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# Article

Ufa Mathematical Journal
Volume 14, Number 3, pp. 60-69

# On $C^1$-convergence of piecewise polynomial solutions to a fourth order variational equation

Klyachin A.A.

DOI:10.13108/2022-14-3-60

In the present work we consider a boundary value problem in a polygonal domain for a fourth order variational equation. We assume that this domain is partitioned into finitely many triangles forming its triangulation. We introduce a class of piecewise polynomial functions of a given degree and for a considered equation we define the notion of a piecewise polynomial solution on a triangle net. We prove a theorem on existence and uniqueness of such solution. Moreover, we establish that under certain conditions for the triangulation of the domain, the second derivatives of the piecewise polynomial solutions are estimated by a constant independent of the fineness of the partition. This fact allows us to prove $C^1$-convergence of piecewise polynomial solutions to the equations as the fineness of grid tends to zero.

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Tate curve

2010 Mathematics Subject Classification: Primary: 14H52 [MSN][ZBL]

A Tate curve is a uniformization of an elliptic curve having stable bad reduction with the help of a $q$-parametrization.

Let $K$ be a local field (e.g., $\mathbb{C}((t))$ or a finite extension of $\mathbb{Q}_p$). Let $E$ be an elliptic curve over $K$ such that it has stable reduction. Then it can have good reduction (i.e. with integral $j$-invariant) or bad reduction (i.e. with non-integral $j$-invariant). In the case of stable bad reduction one can construct an elliptic curve $E_q$ over $K$, which analytically is $K^*/q^{\mathbb{Z}}$ (where $q^{\mathbb{Z}}$ is the subgroup of $K^*$ generated by $q \in K^*$), such that $E$ and $E_q$ are isomorphic over a finite extension of $K$. One of the marvels of this theorem is the fact that the construction of the period $q$ starting from $E$, and the computation of the $j$-value of $E_q$, can be done without denominators (hence can be done in every characteristic): the $j$-value of the Tate curve with period $q$ is a power series in $q$ with coefficients in $\mathbb{Z}$: $$j = q^{-1} + 744 + 196884 q + 21493760 q^2 + \cdots \ .$$

Such formulas can be found in [a4], Chapt. 15; [a8]. See also [a9], A.1.1, and [a10], Appendix C, Sect. 14. In [a2], VII, constructions over $\mathbb{Z}$ are given, with applications to compactifications of moduli schemes of elliptic curves. A generalization to higher-dimensional Abelian varieties over local fields with totally bad, stable reduction was given by H. Morikawa [a5], [a6] and by D. Mumford [a7]. This was generalized by G. Faltings and by C.-L. Chai to the case of stable reduction of Abelian varieties in [a3] and [a1], and it was used in the theory of compactifications of moduli schemes of Abelian varieties.

References

 [a1] C.-L. Chai, "Compactifications of Siegel moduli schemes" , Cambridge Univ. Press (1985) [a2] P. Deligne, M. Rapoport, "Les schémas de modules de courbes elliptiques" P. Deligne (ed.) W. Kuyk (ed.) , Modular Functions II , Lect. notes in math. , 349 , Springer (1973) pp. 143–316 [a3] G. Faltings, "Arithmetische Kompaktifizierung des Modulraums der abelschen Varietäten" F. Hirzebruch (ed.) J. Schwermer (ed.) S. Suter (ed.) , Arbeitstagung Bonn 1984 , Lect. notes in math. , 1111 , Springer (1985) pp. 321–383 [a4] S. Lang, "Elliptic functions" , Addison-Wesley (1973) [a5] H. Morikawa, "On theta functions and abelian varieties over valuation fields of rank one" Nagoya Math. J. , 20 (1962) pp. 1–27 [a6] H. Morikawa, "On theta functions and abelian varieties over valuation fields of rank one" Nagoya Math. J. , 21 (1962) pp. 231–250 [a7] D. Mumford, "An analytic construction of degenerating abelian varieties over complete rings" Compos. Math. , 24 (1972) pp. 129–174; 239–272 [a8] P. Roquette, "Analytic theory of elliptic functions over local fields" , Vandenhoeck & Ruprecht (1970) [a9] J.-P. Serre, "Abelian $\ell$-adic representations and elliptic curves" , Benjamin (1986) (Translated from French) [a10] J.H. Silverman, "The arithmetic of elliptic curves" , Springer (1986) [a11] C.-L. Chai, G. Faltings, "Semiabelian degeneration and compactification" , Forthcoming [a12] S. Bosch, W. Lütkebohmert, M. Raynaud, "Néron models" , Springer (Forthcoming) [b1] Alain Robert, "Elliptic curves. Notes from postgraduate lectures given in Lausanne 1971/72", Lecture Notes in Mathematics 326 Springer (1973) Zbl 0256.14013
How to Cite This Entry:
Tate curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tate_curve&oldid=35795
This article was adapted from an original article by F. Oort (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article


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Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01z316q177z
 Title: Scanning Tunneling Microscopy Studies of the Heavy-fermion Superconductor (100) CeCoIn5 in the Normal Conducting State Authors: Peterson, Gabriel Advisors: Yazdani, Ali Contributors: Bakr, Waseem Department: Physics Class Year: 2014 Abstract: This thesis reports scanning tunneling microscopy (STM) measurements of the heavy-fermion material CeCoIn5 on the previously unmeasured (100) termination surface. This study demonstrates both the experimental feasibility of measuring these surfaces and the observation of interesting electronic properties. The measurements presented here open the possibility for future experiments to go to lower temperature to explore new aspects of the unconventional superconducting nature of CeCoIn5. In the present work, we study (100) CeCoIn5 in the normal conducting phase. Through topographic and spectroscopic measurements, we demonstrate some of the remarkable physical properties of this material. Topographic maps reveal two surfaces, of which one bears strong resemblance to the bulk crystal structure. Spectroscopic measurements on this surface show both angstrom-scale and nanometer-scale spatial modulation of the electronic density of states. The short wavelength modulation corresponds to the local hybridization of f electrons with the conduction electron sea and also demonstrates the twodimensional character of the material. The long wavelength modulation arises from electronic scattering on the surface, and is shown to prefer certain crystallographic directions, a result that further characterizes the importance of layering in the CeCoIn5 system. The wavelength, energy dependence, and direction of the waves gives indirect information about the Fermi surface. Our measurements show several distinct scattering wavevectors that appear qualitatively consistent with calculated band structure. Extent: 63 pages URI: http://arks.princeton.edu/ark:/88435/dsp01z316q177z Type of Material: Princeton University Senior Theses Language: en_US Appears in Collections: Physics, 1936-2016

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# The Coefficients of Linear Regression in R

gmp package, Cholesky and QR decomposition

The following are the R packages used in this post.

library(datasets)
library(magrittr)
library(microbenchmark)
library(gmp)

## Normal equation for the beta vector

Today, I want to talk little bit about the process of getting the coefficients of a linear model in R. Among many methods, our main focus for today is related to the Cholesky decomposition and QR decomposition.

The reason why we are discussing matrix decomposition methods for linear regression is that the coefficients of the regression are the solution of the matrix equation called the Normal equation.

Suppose we have the following $$n$$ observations in a vector $$y$$ and the data matrix $$X$$. If we assume that the vector $$y$$ follows a linear model with noise vector $$\mathbf{e}$$, then the situation can be written as follows:

$\begin{eqnarray*} \mathbf{y}_{n\times1} & = & X_{n\times p}\boldsymbol{\beta}_{p\times1}+\mathbf{e}_{n\times1}\\ & = & \left[\begin{array}{ccccc} \mathbf{1} & \mathbf{x}_{1} & \mathbf{x}_{2} & ... & \mathbf{x}_{p-1}\end{array}\right]\left[\begin{array}{c} \beta_{0}\\ \beta_{1}\\ \vdots\\ \beta_{p-1} \end{array}\right]+\mathbf{e}\\ & = & \beta_{0}\mathbf{1}+\beta_{1}\mathbf{x}_{1}+\beta_{2}\mathbf{x}_{2}+...+\beta_{p-1}\mathbf{x}_{p-1}+\mathbf{e} \end{eqnarray*}$

Note that $$\mathbf{1}$$, $$\mathbf{e}$$ and $$\mathbf{x}_{i}$$, $$i=1,...,p-1$$ are $$n \times 1$$ vectors.

Under this assumption, the coefficients of the linear model, $$\beta$$, which minimize the residual sum of squares (RSS), can be obtained by solving the following Normal eqaution:

$X^{T}X\mathbf{\beta}=X^{T}\mathbf{y}$

The details about the Normal equation can be found at wikipedia.

## longley data set

The Longley data set belongs to the appendix of the paper written by J. W. Longley (1967): An appraisal of least-squares programs from the point of view of the user. The data set is toy data and consists of 7 explanatory variables. This data is famous for having high correlation among the variables when we regress the Employed variable on the rest of the variables. The data set is contained in the datasets package, so let us load the data set.

library(datasets)
longley
kable(longley, format = "html") %>%
kable_styling() %>%
scroll_box(width = "100%", height = "300px")
GNP.deflator GNP Unemployed Armed.Forces Population Year Employed
1947 83.0 234.289 235.6 159.0 107.608 1947 60.323
1948 88.5 259.426 232.5 145.6 108.632 1948 61.122
1949 88.2 258.054 368.2 161.6 109.773 1949 60.171
1950 89.5 284.599 335.1 165.0 110.929 1950 61.187
1951 96.2 328.975 209.9 309.9 112.075 1951 63.221
1952 98.1 346.999 193.2 359.4 113.270 1952 63.639
1953 99.0 365.385 187.0 354.7 115.094 1953 64.989
1954 100.0 363.112 357.8 335.0 116.219 1954 63.761
1955 101.2 397.469 290.4 304.8 117.388 1955 66.019
1956 104.6 419.180 282.2 285.7 118.734 1956 67.857
1957 108.4 442.769 293.6 279.8 120.445 1957 68.169
1958 110.8 444.546 468.1 263.7 121.950 1958 66.513
1959 112.6 482.704 381.3 255.2 123.366 1959 68.655
1960 114.2 502.601 393.1 251.4 125.368 1960 69.564
1961 115.7 518.173 480.6 257.2 127.852 1961 69.331
1962 116.9 554.894 400.7 282.7 130.081 1962 70.551

We can see that the longley data set in R was scaled compared to the original data.

## Calculating accurate coefficients using gmp package

The gmp package uses fractions for its calculation instead of using decimal points. This package allows us to achieve more accurate estimates for the coefficients since it avoids the truncation of floating numbers. By using the solve() function, let us calculate accurate coefficients for the linear regression of the Employed variable on the rest of the variables in the longley data set.

y <- as.vector(longley[,7])
X <- as.matrix(cbind(constant = 1, longley[,-7]))

# install.packages("gmp")
# library(gmp)

# X, y rationals
r_X <- as.bigq(round(1000 * X)) / as.bigq(1000)
r_y <- as.bigq(round(1000 * y)) / as.bigq(1000)
## Big Rational ('bigq') 6 x 7 matrix:
##      [,1] [,2]   [,3]        [,4]    [,5]    [,6]        [,7]
## [1,] 1    83     234289/1000 1178/5  159     13451/125   1947
## [2,] 1    177/2  129713/500  465/2   728/5   13579/125   1948
## [3,] 1    441/5  129027/500  1841/5  808/5   109773/1000 1949
## [4,] 1    179/2  284599/1000 3351/10 165     110929/1000 1950
## [5,] 1    481/5  13159/40    2099/10 3099/10 4483/40     1951
## [6,] 1    981/10 346999/1000 966/5   1797/5  11327/100   1952
# coefficients
cef_exact <- as.double(solve(t(r_X) %*% r_X,
(t(r_X) %*% r_y)))
cef_exact
## [1] -3.482259e+03  1.506187e-02 -3.581918e-02 -2.020230e-02 -1.033227e-02
## [6] -5.110411e-02  1.829151e+00

## Cholesky decomposition

If the matrix $$A$$ satisfies $$x^T \mathbf{A} x > 0$$ for every $$x$$, then the matrix $$A$$ is a positive definite.

The Cholesky decomposition can be applied to symmetric positive definite matrices. If a matrix $$A$$ is a symmetirc positive definite matrix then there exists a lower triangular matrix, $$L$$, such that

$\mathbf{A} = \mathbf{L} \mathbf{L}^T$

Thus, if our data matrix is of full rank, in other words, if the matrix $$X^TX$$ is positive definite, then the Normal equation can be written as follows:

\begin{align*} \mathbf{X}^{T}\mathbf{X}\beta & =\mathbf{X}^{T}y\\ \Rightarrow\mathbf{L}\mathbf{L}^{T}\beta & =\mathbf{X}^{T}y\\ \tag{1}\Rightarrow\mathbf{L}\left(\mathbf{L}^{T}\beta\right) & =\mathbf{X}^{T}y \end{align*}

Note that the matrix $$\mathbf{X}^T \mathbf{X}$$ is always symmetric. Thus, the $$\beta$$ vector can be obtained by the following two steps:

1. Solve Lz=X’y for z (forward solve lower triangular system).
2. Solve L’b=z for b (backward solve upper triangular system).

Is there any benefit of using the upper or lower triangular matrix in solving the equation? Yes. There exist nice and fast functions in R to solve a system of linear equations when the matrix in the equation is an upper or lower triangular marix: backsolve and forwardsolve. Also, look at the algorithm for finding the inverse matrix of triangular matrices of any size $$n$$ by $$n$$. Here are link 1 and link 2. As the number of variables used in linear regressions has increased over the years, finding an efficient way of solving the Normal equation has become a very important topic.

## Implementation in R

Let us confirm whether the choesky decomposition method works using the chol() function in R.

library(matrixcalc)
## Warning: 패키지 'matrixcalc'는 R 버전 4.0.3에서 작성되었습니다

crossprod(X) %>% is.positive.definite
## [1] TRUE
crossprod(X) %>% eigen(symmetric = TRUE) %$% values > 0 ## [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE To check out whether the matrix $$\mathbf{X}^T \mathbf{X}$$ is positive definite or not, we can either use the is.positiv.definite() function in matrixcalc package or use the basic functions in R for checking that the signs of the eigenvalues are positive. Since the matrix is positive definite, we can apply the Cholesky decomposition to $$\mathbf{X}^T \mathbf{X}$$. The following function calculates the regression coefficients using the Cholesky decomposition: # X always includes a constant vector regCef_chol <- function(X, y){ a <- crossprod(X) b <- crossprod(X, y) # cholesky decomposition upper <- chol(a) # we need to do the following # z <- forwardsolve( t(upper), b ) # but transpose can be avoid by using # backsolve & transpose option = TRUE z <- backsolve( upper, b, transpose = TRUE) cef <- as.vector(backsolve(upper, z)) cef } Here are some comments about the regCef_chol() function: 1. In R, you must use the code crossprod(X) rather than the t(X) %*% X becasue of the efficiency. 2. The chol() function returns an upper triangle matrix. Thus, the code was adjusted for using the upper tringle matrix. 3. We need to use the functions forwardsolve and backsolve if we want to implement equation (1). However, the transpose option allows us to use backsolve twice so that we can avoid the transpose. Let us take a look at the following code. a <- crossprod(X) b <- crossprod(X, y) # cholesky decomposition upper <- chol(a) microbenchmark( forwardsolve( t(upper), b ), backsolve( upper, b, transpose = TRUE) ) ## Unit: microseconds ## expr min lq mean median uq max neval ## forwardsolve(t(upper), b) 6.9 7.1 7.810 7.15 7.3 39.6 100 ## backsolve(upper, b, transpose = TRUE) 3.9 4.1 4.278 4.10 4.2 12.5 100 z1 <- forwardsolve( t(upper), b ) z2 <- backsolve( upper, b, transpose = TRUE) z1 == z2 ## [,1] ## [1,] TRUE ## [2,] TRUE ## [3,] TRUE ## [4,] TRUE ## [5,] TRUE ## [6,] TRUE ## [7,] TRUE Let us save the Cholesky decomposition-based coefficients of the regression to the cef_chol variable. cef_chol <- regCef_chol(X, y) cef_chol ## [1] -3.482259e+03 1.506187e-02 -3.581918e-02 -2.020230e-02 -1.033227e-02 ## [6] -5.110411e-02 1.829151e+00 ## QR decomposition For a real-valued $$n \times p$$ matrix $$A$$, where $$n \geq p$$, we can decompose the matrix $$A$$ into the following product of two matrices: $A = QR$ where the matrix $$Q$$ is a $$n \times p$$ matrix with orthonormal columns which satisfies $$Q^TQ=I_p$$. If the matrix $$A$$ is square, then $$Q$$ is orthogonal. Also, the matrix $$R$$ is an $$p \times p$$ upper triangular matrix whose diagonal elements are nonzero, which implies that $$R$$ is nonsingular. The QR decomposition is preferred to the Cholesky decomposition when it comes to the calculation of regression coefficients because 1. The QR decomposition can be applied to matrices of any size, and it usually provides a more stable calculation. 2. More importantly, in the Normal equation solving process, we don’t need to calculate the term, $$\mathbf{X}^T \mathbf{X}$$, which can be seen as follows: $\begin{eqnarray*} X^{T}X\mathbf{\beta} & = & X^{T}\mathbf{y}\\ \Rightarrow\left(QR\right)^{T}\left(QR\right)\mathbf{\beta} & = & \left(QR\right)^{T}\mathbf{y}\\ \Rightarrow R^{T}Q^{T}QR\mathbf{\beta} & = & R^{T}Q^{T}\mathbf{y}\\ \Rightarrow R^{T}R\mathbf{\beta} & = & R^{T}Q^{T}\mathbf{y} \\ \Rightarrow R\mathbf{\beta} & = & Q^{T}\mathbf{y} \end{eqnarray*}$ Note that we used the properties of the matrices $$Q$$ and $$R$$ to reduce the equation above. From the last equation we can see that the beta vector is actually the solution of the linear system $X \mathbf{\beta}=\mathbf{y}$ using the QR factorization of the matrix $$X$$. Thus, to calculate $$\beta$$, we can use backsolve() only one time after applying the QR decomposition to the matrix $$X$$. ### QR decomposition in R If you use the qr() function in R, you will be a little confused when you see the result of the function: QRmat <- qr(X) QRmat ##$qr
##      constant  GNP.deflator           GNP    Unemployed  Armed.Forces
## 1947    -4.00 -4.067250e+02 -1.550794e+03 -1.277325e+03 -1.042675e+03
## 1948     0.25  4.179551e+01  3.817172e+02  2.246174e+02  1.252618e+02
## 1949     0.25  2.331590e-01  4.982290e+01 -3.118589e+01 -2.998495e+01
## 1950     0.25  2.020552e-01 -1.320204e-01 -2.820602e+02  1.644256e+02
## 1951     0.25  4.175090e-02  2.352068e-01 -4.032101e-01 -1.703533e+02
## 1952     0.25 -3.708533e-03  2.301639e-01 -4.987968e-01  2.597864e-01
## 1953     0.25 -2.524195e-02  3.010869e-02 -4.454059e-01  2.499492e-01
## 1954     0.25 -4.916796e-02  2.634768e-01  3.041786e-02  5.401806e-01
## 1955     0.25 -7.787919e-02 -2.008098e-01 -1.549424e-02  2.755704e-01
## 1956     0.25 -1.592276e-01  1.765048e-03 -2.087977e-01 -8.693505e-02
## 1957     0.25 -2.501465e-01  2.417450e-01 -3.582924e-01 -3.417886e-01
## 1958     0.25 -3.075689e-01  6.566704e-01  1.704505e-02 -1.486493e-01
## 1959     0.25 -3.506358e-01  2.287415e-01 -1.269142e-01 -3.868639e-01
## 1960     0.25 -3.889174e-01  1.297815e-01 -7.134241e-02 -4.035762e-01
## 1961     0.25 -4.248064e-01  9.885428e-02  2.227783e-01 -1.470166e-01
## 1962     0.25 -4.535177e-01 -4.128804e-01  1.547782e-01 -1.059541e-01
##         Population          Year
## 1947 -469.69600000 -7818.0000000
## 1948   26.37951035    18.2758880
## 1949    4.19691804     1.7752596
## 1950   -3.18974111    -1.4529877
## 1951    0.04624846    -0.4491762
## 1952    1.46320173    -0.2819006
## 1953   -0.25728181    -0.6693051
## 1954   -0.26118351     0.4588864
## 1955    0.43979880     0.2646956
## 1956    0.33806289     0.3701027
## 1957    0.13690629     0.1755959
## 1958   -0.05114355    -0.1027051
## 1959    0.47762105    -0.3582417
## 1960    0.13635911    -0.1388969
## 1961   -0.35055843    -0.3066874
## 1962   -0.40330323    -0.3490426
##
## $rank ## [1] 7 ## ##$qraux
## [1] 1.250000 1.225981 1.156696 1.349325 1.102230 1.065271 1.421288
##
## $pivot ## [1] 1 2 3 4 5 6 7 ## ## attr(,"class") ## [1] "qr" The help page for the qr() function has the answer for this: The upper triangle contains the $$\bold{R}$$ matrix of the decomposition and the lower triangle contains information on the $$\bold{Q}$$ matrix of the decomposition (stored in compact form). TO get the $$Q$$ and $$R$$ matrices seperately, we need to use the qr.Q() and qr.R() functions, respectively. Qmat <- qr.Q(QRmat) Rmat <- qr.R(QRmat) Qmat; Rmat; ## [,1] [,2] [,3] [,4] [,5] [,6] ## [1,] -0.25 -0.44696790 0.34534075 -0.097267502 0.11348450 0.14346687 ## [2,] -0.25 -0.31537481 -0.15833007 0.074204845 0.54306575 0.27577175 ## [3,] -0.25 -0.32255262 -0.13087500 -0.415649678 -0.03377641 0.05658751 ## [4,] -0.25 -0.29144879 0.16361083 -0.306089180 0.02304976 -0.32175743 ## [5,] -0.25 -0.13114448 -0.17388270 0.302760179 -0.06259505 -0.13059025 ## [6,] -0.25 -0.08568505 -0.16040787 0.396678978 -0.23146214 0.03796899 ## [7,] -0.25 -0.06415163 0.04364140 0.413247504 -0.20796276 0.34643193 ## [8,] -0.25 -0.04022562 -0.18528890 -0.147931978 -0.57608466 0.12886308 ## [9,] -0.25 -0.01151440 0.28432317 0.061965649 -0.25775992 -0.48931159 ## [10,] -0.25 0.06983406 0.09683706 0.176548287 0.05777222 -0.25842925 ## [11,] -0.25 0.16075293 -0.12627901 0.233202984 0.25321471 0.02907702 ## [12,] -0.25 0.21817537 -0.53055357 -0.295032653 -0.04874867 0.03998579 ## [13,] -0.25 0.26124220 -0.09463650 -0.001197881 0.23969677 -0.38761993 ## [14,] -0.25 0.29952382 0.01142409 -0.024274066 0.24921047 -0.06437349 ## [15,] -0.25 0.33541285 0.04900808 -0.310066907 -0.04091063 0.26459047 ## [16,] -0.25 0.36412407 0.56606822 -0.061098580 -0.02019395 0.32933852 ## [,7] ## [1,] -0.00859901 ## [2,] 0.03836384 ## [3,] -0.03603622 ## [4,] -0.01634503 ## [5,] 0.62685034 ## [6,] 0.24825479 ## [7,] -0.29828585 ## [8,] -0.18932241 ## [9,] -0.06276184 ## [10,] -0.39060004 ## [11,] -0.36911636 ## [12,] -0.02273730 ## [13,] 0.16401119 ## [14,] -0.09588745 ## [15,] 0.16626351 ## [16,] 0.24594785 ## constant GNP.deflator GNP Unemployed Armed.Forces Population ## 1947 -4 -406.72500 -1550.7938 -1277.32500 -1042.67500 -469.69600000 ## 1948 0 41.79551 381.7172 224.61743 125.26181 26.37951035 ## 1949 0 0.00000 49.8229 -31.18589 -29.98495 4.19691804 ## 1950 0 0.00000 0.0000 -282.06021 164.42555 -3.18974111 ## 1951 0 0.00000 0.0000 0.00000 -170.35326 0.04624846 ## 1952 0 0.00000 0.0000 0.00000 0.00000 1.46320173 ## 1953 0 0.00000 0.0000 0.00000 0.00000 0.00000000 ## Year ## 1947 -7818.0000000 ## 1948 18.2758880 ## 1949 1.7752596 ## 1950 -1.4529877 ## 1951 -0.4491762 ## 1952 -0.2819006 ## 1953 -0.6693051 Thus, in R we can obtain the QR factorization-based coefficients as follows: as.vector(backsolve(Rmat, crossprod(Qmat, y))) ## [1] -3.482259e+03 1.506187e-02 -3.581918e-02 -2.020230e-02 -1.033227e-02 ## [6] -5.110411e-02 1.829151e+00 However, R provides the following compact solve function for the QR factorization. cef_qr <- qr.solve(X, y) # or solve.qr(QRmat, y) cef_qr ## constant GNP.deflator GNP Unemployed Armed.Forces ## -3.482259e+03 1.506187e-02 -3.581918e-02 -2.020230e-02 -1.033227e-02 ## Population Year ## -5.110411e-02 1.829151e+00 ## Result comparison Let us compare the results between the Cholesky-based and the QR-based coefficients using the gmp package coefficients as a reference value. sum((cef_exact - cef_chol)^2) ## [1] 1.219947e-09 sum((cef_exact - cef_qr)^2) ## [1] 3.494839e-23 We can see that the QR decomposition coefficients are closer to the gmp coefficients than the one based on the Cholesky decomposition. In general, QR is more stable and faster than Cholesky, which is usually said to be unstable when the positive definite matrix has one or more eigenvalues close to zero. As we can see from the following R code, the last eigenvalue is close to zero, which makes the Cholesky-based coefficients unstable. crossprod(X) %>% is.positive.definite ## [1] TRUE crossprod(X) %>% eigen(symmetric = TRUE) %$% values
## [1] 6.665301e+07 2.090730e+05 1.053550e+05 1.803976e+04 2.455730e+01
## [6] 2.015117e+00 1.172178e-07

For these reasons, the lm function in R uses the QR factorization-based calculation for the regression coefficients.

cef_lm <- lm(y~0+X)\$coefficients
cef_lm == cef_qr
##     Xconstant XGNP.deflator          XGNP   XUnemployed XArmed.Forces
##          TRUE          TRUE          TRUE          TRUE          TRUE
##   XPopulation         XYear
##          TRUE          TRUE

### Reference

[1] Luke Tierney, STAT:7400 Computer Intensive Statistics Note: http://homepage.divms.uiowa.edu/~luke/classes/STAT7400/notes.pdf

[2] J. W. Longley (1967) An appraisal of least-squares programs from the point of view of the user. Journal of the American Statistical Association 62, 819–841.

[3] The QR Decomposition of a Matrix in R: http://astrostatistics.psu.edu/su07/R/html/base/html/qr.html

[4] Monahan, John F. A primer on linear models. CRC Press, 2008.

[5] Cholesky Factorization written by Glyn A. Holton

##### Issac Lee
###### PhD candidate, ABD

I believe anyone can learn anything with a proper education process.


text:
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text:
000033004 001__ 33004
000033004 005__ 20181205220013.0
000033004 0247_ $$2doi$$a10.5075/epfl-thesis-2528
000033004 02471 $$2nebis$$a4302004
000033004 037__ $$aTHESIS 000033004 041__$$aeng
000033004 088__ $$a2528 000033004 245__$$aElectron beam induced deposition of rhodium nanostructures
000033004 269__ $$a2002 000033004 260__$$aLausanne$$bEPFL$$c2002
000033004 300__ $$a208 000033004 336__$$aTheses
000033004 502__ $$aHannes Bleuler, Jens Gobrecht, Elyahou Kapon, Hans Koops, Hans Jörg Mathieu 000033004 520__$$aElectron Beam Induced Deposition (EBID) allows deposition of three-dimensional micro- and nano-structures of conductive and insulating materials on a wide range of substrates. The process is based on the decomposition of molecules of a pre-selected precursor by a focused electron beam. In recent decades EBID of several metals, namely Au, W, Cu and Pt, from different families of precursors, has been achieved and the technique has found some application for small-scale production of laboratory devices and for repair of masks and micro-optoelectronic devices. The weak point of the technique is at present the low purity of the deposited material, caused by metal-organic precursors and by the lack of selectivity of the electron-induced decomposition process. This work is dedicated to EBID of Rh nanostructures from the precursor [RhCl(PF3)2]2. High metal content deposits are expected because the precursor does not contain C atoms and because Rh is one of the less reactive metals. [RhCl(PF3)2]2 as EBID precursor has been characterized by vapor pressure, mass spectrometry and surface residence time measurements. The vapor pressure of 7.5 Pa at room temperature reveals that the precursor is sufficiently volatile for room temperature EBID. The knowledge of the vapor pressure allows also to estimate the number of precursor molecules delivered to the reaction area per unit time. Mass spectrometry measurements allow to know the decomposition path of the precursor under electron impact in the gas phase. The measured spectrum indicates that the molecule decomposes by successive loss of PF3 groups, as confirmed by density functional theory calculations. This is compatible with a high metal content deposit. Residence time measurements show that [RhCl(PF3)2]2 does not decompose on stainless steel surfaces. The measured residence time of 2 ms allows to estimate that the activation energy for desorption of [RhCl(PF3)2]2 on stainless steel is about 0.6 eV and that precursor molecules can travel distances in the micrometer range before being desorbed. EBID structures obtained from [RhCl(PF3)2]2 have been characterized with a wide range of techniques for a better knowledge of the material properties and the deposition process. The deposit morphology has been studied by Transmisssion Electron Microscopy (TEM) and Scanning Electron Microscopy (SEM) to characterize the different steps of the deposition process. Crystallographic analysis is carried out by TEM in diffraction mode. Chemical analysis is carried out by Auger Electron Spectroscopy (AES) and Electron Energy Loss Spectroscopy (EELS). Morphological analysis of deposits carried out at different exposure times reveals that the first phase of the growth process, in close proximity to the substrate, is characterized by an increase of the deposit height and the deposit diameter. On the other hand the second phase of the growth process is characterized by increasing height and constant diameter. TEM contrast profiles of dots and Atomic Force Microscopy (AFM) sections of lines have clearly shown that the EBID rate is highest in the center of the beam and decreases in the peripheral regions. Deposition at variable distances from the precursor source allowed to obtain hollow structures, whose morphology reveals that the precursor reaches the reaction area mainly by direct gas phase transport. Structural analysis and TEM revealed that, independently of the deposition conditions, the deposited material is made up of Rh nanocrystals immersed in a lighter amorphous matrix. Chemical analysis by Auger Electron Spectroscopy revealed that, after removal of the C rich contamination layer by Ar ion sputtering, the average composition of the deposits is about: 60 at.% Rh, 20 at.% P, 5 at.% Cl, 7 at.% N, 8 at.% O. The absence of C and the presence of N and O in the deposit bulk have been confirmed by Electron Energy Loss Spectroscopy. This technique allowed also to prove that Rh is dominant also in deposits of sub-micrometer size (not analyzable with AES) and to determine the elemental distribution in the deposit with nanometer resolution. Comparison of the deposit composition and the positive ions detected by mass spectrometry revealed that EBID, compared to low pressure gas phase ionization, involves a higher number of events, i.e. multi-electron decomposition and rearrangements of partially decomposed species.
000033004 700__ $$0(EPFLAUTH)115420$$aCicoira, Fabio$$g115420 000033004 720_2$$0241019$$aHoffmann, Patrik$$edir.$$g113150 000033004 8564_$$s7540811$$uhttps://infoscience.epfl.ch/record/33004/files/EPFL_TH2528.pdf$$yTexte intégral / Full text$$zTexte intégral / Full text 000033004 909C0$$0252008$$pLOA$$xU10346
000033004 909CO $$ooai:infoscience.tind.io:33004$$pthesis$$pDOI2$$pDOI$$pSTI 000033004 918__$$aSTI$$bSTI-SMT$$cIOA
000033004 919__ $$aLOA 000033004 920__$$a2002-3-1$$b2002 000033004 970__$$a2528/THESES
000033004 973__ $$aEPFL$$sPUBLISHED
000033004 980__ aTHESIS


text:
## Introduction

I use LaTeX for writing academic papers – it takes care of the formatting and referencing, so I can concentrate on generating the content. This page is intended to document my experiences with setting up LaTeX for generating PDFs which meet the style guide required for Common Awards (the academic side of the Church of England’s ordination training).

LaTeX is a type-setting suite, which can create typeset documents from a simple marked up text file. It takes care of all the formatting, page breaks, line spaces, margins, and so on, based on the style of document you instruct it to create. There is some abiguity over which reference formats are allowed – the template below does MHRA, but I think it’s moving to Harvard. It should be straightforward to adapt. The only gotcha is that submissions go through TurnItIn, which can be a bit sniffy about pdflatex, so I had to add to generate PDFs with the correct ligature encoding.

It will assume that you already know how to produce a document in LaTeX (or are willing to learn), and can structure your files and folders sensibly! There are many and varied tutorials on the web for how to use LaTeX, e.g. www.andy-roberts.net/writing/latex

If you need convincing, I typed the following text into my template LaTeX File:

An example document, complete with a long citation of Johnson,\cite{Johnson10}
and a shorter (subsequent) one with page numbers.
\citetitle[pp.\@ 1--4]{Johnson10}

Notice also that reflection'' is recognised by copy and paste, so it plays nicely with \textit{TurnItIn}.

\begin{center}
\textbf{Word Count: xx words}
\end{center}

ran LaTex, and and this is what came out of the other end: Student-A-TMM1234-A2

## Installation

This is the easy bit –  download and install ProTeXt as per the instructions on the proTeXt project page.

There are a couple of extra packages which are also needed, but I think these arrive by magic once included in a document.

## Bibliography

I maintain a single BibTex file that’s broadly categorised (“Old Testament”, “Mission”, etc), and lives in the root (top) folder of my ordination training storage space.
All the books/articles which I have read go in here, so I can easily and consistently cite them in my essays.

This is an extract from my bibliography:

@book{Johnson10,
Author = "Luke Timothy Johnson",
Title = "The Writings of the {N}ew {T}estament",
Shorttitle="New Testament",
Publisher="Fortress Press",
Edition=3
}

The above entry will create footnote citations that look like this:

1 Luke Timothy Johnson, The Writings of the New Testament, 3rd edition (Minneapolis:  Fortress Press), p. N.
2 Johnson, New Testament, p.  N.

And then in the bibliography:
Johnson, Luke Timothy, The Writings of the New Testament, 3rd edition (Minneapolis: Fortress Press).

Hopefully you can start to see the advantage of this approach – all I have to do is to in my essay, and these beauties will magically appear, styled according to the reference guidelines, as per the example at the top of the page.

#### Notes on BibTex

1. Lines starting % are comments, and are ignored.
2. Spaces don’t matter.
3. The first line (““) is the citation key, and is how you will reference this book in your essays.
4. You must always include of these fields, except for Edition, which is optional.
5. The braces force the capital letters of New Testament, as sometimes LaTex changes the case.
6. There are lots of entry types. I have only shown a entry, but you also have articles, collections, web pages, …

## LaTeX pragma

I had lots of fun and games getting the styles to work correctly, and have settled on a standard template document which sets me off!

It looks like the following (you can also download it as a LaTeX file via this link: Student-A-TMM1234-A2 — note this will come down as “.txt”, so will need to renamed “.tex”)

The highlighted sections are the bits I have to change when writing a specific essay – everything else just stays the same.

\usepackage{geometry}
% This fontenc ensures ligature's are compatible with text exports.
\usepackage[T1]{fontenc}
\usepackage{fancyhdr}
\pagestyle{fancy}
\lfoot{}
\cfoot{\thepage}
\rfoot{1 January 2000}

\usepackage{tabularx}
\usepackage{graphicx}
\usepackage{float}
\usepackage{setspace}
\usepackage[%
titleformat=italic,% Titles in italic title
commabeforerest,% A comma after title
%ibidem=name&title,%
%authorformat=reducedifibidem,
citefull=first,% The first citing in full form
oxford,% The oxford style
super,% Footnotes
dotafter=bibentry,%
see,% An extra optional argument as a prenote idem
bibformat=ibidem,
human,
]{jurabib}

\renewcommand{\jbtitlefont}{\textit}

\usepackage{attrib}
\def\PreTrib{} \def\PostTrib{} % We'll do our own parens, thanks!

\usepackage{verse}
\usepackage{bibleref}

\def\V#1 {\raisebox{.5ex}{\scriptsize#1}}

\renewcommand{\refname}{Bibliography}

\title{
\large
TMM1234 --- Theology For Beginners \\
Assignment 1 --- Essay on a Theological Topic
\\ Prof. Theophilus}
\author{Name: A. Student}
\date{Due: Noon, 1 January 2000 }
\begin{document}
\maketitle

\thispagestyle{empty}
\newpage
\doublespacing

\section{Introduction}

An example document, complete with a long citation of Johnson,\cite{Johnson10}
and a shorter (subsequent) one with page numbers.
\citetitle[pp.\@ 1--4]{Johnson10}

Notice also that reflection'' is recognised by copy and paste, so it plays nicely with \textit{TurnItIn}.

\begin{center}
\textbf{Word Count: xx words}
\end{center}

\newpage
\singlespacing

\bibliography{theology}
\bibliographystyle{jox} %instead of the normal one, this is jurabib oxford style

\end{document}

#### Notes on LaTex

1. % indicates the start of a comment, and everything after it is ignored.
2. Spaces don’t matter.
3. The block after sets up the page headers and footers.
4. The bit is setting up the citation format. These are the magic runes that get it close enough to count.
5. does the title page.
6. The reference is to the aforementioned bibliography file. It tells BibTex to look for a file called 3 directories up.

Running PdfLaTeX, BibTex, PdfLaTeX, PdfLaTeX produces the delightful PDF above.

#### Word Counts

Just a quick mention of word counts. I use ProTeXt’s analysis as the basis of my word count. It will count every word which isn’t a LaTeX command or a comment, but will count words inside the command. For instance will add 3 words to the total word count. This of course means that out of the box it isn’t an exact word count of the actual words in the document, because some commands can result in more or fewer words in the output than the command held. It should be noted that no word count system actually counts every word in the document in any case – for instance the page number on each page doesn’t contribute to the page count, and (more significantly) neither does the bibliography normally.

This cuts both ways in my template; for example the bibliography will count as two words “theology” and “jox”, whereas by standard it would count as none. On the other hand references will also only count as a single word. That is to say, will count as a single word by ProTeXt’s reckoning (“Johnson10”), but in terms of words in the PDF at the end there will be an additional 14 words in a footnote the first time, and an additional 3 thereafter. On the other hand, if you were using Harvard, it would only add two words “(Johnson, 2010)”. Likewise, the page headers and footers will count once, not once per page.

How you take this into account depends greatly on your (and your institutions) intepretation of the rules and principles of the word count, and that’s not for me to say! I just wanted to highlight it as something to be aware of, particular if you use custom commands/definitions, which might introduce a significant mismatch.

This article, which follows on from my previous article on LaTeX, aims to introduce historians to using the bibliographical aspects of LaTeX. Organising a bibliography by hand can be a real pain, especially when you have to tailor your citation style to where you are aiming for publication. This can lead to writing the bibliography multiple times in different styles. Even using bibliographical database software such as Zotero or Endnote, this can be a pain. However, there is a better way: let LaTeX take care of it.

There is a better way: let LaTeX take care of it.

From the outset, LaTeX has been able to perform bibliography management; however, over the years numerous packages have been developed to enhance the basic bibliography. These packages range from those that add more citation styles to those that completely rethink how bibliographies should be interpreted. Here we introduce BibLaTeX Chicago. This LaTeX package enhances the BibLaTeX package, which is one of the more popular bibliography interpreter packages, by providing the 15th and 16th editions of Chicago Manual of Style.

1. A basic introductory guide to creating a LaTeX document.
2. An overview of using BibLaTeX Chicago.

## Introduction to LaTeX

We begin by covering the basics of writing in LaTeX (for a more in depth guide see the excellent guide by ShareLaTeX) before diving straight into using BibLaTeX Chicago. The LaTeX document can be divided into two sections, the preamble which tells LaTeX about how you want to set-up the document, including which additional packages LaTeX should use, and the main document which is the text to include. The main document is framed by the commands \begin{document} and \end{document}.

In LaTeX, there are a set of special characters which alter the behaviour of subsequent characters. We will briefly cover two of these special characters. The first of these is the backslash. The backslash marks the start of a sequence that is to be treated as a command, such as \begin{} which marks the beginning of a section. The second important character is the percentage symbol. This symbol marks the beginning of a comment, whereby the following text on the line only adds information to the writer and is not interpreted by LaTeX. This leads to the problem of how to include a backslash or percentage symbol in the text. The usual way to display these symbols is to proceed them with a backslash.

## Preamble

The preamble appears before the main document frame. There are a few important features that the preamble must contain.

To begin creating a LaTeX document, you first tell it which document class and which additional options you wish to use. The document class is essentially the style guide LaTeX uses when creating your document. There are hundreds of options for the document class to choose from. If you are using a premade thesis template, they will most likely have set this up for you with their own custom class. Here we will set-up our LaTeX document using the scrartcl class from the Koma script package.

Now type this into the document:

\documentclass{scrartcl}

Following the class declaration, we will load two packages, both of which are automatically installed with a typical TeX Live LaTeX install. LaTeX packages modify the behaviour and add further capabilities to LaTeX. The two packages we will load are inputenc and graphicsx. Packages are loaded into the document using the \usepackage{} or \requirepackage{} commands.

These commands are essentially equivalent and are both seen in various LaTeX documents. Their only difference is in how they handle errors. The name of the package you wish to load is entered between the curly brackets. If the package accepts options to customise its behaviour these are added in between square brackets before the curly brackets.

We will first set up the text encodings of the document by calling the inputenc package. There are a variety of ways in which text characters can be encoded on a computer. Here we will set the encodings option to UTF-8, this is one of the more common text encodings. This line should always be included in a document. If in doubt, set it to utf8 and see what happens.

In this example we will include a figure. LaTeX does not know what to do with figures by default. The graphicsx package is included which tells LaTeX what to do with an image.

\usepackage[utf8]{inputenc} \usepackage{graphicx}

The title and author information for the document are set in the preamble so they can be included in the metadata for the document. However, they will only appear in the document if you tell LaTeX to include them in the main text. Additionally with many of the document classes you can include a date field. This is usually set by default to the current date.

\title{Bibliographies for Historians} \author{Richard Gunning} %date{March 2015}

That concludes setting up the preamble section. Our document now knows everything it needs to compile the document.

## Main Document

The main document section is where you write the text which you want to appear in the final document. This section is started by \begin{document} and closed by \end{document}. In this example we will start by putting the title and author information at the start of the page. To do this we use the command \maketitle.

\begin{document} \maketitle

Now we will add a section header called “Section 1” and add some basic text to this section. There are several levels within a document. The template will do different things with each of these levels. The levels are:

1. chapter
2. section
3. subsection
4. paragraph

In this document, the highest level will be a section.

LaTeX will automatically treat consecutive lines of text as part of the same paragraph. To separate paragraphs you must leave a blank line in between, or start a paragraph with \paragraph{}.

\section{Section 1} There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.

We will now insert a figure into our document. The \begin{figure} command starts a figure environment. This essentially groups the different elements (image(s), caption(s)) that go into the figure together. The [h!] option tells the LaTeX to force the figure to go here. By default LaTeX attempts to find the best placement for the figure. Other options are t or b, which tell LaTeX to put the figure at the top or bottom of a page. A figure will always be placed where it fits in the document best, i.e. it will avoid breaking paragraphs.

\begin{figure}[h!] \centering % This command tells Latex to center align everything in the figure environment \includegraphics[scale=1.7]{universe.jpg} % This command is where you tell Latex which figure file to use and how big it should be. \caption{The Universe} % Here we add a caption to the figure, placed below the image. \label{fig:universe} % This label is not visible. It allows us to reference this image later using the name fig:universe \end{figure}

Finally, for this basic introduction to LaTeX, we will add a second section where we will demonstrate referencing the figure and sections. Adding the \label{} command with a unique name to a section environment enables you to reference that section at a later time. In the previous example we labelled the figure “fig:universe”. Now whenever you want to reference that figure you just call the reference command, \ref{}, with the same unique name. The same can be done to reference sections.

\section{Section 2} \label{section:section2} We will now reference the figure \ref{fig:universe}. This points to the label in the figure environment. We can also label sections so that we can reference them. The label command will always link to the environment it is in, so a figure if placed between \begin{figure} and \end{figure}, and a section if following a \section command. We will now reference section \ref{section:conclusion}.

The final step for creating our document is to tell the document where the end is. We will add the line

\end{document}

to the end of the file.

With that we have created a working document which will compile into a PDF with no errors.

## BibLaTeX Chicago

In the next stage of this document I will present a worked example of using BibLaTeX Chicago Manual of Style.

This part of the text goes into the beginning of your LaTeX file, after you have sent your document class.

The BibLaTeX style asks for several packages to be included to work properly, although it works with just csquotes and Babel. Babel adds multilingual support to LaTeX. Csquotes adds context sensitive quotation facilities. Knowing the details of what they do is not too important, but it is necessary to include these:

\usepackage{etex} \usepackage{etoolbox} \usepackage{keyval} \usepackage{ifthen} \usepackage{url} \usepackage[american]{babel} \usepackage[babel]{csquotes}

The next step is to include the BibLaTeX package, telling it to use Chicago style. BibLaTeX Chicago defaults to using the 16th edition of Chicago Manual of Style, but this can be overridden to the 15th edition if required (change notes to notes15). There are two equivalent methods of loading BibLaTeX Chicago. The notes style of Chicago Manual of Style used here puts citations as footnotes.

\usepackage[notes,backend=biber]{biblatex-chicago}

or

\usepackage[style=chicago-notes,backend=biber]{biblatex}

For a full list of options suppliable to BibLaTeX Chicago, see the manual.

The next important step is to tell LaTeX to use your bibliography file. This can be done using the \addbibresource{} or \bibliography{} commands pointing them to your .bib file.

BibLaTeX uses a style-independent text-based file format for lists of bibliographic items, such as articles, books, and theses. BibTeX bibliography file names usually end in .bib. A BibLaTeX database contained in a .bib file is formed by “entries” (each corresponding to a bibliographical item, e.g. a journal paper or a conference paper) and each entry is formed by “fields” (e.g., “author”, “year”, “title”). Entries have the following format.

Importantly, the first bit of text after the curly bracket is the entry key, so in this case it is {adams1995hitchhiker. This key is used to reference the item later. For multiple bibliography items it is usually best to get a bibliography manager such as Zotero to generate the file.

The last options we will include in the preamble tells LaTeX how the to section the bibliography. Here we are setting the bibliography to sit at the section level.

Calling citations within a LaTeX document is relatively easy. I will focus on adding footnote citations using the \autocite{} command. For the basic command, all that is required is to add the bib entry key between the curly brackets.

I always thought something was fundamentally wrong with the universe'' \autocite{adams1995hitchhiker}.

However, sometimes it may be required to add more than just the basic citation to the footnote. The \autocite{} command accepts two sets of options both in square brackets. The first set of square brackets adds the text that will precede the citation. The second set of square brackets sets a page range. In Chicago Manual of style, page numbers are added after the citation without p or pp. Setting page ranges like this overrides any page ranges in the bibliography file.

Here we will cite pages 1 to 2 \autocite[see][1--2]{adams1995hitchhiker}.

What if you want to reference several works in a single footnote? BibLaTeX Chicago adds a \autocites command. This will accept an unlimited number of citations to include in a single footnote. This command is a bit more complex with its options.

Following the autocites command name are two sets of round brackets. These contain the text that precedes and proceeds the whole footnote. Next are two sets of square brackets followed by a pair of curly brackets. These have the same function as above and add prenote text and page numbers to individual references. You can add as many sets of square and curly brackets as required for your footnote.

Use the \autocites{} command \autocites(This goes before everything)(This goes after everything)[Before the first citation][Page]{Nobody06}[Before the second citation][Page]{adams1995hitchhiker}

Another way of adding citations in footnotes is by using \Cite{} command from within the \footnote{} command. This command allows you write whatever you would like in the footnote, placing the citation(s) where you choose.

\footnote{You can write anything here and include a cite command wherever you want \Cite{Nobody06}}

Finally, we will add the bibliography to the document. A bibliographical file can contain unused entries. By default, these will not be included into the bibliography. To add unused references to the bibliography, include the command \nocite{} with either a specific citation key, comma separated list of citation keys, or a star indicating all entries.

The actual bibliography is included into the document using the \printbibliography command. This command can take further options in square brackets trailing the command. For this example, we will add a second bibliography which only lists the books cited in the document.

\printbibliography % Now if we only want to include some of the bibliography items of a specific type, or split the bibliography by type we can do this. % Here we define a bibliography heading of books, which will be a subsubsection \defbibheading{Books}{\subsubsection*{Books}} % This command can be moved to the preamble \printbibliography[heading=Books,type=book] % This command then tells LaTeX to use the Books bibliography heading, which we've just defined as a subsubsection which only contains items of type book

With the commands shown in this introduction to using BibLaTeX Chicago, we have worked through the various steps involved in creating a LaTeX document and adding citations. A full worked example of this guide can be found here. One of the biggest issues in the humanities is the lack of LaTeX citation styles to choose from. This is a result of the lack of historians making active use of LaTeX. It is hoped that this guide will encourage more historians to begin using LaTeX. As a result of this programmers will develop the tools required by historians. As with my previous article, if you have any questions regarding BibLaTeX, don’t hesitate to get in contact through the comments.

## Bibliography

312954{KEA947HQ};{4H8WZGQ7}harvard1authorASCno1yes629http://historytothepublic.org/wp-content/plugins/zotpress/


text:
# Implement GAaP: Gaussian Aperture and PSF photometry

XMLWordPrintable

#### Details

• Type: Story
• Status: Done
• Resolution: Done
• Fix Version/s: None
• Component/s: None
• Labels:
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#### Description

In addition to Kron and CModel photometry, this ticket proposes to implement GAaP photometry (Kuijken 2008). For extended objects (galaxies), GAaP is not designed to give accurate fluxes, but provides accurate and robust colors. This ticket is a master-ticket to contain multiple tickets related to this work.

#### Activity

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 Summary Implement GAaP: Gaussian Aperture and PSF photometry Add Weak Lensing Metrics to Regular DRP Processing
 Status To Do [ 10001 ] In Progress [ 3 ]
 Summary Add Weak Lensing Metrics to Regular DRP Processing Implement GAaP: Gaussian Aperture and PSF photometry
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This ticket looks more epic-sized than story-sized. Split off a smaller first step if possible

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Yusra AlSayyad added a comment - This ticket looks more epic-sized than story-sized. Split off a smaller first step if possible
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The link to the actual PR (in lsst-dm org) is https://github.com/lsst-dm/meas_extensions_gaap/pull/1

The repository has a mix of C++ and Python files. The C++ files are based heavily off of meas_base/GaussianFlux, with minor modifications. The reason to replicate the code instead of modifying the meas_base repo is because I expect to make further changes and it is best to keep the base code untouched. The gaapFlux implemented in C++ measures the flux with a Gaussian aperture after accounting for the GaussianPsf. The Python layer takes care of Gaussianizing the PSF and passing the Gaussianized exposure to C++ for measurement.

Show
Arun Kannawadi added a comment - - edited The link to the actual PR (in lsst-dm org) is https://github.com/lsst-dm/meas_extensions_gaap/pull/1 The repository has a mix of C++ and Python files. The C++ files are based heavily off of meas_base/GaussianFlux , with minor modifications. The reason to replicate the code instead of modifying the meas_base repo is because I expect to make further changes and it is best to keep the base code untouched. The gaapFlux implemented in C++ measures the flux with a Gaussian aperture after accounting for the GaussianPsf. The Python layer takes care of Gaussianizing the PSF and passing the Gaussianized exposure to C++ for measurement.
 Status In Progress [ 3 ] In Review [ 10004 ]
 Status In Review [ 10004 ] In Progress [ 3 ]
 Link This issue relates to DM-28024 [ DM-28024 ]
 Epic Link DM-26794 [ 439744 ] DM-27954 [ 442727 ]
 Link This issue relates to DM-28740 [ DM-28740 ]
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The PR is to pull the ticket branch into u/kannawad branch as opposed to master in order to create a clean(er) pull request and master has some relic from its initial stages which I am yet to clean up.

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Arun Kannawadi added a comment - The PR is to pull the ticket branch into u/kannawad branch as opposed to master in order to create a clean(er) pull request and master has some relic from its initial stages which I am yet to clean up.
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I tried to get in the review request before Focus Friday, but the origin/master branch and my ticket branch have diverged so much that I decided to pull into a temporary u/kannawad branch before I get everything in order.

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Arun Kannawadi added a comment - I tried to get in the review request before Focus Friday, but the origin/master branch and my ticket branch have diverged so much that I decided to pull into a temporary u/kannawad branch before I get everything in order.
 Reviewers Jim Bosch, Yusra AlSayyad [ jbosch, yusra ] Status In Progress [ 3 ] In Review [ 10004 ]
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Link to the PR where the base branch is master: https://github.com/lsst-dm/meas_extensions_gaap/pull/3

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Arun Kannawadi added a comment - Link to the PR where the base branch is master:  https://github.com/lsst-dm/meas_extensions_gaap/pull/3
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Jim Bosch added a comment -

I'm done reviewing, and while I've left a ton of comments on the PR, overall I think this looks pretty good, and the volume of comments just reflects this being your first big PR and the fact that it's in a sort of crusty part of the codebase (measurement plugins), where our best examples are sort of fossilized relative to modern style.

I've looked through all of this pretty throughly, so one approach to splitting up with the review with Yusra AlSayyad might be for her to take the second pass, after you think you've addressed my comments?

Show
Jim Bosch added a comment - I'm done reviewing, and while I've left a ton of comments on the PR, overall I think this looks pretty good, and the volume of comments just reflects this being your first big PR and the fact that it's in a sort of crusty part of the codebase (measurement plugins), where our best examples are sort of fossilized relative to modern style. I've looked through all of this pretty throughly, so one approach to splitting up with the review with Yusra AlSayyad might be for her to take the second pass, after you think you've addressed my comments?
 Status In Review [ 10004 ] Reviewed [ 10101 ]
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That sounds good to me. I probably won't be pushing any new changes to the Github repo until Monday, so if Yusra likes to have a first pass, she can do that before Monday and after that, it's probably best waiting for me to address Jim's comments.

Show
Arun Kannawadi added a comment - That sounds good to me. I probably won't be pushing any new changes to the Github repo until Monday, so if Yusra likes to have a first pass, she can do that before Monday and after that, it's probably best waiting for me to address Jim's comments.
 Link This issue relates to DM-28925 [ DM-28925 ]
 Link This issue relates to DM-28955 [ DM-28955 ]
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Along with DM-28955, I have modified the plugin so that the modelPsfMatch task is constructed once during the plugin initialization and the alardSigGauss parameter can be called during the run time. I am not creating the PR on DM-28955 at the moment, because the size parameter of the AL Basis can be chosen dynamically using the scaleByFwhm parameter (bad naming that kept it hidden to me all this while). However, it is a different choice from one that Konraad chose to implement in his GAaP code. I'm replicating his choices as closely as possible, but I'd like to investigate how well the PSF Gaussianization is if I simply set scaleByFwhm=True.

Show
Arun Kannawadi added a comment - Along with DM-28955 , I have modified the plugin so that the modelPsfMatch task is constructed once during the plugin initialization and the alardSigGauss parameter can be called during the run time. I am not creating the PR on DM-28955 at the moment, because the size parameter of the AL Basis can be chosen dynamically using the scaleByFwhm parameter (bad naming that kept it hidden to me all this while). However, it is a different choice from one that Konraad chose to implement in his GAaP code. I'm replicating his choices as closely as possible, but I'd like to investigate how well the PSF Gaussianization is if I simply set scaleByFwhm=True.
 Status Reviewed [ 10101 ] In Review [ 10004 ]
 Link This issue is blocked by DM-29256 [ DM-29256 ]
 Reviewers Jim Bosch, Yusra AlSayyad [ jbosch, yusra ] Jim Bosch [ jbosch ]
Hide

Posted some comments on github and removed myself from the reviewer list. Main complaint is that it doesn't build.

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Yusra AlSayyad added a comment - Posted some comments on github and removed myself from the reviewer list. Main complaint is that it doesn't build.
Hide

I've fixed the build issue and made sure it builds successfully in a cloned repo as well. I made one other commit during pair coding with Jim this Wednesday. I think I've also addressed all other minor comments. I'll merge it to master on Monday and leave any remaining comments for smaller tickets later. This is going to remain an lsst-dm repo for some more time, at least until I wrap up DM-28955.

Show
Arun Kannawadi added a comment - I've fixed the build issue and made sure it builds successfully in a cloned repo as well. I made one other commit during pair coding with Jim this Wednesday. I think I've also addressed all other minor comments. I'll merge it to master on Monday and leave any remaining comments for smaller tickets later. This is going to remain an lsst-dm repo for some more time, at least until I wrap up DM-28955 .
 Status In Review [ 10004 ] Reviewed [ 10101 ]
 Resolution Done [ 10000 ] Status Reviewed [ 10101 ] Done [ 10002 ]

#### People

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text:
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

# How can you find the density of Iron without looking up on the internet?

The formula for is: rho = m/V => where rho is the density in either kg/L or the variant conversions - g/mL, etc, mass is the mass of the object in kg or other variants, and V is the volume in L or other variants.

Mass of an object can be measured in either g or kg by using a scale.

Volume of the object can either be calculated by V=lwh or through a water displacement experiment - getting a beaker of water and placing the object in the beaker. Subtract the new measure of water with the original measure. The difference is the volume of the object.

In the video, the density was found online, and the volume was found by subtracting the final volume of water (425 mL) by the initial volume (150 mL), equaling 275 mL - the volume of the object.

When calculating the mass (in the video), the mass will come out in grams because of the relationship between mass and volume. You can convert the variable to another, like mL -> L, but then your mass will be in kg.

Hope this helps :)


text:
# 31.7 Tunneling  (Page 2/5)

 Page 2 / 5

Good ideas explain more than one thing. In addition to qualitatively explaining how the four nucleons in an $\alpha$ particle can get out of the nucleus, the detailed theory also explains quantitatively the half-life of various nuclei that undergo $\alpha$ decay. This description is what Gamow and others devised, and it works for $\alpha$ decay half-lives that vary by 17 orders of magnitude. Experiments have shown that the more energetic the $\alpha$ decay of a particular nuclide is, the shorter is its half-life. Tunneling explains this in the following manner: For the decay to be more energetic, the nucleons must have more energy in the nucleus and should be able to ascend a little closer to the rim. The barrier is therefore not as thick for more energetic decay, and the exponential decrease of the wave function inside the barrier is not as great. Thus the probability of finding the particle outside the barrier is greater, and the half-life is shorter.

Tunneling as an effect also occurs in quantum mechanical systems other than nuclei. Electrons trapped in solids can tunnel from one object to another if the barrier between the objects is thin enough. The process is the same in principle as described for $\alpha$ decay. It is far more likely for a thin barrier than a thick one. Scanning tunneling electron microscopes function on this principle. The current of electrons that travels between a probe and a sample tunnels through a barrier and is very sensitive to its thickness, allowing detection of individual atoms as shown in [link] .

## Phet explorations: quantum tunneling and wave packets

Watch quantum "particles" tunnel through barriers. Explore the properties of the wave functions that describe these particles.

## Section summary

• Tunneling is a quantum mechanical process of potential energy barrier penetration. The concept was first applied to explain $\alpha$ decay, but tunneling is found to occur in other quantum mechanical systems.

## Conceptual questions

A physics student caught breaking conservation laws is imprisoned. She leans against the cell wall hoping to tunnel out quantum mechanically. Explain why her chances are negligible. (This is so in any classical situation.)

When a nucleus $\alpha$ decays, does the $\alpha$ particle move continuously from inside the nucleus to outside? That is, does it travel each point along an imaginary line from inside to out? Explain.

## Problems-exercises

Derive an approximate relationship between the energy of $\alpha$ decay and half-life using the following data. It may be useful to graph the log of ${t}_{1/2}$ against ${E}_{\alpha }$ to find some straight-line relationship.

Energy and half-life for $\alpha$ Decay
Nuclide ${E}_{\text{α}}\phantom{\rule{0.25em}{0ex}}\text{(MeV)}$ ${t}_{\text{1/2}}$
${}^{\text{216}}\text{Ra}$ $\text{9.5}$ $\text{0.18 μs}$
${}^{\text{194}}\text{Po}$ $\text{7.0}$ $\text{0.7 s}$
${}^{\text{240}}\text{Cm}$ $\text{6.4}$ $\text{27 d}$
${}^{\text{226}}\text{Ra}$ $\text{4.91}$ $\text{1600 y}$
${}^{\text{232}}\text{Th}$ $\text{4.1}$ $1.4×{\text{10}}^{\text{10}}\phantom{\rule{0.25em}{0ex}}\text{y}$

Integrated Concepts

A 2.00-T magnetic field is applied perpendicular to the path of charged particles in a bubble chamber. What is the radius of curvature of the path of a 10 MeV proton in this field? Neglect any slowing along its path.

22.8 cm

(a) Write the decay equation for the $\alpha$ decay of ${}^{235}\text{U}$ . (b) What energy is released in this decay? The mass of the daughter nuclide is 231.036298 u. (c) Assuming the residual nucleus is formed in its ground state, how much energy goes to the $\alpha$ particle?

(a) ${}_{92}^{235}{\text{U}}_{143}\to {}_{\text{90}}^{\text{231}}{\text{Th}}_{\text{141}}+{}_{2}^{4}{\text{He}}_{\text{2}}$

(b) 4.679 MeV

(c) 4.599 MeV

Unreasonable Results

The relatively scarce naturally occurring calcium isotope ${}^{48}\text{Ca}$ has a half-life of about $2×{10}^{16}\phantom{\rule{0.25em}{0ex}}\text{y}$ . (a) A small sample of this isotope is labeled as having an activity of 1.0 Ci. What is the mass of the ${}^{48}\text{Ca}$ in the sample? (b) What is unreasonable about this result? (c) What assumption is responsible?

Unreasonable Results

A physicist scatters $\gamma$ rays from a substance and sees evidence of a nucleus $7.5×{10}^{–13}\phantom{\rule{0.25em}{0ex}}\text{m}$ in radius. (a) Find the atomic mass of such a nucleus. (b) What is unreasonable about this result? (c) What is unreasonable about the assumption?

a) $2.4×{\text{10}}^{8}$ u

(b) The greatest known atomic masses are about 260. This result found in (a) is extremely large.

(c) The assumed radius is much too large to be reasonable.

Unreasonable Results

A frazzled theoretical physicist reckons that all conservation laws are obeyed in the decay of a proton into a neutron, positron, and neutrino (as in ${\beta }^{+}$ decay of a nucleus) and sends a paper to a journal to announce the reaction as a possible end of the universe due to the spontaneous decay of protons. (a) What energy is released in this decay? (b) What is unreasonable about this result? (c) What assumption is responsible?

(a) $–1.805 MeV$

(b) Negative energy implies energy input is necessary and the reaction cannot be spontaneous.

(c) Although all conversation laws are obeyed, energy must be supplied, so the assumption of spontaneous decay is incorrect.

Consider the decay of radioactive substances in the Earth’s interior. The energy emitted is converted to thermal energy that reaches the earth’s surface and is radiated away into cold dark space. Construct a problem in which you estimate the activity in a cubic meter of earth rock? And then calculate the power generated. Calculate how much power must cross each square meter of the Earth’s surface if the power is dissipated at the same rate as it is generated. Among the things to consider are the activity per cubic meter, the energy per decay, and the size of the Earth.

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vf=vi+at vf=2.5+ 0.67*10 vf= 2.5 + 6.7 vf = 9.2
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s = vi t +1/2at sq s=58.5 s=v av X t vf= 9.2
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Note: LINEAR MOMENTUM Linear momentum is defined as the product of a system’s mass multiplied by its velocity: size 12{p=mv} {}
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Mohit


text:
# cf.Field.auxiliary_coordinate¶

Field.auxiliary_coordinate(identity, default=ValueError(), key=False)[source]

Return an auxiliary coordinate construct, or its key.

New in version 3.0.0.

Parameters: identity: Select the auxiliary coordinate construct by one of: The identity or key of an auxiliary coordinate construct. The identity or key of a domain axis construct that is spanned by a unique 1-d auxiliary coordinate construct’s data. The position, in the field construct’s data, of a domain axis construct that is spanned by a unique 1-d auxiliary coordinate construct’s data. A construct identity is specified by a string (e.g. 'latitude', 'long_name=time', 'ncvar%lat', etc.); a Query object (e.g. cf.eq('longitude')); or a compiled regular expression (e.g. re.compile('^atmosphere')) that selects the relevant constructs whose identities match via re.search. A construct has a number of identities, and is selected if any of them match any of those provided. A construct’s identities are those returned by its identities method. In the following example, the construct x has six identities: >>> x.identities() ['time', 'long_name=Time', 'foo=bar', 'standard_name=time', 'ncvar%t', T']  A construct key may optionally have the 'key%' prefix. For example 'auxiliarycoordinate2' and 'key%auxiliarycoordinate2' are both acceptable keys. A position of a domain axis construct in the field construct’s data is specified by an integer index. Note that in the output of a print call or dump method, a construct is always described by one of its identities, and so this description may always be used as an identity argument. Parameter example: identity='Y' Parameter example: identity='latitude' Parameter example: identity='long_name=Latitude' Parameter example: identity='auxiliarycoordinate1' Parameter example: identity='domainaxis2' Parameter example: identity='ncdim%y' Parameter example: identity=0 key: bool, optional If True then return the selected construct key. By default the construct itself is returned. default: optional Return the value of the default parameter if a construct can not be found. If set to an Exception instance then it will be raised instead. AuxiliaryCoordinate or str The selected auxiliary coordinate construct, or its key.

Examples:

TODO


text:
# Number Theory using an induction proof

Prove that the inequality $$\left(1+\frac{1}{n}\right)^n > \frac{9}{4}$$ holds for all integers $n$ beyond a certain point.

I must show that it is true for all $n>3$. but I am having difficult time showing $P(n+1)$ is true using induction. The book give hint: $$[1+(n+1)^{-1}]^{n+1} = \left[\frac{n+2}{n+1}\right]\left[\frac{n+1}{n}\right]^n \left[1-(n+1)^{-2}]^n\right]$$

• What have you tried so far? Can you write out the steps up to the point you get stuck? – graydad Feb 26 '15 at 15:31
• You can use binomial theorem if you can't get induction to work – AvZ Feb 26 '15 at 15:34

We can write $$\frac{(n+1)^n}{n^n}$$ By binomial theorem, $$=\frac{{n\choose 0}n^n+{n\choose 1}n^{n-1}+\cdots+{n\choose n}n^0}{n^n}$$ $$={n\choose 0}+{n\choose 1}n^{-1}+\cdots+{n\choose n}n^{-n}$$ $$=1+\frac{n!}{n(n-1)!(1)!}+\frac{n!}{n^2(n-2)!(2)!}+\cdots+\frac{n!}{n^n(0)!(n)!}$$ $$=1+1+\frac{n-1}{2n}+\cdots+\frac{1}{n^n}$$ As you can see, $$1+1+\frac{n-1}{2n}>\frac{9}{4}\forall n>3,n\in\mathbb N$$ (as $$\frac{n-1}{2n}$$ is strictly increasing for all positive numbers and $$2+\frac{3-1}{2(3)}>\frac{9}{4}$$) $$\implies 2+\frac{n-1}{2n}>\frac{9}{4}$$ The first three terms are already greater than $$\frac{9}{4}$$ and rest of the terms are positive, hence proved.
$$P(4)$$ is true for $$n=4$$ as$$\frac{19}{8}>\frac{9}{4}$$
Let $$P(k)$$ be true.
Then $$P(k+1)\implies 2+\frac{k}{2k+2}>\frac{9}{4}$$ We have to prove $$P(k+1)$$ is true $$\frac{k}{2k+2}>\frac{1}{4}$$ $$4k>2k+2$$ $$k>1$$ This is obviously true as we have taken $$k>3$$.
We now have the first three terms always greater than $$\frac{9}{4}$$. Since rest all terms are positive, hence proved by induction.


text:
# Find the exact value of each of the remaining trigonometric function of theta.displaystyle

Find the exact value of each of the remaining trigonometric function of $$\theta.$$
$$\displaystyle \cos{\theta}=\frac{24}{{25}},{270}^{\circ}<\theta<{360}^{\circ}$$
$$\displaystyle \sin{\theta}=?$$
$$\displaystyle \tan{\theta}=?$$
$$\displaystyle \sec{\theta}=?$$
$$\displaystyle \csc{\theta}=?$$
$$\displaystyle \cot{\theta}=?$$

• Questions are typically answered in as fast as 30 minutes

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Step 1
Given:
$$\displaystyle \cos{\theta}=\frac{24}{{25}},{270}^{\circ}<\theta<{360}^{\circ}$$
So the angle lies in fourth quadrant. In the fourth quadrant cosine and secant are positive all others are negative.
Pythagorean identity is used for finding sine value
$$\displaystyle{{\sin}^{2}\theta}+{{\cos}^{2}\theta}={1}$$
$$\displaystyle \sin{\theta}=\pm\sqrt{{{1}-{{\cos}^{2}\theta}}}$$
$$\displaystyle=\pm\sqrt{{{1}-{\left(\frac{24}{{25}}\right)}^{2}}}$$
$$\displaystyle=\pm\sqrt{{\frac{{{25}^{2}-{24}^{2}}}{{25}^{2}}}}$$
$$\displaystyle=-\frac{7}{{25}}$$
The sine value is negative since angle is in third quadrant.
Step 2
The tan value can be found as
$$\displaystyle \tan{\theta}=\frac{{ \sin{\theta}}}{{ \cos{\theta}}}$$
$$\displaystyle=\frac{{-\frac{7}{{25}}}}{{\frac{24}{{25}}}}$$
$$\displaystyle=-\frac{7}{{24}}$$
Step 3
Secant is reciprocal of cosine.
$$\displaystyle \sec{\theta}=\frac{1}{{ \cos{\theta}}}$$
$$\displaystyle=\frac{1}{{\frac{24}{{25}}}}$$
$$\displaystyle=\frac{25}{{24}}$$
Step 4
Cosecant is the reciprocal of sine.
$$\displaystyle \csc{\theta}=\frac{1}{{ \sin{\theta}}}$$
$$\displaystyle=\frac{1}{{-\frac{7}{{25}}}}$$
$$\displaystyle=-\frac{25}{{7}}$$
Step 5
Cot is the reciprocal of tan
$$\displaystyle \cot{\theta}=\frac{1}{{ \tan{\theta}}}$$
$$\displaystyle=\frac{1}{{-\frac{7}{{24}}}}$$
$$\displaystyle=-\frac{24}{{7}}$$


text:
# Lesson 10Meaningful PossibilitiesSolidify Understanding

Use the given matrices to perform the indicated operations.

## Set

### 7.

A farmer is harvesting his field of potatoes and wonders what the average weight of the potatoes is. He decides to randomly select and finds that the average weight of those is , and they have a standard deviation of . Give an interval of plausible values for the actual weight of all the potatoes in the field.

### 8.

If the farmer were to double the sample size, what would happen to the interval of plausible values? Would the interval increase, decrease, or stay the same? Why?

### 9.

Based on your interval of plausible values, is it likely that the potatoes for the entire field weigh more than on average? Why or why not?

### 10.

Suppose the actual weight of the potatoes in the field is with a standard deviation of . If the farmer were to create a sampling distribution by taking repeated samples of size and identify the average weight of potatoes in each sample, what would be the shape, center, and standard deviation of this distribution?

### 11.

If the actual weight of potatoes in the field is really , what is the probability of taking a sample of and finding that they weigh or fewer pounds?

## Go

Fill in the blanks on the sentences below.

### 12.

The expression is called a perfect square trinomial. I can recognize it because the first and last terms will always be perfect . The middle term will be times the and . There will always be an sign before the last term. The expression factors as .

### 13.

The expression is called the difference of squares. I can recognize it because it’s a binomial and the first and last terms are perfect . The sign between the first term and the last term is always a .

The expression factors as .

### 14.

The expression is called the sum of cubes. I can recognize it because it’s a binomial and the first and last terms are . The expression factors into a binomial and a trinomial. The sign between the terms in the binomial is the as the sign in the expression. The first sign in the trinomial is the of the sign in the binomial. That’s why all of the middle terms cancel when multiplying. The last sign in the trinomial is always . The expression factors as .

Factor using what you know about special products.


text:
Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals → CJM
Abstract view

On Polynomial Invariants of Exceptional Simple Algebraic Groups

Published:1999-06-01
Printed: Jun 1999
• A. Elduque
• A. V. Iltyakov
 Format: HTML LaTeX MathJax PDF PostScript

Abstract

We study polynomial invariants of systems of vectors with respect to exceptional simple algebraic groups in their minimal linear representations. For each type we prove that the algebra of invariants is integral over the subalgebra of trace polynomials for a suitable algebraic system (\cf\ \cite{Schw1}, \cite{Schw2}, \cite{Ilt}).
 MSC Classifications: 15A72 - Vector and tensor algebra, theory of invariants [See also 13A50, 14L24] 17C20 - Simple, semisimple algebras

© Canadian Mathematical Society, 2014 : https://cms.math.ca/