of Spatial Coordinates in R3
Since w is connected to angle
\phi
w={\phi }_{t}
, we will consider solutions
\theta =\theta \left(\phi \right)
・ With
{w}^{2}={\phi }_{tt}
i.e. angular acceleration and
\theta
proportional to angle
\phi
(2) gives hyperbolic solutions as functions of angles.
{d}_{t}\eta =
const there are solutions to (1);
\theta =C\mathrm{exp}\left(-a\phi \right)
where r,
\phi
are cylindrical coordinates, and
{\left(a/r\right)}^{2}\kappa =\rho {d}_{t}\eta -b
2\mu {w}^{2}=-s+b\theta
{d}_{t}\eta =0
, the solution parameters a, b can be expressed in statistical variables;
b=k/{m}_{w}
{\left(a/r\right)}^{2}\kappa =-b
・ Finally, a case with small scale harmonic oscillator solutions will be derived.
\Delta \theta
\theta
, a solution is
\theta =D\mathrm{sinh}\left(a\phi \right)
where D and a are constants with
{w}^{2}={\phi }_{tt}
, insertion in (1) gives;
2\mu {\phi }_{tt}=-s+\left(\rho {d}_{t}\eta -\kappa {\left(a/r\right)}^{2}\right)\theta
. Linearised Taylor expansion of
\theta
in gives now a harmonic oscillator for a sub-scale coordinate angle
\phi \left(t\right)
, and thus, the Taylor expanded
\theta \left(\phi \right)
. Hereby, from a spatial representation of temperature as sinus-hyperbolicus
\left(\phi \right)
, we obtained (on a sub-level) a time dependent harmonic oscillator for a sub-angle, i.e.
\phi ={\phi }_{0}\mathrm{sin}\left({\Omega }_{0}t\right)
{\Omega }_{0}
is constant depending on material parameters, coordinate r and
\rho {d}_{t}\eta
A plausible application is the vascular system of fishes in the North Sea where it is cold. At the outside boundary of the fish, the flow creates whirl. This could be copied inside, but such a remote connection of shapes is not necessary to invoke in this modeling. It is sufficient to assume that any large scale kinetic energy from outside can be transferred to interior fluid with a density and a vorticity flow.
Another application, where whirls transform into other energy is when fishes move up in a fall, c.f. Figure 1 and [3] . First, we may consider it as a quantised system with two states, namely down with whirls of energy V, and on its way almost up with energy V-Vpot. Looking in more details, it appears that the whirls provide an elastic ground to bump from. Then the description will materialize in spatial coordinates, but the input may remain similar with energies. Functional expressions while depending on coordinates in R3 is explained in [1] , as exemplified above.
The model shows how whirls can transform into a thermal energy-form determined by a functional relation for temperature. Solutions are evaluated in cylindrical coordinates, such that temperature is a function of angle.
Figure 1. Waterfalls and a whirly waterfall with Salmons swimming/jumping up.
The possibility for fishes of turning a larger scale kinetic energy into heat inside a system is probably the most important application in this respect.
To Dr Chaffin and Dr Pudikievicz for providing guidance and references.
[1] Starke, R. and Schober, G.A.H. (2016) Ab Initio Materials Physics and Microscopic Electromagnetism of Media. Subsection 4.1. Short Review of Thermodynamics. arXiv:1606.00445v2.
[2] Freund, L.B. (2014) Entropic Forces in the Mechanics of Solids. Procedia IUTAM, 10, 115-124.
[3] https://www.youtube.com/watch?v=iM0mn5unvoM
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Using Units - Maple Help
Home : Support : Online Help : Science and Engineering : Units : Using Units
Using Units and Dimensions
Example: How much space does 1 billion dollars worth of gold occupy?
Maple provides a comprehensive package for managing units and dimensions. Problems in science and engineering can now be fully managed with appropriate dimensions in any modern unit system (and even some historical systems!), including MKS, FPS, CGS, atomic, and more. Over 500 standard units are recognized by the Maple Units package.
This document explains how the Units package is structured, how to do simple and complex conversions between units, and how to use units in a Maple document. It also gives tips and examples on how to effectively use the different notation styles, how to change the default measurement system, and how to customize defaults in an existing measurement system.
For an introductory tutorial on units, see the Units & Tolerances. For more detailed help on the support for units and dimensions in Maple, see the Units package.
You can enter units by clicking the appropriate symbol in the Units palette on the left or by using the Units hot key:
- Ctrl + Shift + U : Windows, Linux
- Command + Shift + U : Mac
- Alt + Shift + U : Linux
Default Environment: For basic units support, you do not need to load the Units package. Maple will allow you to assign units to variables and expressions through the Units palette and perform calculations with them. The unit names are treated as non-assigned symbols and will be manipulated to give the correct combination of units in your results.
However, you can perform unit conversions and simplifications on your expressions by using the Context Panel.
For more details on the Default units environment, click here.
Default Environment Examples
\mathrm{restart}
\mathrm{mass}:=90.4⟦\mathrm{kg}⟧:
l:=34⟦\mathrm{cm}⟧:
t:=24.5⟦s⟧:
\mathrm{f1}≔\frac{\mathrm{mass}\cdot l}{{t}^{2}}
\frac{\textcolor[rgb]{0,0,1}{5.120533111}⟦\textcolor[rgb]{0,0,1}{\mathrm{kg}}⟧⟦\textcolor[rgb]{0,0,1}{\mathrm{cm}}⟧}{{⟦\textcolor[rgb]{0,0,1}{s}⟧}^{\textcolor[rgb]{0,0,1}{2}}}
From the Context Panel, select Units>Simplify to give:
\textcolor[rgb]{0,0,1}{0.05120533111}⟦\textcolor[rgb]{0,0,1}{N}⟧
Simple Environment: Establishes an environment in which some functions, including the ones for basic arithmetic, are modified to accept input with units.
In contrast to the Natural and Standard environments, unassigned variables are not automatically assumed to represent unit-free quantities. For example,
5⟦m⟧+x
x
x
\left(5⟦m⟧+x\right)\cdot \left(6⟦s⟧+x\right)
x
x
By default, the Simple Environment is loaded automatically when you load the Units package.
You assign units to variables or expressions either using the Unit function or the Units palette on the left. If you do not find the unit you need, simply select the
⟦\textcolor[rgb]{0,0,1}{\mathrm{unit}}⟧
symbol and enter the unit name into the placeholder.
For more details on Units[Simple], click here.
Simple Environment Examples
\mathrm{restart}
\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Simple}\right]\right):
3⟦\mathrm{cm}⟧+2⟦m⟧
\frac{\textcolor[rgb]{0,0,1}{203}}{\textcolor[rgb]{0,0,1}{100}}⟦\textcolor[rgb]{0,0,1}{m}⟧
\mathrm{mass}:=90.4⟦\mathrm{kg}⟧:
l:=34⟦\mathrm{cm}⟧:
t:=24.5⟦s⟧:
\mathrm{f3}≔\frac{\mathrm{mass}\cdot l}{{t}^{2}}+x
\textcolor[rgb]{0,0,1}{0.05120533110}⟦\textcolor[rgb]{0,0,1}{N}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}
\frac{ⅆ}{ⅆ\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}s}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{s}^{2}
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{s}
Standard Environment: Allows you to take full advantage of the units and dimensions support in the Units package. Unit names are separated from variable names so that you can still use variables with the same name for other computations.
⟦\textcolor[rgb]{0,0,1}{\mathrm{unit}}⟧
For more details on Units[Standard], click here.
Standard Environment Examples
\mathrm{restart}
\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Standard}\right]\right):
\mathrm{mass}:=90.4⟦\mathrm{kg}⟧:
l:=34⟦\mathrm{cm}⟧:
t:=24.5⟦s⟧:
\mathrm{f3}≔\frac{\mathrm{mass}\cdot l}{{t}^{2}}
\textcolor[rgb]{0,0,1}{0.05120533110}⟦\textcolor[rgb]{0,0,1}{N}⟧
\frac{ⅆ}{ⅆ\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}s}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{s}^{2}
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{s}
Natural Environment: You can express unit variables as ordinary variables without special notation.
This convenient notation is very readable, and is suitable for many situations, including interactive calculations where you have a limited number of user-defined variable names.
The Natural units notation is suitable for situations in which you are confident you will not create naming conflicts or expressions that appear ambiguous. Because the Units package supports many units, many names are "taken" when the Natural notation is used. This list will contain many unit names which you may not even be aware are units, for example, f, year, barn, and Pa. Using these names as variables could inadvertently create problems with your calculations.
For more details on Units[Natural], click here.
Natural Environment Examples
\mathrm{restart}
\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Natural}\right]\right):
4.2\cdot m+32\cdot \mathrm{cm}
\textcolor[rgb]{0,0,1}{4.520000000}⟦\textcolor[rgb]{0,0,1}{m}⟧
\mathrm{f2}≔\frac{\mathrm{mass}\cdot l}{{t}^{2}}
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{1000000000}}\textcolor[rgb]{0,0,1}{\mathrm{mass}}⟦\frac{{\textcolor[rgb]{0,0,1}{m}}^{\textcolor[rgb]{0,0,1}{3}}}{{\textcolor[rgb]{0,0,1}{\mathrm{kg}}}^{\textcolor[rgb]{0,0,1}{2}}}⟧
But, because s is defined as seconds in the Natural units environment, if you attempt to use it in a symbolic calculation, it results in an error.
\frac{ⅆ}{ⅆs\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{s}^{2}
Error, (in Units:-Standard:-diff) wrong number (or type) of parameters in function Units:-Standard:-diff
Units Package: Loading the Units package does not enable a separate units environment. Instead, it loads one of the three non-default environments listed above; by default the Simple Environment. In addition, the Units package provides functions that allow you to make changes to the units support: add new units, rename existing ones, even create your own systems of units.
Examples using the Units package
Maple automatically converts the answer to a default unit system and unit. SI is the default unit system used by the Units package.
\mathrm{restart}
\mathrm{with}\left(\mathrm{Units}\right):
\mathrm{myDist} :=\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}4⟦\mathrm{ft}⟧\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}+\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}3.5⟦\mathrm{inches}⟧
\textcolor[rgb]{0,0,1}{1.308100000}⟦\textcolor[rgb]{0,0,1}{m}⟧
You can easily express the above in the standard US (FPS: foot-pound-second) system using the convert command.
\mathrm{myDist}≔\mathrm{convert}\left( \mathrm{myDist}, \mathrm{system}, \mathrm{FPS}\right)
\textcolor[rgb]{0,0,1}{4.291666667}⟦\textcolor[rgb]{0,0,1}{\mathrm{ft}}⟧
The default unit system can be changed using the Units[UseSystem] command.
\mathrm{Units}\left[\mathrm{UseSystem}\right]\left(\mathrm{FPS}\right):
\mathrm{myDist} ≔4⟦\mathrm{km}⟧\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}+\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}3.5⟦\mathrm{mile}⟧
\textcolor[rgb]{0,0,1}{31603.35958}⟦\textcolor[rgb]{0,0,1}{\mathrm{ft}}⟧
Finding what Unit Systems are Available
GetSystems() returns an expression sequence of all systems of units.
\mathrm{restart}
\mathrm{with}\left(\mathrm{Units}\right):
\mathrm{GetSystems}\left(\right)
\textcolor[rgb]{0,0,1}{\mathrm{Atomic}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{CGS}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{EMU}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ESU}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{FPS}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{MKS}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{MTS}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{SI}}
Redefining Default Units inside a Measurement System
Inside each measurement system, each dimension has a specified default unit. For example, all length calculations done inside the SI system return answers in meters. It is possible to change the default unit for a particular dimension, which will be respected even after combining and simplifying units, while leaving the rest of the measurement system unchanged.
\mathrm{with}\left(\mathrm{Units}\right):
\mathrm{AddSystem}\left(\mathrm{NewSI},\mathrm{GetSystem}\left(\mathrm{SI}\right),\mathrm{mm}\right):
\mathrm{UseSystem}\left(\mathrm{NewSI}\right):
\mathrm{Unit}\left(50000.⟦{\mathrm{cm}}^{2}⟧\right),\mathrm{Unit}\left(3.⟦\mathrm{cm}⟧\right),\mathrm{Unit}\left(11.⟦{\mathrm{ft}}^{3}⟧\right)
\textcolor[rgb]{0,0,1}{50000.}⟦{\textcolor[rgb]{0,0,1}{\mathrm{cm}}}^{\textcolor[rgb]{0,0,1}{2}}⟧\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3.}⟦\textcolor[rgb]{0,0,1}{\mathrm{cm}}⟧\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11.}⟦{\textcolor[rgb]{0,0,1}{\mathrm{ft}}}^{\textcolor[rgb]{0,0,1}{3}}⟧
For simple conversions between units, the convenient Units Converter is available.
From Tools menu, select Assistants>Units Converter or in Maple help, type assistants/Units. No knowledge of the Units package is required. All the available dimensions and units are selected from a list, so you know instantly what is available with no worries over choosing the proper spelling or abbreviation.
The Maple convert command can be used directly for unit conversions without the
\mathrm{with}\left(\mathrm{Units}\right)
To determine the conversion factor, use an unassigned variable in the convert command. For example, to see the conversion formula from calories to joules:
\mathrm{restart}
\mathrm{convert}(\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}1.0,\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{units},\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{ft},\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}m)
\textcolor[rgb]{0,0,1}{0.3048000000}
\mathrm{convert}(x,\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{units},\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{cal},\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}J)
\frac{\textcolor[rgb]{0,0,1}{523}\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{125}}
The units conversion for temperature converts changes (for example, a temperature change of 5° Celsius is the same as a change of 9° Fahrenheit). To convert absolute temperatures from one scale to another, use the temperature option to convert.
\mathrm{convert}(5,\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{units},\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{degC},\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{degF})
\textcolor[rgb]{0,0,1}{9}
\mathrm{convert}(0,\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{temperature},\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{degC},\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{degF})
\textcolor[rgb]{0,0,1}{32}
All trigonometric functions in Maple expect angles in radians. In the Units[Simple] mode, you can simply enter the angle in any units and Maple automatically performs the conversion.
\mathrm{with}\left(\mathrm{Units}\right):
\mathrm{α}≔45.⟦\mathrm{deg}⟧:
\mathrm{cos}\left(\mathrm{α}\right)
\textcolor[rgb]{0,0,1}{0.7071067811}
Displaying results in the units you want
Sometimes, you will need to change the units of your result from the units computed by Maple. To do this, from the Context Panel for the result, select Units>Replace Units, and then enter your desired units.
If you enter units with incompatible dimensions, you will get an error message.
\mathrm{with}\left(\mathrm{Units}\right):
\mathrm{distance}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}:=\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}5.0⟦m⟧:
\mathrm{elapsedtime}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}:=\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}10⟦s⟧:
\mathrm{speed}:=\frac{\mathrm{distance}}{\mathrm{elapsedtime}}
\textcolor[rgb]{0,0,1}{0.5000000000}⟦\frac{\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{s}}⟧
To convert to miles per hour, select Units>Replace Units, and then enter your desired units:
\textcolor[rgb]{0,0,1}{1.118468146}⟦\frac{\textcolor[rgb]{0,0,1}{\mathrm{mi}}}{\textcolor[rgb]{0,0,1}{h}}⟧
Initialize the Units package.
\colorbox[rgb]{0,0,0}{$\mathrm{restart}$}
\mathrm{with}\left(\mathrm{Units}\right):
\mathrm{with} \left(\mathrm{ScientificConstants}\right):
First, define the dollar value of the gold, cost per troy ounce (assume $268), and the density.
\mathrm{val}:={10}^{9}⟦\mathrm{USD}⟧:
\mathrm{costPerOz}:=268.00⟦\frac{\mathrm{USD}}{\mathrm{oz}\left[\mathrm{troy}\right]}⟧:
\mathrm{density}:=19.3⟦\frac{g}{{\mathrm{cm}}^{3}}⟧
\textcolor[rgb]{0,0,1}{19.3}⟦\frac{\textcolor[rgb]{0,0,1}{g}}{{\textcolor[rgb]{0,0,1}{\mathrm{cm}}}^{\textcolor[rgb]{0,0,1}{3}}}⟧
The resulting mass and volume are:
\mathrm{mass}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}:=\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\frac{\mathrm{val}}{\mathrm{costPerOz}}
\textcolor[rgb]{0,0,1}{1.160577493}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{5}}⟦\textcolor[rgb]{0,0,1}{\mathrm{kg}}⟧
\mathrm{volume}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}:=\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\frac{\mathrm{mass}}{\mathrm{density}}
\textcolor[rgb]{0,0,1}{6.013354886}⟦{\textcolor[rgb]{0,0,1}{m}}^{\textcolor[rgb]{0,0,1}{3}}⟧
What is the length of one side of a cube having this volume?
\mathrm{LengthSide}:=\sqrt[3]{\mathrm{volume}}
\textcolor[rgb]{0,0,1}{1.818467785}⟦\textcolor[rgb]{0,0,1}{m}⟧
Units Converter, Units Package Index
|
Analyzing genomic data for finding those gene variations which are responsible for hereditary diseases is one of the great challenges in modern bioinformatics. In many living beings (including the human), every gene is present in two copies, inherited from the two parents, the so-called haplotypes. In this paper, we propose a simple combinatorial model for classifying the set of haplotypes in a population according to their responsibility for a certain genetic disease. This model is based on the minimum-ones 2SAT problem with uniform clauses. The minimum-ones 2SAT problem asks for a satisfying assignment to a satisfiable formula in 2CNF which sets a minimum number of variables to true. This problem is well-known to be
\mathrm{𝒩𝒫}
-hard, even in the case where all clauses are uniform, i.e., do not contain a positive and a negative literal. We analyze the approximability and present the first non-trivial exact algorithm for the uniform minimum-ones 2SAT problem with a running time of
𝒪
(1.21061n) on a 2SAT formula with n variables. We also show that the problem is fixed-parameter tractable by showing that our algorithm can be adapted to verify in
{𝒪}^{*}
(2k) time whether an assignment with at most k true variables exists.
Classification : 68Q25, 68R10, 92D20
Mots clés : exact algorithms, fixed-parameter algorithms, minimum-ones 2SAT, haplotypes
author = {B\"ockenhauer, Hans-Joachim and Fori\v{s}ek, Michal and Oravec, J\'an and Steffen, Bj\"orn and Steinh\"ofel, Kathleen and Steinov\'a, Monika},
title = {The uniform minimum-ones {2SAT} problem and its application to haplotype classification},
AU - Böckenhauer, Hans-Joachim
AU - Forišek, Michal
AU - Oravec, Ján
AU - Steffen, Björn
AU - Steinhöfel, Kathleen
AU - Steinová, Monika
TI - The uniform minimum-ones 2SAT problem and its application to haplotype classification
Böckenhauer, Hans-Joachim; Forišek, Michal; Oravec, Ján; Steffen, Björn; Steinhöfel, Kathleen; Steinová, Monika. The uniform minimum-ones 2SAT problem and its application to haplotype classification. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 3, pp. 363-377. doi : 10.1051/ita/2010018. http://www.numdam.org/articles/10.1051/ita/2010018/
[1] B. Aspvall, M.F. Plass and R.E. Tarjan, A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf. Proc. Lett. 8 (1979) 121-123. | Zbl 0398.68042
[2] H.-J. Böckenhauer and D. Bongartz, Algorithmic Aspects of Bioinformatics. Natural Computing Series, Springer-Verlag (2007). | Zbl 1131.68579
[3] P. Bonizzoni, G.D. Vedova, R. Dondi and J. Li, The haplotyping problem: An overview of computational models and solutions. J. Comput. Sci. Technol. 18 (2003) 675-688. | Zbl 1083.68579
[4] J. Chen, I. Kanj and W. Jia, Vertex cover: further observations and further improvements. J. Algorithms 41 (2001) 280-301. | Zbl 1017.68087
[5] I. Dinur and S. Safra, On the hardness of approximating minimum vertex cover. Ann. Math. 162 (2005) 439-485. | Zbl 1084.68051
[6] D. Gusfield and L. Pitt, A bounded approximation for the minimum cost 2-sat problem. Algorithmica 8 (1992) 103-117. | Zbl 0753.68048
[7] J. Hromkovič, Algorithmics for Hard Problems. Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics. Texts in Theoretical Computer Science, An EATCS Series, Springer-Verlag, Berlin (2003). | Zbl 1069.68642
[8] G. Karakostas, A better approximation ratio for the vertex cover problem. Technical Report TR04-084, ECCC (2004). | Zbl 1085.68112
[9] S. Khot and O. Regev, Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci. 74 (2008) 335-349. | Zbl 1133.68061
[10] J. Kiniwa, Approximation of self-stabilizing vertex cover less than 2, in Self Stabilizing Systems (2005) 171-182. | Zbl 1172.68682
[11] E.L. Lawler and D.E. Wood, Branch-and-bound methods: A survey. Operat. Res. 14 (1966) 699-719. | Zbl 0143.42501
[12] J. Li and T. Jiang, A survey on haplotyping algorithms for tightly linked markers. J. Bioinf. Comput. Biol. 6 (2008) 241-259.
[13] J.M. Robson, Finding a maximum independent set in time O(2n/4). Technical Report 1251-01, LaBRI, Université Bordeaux I (2001).
|
Effect of a Comprehensive Health Education Program to Increase Physical Activity among Primary School Students in China
Ling Qian1, Lok-Wa Yuen2, Yonghua Feng2, Ian M. Newman2*, Duane F. Shell2, Weijing Du1
1China Center for Health Education, Ministry of Health, Beijing, China
2Department of Educational Psychology, University of Nebraska-Lincoln, Lincoln, Nebraska, USA
This work is licensed under the Creative Commons Attribution International License(CC BY 4.0).
China’s National Physical Fitness and Health Surveillance(NPFHS)survey revealed high levels of sedentary behavior among primary school-aged children. Sedentary behavior is linked to both short-term and long-term physical and mental health conditions. A comprehensive school health education program was designed and its effectiveness to raise physical activity levels in Grade 4 primary school children was evaluated. Twelve schools(6 of program, 6 of control)from six cities in two economically different provinces were selected. Students at program schools received physical activity instruction both in and out of school. Control schools carried on with their usual level of physical activity for students, as required by national educational standards. Program effectiveness was assessed by comparing students’ physical activity behaviors at pre- and post-program, and by comparing students’ physical activity scores at program schools with students’ scores at control schools. The pre-program survey of students’ behaviors was done at the end of Grade 3, and the post-program survey was done at the end of Grade 4. Multi-level modeling was used to evaluate program effectiveness to allow for missing data. Results from 4472 students showed at pre-program there was no difference in control and program schools’ student physical activity scores. At post-program students in program schools did significantly more physical activity compared to students in control schools. Students in program schools were more physically active after the comprehensive school health education program. A school-based comprehensive health education program would effectively increase children’s physical activity level in China. There is a potential to reduce sedentary behavior among children by implementing school programs that are environmental in nature; that is, activities involve not only the students, but also the school administration, teachers, parents, and community members.
Obesity, Child, Sedentary, Environmental, Health Promotion, Multilevel Modeling
Qian, L., Yuen, L.-W., Feng, Y. H., Newman, I. M., Shell, D. F., & Du, W. J.(2018). Effect of a Comprehensive Health Education Program to Increase Physical Activity among Primary School Students in China. Advances in Physical Education, 8, 193-204. https://doi.org/10.4236/ape.2018.82018
1. Chen, Y., Zheng, Z., Yi, J., & Yao, S.(2014). Associations between Physical Inactivity and Sedentary Behaviors among Adolescents in 10 Cities in China. BMC Public Health, 14, 744. https://doi.org/10.1186/1471-2458-14-744 [Paper reference 1]
2. Chinese Nutrition Society(2010). Dietary Guidelines for Chinese Residents. Lhasa: Tibetan People’s Press.(In Chinese)[Paper reference 1]
3. Cui, Z., Hardy, L. L., Dibley, M. J., & Bauman, A.(2011). Temporal Trends and Recent Correlates in Sedentary Behaviours in Chinese Children. The International Journal of Behavioral Nutrition and Physical Activity, 8, 93. [Paper reference 2]
4. https://doi.org/10.1186/1479-5868-8-93 [Paper reference 1]
5. Duan, J., Hu, H., Wang, G., & Arao, T.(2015). Study on Current Levels of Physical Activity and Sedentary Behavior among Middle School Students in Beijing, China. PLoS One, 10, e0133544. https://doi.org/10.1371/journal.pone.0133544 [Paper reference 1]
6. Janssen, O., Katzmarzyk, P. T., Boyce, W. F., Vereecken, C., Mulvihill, C., Roberts, C. et al.(2005). Comparison of Overweight and Obesity Prevalence in School-Aged Youth from 34 Countries and Their Relationships with Physical Activity and Dietary Patterns. Obesity Reviews, 6, 123-132. https://doi.org/10.1111/j.1467-789X.2005.00176.x [Paper reference 1]
7. Li, H., Lin, S., Guo, H., Huang, Y., Wu, L., Zhang, Z., Ma, J., & Wang, H.(2014). Effectiveness of a School-Based Physical Activity Intervention on Obesity in School Children: A Nonrandomized Controlled Trial. BMC Public Health, 14, 1282. [Paper reference 1]
8. https://doi.org/10.1186/1471-2458-14-1282 [Paper reference 1]
9. Li, M., Dibley, M. J., Sibbritt, D., & Yan, H.(2006). Factors Associated with Adolescents' Physical Inactivity in Xi’an City, China. Medicine and Science in Sports and Exercise, 38, 2075-2085. https://doi.org/10.1249/01.mss.0000233802.54529.87 [Paper reference 1]
10. Li, Y.-P., Hu, X.-Q., Schouten, E. G., Liu, A.-L., Du, S.-M., Li, L.-Z. et al.(2010). Report on Childhood Obesity in China(8): Effects and Sustainability of Physical Activity Intervention on Body Composition of Chinese Youth. Biomedical and Environmental Sciences, 23, 180-187. https://doi.org/10.1016/S0895-3988(10)60050-5 [Paper reference 1]
11. Liu, A.-L., Hu, X.-Q., Ma, G.-S., Cui, Z.-H., Pan, Y.-P., Chang, S.-Y. et al.(2007). Report on Childhood Obesity in China(6)Evaluation of a Classroom-Based Physical Activity Promotion Program. Biomedical and Environmental Sciences, 20, 19-23. [Paper reference 1]
12. National Bureau of Statistics of China(n.d.). China Regional Economic Development Visual Statistics Website. http://data.stats.gov.cn/english/swf.htm?m=turnto&id=3 [Paper reference 3]
13. Penedo, F. J., & Dahn, J. R.(2005). Exercise and Well-Being: A Review of Mental and Physical Health Benefits Associated with Physical Activity. Current Opinion in Psy-chiatry, 18, 189-193. https://doi.org/10.1097/00001504-200503000-00013 [Paper reference 1]
14. Qian, L., Newman, I. M., Shell, D. F., & Cheng, M. J.(2012)Reducing Overweight and Obesity among Elementary Students in Wuhan, China. International Electronic Journal for Health Education, 15, 62-71. [Paper reference 1]
15. https://digitalcommons.unl.edu/edpsychpapers/242/ [Paper reference 2]
16. Qian, L., Newman, I. M., Yuen, L.-W., Du, W., & Shell, D. F.(2018). Effects of a Comprehensive Nutrition Education Program to Change Grade School Students’ Eating Behaviors in China. Unpublished Draft Manuscript. [Paper reference 1]
17. Qin, L.(2014). Evaluation on the Exercise and Nutrition Intervention among Over-weight and Obesity Adolescents. Xian daiyu fang yixue [Modern Preventive Medi-cine], 41, 1597-1604.(In Chinese)[Paper reference 3]
18. Rasberry, C. N., Lee, S. M., Robin, L., Laris, B. A., Russell, L .A., Coyle, K. K., & Nihiser, A. J.(2011). The Association between School-Based Physical Activity, Including Physical Education, and Academic Performance: A Systematic Review of the Literature. Preventive Medicine, 52, S10-S20. https://doi.org/10.1016/j.ypmed.2011.01.027 [Paper reference 1]
19. Raudenbush, S. W., & Bryk, A. S.(2002). Hierarchical Linear Models: Applications and Data Analysis Methods(2nd ed.). Thousand Oaks, CA: Sage Publishing. [Paper reference 2]
20. Ren, H., Zhou, Z. Liu, W., Wang, X., & Yin, Z.(2016). Excessive Homework, Inadequate Sleep, Physical Inactivity and Screen Viewing Time Are Major Contributors to High Pediatric Obesity. Acta Paediatrica, 106, 120-127.
21. Song, Y., Wang, H., Ma, J., & Wang, Z.(2013). Secular Trends of Obesity Prevalence in Urban Chinese Children from 1985 to 2010: Gender Disparity. PLoS ONE, 8, e53069. [Paper reference 2]
22. https://doi.org/10.1371/journal.pone.0053069 [Paper reference 1]
23. Song, Y., Zhang, X., Yang, T., Zhang, B., Dong, B., & Ma, J.(2012). Current Situation and Cause Analysis of Physical Activity in Chinese Primary and Middle School Students in 2010. Chinese Journal of Peking University, 44, 347-354.(In Chinese)[Paper reference 4]
24. Wang, L., Sun, J., & Zhao, S.(2016). Parental Influence on the Physical Activity of Chinese Children Do Gender Differences Occur? European Physical Education Review, 23, 110-126.
25. Wang, X. Q., Hui, Z. Z., Terry, P. D., Ma, M., Cheng, L., Deng, F., Gu, W., & Zhang, B.(2016). Correlates of Insufficient Physical Activity among Junior High School Stu-dents: A Cross-Sectional Study in Xi’an, China. International Journal of Environmental Research and Public Health, 13, 397. https://doi.org/10.3390/ijerph13040397
26. Wang, X., Liu, Q., Ren, Y., Lv, J., & Li, L.(2015). Family Influences on Physical Activity and Sedentary Behaviours in Chinese Junior High School Students: A Cross-Sectional Study Health Behavior, Health Promotion and Society. BMC Public Health, 15, 287.
28. Zhang, X., Song, Y., Yang, T., Zhang, B., Dong, B., & Ma, J.(2012). Analysis of Current Situation of Physical Activity and Influencing Factors in Chinese Primary and Middle School Students in 2010. Chinese Journal of Preventive Medicine, 46, 781-788.(In Chinese)
\begin{array}{l}{Y}_{ijkl}={\gamma }_{0000}+{\gamma }_{1000}{\text{Time}}_{ijkl}+{\gamma }_{0001}{\text{Intervention}}_{l}+{\gamma }_{0002}{\text{Province}}_{l}\\ \text{}+{\gamma }_{1001}{\text{Time}}_{ijkl}\times {\text{Intervention}}_{l}+{e}_{ijkl}+{u}_{0jkl}+{u}_{1jkl}{\text{Time}}_{ijkl}\\ \text{}+{u}_{00kl}^{\mathrm{Pr}e}{d}_{ijkl}^{\mathrm{Pr}e}+{u}_{00kl}^{Post}{d}_{ijkl}^{Post}+{u}_{000l}+{u}_{100l}{\text{Time}}_{ijkl}\end{array}
{Y}_{ijkl}
{i}^{th}
{k}^{th}
{l}^{th}
{\text{Time}}_{ijkl}
{\text{Intervention}}_{l}
{\text{Province}}_{l}
{\gamma }_{1000}
{\gamma }_{0001}
{\gamma }_{0002}
{\gamma }_{1001}
{e}_{ijkl}\sim N\left(0,{\sigma }^{2}\right)
{u}_{0jkl}\sim N\left(0,{\tau }_{0}^{j}\right)
{u}_{1jkl}~N\left(0,{\tau }_{1}^{j}\right)
{u}_{00kl}^{\text{Pre}}\sim N\left(0,{\tau }_{0}^{\text{Pre},k}\right)
{u}_{00kl}^{Post}\sim N\left(0,{\tau }_{0}^{Post,k}\right)
{d}_{ijkl}^{Pre}
{d}_{ijkl}^{Post}
{u}_{000l}\sim N\left(0,{\tau }_{0}^{l}\right)
{u}_{100l}\sim N\left(0,{\tau }_{1}^{l}\right)
|
q
q
p
such that FFT-based polynomial arithmetic can be used for this actual computation. The higher the degrees of f and rc are, the larger must be
{2}^{e}
p-1
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{FastArithmeticTools}\right):
\mathrm{with}\left(\mathrm{ChainTools}\right):
p≔962592769:
\mathrm{vars}≔[y,x]:
R≔\mathrm{PolynomialRing}\left(\mathrm{vars},p\right):
\mathrm{f1}≔x\left({y}^{2}+y+1\right)+2:
\mathrm{f2}≔\left(x+1\right)\left({y}^{2}+y+1\right)+{x}^{3}+x+1:
\mathrm{SCube}≔\mathrm{SubresultantChainSpecializationCube}\left(\mathrm{f1},\mathrm{f2},y,R,1\right)
\textcolor[rgb]{0,0,1}{\mathrm{SCube}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{subresultant_chain_specialization_cube}}
\mathrm{r2}≔\mathrm{ResultantBySpecializationCube}\left(\mathrm{f1},\mathrm{f2},x,\mathrm{SCube},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{r2}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{962592767}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{962592766}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{962592767}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{962592766}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}
\mathrm{Gcd}\left(\mathrm{r2},x\left(x+1\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p
\textcolor[rgb]{0,0,1}{1}
\mathrm{rc}≔\mathrm{Chain}\left([\mathrm{r2}],\mathrm{Empty}\left(R\right),R\right)
\textcolor[rgb]{0,0,1}{\mathrm{rc}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}
\mathrm{g2}≔\mathrm{RegularGcdBySpecializationCube}\left(\mathrm{f1},\mathrm{f2},\mathrm{rc},\mathrm{SCube},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{g2}}\textcolor[rgb]{0,0,1}{≔}[[{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]\textcolor[rgb]{0,0,1}{,}[{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]]
\mathrm{NormalizePolynomialDim0}\left(\mathrm{g2}[1][1],\mathrm{g2}[1][2],R\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}
|
Piezoelectric sensor - 3D CAD Models & 2D Drawings
Piezoelectric sensor (7457 views - Mechanical Engineering)
A piezoelectric sensor is a device that uses the piezoelectric effect, to measure changes in pressure, acceleration, temperature, strain, or force by converting them to an electrical charge. The prefix piezo- is Greek for 'press' or 'squeeze'.
3D CAD Models - Piezoelectric sensor
Licensed under Creative Commons Attribution-Share Alike 3.0 (Tizeff).
2.1 Transverse effect
2.2 Longitudinal effect
2.3 Shear effect
5 Sensing materials
Piezoelectric sensors are versatile tools for the measurement of various processes. They are used for quality assurance, process control, and for research and development in many industries. Pierre Curie discovered the piezoelectric effect in 1880, but only in the 1950s did manufacturers begin to use the piezoelectric effect in industrial sensing applications. Since then, this measuring principle has been increasingly used, and has become a mature technology with excellent inherent reliability.
They have been successfully used in various applications, such as in medical, aerospace, nuclear instrumentation, and as a tilt sensor in consumer electronics[1] or a pressure sensor in the touch pads of mobile phones. In the automotive industry, piezoelectric elements are used to monitor combustion when developing internal combustion engines. The sensors are either directly mounted into additional holes into the cylinder head or the spark/glow plug is equipped with a built-in miniature piezoelectric sensor.[2]
The rise of piezoelectric technology is directly related to a set of inherent advantages. The high modulus of elasticity of many piezoelectric materials is comparable to that of many metals and goes up to 106 N/m².[citation needed] Even though piezoelectric sensors are electromechanical systems that react to compression, the sensing elements show almost zero deflection. This gives piezoelectric sensors ruggedness, an extremely high natural frequency and an excellent linearity over a wide amplitude range. Additionally, piezoelectric technology is insensitive to electromagnetic fields and radiation, enabling measurements under harsh conditions. Some materials used (especially gallium phosphate or tourmaline) are extremely stable at high temperatures, enabling sensors to have a working range of up to 1000 °C. Tourmaline shows pyroelectricity in addition to the piezoelectric effect; this is the ability to generate an electrical signal when the temperature of the crystal changes. This effect is also common to piezoceramic materials. Gautschi in Piezoelectric Sensorics (2002) offers this comparison table of characteristics of piezo sensor materials vs other types:[3]
Sensitivity [V/µε]
Span to
Piezoelectric 5.0 0.00001 100,000,000
Piezoresistive 0.0001 0.0001 2,500,000
Inductive 0.001 0.0005 2,000,000
Capacitive 0.005 0.0001 750,000
Resistive 0.000005 0.01 50,000
One disadvantage of piezoelectric sensors is that they cannot be used for truly static measurements. A static force results in a fixed amount of charge on the piezoelectric material. In conventional readout electronics, imperfect insulating materials and reduction in internal sensor resistance causes a constant loss of electrons and yields a decreasing signal. Elevated temperatures cause an additional drop in internal resistance and sensitivity. The main effect on the piezoelectric effect is that with increasing pressure loads and temperature, the sensitivity reduces due to twin formation. While quartz sensors must be cooled during measurements at temperatures above 300 °C, special types of crystals like GaPO4 gallium phosphate show no twin formation up to the melting point of the material itself.
However, it is not true that piezoelectric sensors can only be used for very fast processes or at ambient conditions. In fact, numerous piezoelectric applications produce quasi-static measurements, and other applications work in temperatures higher than 500 °C.
Piezoelectric sensors can also be used to determine aromas in the air by simultaneously measuring resonance and capacitance. Computer controlled electronics vastly increase the range of potential applications for piezoelectric sensors.[4]
Piezoelectric sensors are also seen in nature. The collagen in bone is piezoelectric, and is thought by some to act as a biological force sensor.[5][6]
The way a piezoelectric material is cut produces three main operational modes:
A force applied along a neutral axis (y) displaces charges along the (x) direction, perpendicular to the line of force. The amount of charge (
{\displaystyle C_{x}}
) depends on the geometrical dimensions of the respective piezoelectric element. When dimensions
{\displaystyle a,b,c}
apply,
{\displaystyle C_{x}=d_{xy}F_{y}b/a~}
{\displaystyle a}
is the dimension in line with the neutral axis,
{\displaystyle b}
is in line with the charge generating axis an{\displaystyle d}
is the corresponding piezoelectric coefficient.[3]
The amount of charge displaced is strictly proportional to the applied force and independent of the piezoelectric element size and shape. Putting several elements mechanically in series and electrically in parallel is the only way to increase the charge output. The resulting charge is
{\displaystyle C_{x}=d_{xx}F_{x}n~}
{\displaystyle d_{xx}}
is the piezoelectric coefficient for a charge in x-direction released by forces applied along x-direction (in pC/N).
{\displaystyle F_{x}}
is the applied Force in x-direction [N] and
{\displaystyle n}
corresponds to the number of stacked elements.
The charges produced are strictly proportional to the applied forces and independent of the element size and shape. For
{\displaystyle n}
elements mechanically in series and electrically in parallel the charge is
{\displaystyle C_{x}=2d_{xx}F_{x}n}
In contrast to the longitudinal and shear effects, the transverse effect make it possible to fine-tune sensitivity on the applied force and element dimension.
A piezoelectric transducer has very high DC output impedance and can be modeled as a proportional voltage source and filter network. The voltage V at the source is directly proportional to the applied force, pressure, or strain.[7] The output signal is then related to this mechanical force as if it had passed through the equivalent circuit.
A detailed model includes the effects of the sensor's mechanical construction and other non-idealities.[8] The inductance Lm is due to the seismic mass and inertia of the sensor itself. Ce is inversely proportional to the mechanical elasticity of the sensor. C0 represents the static capacitance of the transducer, resulting from an inertial mass of infinite size.[8] Ri is the insulation leakage resistance of the transducer element. If the sensor is connected to a load resistance, this also acts in parallel with the insulation resistance, both increasing the high-pass cutoff frequency.
For use as a sensor, the flat region of the frequency response plot is typically used, between the high-pass cutoff and the resonant peak. The load and leakage resistance must be large enough that low frequencies of interest are not lost. A simplified equivalent circuit model can be used in this region, in which Cs represents the capacitance of the sensor surface itself, determined by the standard formula for capacitance of parallel plates.[8][9] It can also be modeled as a charge source in parallel with the source capacitance, with the charge directly proportional to the applied force, as above.[7]
Based on piezoelectric technology various physical quantities can be measured; the most common are pressure and acceleration. For pressure sensors, a thin membrane and a massive base is used, ensuring that an applied pressure specifically loads the elements in one direction. For accelerometers, a seismic mass is attached to the crystal elements. When the accelerometer experiences a motion, the invariant seismic mass loads the elements according to Newton's second law of motion
{\displaystyle F=ma}
The main difference in working principle between these two cases is the way they apply forces to the sensing elements. In a pressure sensor, a thin membrane transfers the force to the elements, while in accelerometers an attached seismic mass applies the forces.
Sensors often tend to be sensitive to more than one physical quantity. Pressure sensors show false signal when they are exposed to vibrations. Sophisticated pressure sensors therefore use acceleration compensation elements in addition to the pressure sensing elements. By carefully matching those elements, the acceleration signal (released from the compensation element) is subtracted from the combined signal of pressure and acceleration to derive the true pressure information.
Vibration sensors can also harvest otherwise wasted energy from mechanical vibrations. This is accomplished by using piezoelectric materials to convert mechanical strain into usable electrical energy.[10]
Two main groups of materials are used for piezoelectric sensors: piezoelectric ceramics and single crystal materials. The ceramic materials (such as PZT ceramic) have a piezoelectric constant/sensitivity that is roughly two orders of magnitude higher than those of the natural single crystal materials and can be produced by inexpensive sintering processes. The piezoeffect in piezoceramics is "trained", so their high sensitivity degrades over time. This degradation is highly correlated with increased temperature.
The less-sensitive, natural, single-crystal materials (gallium phosphate, quartz, tourmaline) have a higher – when carefully handled, almost unlimited – long term stability. There are also new single-crystal materials commercially available such as Lead Magnesium Niobate-Lead Titanate (PMN-PT). These materials offer improved sensitivity over PZT but have a lower maximum operating temperature and are currently more expensive to manufacture.[citation needed]
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Technical Issues - EconGames
Typesetting math is not configured yet.
{\displaystyle e^{\pi i}=-1}
Emails are not configured yet. Do we need them, for getting informed about updates and such?
Add URL rewriting (so that page addresses look like on wikipedia)
SVG + MathML rendering of math requires a new feature, see https://phabricator.wikimedia.org/T155201 This has been resolved starting with MW 1.31
Mobile version of the website is not configured
There is not template for journal citations (can be copied from wikipedia, albeit wikipedia version requires Lua, see https://en.wikipedia.org/wiki/Template:Cite_journal)
News page should be replaced with a mailing list. Discussion pages are good for a large wiki, but for a smaller community a mailing list is more convenient, imho. It would be best to integrate it with the wiki, say add "Mailinglists" tab next to "Watchlist", so everyone can easily subscribe, unsubscribe and add new mailining lists. (There is no ready made extension it seems.)
Would be nice to have Affiliation and Homepage fields in user's preferences; this info can then be used for the list of participants.
Windows and Linux versions of Python seem to generate different random numbers, so the results are not precisely reproducible on all computers.
Retrieved from "https://econgames.org/w/index.php?title=Technical_Issues&oldid=107"
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Morera type problems in Clifford analysis | EMS Press
Universidad Nacional Autónoma de México, Cuernavaca, Mexico
The Pompeiu and the Morera problems have been studied in many contexts and generality. For example in different spaces with different groups locally without an invariant measure etc. The variations obtained exhibit the fascination of these problems. In this paper we present a new aspect: we study the case in which the functions have values over a Clifford Algebra. We show that in this context it is completely natural to consider the Morera problem and its variations. Specically we show the equivalence between the Morera problem in Clifford analysis and Pompeiu problem for surfaces in
\mathbb R^n
. We also show an invariance theorem. The noncommutativity of the Clifford algebras brings in some peculiarities. Our main result is a theorem showing that the vanishing of the first moments of a Clifford valued function implies Clifford analyticity. The proof depends on results which show that a particular matrix system of convolution equations admits spectral synthesis.
Emilio Marmolejo Olea, Morera type problems in Clifford analysis. Rev. Mat. Iberoam. 17 (2001), no. 3, pp. 559–585
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Wave Equation for the Superluminal Photon
Probability of a Photon to Tunnel through the Light Barrier
RESEARCH ANALYST | VOLUME 2 | ISSUE 1 | DOI: 10.36959/665/320 OPEN ACCESS
Takaaki Musha*
Takaaki Musha 1*
Advanced Science-Technology Research Organization, Yokohama, Japan
Musha T (2019) Superluminal Speed of Photons in the Electromagnetic Near-Field. Recent Adv Photonics Opt 2(1):36-39.
The possible existence of superluminal particles, which are forbidden by well-known laws of physics, has been studied by many physicists. Some of them confirmed the superluminal speed by their experiments. By using Klein-Gordon wave equation for photons, the author shows that the photon travels at a superluminal speed in an electromagnetic near field of the source and they reduce to the speed of light as they propagate into the far field.
Photon, Near-field, Superluminal speed, Electromagnetic radiation
E. Recami claimed in his paper [1] that tunneling photons traveling in evanescent mode can move with superluminal group speed inside the barrier. Chu and S. Wong at AT&T Bell Labs measured superluminal velocities for light traveling through the absorbing material [2]. Furthermore, Steinberg, Kwait and Chiao devised an experiment measuring the tunneling time for visible light through an optical filter, consisting of a multilayer coating about m thick, and confirmed superluminal speed [3]. The results obtained by Steinberg and co-workers have shown that the photons seemed to have traveled at 1.7 times the speed of light. Recent optical experiments at Princeton NEC have verified that superluminal pulse propagation can occur in transparent media [4]. These results indicate that the process of tunneling in quantum physics is indeed superluminal, as claimed by E. Recami. Caligiuri and Musha also confirmed the existence of superluminal photons in brain microtubules theoretically [5].
The photon is a type of elementary particle, the quantum of the electromagnetic field including electromagnetic radiation such as light, and the force carrier for the electromagnetic force. The photon has zero rest mass and always moves at the speed of light in vacuum. The single photons travel through space at the group velocity of light. The group velocity of the light wave is thought of as the velocity at which energy or information is conveyed along a wave. The electromagnetic field by photons consists of the near-field and far field around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Far-field strength decreases inversely with distance from the source, resulting in an inverse-square law for the radiated power intensity of electromagnetic radiation. By contrast, near-field E and B strength decrease more rapidly with distance: The one part decreases by the inverse-distance squared, the other part by an inverse cubed law, resulting in a diminished power in the parts of the electric field by an inverse fourth-power and sixth-power, respectively.
The author published a paper that claimed a tiny particle compared with electrons like neutrinos might be generated as a superluminal particle [6]. In this paper, by applying Klein-Gordon equation, it is shown that photons are generated as a superluminal particle and moves at a superluminal speed (i.e. superluminal group velocity of the light wave) in the electromagnetic near-field from the source.
From the wave function mathematical description of a particle, taking account the special theory of relativity, the Klein-Gordon equation is considered:
i\hslash \frac{\partial \psi }{\partial t}\text{ = }H\psi \text{ (1)}
Where H is a Hamiltonian given by
H\text{ = }\sqrt{{p}^{2}{c}^{2}+{m}^{2}{c}^{4}}
( p : Momentum of the particle, m : Effective mass) and
\psi
is a wave function of a particle. The following equation can be obtained for the accelerating particle [7]
\frac{\partial \psi }{\partial p}\text{ = }-\frac{i}{ma\hslash }\sqrt{{p}^{2}{c}^{2}+{m}^{2}{c}^{4}}\psi \text{ (2)}
\hslash
is the Plank constant divided by
2\pi
, c is the speed of light, and a is a proper acceleration given by
p\text{ = }mat
From Eq. (2), we have the solution given by [7]
\psi \text{ = }C\cdot \mathrm{exp}\left[-i\frac{c}{2ma\hslash }\left(p\sqrt{{p}^{2}+{m}^{2}{c}^{2}}+{m}^{2}{c}^{2}\mathrm{log}\left(p+\sqrt{{p}^{2}+{m}^{2}{c}^{2}}\right)\right)\right]\text{ (3)}
By introducing the particle linear momentum
p\text{ = }\frac{mv}{\sqrt{1-{v}^{2}/{c}^{2}}}
, and the energy
E\text{ = }\frac{m{c}^{2}}{\sqrt{1-{v}^{2}/{c}^{2}}}
\psi \text{ = }C\cdot \mathrm{exp}\left[-i\frac{Ec}{2a\hslash }\sqrt{1-{\beta }^{2}}\left(\frac{\beta }{1-{\beta }^{2}}+\mathrm{log}\left(\frac{E}{c}\right)+\mathrm{log}\left(1+\beta \right)\right)\right]\text{ (4)}
\beta \text{ = }v/c
The proper acceleration of a particle is determined as if the photon is generated in a quantum region, with size
l
{m}_{e}\text{ = }\Delta E/{c}^{2}
(equivalent mass of a photon) and
\Delta t\text{ = }l/c
, and in addition taking care of the uncertainty principle for momentum
\Delta p\cdot l\approx \hslash
, and energy
\Delta E\text{ = }\Delta p\cdot c
, then the proper acceleration of the photon is estimated [9]:
a\approx \frac{1}{{m}_{e}}\frac{\Delta p}{\Delta t}\text{ = }\frac{{c}^{2}}{l}\text{ (5)}
When we let
E\text{ = }\hslash \omega
, we obtain wave functions for subluminal speed as [10] (
v<c
\psi \text{ = }C\cdot \mathrm{exp}\left[-i\frac{{\omega }_{}l}{2c}\sqrt{1-{\beta }^{2}}\left(\frac{\beta }{1-{\beta }^{2}}+\mathrm{log}\left(\hslash \omega /c\right)+\mathrm{log}\left(1+\beta \right)\right)\right]\text{ (6}\text{.1)}
and for superluminal speed (
v>c
{\psi }_{*}\text{ = }C\cdot \mathrm{exp}\left[-\frac{{\omega }_{}l}{2c}\sqrt{{\beta }^{2}-1}\left(\frac{\beta }{{\beta }^{2}-1}-\mathrm{log}\left(\hslash \omega /c\right)-\mathrm{log}\left(1+\beta \right)\right)\right]\text{ (6}\text{.2)}
\omega
is an angular frequency of the particle and
\beta
is the ratio of the particle velocity divided by the light speed.
Figure 1 shows the wave function for the highly accelerated particle, which shows the possibility of tunnelling through the light barrier.
If we consider the light barrier as a potential barrier, the probability of a particle to tunnel through the light barrier can be estimated in the WKB approximation and it is given by
T\approx {{|{\psi }_{*}|}^{2}/|\psi |}^{2}\text{ = }\mathrm{exp}\left[-\frac{{\omega }_{}l}{c}\sqrt{{\beta }^{2}-1}\left(\frac{\beta }{{\beta }^{2}-1}-\mathrm{log}\left(\hslash \omega /c\right)-\mathrm{log}\left(1+\beta \right)\right)\right]\text{ (7)}
Supposing that the size of the quantum region
l
is approximately equal to the Plank length
{l}_{p}
T\left(\omega \right)\approx \mathrm{exp}\left[-\frac{\omega }{c}{l}_{p}\sqrt{3}\left(\frac{2}{3}-\mathrm{log}\left(\hslash \omega /c\right)-\mathrm{log}3\right)\right]\text{ (8)}
Obtained after inserting
\beta \approx 2
, a value that can be estimated from the uncertainty principle [11].
Then Equation (8) can be rewritten under the form
T\left(\omega \right)\approx \mathrm{exp}\left[-{l}_{p}\omega \left(\gamma -\epsilon \mathrm{log}\omega \right)\right]\text{ (9)}
\gamma \text{ = }\frac{2-3\mathrm{log}\left(\hslash /c\right)-3\mathrm{log}3}{\sqrt{3}c}\approx 5.62×{10}^{-7}\text{ (10}\text{.1)}
\epsilon \text{ = }\sqrt{3}/c\approx 5.77×{10}^{-9}\text{ (10}\text{.2)}
From Equation (9), we can estimate the probability of a photon to tunnel through the light barrier, represented graphically in Figure 2. In this figure, the horizontal line is for the radial frequency of the photon and the vertical line is for the probability of photons to tunnel through the light barrier.
From this figure, the probability of a photon in a superluminal state almost becomes unity up to the Plank frequency (
1.85×{10}^{43}
Hz), and it is seen that the photon is generated as a superluminal particle. From Figure 1, it can be seen that a tunneling particle through the light barrier returns to the original state within a finite length of time, which can be estimated by the uncertainty principle. The difference of momentum
\Delta p
of the photon in an original state at the source and the photon in a superluminal state can be estimated to be
\Delta p\text{ = }p
, where p is the momentum of the photon in a superluminal state. As the momentum of the photon can be described as
p\text{ = }\hslash \cdot k\text{ = }h/\lambda
, then from the uncertainty principle,
\Delta p\cdot \Delta x\approx \hslash
\Delta x\approx \frac{\hslash }{p}\text{ = }\frac{\lambda }{2\pi }\text{ (11)}
Which represents the traveling distance of a photon in a superluminal state.
This value coincides with the region of the near field of the electromagnetic wave as shown in Figure 3 and we can conclude that photons moves in a near field from the source at the superluminal speed and reduce to the speed of light (i.e. group velocity of the light wave) as they propagate into the far field.
W. D. Walkers obtained the same result for the electromagnetic wave by using the electromagnetic theory [12-14], and he claimed that it should be possible to reduce the time delay by monitoring lower frequency EM field in the electromagnetic near-field.
From the above theoretical analysis, it is seen that photons propagate at superluminal speed in the electromagnetic near-field from the source and reduce to the speed of light as they propagate into the far field. This phenomenon can be applied for speeding up the processing time of photonic computer systems.
The author appreciates Prof. Mario J. Pinheiro at University of Lisbon (IST) to help him to revise this paper.
Recami E (2001) A bird's-eye view of the experimental status-of-the-art for superlsuminal motions. Foundation of Physics 32: 1119-1135.
Brown J (1995) Faster than the speed of light. New Scientist 146: 26-30.
Steinberg AM, Kwait PG, Chiao RY (1993) Measurement of the single-photon tunneling time. Physical Review Letters 71: 708-711.
Wang LJ, Kuzmich A, Dogariu A (2000) Gain-assisted superluminal light propagation. Nature 406: 277-279.
Caligiuri LM, Musha T (2014) Quantum vacuum dynamics, coherence, superluminal photons and hypercomputation in brain microtubules. Applied Numerical Mathematics and Scientific Computation 105-115.
Musha T (1998) Possible existence of faster-than-light phenomena for highly accelerated elementary particles. Speculations in Science and Technology 21: 29-36.
Musha T (2005) Superluminal effect for quantum computation that utilizes tunneling photons. Physics Essays 18: 525-529.
Jyukov AI (1961) Introduction to the theory of relativity. National Culture Physics Mathematics Library, Moscow.
Caianiello ER (1984) Maximal acceleration as a consequence of Heisenberg's uncertainty relations. Lettere Al Nuovo Cimento 41: 370-372.
Musha T (2009) Thermal radiation generated inside the Sun due to the Cherenkov radiation from ZPF field. Far East Journal of Applied Mathematics 37: 229-235.
Musha T (2001) Cherenkov radiation from faster-than-light photons created in a ZPF background. Journal of Theoretics 3.
Walker WD (2000) Experimental evidence of near field superluminally propagating electromagnetic fields. Viger III Symposium "Gravitation and Cosmology", Berkley, California, USA, 21-25.
Walker WD (1999) Superluminal near-field dipole electromagnetic fields. International Workshop Lorentz Group CPT and Neutrinos, World Scientific, Zacatecas, Mexico.
Walkers WD (1998) Superluminal propagation speed of longitudinally oscillating electrical fields. Conference on Causality and Locality in Modern Physics, Toronto, Canada.
Takaaki Musham, Advanced Science-Technology Research Organization, Yokohama, Japan.
© 2019 Musha T. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Figure 1: The wave function of a tunneling particle through the light barrier.
Figure 2: Probability of a photon to tunnel through the light barrier.
Figure 3: The near and far electromagnetic fields.
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!!12 Upper Gower Stt!!
Monday January 1st. 18391 | And the first of Our Marriage
Many thanks for your two most kind, dear, & affectionate letters, which I received this morning.— I will finish this letter tomorrow. I sit down just to date & begin it, that I may enjoy the infinite satisfaction of writing to my own dear wife, that is to be, the very first evening of my entering our house. After writing to you on Saturday evening, I thought much of the happy future & in consequence did not close my eyes till long past two oclock, awoke at five & could not go to sleep.— got up, & set to work with the good resolution of spending a quiet day.— about eleven oclock found that would never do, so rang for Covington & said “I am very sorry to spoil your Sunday, but begin packing up I must, as I cannot rest”:— “Pack up, Sir, what for?” said Mr Covington with his eyes open with astonishment, as it was the first notice he had received of my flitting. So we arranged some ⟨of⟩2 the specimens of Natural History, but did no real packing up.— I, however, sorted a multitude of papers.—
This morning however, we began early & in earnest, & I may be allowed to boast, when I say that by half past three we had two large vans full of goods, well & carefully packed.— by six oclock we had them all safe here.— There is nothing left but some few dozen drawers of shells, which must be carried by Hand.— I was astounded, & so was Erasmus at the bulk of my luggage & the Porters were even more so at the weight of those containing my Geological Specimens.— The dining room, hall, & my own room are crammed & piled with goods— One servants room up stairs, & my own charming room below will hold all most admirably.— There never was so good a house for me, & I devoutly trust you will approve of it equally.— the little garden is worth its weight in gold.— About eight oclock, the old lady here, cooked me some eggs & bacon, (as I had no dinner), & with some tea, I felt supremely comfortable; How I wish my own dear lady had been here.— My room is so quiet, that the contrast to Marlborough is as remarkable, as it is delightful.— It is now near nine, & I will write no more, as I am thoroughily tired in the legs,—but wish you a good night, my own good dear Emma.— C. D.—
Tuesday morning— Once ⟨mo⟩re I must thank you for your letters, which I have just read. — — I have been busy at work all morning, & have made my own room quite charming so comfortable.— the only difficulty is, that I have not things enough!! to put in all the drawers & corners. Erasmus has just been here, & has properly admired & declares that the house in many respects is even better than Tavistock Square.— He has me to dine with him to day at
\frac{1}{2}
past four to meet the Hensleighs & Carlyles, which I shall ⟨ ⟩ do, as I am anxious to see Fanny to have some Maid-servant talk. After due deliberation, & having received your letters on Monday I write to Margaret.—to take her With respect to the sheets you had better send them direct here 12. Upper Gower ⟨S⟩t.— send them by water for there is no such great hurry for them, as I find I have enough old ones from Cambridge for Covington & Margaret for one week, & I have borrowed a pair from Erasmus for myself.— This vile paper, which was the only sort I could yesterday get out, puts my fingers on an edge, it is so rough.—
I can neither write nor think about anything, but the house. I am in such spirits at our good fortune. Erasmus & Coy used to be always talking of the immense advantage of Chester Square being so near the Park.— would you believe it, I find by the Compasses, we are as near, within a hundred yards of Regents Park, as Chester Square is of Green Park!! I quite agree with you, that this house is far pleasanter than Gordon Square. In two more days, I shall be quite settled & this change from mental & bodily work, will I do not doubt rest me, so that I trust to be able to finish my Glen Roy Paper & enjoy my Country Holiday, with a clear conscience.—
I will not write anymore at present, but will write very ⟨soon⟩ again.— I want just to scribble one line home, & then go & get the Respirator for Uncle Jos
Most affectionly Yours | Chas Darwin
P.S. I have heard of a Cook, who promises well through Erasmus’s maid Sarah, I mean to get Fanny, as your representative, to see her.—
1 January 1839 was a Tuesday. As the letter makes clear, CD began it on the evening of Monday, 31 December.
Two excisions were made in this letter by Emma Wedgwood, see letter from Emma Wedgwood, [3 January 1839], and letter to Emma Wedgwood, [6–7 January 1839]. The four sets of ⟨ ⟩ in the transcription indicate alternately the recto and verso of the excisions.
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What Divides Us | Cathologies
January 2, 2021 January 2, 2021 cathologies_wvbrzv
Ah, ha…gotcha! You thought this post was going to be about politics, I’ll bet, but it is really about something a bit more…mathematical.
Back in school, you were probably taught a few easy tricks to test for divisibility of one number by another number. For instance, if the last digit is even, the number is divisible by 2; if the last number ends in a zero, it is divisible by 10; if the last number is divisible by 5, so is the number; if the individual digits of the number add up to a number divisible by 3, then, so is the number (this works for 9, as well).
There were even more obscure divisibility tests, such as: to test to see if a number is divisible by 11, add the odd-place digits and subtract the even-place digits. If the result is divisible by 11, then, so was the original number. For example: is 286 divisible by 11? Well, the first digit is 6 and the third digit is 2, so 6+2=8 and the second digit is 8, so 8-8=0 and 0 counts as a number divisible by 11, according to the method, so 286 is, theoretically, divisible by 11 and, indeed, 26 x 11 = 286. Let’s take a harder example: 1,358,016.
so, the number is divisible by 11 (in fact, it is 11 x 123,456).
Why these tests work is usually taught in an elective course on elementary number theory to undergraduate mathematics majors.
Back around 2000 A. D., I was teaching math to 7th- and 8th-graders in a private Catholic school and I decided to teach the class divisibility tests. I started to play around with them and I developed a comprehensive system for testing divisibility by prime numbers. Why primes? This is because it can be rigorously proven that all numbers are either primes (having no divisors except themselves and 1) or may be factored into a series of primes that, when multiplied together, give the, “composite,” number. This is called the Fundamental Theorem of Arithmetic. For example, the number 5040 =
{2}^{4} x {3}^{2} x 5 x 7
. The first 10 prime numbers are:
What started this post is a dream I had, last night, involving the numbers 161 and 63 (don’t ask me why) and I decided to see if 161 were prime (63 is not, as the digits add up to 9, so the number is divisible by 9). As I was lying down, I tried to recall my earlier work and I rediscovered, pretty quickly, the divisibility test for 7 and I tried it on the number and the test result showed that 161 was divisible by 7 (it is 23 x 7), so, it is not a prime.
We have already seen divisibility tests for a few primes: 2, 3, 5, 11. What about some of the other primes, like 7 or 13? Are there tests for these? It turns out that there are and it shows the depths that even simple math can reveal. Here are some of the tests I devised back in 2000, but never published (I did write an article for a journal on math education, but I never sent it in):
Multiply the last digit by 2; subtract it from the rest; if the result is divisible by 7, so is the original number; if necessary, repeat until a small enough number is obtained.
Multiply the last digit by 9; subtract it from the rest of the digits; if the result is divisible by 13, then, so is the original number; if necessary, repeat the process until a small enough number is obtained.
so, 59,774 is divisible by 13.
Multiply the last digit by 5 and subtract from the rest. Repeat, if necessary.
34 is divisible by 17, so 424,405 is divisible by 17 (it is 24,965 x 17).
Multiply the last digit by 2; add it to the rest of the digits; if the resulting number is divisible by 19, so is the original number.
26+12=38=2×19
so, the number is divisible by 19.
Why did we switch from subtracting to adding? The mystery deepens (more, later)!
Multiply the last digit by 7, add to the rest of the numbers. Repeat, if necessary.
46=23×2, so, the original number is divisible by 23.
Multiply the last digit by 3 and add to the remaining digits. If the number is divisible by 29, so was the original number. Repeat, if necessary.
so, the original number is divisible by 29.
There are two questions to ask: 1) is this, “multiply by a seed number and add/subtract until reduced,” generalizable, 2) why was there a switch from subtraction to addition at 19?
The second question is easy to answer if we notice that we could have switched to subtraction, for, say, 19, if we had used multiply the last digit by 17 and then subtracted to reduction:
11-68=-57=-3×19
so, again, the number is divisible by 19.
Notice, that 17+2=19, so the two seeds add up to the divisor. Is this generalizable? Let’s look at 23. We used a seed of 7 and addition. What if we switched and used 23-7=16 as a seed and subtraction. Does this work? Let’s see:
So far, our hypothesis seems to work. This conjecture is:
Seed/Subtract+ Seed/Add = Divisor
Let’s make a table to summarize the hypothetical results, so far:
Divisor Seed/Subtract Seed/Add
2 Standard Test Standard Test
11 Standard Test Standard Test
According to this hypothesis, times 5 and add should work for division by 7. Does it:
21+35=56=7×8,
so, the number is divisible by 7.
It should be apparent that we chose the smaller of the two seeds in each of our examples, so addition/subtraction is determined by that fact. If we write +5 to indicate seed/addition and -2 to indicate seed/subtraction, then we can make the chart, above, more streamlined. The smaller test is indicated.
Divisor Small Seed Large Seed
2 S. T. S. T.
11 S. T. S. T.
Could there be a different test for 11 based on the seed method? An obvious choice would be a seed of -1 or +10:
59-4=55=5×11
so, 601,678 is divisible by 11.
We can add seeds for the rest of the numbers that use other techniques. The resulting table is:
2 No Seed No Seed
It is easy to see that no seed is possible for 2, because of a parity (even/odd) problem. If the left-over is odd, but the last digit is even, the result will always be odd, which will invalidate the test. For example, there is no seed that will work for 34, because the number to be added or subtracted will always be even, because an even number (in this case, 4) time any number returns an even number and when subtracted or added to 3 will always return an odd number. This, “two problem,” may be at the heart of finding a universal test for prime divisors.
In any case, what is behind this method and is it generalizable to any prime? The secret to finding the seeds is to write the original number as a simple Diophantine equation, for example, with 29:
2 + d*9= 29
The value of d is, obviously, 3, and the leftover should be added. We can call the smaller seed, S, and the larger seed the compliment, Ś. Thus, Conjecture 1 may be re-written as:
S + Ś=d
Does this method always work and is it unique, in that does a given seed work only for a unique prime? That will be the subject of a future post. I suspect that the answer is, yes, but, possibly for only one of the two forms of the seed.
Previous Post O Adonai. Thoughts for December 18
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Existence and Global Behavior of Positive Solutions for Some Fourth-Order Boundary Value Problems
2014 Existence and Global Behavior of Positive Solutions for Some Fourth-Order Boundary Value Problems
Ramzi S. Alsaedi
We establish the existence and uniqueness of a positive solution to the following fourth-order value problem:
{u}^{\left(4\right)}\left(x\right)=a\left(x\right){u}^{\sigma }\left(x\right)
x\in \left(0,1\right)
u\left(0\right)=u\left(1\right)=u\mathrm{\text{'}}\left(0\right)=u\mathrm{\text{'}}\left(1\right)=0
\sigma \in \left(-1,1\right)
a
is a nonnegative continuous function on (0, 1) that may be singular at
x=0
x=1
. We also give the global behavior of such a solution.
Ramzi S. Alsaedi. "Existence and Global Behavior of Positive Solutions for Some Fourth-Order Boundary Value Problems." Abstr. Appl. Anal. 2014 (SI54) 1 - 5, 2014. https://doi.org/10.1155/2014/657926
Ramzi S. Alsaedi "Existence and Global Behavior of Positive Solutions for Some Fourth-Order Boundary Value Problems," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI54), 1-5, (2014)
|
2014 A Predictor-Corrector Method for Solving Equilibrium Problems
Zong-Ke Bao, Ming Huang, Xi-Qiang Xia
We suggest and analyze a predictor-corrector method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle. In the main algorithm each stage of computation requires two proximal steps. One step serves to predict the next point; the other helps to correct the new prediction. At the same time, we present convergence analysis under perfect foresight and imperfect one. In particular, we introduce a stopping criterion which gives rise to
\mathrm{\Delta }
-stationary points. Moreover, we apply this algorithm for solving the particular case: variational inequalities.
Zong-Ke Bao. Ming Huang. Xi-Qiang Xia. "A Predictor-Corrector Method for Solving Equilibrium Problems." Abstr. Appl. Anal. 2014 (SI71) 1 - 11, 2014. https://doi.org/10.1155/2014/313217
Zong-Ke Bao, Ming Huang, Xi-Qiang Xia "A Predictor-Corrector Method for Solving Equilibrium Problems," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI71), 1-11, (2014)
|
Impulse response length - MATLAB impzlength - MathWorks Switzerland
IIR Filter Effective Impulse Response Length — Coefficients
IIR Filter Effective Impulse Response Length — Second-Order Sections
IIR Filter Effective Impulse Response Length — Digital Filter
len = impzlength(b,a)
len = impzlength(sos)
len = impzlength(d)
len = impzlength(___,tol)
len = impzlength(b,a) returns the impulse response length for the causal discrete-time filter with the rational system function specified by the numerator, b, and denominator, a, polynomials in z–1. For stable IIR filters, len is the effective impulse response sequence length. Terms in the IIR filter’s impulse response after the len-th term are essentially zero.
len = impzlength(sos) returns the effective impulse response length for the IIR filter specified by the second order sections matrix, sos. sos is a K-by-6 matrix, where the number of sections, K, must be greater than or equal to 2. If the number of sections is less than 2, impzlength considers the input to be the numerator vector, b. Each row of sos corresponds to the coefficients of a second order (biquad) filter. The ith row of the sos matrix corresponds to [bi(1) bi(2) bi(3) ai(1) ai(2) ai(3)].
len = impzlength(d) returns the impulse response length for the digital filter, d. Use designfilt to generate d based on frequency-response specifications.
len = impzlength(___,tol) specifies a tolerance for estimating the effective length of an IIR filter’s impulse response. By default, tol is 5e-5. Increasing the value of tol estimates a shorter effective length for an IIR filter’s impulse response. Decreasing the value of tol produces a longer effective length for an IIR filter’s impulse response.
Create a lowpass allpole IIR filter with a pole at 0.9. Calculate the effective impulse response length. Obtain the impulse response. Plot the result.
stem(t,h)
h(len)
Design a 4th-order lowpass elliptic filter with a cutoff frequency of 0.4π rad/sample. Specify 1 dB of passband ripple and 60 dB of stopband attenuation. Design the filter in pole-zero-gain form and obtain the second-order section matrix using zp2sos. Determine the effective impulse response sequence length from the second-order section matrix.
[z,p,k] = ellip(4,1,60,.4);
Use designfilt to design a 4th-order lowpass elliptic filter with normalized passband frequency 0.4π rad/sample. Specify 1 dB of passband ripple and 60 dB of stopband attenuation. Determine the effective impulse response sequence length and visualize it.
impz(d)
Numerator coefficients, specified as a scalar (allpole filter) or a vector.
Denominator coefficients, specified as a scalar (FIR filter) or vector.
sos — Matrix of second order sections
Matrix of second order sections, specified as a K-by-6 matrix. The system function of the K-th biquad filter has the rational Z-transform
{H}_{k}\left(z\right)=\frac{{B}_{k}\left(1\right)+{B}_{k}\left(2\right){z}^{-1}+{B}_{k}\left(3\right){z}^{-2}}{{A}_{k}\left(1\right)+{A}_{k}\left(2\right){z}^{-1}+{A}_{k}\left(3\right){z}^{-2}}.
\left[\begin{array}{cccccc}{B}_{k}\left(1\right)\text{ }& {B}_{k}\left(2\right)\text{ }& {B}_{k}\left(3\right)& {A}_{k}\left(1\right)\text{ }& {A}_{k}\left(2\right)& {A}_{k}\left(3\right)\end{array}\right]
z={e}^{j2\pi f}.
tol — Tolerance for IIR filter effective impulse response length
Tolerance for IIR filter effective impulse response length, specified as a positive number. The tolerance determines the term in the absolutely summable sequence after which subsequent terms are considered to be 0. The default tolerance is 5e-5. Increasing the tolerance returns a shorter effective impulse response sequence length. Decreasing the tolerance returns a longer effective impulse response sequence length.
len — Length of impulse response
Length of the impulse response, specified as a positive integer. For stable IIR filters with absolutely summable impulse responses, impzlength returns an effective length for the impulse response beyond which the coefficients are essentially zero. You can control this cutoff point by specifying the optional tol input argument.
To compute the impulse response for an FIR filter, impzlength uses the length of b. For IIR filters, the function first finds the poles of the transfer function using roots.
If the filter is unstable, the length extends to the point at which the term from the largest pole reaches 106 times its original value.
If the filter is stable, the length extends to the point at which the term from the largest-amplitude pole is tol times its original amplitude.
If the filter is oscillatory, with poles on the unit circle only, then impzlength computes five periods of the slowest oscillation.
If the filter has both oscillatory and damped terms, the length extends to the greater of these values:
Five periods of the slowest oscillation.
The point at which the term due to the largest pole is tol times its original amplitude.
If the first input to impzlength is a variable-size matrix at compile time, then it must not become a vector at runtime.
designfilt | digitalFilter | impz | zp2sos
|
Rotation invariant subspaces of Besov and Triebel-Lizorkin space: compactness of embeddings, smoothness and decay of functions | EMS Press
H
be a closed subgroup of the group of rotation of
\mathbb{R}^n
. The subspaces of distributions of Besov-Lizorkin-Triebel type invariant with respect to natural action of
H
are investigated. We give sufficient and necessary conditions for the compactness of the Sobolev-type embeddings. It is also proved that
H
-invariance of function implies its decay properties at infinity as well as the better local smoothness. This extends the classical Strauss lemma. The main tool in our investigations is an adapted atomic decomposition.
Leszek Skrzypczak, Rotation invariant subspaces of Besov and Triebel-Lizorkin space: compactness of embeddings, smoothness and decay of functions. Rev. Mat. Iberoam. 18 (2002), no. 2, pp. 267–299
|
142,857 - Wikipedia
"1/7 (number)" redirects here. For the number 7, see 7 (number).
Main articles: Repeating decimal and Cyclic number
one hundred forty-two thousand eight hundred fifty-seven
(one hundred forty-two thousand eight hundred fifty-seventh)
1, 3, 9, 11, 13, 27, 33, 37, 39, 99, 111, 117, 143, 297, 333, 351, 407, 429, 481, 999, 1221, 1287, 1443, 3663, 3861, 4329, 5291, 10989, 12987, 15873, 47619, 142857
{\displaystyle {\stackrel {\iota \delta }{\mathrm {M} }}}
͵βωνζ´
142857, the six repeating digits of 1/7 (0.142857), is the best-known cyclic number in base 10.[1][2][3][4] If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2/7, 3/7, 4/7, 5/7, or 6/7 respectively.
142,857 is a Kaprekar number.[5]
2 1/7 as an infinite sum
4 Connection to the enneagram
If multiplying by an integer greater than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits (ones through hundred thousands) to the remaining digits and repeating this process until only six digits are left, it will result in a cyclic permutation of 142857:[citation needed]
Multiplying by a multiple of 7 will result in 999999 through this process:
If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number.[citation needed]
It is the repeating part in the decimal expansion of the rational number 1/7 = 0.142857. Thus, multiples of 1/7 are simply repeated copies of the corresponding multiples of 142857:
7/7 = 0.999999 = 1
1/7 as an infinite sum[edit]
There is an interesting pattern of doubling, shifting and addition that gives 1/7.
{\displaystyle {\begin{aligned}{\frac {1}{7}}&=0.142857142857142857\ldots \\[6pt]&=0.14+0.0028+0.000056+0.00000112+0.0000000224+0.000000000448+0.00000000000896+\cdots \\[6pt]&={\frac {14}{100}}+{\frac {28}{100^{2}}}+{\frac {56}{100^{3}}}+{\frac {112}{100^{4}}}+{\frac {224}{100^{5}}}+\cdots +{\frac {7\times 2^{N}}{100^{N}}}+\cdots \\[6pt]&=\left({\frac {7}{50}}+{\frac {7}{50^{2}}}+{\frac {7}{50^{3}}}+{\frac {7}{50^{4}}}+{\frac {7}{50^{5}}}+\cdots +{\frac {7}{50^{N}}}+\cdots \right)\\[6pt]&=\sum _{k=1}^{\infty }{\frac {7}{50^{k}}}\end{aligned}}}
Each term is double the prior term shifted two places to the right. This is can be proved by applying the identity for the sum of a geometric sequence:
{\displaystyle \sum _{k=1}^{\infty }{\frac {7}{50^{k}}}=7\cdot \sum _{k=1}^{\infty }\left({\frac {1}{50}}\right)^{k}=7\cdot {\frac {1}{50-1}}={\frac {7}{49}}={\frac {1}{7}}}
Another infinite sum is
{\displaystyle {\begin{aligned}{\frac {1}{7}}&=0.1+0.03+0.009+0.0027+0.00081+0.000243+0.0000729+\cdots \\[6pt]&={\frac {3^{0}}{10^{1}}}+{\frac {3^{1}}{10^{2}}}+{\frac {3^{2}}{10^{3}}}+{\frac {3^{3}}{10^{4}}}+{\frac {3^{4}}{10^{5}}}+\cdots +{\frac {3^{N-1}}{10^{N}}}+\cdots \\[6pt]&=\sum _{k=1}^{\infty }{\frac {3^{k-1}}{10^{k}}}=3^{-1}\cdot \sum _{k=1}^{\infty }\left({\frac {3}{10}}\right)^{k}={\frac {1}{3}}\cdot {\frac {3}{10-3}}={\frac {1}{7}}\end{aligned}}}
Other bases[edit]
In some other bases, six-digit numbers with similar properties exist, given by base6 − 1/7.[citation needed] For example, in base 12 it is 186A35 and base 24 3A6KDH.
Connection to the enneagram[edit]
The 142857 number sequence is used in the enneagram figure, a symbol of the Gurdjieff Work used to explain and visualize the dynamics of the interaction between the two great laws of the Universe (according to G. I. Gurdjieff), the Law of Three and the Law of Seven. The movement of the numbers of 142857 divided by 1/7, 2/7. etc., and the subsequent movement of the enneagram, are portrayed in Gurdjieff's sacred dances known as the movements.[6]
The 142857 number sequence is also found in several decimals in which the denominator has a factor of 7. In the examples below, the numerators are all 1, however there are instances where it does not have to be, such as 2/7 (0.285714).
For example, consider the fractions and equivalent decimal values listed below:
1/28 = 0.03571428...
The above decimals follow the 142857 rotational sequence. There are fractions in which the denominator has a factor of 7, such as 1/21 and 1/42, that do not follow this sequence and have other values in their decimal digits.
^ "Cyclic number". The Internet Encyclopedia of Science. Archived from the original on 2007-09-29.
^ Ecker, Michael W. (March 1983). "The Alluring Lore of Cyclic Numbers". The Two-Year College Mathematics Journal. 14 (2): 105–109. doi:10.2307/3026586. JSTOR 3026586.
^ "Cyclic number". PlanetMath. Archived from the original on 2007-07-14.
^ Hogan, Kathryn (August 2005). "Go figure (cyclic numbers)". Australian Doctor. Archived from the original on 2007-12-24.
^ "Sloane's A006886: Kaprekar numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-03.
^ Ouspensky, P. D. (1947). "Chapter XVIII". In Search of the Miraculous: Fragments of an Unknown Teaching. London: Routledge.
Leslie, John (1820). The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of…. Longman, Hurst, Rees, Orme, and Brown. ISBN 1-4020-1546-1.
Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (revised ed.). London: Penguin Group. pp. 171–175. ISBN 978-0-140-26149-3.
Tahan, Malba (1938). The Man Who Counted.
Retrieved from "https://en.wikipedia.org/w/index.php?title=142,857&oldid=1085871619"
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On independent times and positions for Brownian motions | EMS Press
(B_t ; t \ge 0)
\big(\mbox{resp. }((X_t, Y_t) ; t \ge 0)\big)
be a one (resp. two) dimensional Brownian motion started at 0. Let
T
be a stopping time such that
(B_{t \wedge T} ; t \ge 0)
\big(resp.
(X_{t \wedge T} ; t \ge 0) ; (Y_{t \wedge T} ; t \ge 0)\big)
is uniformly integrable. The main results obtained in the paper are: \begin{itemize} \item[1)] if
T
B_T
T
has all exponential moments, then
T
is constant. \item[2)] If
X_T
Y_T
are independent and have all exponential moments, then
X_T
Y_T
are Gaussian. \end{itemize} We also give a number of examples of stopping times
T
, with only some exponential moments, such that
T
B_T
are independent, and similarly for
X_T
Y_T
. We also exhibit bounded non-constant stopping times
T
X_T
Y_T
are independent and Gaussian.
Bernard de Meyer, Bernard Roynette, Pierre Vallois, Marc Yor, On independent times and positions for Brownian motions. Rev. Mat. Iberoam. 18 (2002), no. 3, pp. 541–586
|
The table below lists the time and elevation of a submarine as it dives towards the ocean floor.
10
30
60
90
120
−150
−450
−900
−1350
−1800
Why is the elevation given as a negative number?
This problem is about a submarine.
Submarines usually travel . . .
What happens to the elevation of the submarine as the time increases?
The elevation is getting more negative, which means the submarine is going deeper.
Using the pattern in the table, what is the elevation of the submarine after 3 hours? After 45 minutes? After 0 minutes?
After 3 hours (180 minutes) the elevation is
−2700
After 45 minutes the elevation is
−675
After 0 minutes, the elevation is
0
Graph the relationship in the table.
Plot the points. Put time on the
x
-axis and elevation on the
y
-axis. One point would be at
(10, −150)
Review the Math Notes box in this lesson. Is this relationship proportional? Why or why not? If the relationship is proportional, is it increasing or decreasing? If it is not proportional, explain why not.
As you review the Math Notes box, which type of graph does your graph look most like?
|
\left(1, 6\right)
\left(4, 48\right)
, find an equation of the function. Refer to the Math Notes box in this lesson if you need help.
To find an exponential function that goes through two given points, create a system of equations by substituting one (
x,y
) point into
y=ab^x
, then substituting the other point. Rewrite both equations in “
a=
” form. Solve the system with the equal values method to find
and
b
and now you can write the equation.
For example, find an exponential function that passes through
(2,14)
(5,112)
. Create a system of equations by substituting (
x,y
=(2,14)
y=ab^x
, and then substituting again with (
x,y
=(5,112)
14=ab^2
112=ab^5
a=\frac{14}{b^2}
a=\frac{112}{b^5}
Use the equal values method to find
b
\left. \begin{array}{l}{ \frac { 14 } { b ^ { 2 } } = \frac { 112 } { b ^ { 5 } } }\\{ 14 b ^ { 5 } = 112 b ^ { 2 } }\\{ 14 \cdot \frac { b ^ { 5 } } { b ^ { 2 } } = 112 }\\{ 14 b ^ { 3 } = 112 }\\{ b ^ { 3 } = \frac { 112 } { 14 } = 8}\\{ b = 2 }\end{array} \right.
\Huge\nearrow
Use either original equation to find
a
\left. \begin{array} { l } { 14 = a b ^ { 2 } } \\ { 14 = a ( 2 ) ^ { 2 } } \\ { \frac { 14 } { 4 } = a } \\ { 3.5 = a } \end{array} \right.
The equation of the exponential function that passes through the two given points is
y=3.5·2^x
a=3
|
§ What is a syzygy?
Word comes from greek word for "yoke" . If we have two oxen pulling, we yoke them together to make it easier for them to pull.
§ The ring of invariants
Rotations of
\mathbb R^3
: We have a group
SO(3)
which is acting on a vector space
\mathbb R^3
. This preserves the length, so it preserves the polynomial
x^2 + y^2 + z^2
. This polynomial
x^2 + y^2 + z^2
is said to be the invariant polynomial of the group
SO(3)
acting on the vector space
\mathbb R^3
But what does
x, y, z
even mean? well, they are linear function
x, y, z: \mathbb R^3 \rightarrow \mathbb R
x^2 + y^2 + z^2
is a "polynomial" of these linear functions.
§ How does a group act on polynomials?
G
V
, how does
G
act on the polynomial functions
V \rightarrow \mathbb R
In general, if we have a function
f: X \rightarrow Y
g
X
Y
(in our case,
G
acts trivially on
Y=\mathbb R
), what is
g(f)
(gf)(x) \equiv g (f(g^{-1}(x)))
g^{-1}
? We should write
(gf)(gx) = g(fx)
. This is like
g(ab) = g(a) g(b)
. We want to get
(gf)(x) = (gf)(g(g^{-1}x) = g(f(g^{-1}x))
If we miss out
g^{-1}
we get a mess. Let's temporarily define
(gf)(x) = f(g(x))
(gh)f(x) = f(ghx)
. But we can also take this as
(gh)(f(x)) = g((hf)(x)) = (hf)(gx) = f(hgx)
. This is absurd as it gives
f(ghx) = f(hgx)
§ Determinants
SL_n(k)
k^n
, it acts transitively, so there's no interesting non-constant invariants. On the other hand, we can have
SL_n(k)
\oplus_{i=1}^n k^n
n=2
\begin{bmatrix} a & b \\ c & d \end{bmatrix}
\begin{bmatrix} x_1 & y_1 \\ x_2 & y_2 \end{bmatrix}
This action preserves the polynomial
x_1 y_2 - x_2 y_1
, aka the determinant. anything that ends with an "-ant" tends to be an "invari-ant" (resultant, discriminant)
S_n
\mathbb C^n
by permuting coordinates.
Polynomials are functions
\mathbb C[x_1, \dots, x_n]
. Symmetric group acts on polynomials by permuting
x_1, \dots, x_n
. What are the invariant polynomials?
e_1 \equiv x_1 + x_2 + \dots x_n
e_2 \equiv x_1 x_2 + x_1 x_3 + \dots + x_{n-1} x_n
e_n \equiv x_1 x_2 \dots x_n
These are the famous elementary symmetric functions. If we think of
(y - x_1) (y - x_2) \dots (y - x_n) = y^n - e_1 y^{n-1} + \dots e_n
The basic theory of symmetric functions says that every invariant polynomial in
x_1, \dots x_n
e_1, \dots, e_n
§ Proof of elementary theorem
Define an ordering on the monomials; order by lex order. Define
x_1^{m_1} x_2^{m_2} > x_1^{n_1} x_2^{n_2} \dots
m_1 > n_1
m_1 = n_1 \land m_2 > n_2
m_1 = n_1 \land m_2 = n_2 \land m_3 > n_3
and so on. Suppose
f \in \mathbb C[x_1, \dots, x_n]
is invariant. Look at the biggest monomial in
f
. Suppose it is
x_1^{n_1} x_2^{n_2} \dots
. We subtract:
\begin{aligned} P \equiv &(x_1 + x_2 \dots)^{n_1 - n_2} \\ &\times (x_1 x_2 + x_1 x_2 \dots)^{n_2 - n_3} \\ &\times (x_1 x_2 x_3 + x_1 x_2 x_4 \dots)^{n_3 - n_4} \\ \end{aligned}
This kills of the biggest monomial in
f
. If
is symmetric, Then we can order the term we choose such that
n_1 \geq n_2 \geq n_3 \dots
. We need this to keep the terms
(n_1 - n_2), (n_2 - n_3), \dots
to be positive. So we have now killed off the largest term of
. Keep doing this to kill of
completely. This means that the invariants of
S_n
\mathbb C^n
are a finitely generated algebra over
\mathbb C
. So we have a finite number of generating invariants such that every invariant can be written as a polynomial of the generating invariants with coefficients in
\mathbb C
. This is the first non-trivial example of invariants being finitely generated. The algebra of invariants is a polynomial ring over
e_1, \dots, e_n
. This means that there are no non-trivial-relations between
e_1, e_2, \dots, e_n
. This is unusual; usually the ring of generators will be complicated. This simiplicity tends to happen if
G
is a reflection group. We haven't seen what a syzygy is yet; We'll come to that.
§ Complicated ring of invariants
A_n
(even permutations). Consider the polynomial
\Delta \equiv \prod_{i < j} (x_i - x_j)
This is called as the discriminant. This looks like
(x_1 - x_2)
(x_1 - x_2)(x_1 - x_3)(x_2 - x_3)
, etc. When
S_n
\Delta
, it either keeps the sign the same or changes the sign.
A_n
S_n
that keeps the sign fixed. What are the invariants of
A_n
? It's going to be all the invariants of
S_n
e_1, \dots, e_n
\Delta
(because we defined
A_n
to stabilize
\Delta
). There are no relations between
e_1, \dots, e_n
. But there are relations between
\Delta^2
e_1, \dots, e_n
\Delta^2
is a symmetric polynomial. Working this out for
n=2
\Delta^2 = (x_1 - x_2)^2 = (x_1 + x_2)^2 - 4 x_1 x_2 = e_1^2 - 4 e_1 e_2
. When
gets larger, we can still express
\Delta^2
in terms of the symmetric polynomials, but it's frightfully complicated. This phenomenon is an example of a Syzygy. For
A_n
, the ring of invariants is finitely generated by
(e_1, \dots, e_n, \Delta)
. There is a non-trivial relation where
\Delta^2 - poly(e_1, \dots, e_n) = 0
. So this ring is not a polynomial ring. This is a first-order Syzygy. Things can get more complicated!
§ Second order Syzygy
Z/3Z
\mathbb C^2
s
be the generator of
Z/3Z
. We define the action as
s(x, y) = (\omega x, \omega y)
\omega
is the cube root of unity. We have
x^ay^b
is invariant if
(a + b)
3
, since we will just get
\omega^3 = 1
. So the ring is generated by the monomials
(z_0, z_1, z_2, z_3) \equiv (x^3, x^2y, xy^2, y^3)
. Clearly, these have relations between them. For example:
z_0 z_2 = x^4y^2 = z_1^2
z_0 z_2 - z_1^2 = 0
z_1 z_3 x^2y^4 = z_2^2
z_1 z_3 - z_2^2 = 0
z_0 z_3 = x^3y^3 = z_1 z_2
z_0 z_3 - z_1 z_2 = 0
We have 3 first-order syzygies as written above. Things are more complicated than that. We can write the syzygies as:
p_1 \equiv z_0 z_2 - z_1^2
p_2 \equiv z_1 z_3 - z_2^2
p_3 \equiv z_0 z_3 - z_1 z_2
z_0 z_2
p_1
. Let's try to cancel it with the
z_2^2
p_2
. So we consider:
\begin{aligned} & z_2 p_1 + z_0 p_2 \\ &= z_2 (z_0 z_2 - z_1^2) + z_0 (z_1 z_3 - z_2^2) \\ &= (z_0 z_2^2 - z_2 z_1^2) + (z_0 z_1 z_3 - z_0 z_2^2) \\ &= z_0 z_1 z_3 - z_2 z_1^2 \\ &= z_1(z_0 z_3 - z_1 z_2) \\ &= z_1 p_3 \end{aligned}
So we have non-trivial relations between
p_1, p_2, p_3
! This is a second order syzygy, a sygyzy between syzygies. We have a ring
R \equiv k[z_0, z_1, z_2, z_3]
. We have a map
R \rightarrow \texttt{invariants}
. This has a nontrivial kernel, and this kernel is spanned by
(p_1, p_2, p_3) \simeq R^3
. But this itself has a kernel
q = z_1 p_1 + z_2 p_2 + z_3 p_3
. So there's an exact sequence:
\begin{aligned} 0 \rightarrow R^1 \rightarrow R^3 \rightarrow R=k[z_0, z_1, z_2, z_3] \rightarrow \texttt{invariants} \end{aligned}
In general, we get an invariant ring of linear maps that are invariant under the group action. We have polynomials
R \equiv k[z_0, z_1, \dots]
that map onto the invariant ring. We have relationships between the
z_0, \dots, z_n
. This gives us a sequence of syzygies. We have many questions:
R
finitely generated as a
k
algebra? Can we find a finite number of generators?
R^m
finitely generated (the syzygies as an
R
-MODULE)? To recall the difference, see that
k[x]
is finitely generated as an ALGEBRA by
(k, x)
since we can multiply the
x
s. It's not finitely generated as a MODULE as we need to take all powers of
x
(x^0, x^1, \dots)
Is this SEQUENCE of sygyzy modulues FINITE?
Hilbert showed that the answer is YES if
G
is reductive and
k
has characteristic zero. We will do a special case of
G
finite group.
We can see why a syzygy is called such; The second order sygyzy "yokes" the first order sygyzy. It ties together the polynomials in the first order syzygy the same way oxen are yoked by a syzygy.
§ Is inclusion/exclusion a syzygy?
I feel it is, since each level of the inclusion/exclusion arises as a "yoke" on the previous level. I wonder how to make this precise.
Richard E Borcherds: Commutative Algebra, lecture 3
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Succinct data structure - Wikipedia
In computer science, a succinct data structure is a data structure which uses an amount of space that is "close" to the information-theoretic lower bound, but (unlike other compressed representations) still allows for efficient query operations. The concept was originally introduced by Jacobson[1] to encode bit vectors, (unlabeled) trees, and planar graphs. Unlike general lossless data compression algorithms, succinct data structures retain the ability to use them in-place, without decompressing them first. A related notion is that of a compressed data structure, in which the size of the data structure depends upon the particular data being represented.
{\displaystyle Z}
is the information-theoretical optimal number of bits needed to store some data. A representation of this data is called:
implicit if it takes
{\displaystyle Z+O(1)}
bits of space,
succinct if it takes
{\displaystyle Z+o(Z)}
bits of space, and
compact if it takes
{\displaystyle O(Z)}
bits of space.
For example, a data structure that uses
{\displaystyle 2Z}
bits of storage is compact,
{\displaystyle Z+{\sqrt {Z}}}
bits is succinct,
{\displaystyle Z+\lg Z}
bits is also succinct, and
{\displaystyle Z+3}
bits is implicit.
Implicit structures are thus usually reduced to storing information using some permutation of the input data; the most well-known example of this is the heap.
1 Succinct dictionaries
1.1 Entropy-compressed dictionaries
Succinct dictionaries[edit]
Succinct indexable dictionaries, also called rank/select dictionaries, form the basis of a number of succinct representation techniques, including binary trees,
{\displaystyle k}
-ary trees and multisets,[2] as well as suffix trees and arrays.[3] The basic problem is to store a subset
{\displaystyle S}
of a universe
{\displaystyle U=[0\dots n)=\{0,1,\dots ,n-1\}}
, usually represented as a bit array
{\displaystyle B[0\dots n)}
{\displaystyle B[i]=1}
{\displaystyle i\in S.}
An indexable dictionary supports the usual methods on dictionaries (queries, and insertions/deletions in the dynamic case) as well as the following operations:
{\displaystyle \mathbf {rank} _{q}(x)=|\{k\in [0\dots x]:B[k]=q\}|}
{\displaystyle \mathbf {select} _{q}(x)=\min\{k\in [0\dots n):\mathbf {rank} _{q}(k)=x\}}
{\displaystyle q\in \{0,1\}}
{\displaystyle \mathbf {rank} _{q}(x)}
returns the number of elements equal to
{\displaystyle q}
up to position
{\displaystyle x}
{\displaystyle \mathbf {select} _{q}(x)}
returns the position of the
{\displaystyle x}
-th occurrence of
{\displaystyle q}
There is a simple representation[4] which uses
{\displaystyle n+o(n)}
bits of storage space (the original bit array and an
{\displaystyle o(n)}
auxiliary structure) and supports rank and select in constant time. It uses an idea similar to that for range-minimum queries; there are a constant number of recursions before stopping at a subproblem of a limited size. The bit array
{\displaystyle B}
is partitioned into large blocks of size
{\displaystyle l=\lg ^{2}n}
bits and small blocks of size
{\displaystyle s=\lg n/2}
bits. For each large block, the rank of its first bit is stored in a separate table
{\displaystyle R_{l}[0\dots n/l)}
; each such entry takes
{\displaystyle \lg n}
bits for a total of
{\displaystyle (n/l)\lg n=n/\lg n}
bits of storage. Within a large block, another directory
{\displaystyle R_{s}[0\dots l/s)}
stores the rank of each of the
{\displaystyle l/s=2\lg n}
small blocks it contains. The difference here is that it only needs
{\displaystyle \lg l=\lg \lg ^{2}n=2\lg \lg n}
bits for each entry, since only the differences from the rank of the first bit in the containing large block need to be stored. Thus, this table takes a total of
{\displaystyle (n/s)\lg l=4n\lg \lg n/\lg n}
bits. A lookup table
{\displaystyle R_{p}}
can then be used that stores the answer to every possible rank query on a bit string of length
{\displaystyle s}
{\displaystyle i\in [0,s)}
; this requires
{\displaystyle 2^{s}s\lg s=O({\sqrt {n}}\lg n\lg \lg n)}
bits of storage space. Thus, since each of these auxiliary tables take
{\displaystyle o(n)}
space, this data structure supports rank queries in
{\displaystyle O(1)}
{\displaystyle n+o(n)}
To answer a query for
{\displaystyle \mathbf {rank} _{1}(x)}
in constant time, a constant time algorithm computes:
{\displaystyle \mathbf {rank} _{1}(x)=R_{l}[\lfloor x/l\rfloor ]+R_{s}[\lfloor x/s\rfloor ]+R_{p}[x\lfloor x/s\rfloor ,x{\text{ mod }}s]}
In practice, the lookup table
{\displaystyle R_{p}}
can be replaced by bitwise operations and smaller tables that can be used to find the number of bits set in the small blocks. This is often beneficial, since succinct data structures find their uses in large data sets, in which case cache misses become much more frequent and the chances of the lookup table being evicted from closer CPU caches becomes higher.[5] Select queries can be easily supported by doing a binary search on the same auxiliary structure used for rank; however, this takes
{\displaystyle O(\lg n)}
time in the worst case. A more complicated structure using
{\displaystyle 3n/\lg \lg n+O({\sqrt {n}}\lg n\lg \lg n)=o(n)}
bits of additional storage can be used to support select in constant time.[6] In practice, many of these solutions have hidden constants in the
{\displaystyle O(\cdot )}
notation which dominate before any asymptotic advantage becomes apparent; implementations using broadword operations and word-aligned blocks often perform better in practice.[7]
Entropy-compressed dictionaries[edit]
{\displaystyle n+o(n)}
space approach can be improved by noting that there are
{\displaystyle \textstyle {\binom {n}{m}}}
{\displaystyle m}
{\displaystyle [n)}
(or binary strings of length
{\displaystyle n}
with exactly
{\displaystyle m}
1’s), and thus
{\displaystyle \textstyle {\mathcal {B}}(m,n)=\lceil \lg {\binom {n}{m}}\rceil }
is an information theoretic lower bound on the number of bits needed to store
{\displaystyle B}
. There is a succinct (static) dictionary which attains this bound, namely using
{\displaystyle {\mathcal {B}}(m,n)+o({\mathcal {B}}(m,n))}
space.[8] This structure can be extended to support rank and select queries and takes
{\displaystyle {\mathcal {B}}(m,n)+O(m+n\lg \lg n/\lg n)}
space.[2] Correct rank queries in this structure are however limited to elements contained in the set, analogous to how minimal perfect hashing functions work. This bound can be reduced to a space/time tradeoff by reducing the storage space of the dictionary to
{\displaystyle {\mathcal {B}}(m,n)+O(nt^{t}/\lg ^{t}n+n^{3/4})}
with queries taking
{\displaystyle O(t)}
A null-terminated string (C string) takes Z + 1 space, and is thus implicit. A string with an arbitrary length (Pascal string) takes Z + log(Z) space, and is thus succinct. If there is a maximum length – which is the case in practice, since 232 = 4 GiB of data is a very long string, and 264 = 16 EiB of data is larger than any string in practice – then a string with a length is also implicit, taking Z + k space, where k is the number of data to represent the maximum length (e.g., 64 bits).
When a sequence of variable-length items (such as strings) needs to be encoded, there are various possibilities. A direct approach is to store a length and an item in each record – these can then be placed one after another. This allows efficient next, but not finding the kth item. An alternative is to place the items in order with a delimiter (e.g., null-terminated string). This uses a delimiter instead of a length, and is substantially slower, since the entire sequence must be scanned for delimiters. Both of these are space-efficient. An alternative approach is out-of-band separation: the items can simply be placed one after another, with no delimiters. Item bounds can then be stored as a sequence of length, or better, offsets into this sequence. Alternatively, a separate binary string consisting of 1s in the positions where an item begins, and 0s everywhere else is encoded along with it. Given this string, the
{\displaystyle select}
function can quickly determine where each item begins, given its index.[10] This is compact but not succinct, as it takes 2Z space, which is O(Z).
Another example is the representation of a binary tree: an arbitrary binary tree o{\displaystyle n}
nodes can be represented in
{\displaystyle 2n+o(n)}
bits while supporting a variety of operations on any node, which includes finding its parent, its left and right child, and returning the size of its subtree, each in constant time. The number of different binary trees o{\displaystyle n}
{\displaystyle {\tbinom {2n}{n}}}
{\displaystyle /(n+1)}
. For large
{\displaystyle n}
{\displaystyle 4^{n}}
{\displaystyle \log _{2}(4^{n})=2n}
bits to encode it. A succinct binary tree therefore would occupy only
{\displaystyle 2}
bits per node.
^ Jacobson, G. J (1988). Succinct static data structures (Ph.D. thesis). Pittsburgh, PA: Carnegie Mellon University.
^ a b Raman, R.; V. Raman; S. S Rao (2002). "Succinct indexable dictionaries with applications to encoding k-ary trees and multisets". Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms. pp. 233–242. arXiv:0705.0552. CiteSeerX 10.1.1.246.3123. doi:10.1145/1290672.1290680. ISBN 0-89871-513-X.
^ Sadakane, K.; R. Grossi (2006). "Squeezing succinct data structures into entropy bounds" (PDF). Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm. pp. 1230–1239. ISBN 0-89871-605-5. Archived from the original (PDF) on 2011-09-29.
^ Jacobson, G. (1 November 1989). Space-efficient static trees and graphs (PDF). 30th IEEE Symposium on Foundations of Computer Science. doi:10.1109/SFCS.1989.63533. Archived from the original (PDF) on 2016-03-12.
^ González, R.; S. Grabowski; V. Mäkinen; G. Navarro (2005). "Practical implementation of rank and select queries" (PDF). Poster Proceedings Volume of 4th Workshop on Efficient and Experimental Algorithms (WEA). pp. 27–38.
^ Clark, David (1996). Compact pat trees (PDF) (Ph.D. thesis). University of Waterloo.
^ Vigna, S. (2008). Broadword implementation of rank/select queries (PDF). Experimental Algorithms. Lecture Notes in Computer Science. Vol. 5038. pp. 154–168. CiteSeerX 10.1.1.649.8950. doi:10.1007/978-3-540-68552-4_12. ISBN 978-3-540-68548-7.
^ Brodnik, A.; J. I Munro (1999). "Membership in constant time and almost-minimum space" (PDF). SIAM J. Comput. 28 (5): 1627–1640. CiteSeerX 10.1.1.530.9223. doi:10.1137/S0097539795294165.
^ Pătraşcu, M. (2008). "Succincter" (PDF). Foundations of Computer Science, 2008. FOCS'08. IEEE 49th Annual IEEE Symposium on. pp. 305–313.
^ Belazzougui, Djamal. "Hash, displace, and compress" (PDF).
Retrieved from "https://en.wikipedia.org/w/index.php?title=Succinct_data_structure&oldid=1007270369"
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A Study on the Largest Hydraulic Fracturing Induced Earthquake in Canada: Numerical Modeling and Triggering Mechanism | Bulletin of the Seismological Society of America | GeoScienceWorld
Bei Wang;
School of Earth and Ocean Sciences, University of Victoria, Victoria, Canada
Alessandro Verdecchia;
Alessandro Verdecchia
Department of Earth and Planetary Sciences, McGill University, Montreal, Quebec, Canada
Honn Kao *
Corresponding author: honn.kao@canada.ca
Ruhr‐University Bochum, Institute of Geology, Mineralogy, and Geophysics, Bochum, Germany
Bei Wang, Alessandro Verdecchia, Honn Kao, Rebecca M. Harrington, Yajing Liu, Hongyu Yu; A Study on the Largest Hydraulic Fracturing Induced Earthquake in Canada: Numerical Modeling and Triggering Mechanism. Bulletin of the Seismological Society of America 2021;; 111 (3): 1392–1404. doi: https://doi.org/10.1785/0120200251
Mw
4.6 earthquake that occurred on 17 August 2015 northwest of Fort St. John, British Columbia, is considered the largest hydraulic‐fracturing‐induced event in Canada, based on its spatiotemporal relationship with respect to nearby injection operations. There is a
∼5 day
delay of this
Mw
4.6 mainshock from the onset of fluid injection at the closest well pad (W1). In contrast, other two nearby injection wells (W2 and W3) have almost instantaneous seismic responses. In this study, we first take a forward numerical approach to investigate the causative mechanisms for the
Mw
4.6 event. Specifically, three finite‐element 3D poroelastic models of various permeability structures and presence or absence of hydraulic conduits are constructed, to calculate the coupled evolution of elastic stress and pore pressure caused by multistage fluid injections. Our simulation results suggest that pore pressure increase associated with the migration of injected fluid is required to accumulate sufficient stress perturbations to trigger this
Mw
4.6 earthquake. In contrast, the elastic stress perturbation caused by rock matrix deformation alone is not the main cause. Furthermore, injection and seismicity at W1 may have altered the local stress field and brought local faults closer to failure at sites W2 and W3. This process could probably shorten the seismic response time and, thus, explain the observed simultaneous appearance of injection and induced seismicity at W2 and W3.
Fort Saint John earthquake 2015
Ground Motion from
M
1.5 to 3.8 Induced Earthquakes at Hypocentral Distance < 45 km in the Montney Play of Northeast British Columbia, Canada
Stress Drop Variations of Induced Earthquakes near the Dallas–Fort Worth Airport, Texas
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(Redirected from Metric screw thread)
Hardware threading standard
Find sources: "ISO metric screw thread" – news · newspapers · books · scholar · JSTOR (July 2009) (Learn how and when to remove this template message)
The ISO metric screw thread is the most commonly used type of general-purpose screw thread worldwide.[1] They were one of the first international standards agreed when the International Organization for Standardization (ISO) was set up in 1947.[citation needed]
The "M" designation for metric screws indicates the nominal outer diameter of the screw thread, in millimetres. This is also referred to as the "major" diameter in the information below. It indicates the diameter of smooth-walled hole that a male thread (e.g. on a bolt) will pass through easily to create a well-located connection to an internally threaded component (e.g. a nut) on the other side. That is, an M6 screw has a nominal outer diameter of 6 millimetres and will therefore be a well-located, co-axial fit in a hole drilled to 6 mm diameter.
3 Preferred sizes
4 Spanner (wrench) sizes
Basic profile of all ISO metric screw threads, where the male part has the external thread
The design principles of ISO general-purpose metric screw threads ("M" series threads) are defined in international standard ISO 68-1.[2] Each thread is characterized by its major diameter, D (Dmaj in the diagram), and its pitch, P. ISO metric threads consist of a symmetric V-shaped thread. In the plane of the thread axis, the flanks of the V have an angle of 60° to each other. The thread depth is 0.54125 × pitch. The outermost 1⁄8 and the innermost 1⁄4 of the height H of the V-shape are cut off from the profile.
The relationship between the height H and the pitch P is found using the following equation where θ is half the included angle of the thread, in this case 30°:[3]
{\displaystyle H={\frac {1}{2\tan \theta }}\cdot P={\frac {\sqrt {3}}{2}}\cdot P\approx 0.866025\cdot P}
{\displaystyle P=2\tan \theta \cdot H={\frac {2}{\sqrt {3}}}\cdot H\approx 1.154701\cdot H}
In an external (male) thread (e.g., on a bolt), the major diameter Dmaj and the minor diameter Dmin define maximum dimensions of the thread. This means that the external thread must end flat at Dmaj, but can be rounded out below the minor diameter Dmin. Conversely, in an internal (female) thread (e.g., in a nut), the major and minor diameters are minimum dimensions; therefore the thread profile must end flat at Dmin but may be rounded out beyond Dmaj. In practice this means that one can measure the diameter over the threads of a bolt to find the nominal diameter Dmaj, and the inner diameter of a nut is Dmin.
{\displaystyle {\begin{aligned}D_{\text{min}}&=D_{\text{maj}}-2\cdot {\frac {5}{8}}\cdot H=D_{\text{maj}}-{\frac {5{\sqrt {3}}}{8}}\cdot P\approx D_{\text{maj}}-1.082532\cdot P\\[3pt]D_{\text{p}}&=D_{\text{maj}}-2\cdot {\frac {3}{8}}\cdot H=D_{\text{maj}}-{\frac {3{\sqrt {3}}}{8}}\cdot P\approx D_{\text{maj}}-0.649519\cdot P\end{aligned}}}
A metric ISO screw thread is designated by the letter M followed by the value of the nominal diameter D (the maximum thread diameter) and the pitch P, both expressed in millimetres and separated by the multiplication sign, × (e.g., M8×1.25). If the pitch is the normally used "coarse" pitch listed in ISO 261 or ISO 262, it can be omitted (e.g., M8).[4]: 17
The length of a machine screw or bolt is indicated by a following × and the length expressed in millimetres (e.g., M8×1.25×30 or M8×30).[citation needed]
Tolerance classes defined in ISO 965-1 can be appended to these designations, if required (e.g., M500– 6g in external threads). External threads are designated by lowercase letter, g or h. Internal threads are designated by upper case letters, G or H.[4]: 17
Preferred sizes[edit]
ISO 261 specifies a detailed list of preferred combinations of outer diameter D and pitch P for ISO metric screw threads.[5][6] ISO 262 specifies a shorter list of thread dimensions – a subset of ISO 261.[7]
ISO 262 selected sizes for screws, bolts and nuts
1 R10 0.25 0.2 ︙
1.2 R10 0.25 0.2 16 R10 2 1.5
1.4 R20 0.3 0.2 18 R20 2.5 2 or 1.5
1.6 R10 0.35 0.2 20 R10 2.5 2 or 1.5
2 R10 0.4 0.25 24 R10 3 2
2.5 R10 0.45 0.35 27 R20 3 2
3 R10 0.5 0.35 30 R10 3.5 2
3.5 R20 0.6 0.35 33 R20 3.5 2
4 R10 0.7 0.5 36 R10 4 3
5.5[a][8] R20 0.9 0.5 42 R10 4.5 3
6 R10 1 0.75 45 R20 4.5 3
7 R20 1 0.75 48 R10 5 3
8 R10 1.25 1 or 0.75 52 R20 5 4
10 R10 1.5 1.25 or 1 56 R10 5.5 4
12 R10 1.75 1.5 or 1.25 60 R20 5.5 4
14 R20 2 1.5 64 R10 6 4
^ Also DIN13[6][dubious – discuss]
The R10 series is from ISO 3, and the R20 series are rounded off values from ISO 3.[clarification needed][5]
The coarse pitch is the commonly used default pitch for a given diameter. In addition, one or two smaller fine pitches are defined, for use in applications where the height of the normal coarse pitch would be unsuitable (e.g., threads in thin-walled pipes). The terms coarse and fine have (in this context) no relation to the manufacturing quality of the thread.
In addition to coarse and fine threads, there is another division of extra fine, or superfine threads, with a very fine pitch thread. Superfine pitch metric threads are occasionally used in automotive components, such as suspension struts, and are commonly used in the aviation manufacturing industry. This is because extra fine threads are more resistant to coming loose from vibrations.[9] Fine and superfine threads also have a greater minor diameter than coarse threads, which means the bolt or stud has a greater cross-sectional area (and therefore greater load-carrying capability) for the same nominal diameter.
Spanner (wrench) sizes[edit]
Below are some common spanner (wrench) sizes for metric screw threads. Hex head widths (width across flats, spanner size) are for DIN 934 hex nuts and hex head bolts. Other (usually smaller) sizes may occur for reasons of weight and cost reduction.
Spanner (wrench) size (mm)
Hex nut,
flat-head cap screw
or grub,
M1 - 2.5 - - -
M1.2 - 3 - - -
M1.4 - 3 1.25 - 0.7
M2 4 1.5 1.25 0.9
M2.5 5 2 1.5 1.3
M3.5 6 - - -
M7 11 - - -
M14 21 22 10 - -
M18 27 14 12 -
M39 60 - - -
ISO 68-1: ISO general purpose screw threads — Basic profile — Metric screw threads.
ISO 261: ISO general purpose metric screw threads — General plan.
ISO 262: ISO general purpose metric screw threads — Selected sizes for screws, bolts and nuts.
ISO 965: ISO general purpose metric screw threads — Tolerances[4]
ISO 965-1: Principles and basic data
ISO 965-2: Limits of sizes for general purpose external and internal screw threads.
ISO 965-3: Deviations for constructional screw threads
ISO 965-4: Limits of sizes for hot-dip galvanized external screw threads to mate with internal screw threads tapped with tolerance position H or G after galvanizing
ISO 965-5: Limits of sizes for internal screw threads to mate with hot-dip galvanized external screw threads with maximum size of tolerance position h before galvanizing
BS 3643: ISO metric screw threads
ANSI/ASME B1.13M: Metric Screw Threads: M Profile
ANSI/ASME B4.2-1978 (R2009): Preferred Metric Limits and Fits
Page 519 DIN13
British Standard Cycle (BSC)
British Standard Whitworth (BSW) – a British thread standard with 55° profile.
Photographic Filter thread
Unified Thread Standard (UTS, UNC, UNF, UNEF and UNS) – a US/Canadian/British thread standard that uses the same 60° profile as metric threads, but an inch-based set of diameter/pitch combinations.
^ ISO 68-1:1998 ISO general purpose screw threads – Basic profile – Part 1: Metric screw threads. International Organization for Standardization.
^ Oberg et al. 2000, p. 1706
^ a b c ISO 965-1:2013 ISO general purpose metric screw threads — Tolerances — Part 1: Principles and basic data. International Organization for Standardization. 15 Sep 2013.
^ a b ISO 261:1998 ISO general purpose metric screw threads – General plan. International Organization for Standardization. 17 Dec 1998.
^ a b ISO & DIN13
^ ISO 262:1998 ISO general purpose metric screw threads – Selected sizes for screws, bolts and nuts. International Organization for Standardization. 17 Dec 1998.
^ M5.5 x 0.9
^ "Final report" (PDF). ntrs.nasa.gov. Archived from the original (PDF) on 2017-03-14. Retrieved 2017-07-07.
Metric screw thread dimensions and tolerances
Detailed metric thread dimensions
Diagrams and tables of many screwthread series. In German
DIN 931: M1,6 to M39 Hexagon head bolts (Product grades A and B)(1987)
IS 9519: Fasteners - Hexagon products - Width across flats, Indian standard (2013)
Retrieved from "https://en.wikipedia.org/w/index.php?title=ISO_metric_screw_thread&oldid=1083039045"
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§ Hypothesis Testing
§ Mnemonic for type I versus type II errors
Once something becomes "truth", challenging the status quo and making it "false" is very hard. (see: disinformation).
Thus, Science must have high barriers for accepting hypothesis as true.
That is, we must have high barries for incorrectly rejecting the null (that nothing happened).
This error is called as type I error, and is denoted by
\alpha
(more important error).
The other type of error, where something is true, but we conclude it is false is less important. Some grad student can run the experiment again with better experimental design and prove it's true later if need be.
Our goal is to protect science from entrenching/enshrining "wrong" facts as true. Thus, we control type I errors.
Our goal is to "reject" current theories (the null) and create "new theories" (the alternative). Thus, in statistics, we setup our tests with the goal of enabling us to "reject the null".
§ Mnemonic for remembering the procedure
H_0
is the null hypothesis (null for zero). They are presumed innocent until proven guilty.
H_0
is judged guilty, we reject them (from society) and send them to the gulag.
H_0
is judged not guilty, we retain them (in society).
We are the prosecution, who are trying to reject
H_0
(from society) to send them to the gulag.
The scientific /statistical process is the Judiciary which is attempting to keep the structure of "innocent until proven guilty" for
H_0
We run experiments, and we find out how likely it is that
H_0
is guilty based on our experiments.
We calculate an error
\alpha
, which is the probably we screw up the fundamental truth of the court: we must not send an innocent man to the gulag. Thus,
\alpha
it the probability that
H_0
is innocent (ie, true) but we reject it (to the gulag).
§ P value, Neyman interpretation
Now, suppose we wish to send
H_0
to the gulag, because we're soviet union like that. What's the probability we're wrong in doing so? (That is, what is the probability that us sending
H_0
is innocent and we are condemning them incorrectly to a life in the gulag)? that's the
p
value. We estimate this based on our expeiment, of course.
Remember, we can never speak of the "probability of
H_0
being true/false", because
H_0
is true or is false [frequentist ]. There is no probability.
§ P value, Fisher interpretation
The critical region of the test corresponds to those values of the test statistic that would lead us to reject null hypothesis (and send it to the gulag).
Thus, the critical region is also sometimes called the "rejection region", since we reject
H_0
from society if the test statistic lies in this region.
The rejection region is usually corresponds to the tails of the sampling distribution.
The reason for that is that a good critical region almost always corresponds to those values of the test statistic that are least likely to be observed if the null hypothesis is true. This will be the "tails" / "non central tendency" if a test is good.
In this situation, we define the
p
value to be the probability we would have observed a test statistic that is at least as extreme as the one we did get. P(new test stat >= cur test stat).
??? I don't get it.
§ P value, completely wrong edition
"Probability that the null hypothesis is true" --- WRONG
compare to "probability us rejecting the null hypothesis is wrong" -- CORRECT. The probability is in US being wrong, and has NOTHING to do with the truth or falsity of the null hypothesis itself .
§ Power of the test
\beta
H_0
was guilty, but we chose to retain them into society instead.
The less we do this (ie, the larger is
1 - \beta
), the more "power" our test has.
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15 October 2018 Effective finiteness of irreducible Heegaard splittings of non-Haken
3
Tobias Holck Colding, David Gabai
The main result is a short effective proof of Tao Li’s theorem that a closed non-Haken hyperbolic
3
N
has at most finitely many irreducible Heegaard splittings. Along the way we show that
N
has finitely many branched surfaces of pinched negative sectional curvature carrying all closed index-
\le 1
minimal surfaces. This effective result, together with the sequel with Daniel Ketover, solves the classification problem for Heegaard splittings of non-Haken hyperbolic
3
Tobias Holck Colding. David Gabai. "Effective finiteness of irreducible Heegaard splittings of non-Haken
3
-manifolds." Duke Math. J. 167 (15) 2793 - 2832, 15 October 2018. https://doi.org/10.1215/00127094-2018-0022
Keywords: 3-manifold , Heegaard splitting , Hyperbolic , lamination , minimal surface
Tobias Holck Colding, David Gabai "Effective finiteness of irreducible Heegaard splittings of non-Haken
3
-manifolds," Duke Mathematical Journal, Duke Math. J. 167(15), 2793-2832, (15 October 2018)
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From Quasi-Entropy to Various Quantum Information Quantities | EMS Press
The subject is the applications of the use of quasi-entropy in finite dimensional spaces to many important quantities in quantum information. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the
f
-divergence, monotone metrics, covariance and the
\chi^2
-divergence are the most important particular cases. The extension of monotone metrics to those with two parameters is a new concept. Monotone metrics are also characterized by their joint convexity property.
Fumio Hiai, Dénes Petz, From Quasi-Entropy to Various Quantum Information Quantities. Publ. Res. Inst. Math. Sci. 48 (2012), no. 3, pp. 525–542
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Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials | EMS Press
Université Bordeaux 1, Talence, France
Pablo L. Miranda
We consider the unperturbed operator
H_0 : = (-i \nabla - A)^2 + W
, self-adjoint in
L^2(\mathbb{R}^2)
A
is a magnetic potential which generates a constant magnetic field
b>0
, and the edge potential
W
is a non-decreasing non constant bounded function depending only on the first coordinate
x \in \mathbb{R}
(x,y) \in \mathbb{R}^2
. Then the spectrum of
H_0
has a band structure and is absolutely continuous; moreover, the assumption
\lim_{x \to \infty}(W(x) - W(-x)) < 2b
implies the existence of infinitely many spectral gaps for
H_0
. We consider the perturbed operators
H_{\pm} = H_0 \pm V
where the electric potential
V \in L^{\infty}(\mathbb{R}^2)
is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of
H_\pm
in the spectral gaps of
H_0
. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian involves a pseudo-differential operator with generalized anti-Wick symbol equal to
V
. Further, we restrict our attention on perturbations
V
of compact support and constant sign. We establish a geometric condition on the support of
V
which guarantees the finiteness of the eigenvalues of
H_{\pm}
in any spectral gap of
H_0
. In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of
H_+
H_-
) to the left (resp. right) edge of a given spectral gap, is Gaussian.
Vincent Bruneau, Pablo L. Miranda, Georgi Raikov, Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials. J. Spectr. Theory 1 (2011), no. 3, pp. 237–272
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On Associated and Co-Associated Complex Differential Operators | EMS Press
On Associated and Co-Associated Complex Differential Operators
R. Heersink
The paper deals with initial value problems of the form
\frac{\partial u}{\partial t} = \mathcal L u, \ \ \ u = u_0 \ \mathrm {for} \ t = 0
[0,T] \times G \subset \mathbb R^+_0 \times \mathbb R^n
\mathcal L
is a linear first order differential operator. The desired solutions will be sought in function spaces defined as kernel of a linear differential operator
l
being associated to
\mathcal L
. Mainly two assumptions are required for such initial value problems to be solvable: Firstly, the operators have to be associated, i.e.
lu = 0
l(\mathcal L u) = 0
. Secondly, an interior estimate
\| \mathcal L u \|_{G'} ≤ c(G,G’) \| u \|_G
G' \subset G
) must be true. Moreover, operators
\mathcal L
are investigated possessing a family of associated operators
l_k
(which then are said to be co-associated).
The present paper surveys the use of associated and co-associated differential operators for solving initial value problems of the above (Cauchy-Kovalevskaya) type. Discussing interior estimates as starting point for the construction of related scales of Banach spaces, the paper sets up a possible framework for further generalizations. E.g., that way a theorem of Cauchy-Kovalevskaya type with initial functions satisfying a differential equation of an arbitrary order k (with not necessarily analytic coefficients) is obtained.
R. Heersink, Wolfgang Tutschke, On Associated and Co-Associated Complex Differential Operators. Z. Anal. Anwend. 14 (1995), no. 2, pp. 249–257
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PS=\frac{U}{{n}_{1}*{n}_{2}}
z=\frac{U-\frac{{n}_{1}*{n}_{2}}{2}}{\sqrt{\frac{{n}_{1}*{n}_{2}*\left({n}_{1}+{n}_{2}+1\right)}{12}}}
Despite their advantages, SVMs are highly sensitive to class imbalance. Seperating hyperplane may be skewed towards the minority class, which in return cause a degraded performance for the minority class [41]. A second possible explanation would be that the amount of support vectors may be imbalanced. In this case, the neighbourhood of a test instance located close to the boundary is more likely to be dominated by negative support vectors, which makes decision function more likely to classify an instance located close to the boundary as a negative instance [41]. In order to overcome this problem, we have adjusted the class weights when training with LibSVM, which adjusts C parameter of the given class as
weight*C
. We have used 1:5 as class weights in order to overcome class imbalance without resampling.
MCC=\frac{\left(TP*TN\right)-\left(FN*FP\right)}{\sqrt{\left(TP+FN\right)*\left(TN+FP\right)*\left(TP+FP\right)*\left(TN+FN\right)}}
Sn=\frac{TP}{TP+FN}
Sp=\frac{TN}{TN+FP}
Acc=\frac{TP+TN}{TP+FP+TN+FN}
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Why is ans d In amperes law I is enclosed current but here i1and i2 is not enclosed - Physics - Magnetism And Matter - 14358385 | Meritnation.com
Why is ans d?? In amperes law I is enclosed current but here i1and i2 is not enclosed
Consider the amperian loop ABCA, then from Ampere's circuital law:
\oint B.dl = {\mu }_{o}\left({i}_{1}+{i}_{3}\right)
Since, i1 and i3 are both in the direction of the area vector for the surface ABCA.
Consider the Amperian loop ACDA, then from Ampere's circuital law:
\oint B.dl = {\mu }_{o}\left({i}_{2}-{i}_{3}\right)
Since, i2 is in the direction and i3 is opposite to the direction of the area vector for the surface ACDA .
For the entire loop ABCDA:
\oint B.dl = {\mu }_{o}\left({i}_{1}+{i}_{3}\right)+{\mu }_{o}\left({i}_{2}-{i}_{3}\right)={\mu }_{o}\left({i}_{1}+{i}_{2}\right)
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Weighted inequalities for holomorphic functional calculi of operators with heat kernel bounds
October, 2005 Weighted inequalities for holomorphic functional calculi of operators with heat kernel bounds
Xuan Thinh DUONG, Lixin YAN
𝒳
be a space of homogeneous type. Assume that
L
has a bounded holomorphic functional calculus on
{L}^{2}\left(\mathrm{\Omega }\right)
L
generates a semigroup with suitable upper bounds on its heat kernels where
\mathrm{\Omega }
is a measurable subset of
𝒳
. For appropriate bounded holomorphic functions
b
, we can define the operators
b\left(L\right)
{L}^{p}\left(\mathrm{\Omega }\right)
1\le p\le \mathrm{\infty }
. We establish conditions on positive weight functions
u,v
p
1<p<\mathrm{\infty }
{c}_{p}
{\int }_{\mathrm{\Omega }}|b\left(L\right)f\left(x\right){|}^{p}u\left(x\right)d\mu \left(x\right)\le {c}_{p}||b|{|}_{\mathrm{\infty }}^{p}{\int }_{\mathrm{\Omega }}|f\left(x\right){|}^{p}v\left(x\right)d\mu \left(x\right)
f\in {L}^{p}\left(vd\mu \right)
Applications include two-weight
{L}^{p}
inequalities for Schrödinger operators with non-negative potentials on
{\mathbf{R}}^{n}
and divergence form operators on irregular domains of
{\mathbf{R}}^{n}
Xuan Thinh DUONG. Lixin YAN. "Weighted inequalities for holomorphic functional calculi of operators with heat kernel bounds." J. Math. Soc. Japan 57 (4) 1129 - 1152, October, 2005. https://doi.org/10.2969/jmsj/1150287306
Keywords: elliptic operator , holomorphic functional calculus , semigroup kernel , singular integral operator , space of homogeneous type , weights
Xuan Thinh DUONG, Lixin YAN "Weighted inequalities for holomorphic functional calculi of operators with heat kernel bounds," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 57(4), 1129-1152, (October, 2005)
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Lacey and Haley are rewriting expressions in an equivalent, simpler form.
Haley simplified
x^3 · x^2
x^5
. Lacey simplified
x^3 + x^2
and got the same result! However, their teacher told them that only one simplification is correct. Who simplified correctly and how do you know?
Haley is correct. Haley used the product rule with exponents.
Since each factor has the same base, we simply add the exponents.
Haley simplifies
3^5 \cdot 4^5
and gets the result
12^{10}
, but Lacey is not sure. Is Haley correct? Be sure to justify your answer.
If Haley is correct, then
2^1\cdot3^1 = 6^2
2\cdot3 = 36!
Clearly this counterexample provides a hint of why Haley is wrong.
|
Birkhoff normal form for the nonlinear Schrödinger equation | EMS Press
Birkhoff normal form for the nonlinear Schrödinger equation
This paper is intended to highlight the differences between the nonlinear Schr\"odinger equation (NLS) posed on a compact manifold (such as a torus
\T^d
) in contrast to being posed on noncompact regions such as on all of
\R^d
. The point is to indicate a number of specific facts about the behavior of solutions in the former situation, in which they have the possibility for recurrence, and the latter, in which solutions have the tendency to disperse. This is the topic of the short article by McKean \cite{McKean97}, in which the issue of resonance for partial differential evolution equations is discussed. The aspect of this question that we describe in the present paper is that there are different normal forms for these two cases, which rephrases the question as to which of the nonlinear terms are the resonant terms, and what is the appropriate Birkhoff normal form for the NLS. We show that, at least in a neighborhood of zero of an appropriate Hilbert space, the fourth order Birkhoff normal form transformation for the NLS equation is able to eliminate all of the nonresonant terms of the Hamiltonian, and as well, all of the resonant terms. The result is a prognosis, to the negative, for the formal theory of wave turbulence for Hamiltonian partial differential equations posed in Sobolev spaces over
\R^d
^*
Walter Craig, Alessandro Selvitella, Yun Wang, Birkhoff normal form for the nonlinear Schrödinger equation. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 24 (2013), no. 2, pp. 215–228
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