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# inverse trigonometric functions derivatives

AP.CALC: FUN‑3 (EU), FUN‑3.E (LO), FUN‑3.E.2 (EK) Google Classroom Facebook Twitter. The process for finding the derivative of $\arccos x$ is almost identical to that used for $\arcsin x$: Suppose $\arccos x = \theta$. Inverse Trigonometric Functions - Derivatives - Harder Example. Arccosine 3. Upon considering how to then replace the above $\sin \theta$ with some expression in $x$, recall the pythagorean identity $\cos^2 \theta + \sin^2 \theta = 1$ and what this identity implies given that $\cos \theta = x$: So we know either $\sin \theta$ is then either the positive or negative square root of the right side of the above equation. We'll assume you're ok with this, but you can opt-out if you wish. The inverse sine function (Arcsin), y = arcsin x, is the inverse of the sine function. Derivatives of Inverse Trigonometric Functions We can use implicit differentiation to find the formulas for the derivatives of the inverse trigonometric functions, as the following examples suggest: Finding the Derivative of Inverse Sine Function, d d x (arcsin Presuming that the range of the secant function is given by $(0, \pi)$, we note that $\theta$ must be either in quadrant I or II. The usual approach is to pick out some collection of angles that produce all possible values exactly once. Formula for the Derivative of Inverse Secant Function. If $$f\left( x \right)$$ and $$g\left( x \right)$$ are inverse functions then, It has plenty of examples and worked-out practice problems. Because each of the above-listed functions is one-to-one, each has an inverse function. Practice your math skills and learn step by step with our math solver. Important Sets of Results and their Applications The derivatives of the inverse trigonometric functions can be obtained using the inverse function theorem. 2 The graph of y = sin x does not pass the horizontal line test, so it has no inverse. Thus, For example, the sine function $$x = \varphi \left( y \right)$$ $$= \sin y$$ is the inverse function for $$y = f\left( x \right)$$ $$= \arcsin x.$$ Then the derivative of $$y = \arcsin x$$ is given by, ${{\left( {\arcsin x} \right)^\prime } = f’\left( x \right) = \frac{1}{{\varphi’\left( y \right)}} }= {\frac{1}{{{{\left( {\sin y} \right)}^\prime }}} }= {\frac{1}{{\cos y}} }= {\frac{1}{{\sqrt {1 – {\sin^2}y} }} }= {\frac{1}{{\sqrt {1 – {\sin^2}\left( {\arcsin x} \right)} }} }= {\frac{1}{{\sqrt {1 – {x^2}} }}\;\;}\kern-0.3pt{\left( { – 1 < x < 1} \right).}$. Necessary cookies are absolutely essential for the website to function properly. Inverse Trigonometry Functions and Their Derivatives. And To solve the related problems. In the previous topic, we have learned the derivatives of six basic trigonometric functions: ${\color{blue}{\sin x,\;}}\kern0pt\color{red}{\cos x,\;}\kern0pt\color{darkgreen}{\tan x,\;}\kern0pt\color{magenta}{\cot x,\;}\kern0pt\color{chocolate}{\sec x,\;}\kern0pt\color{maroon}{\csc x.\;}$, In this section, we are going to look at the derivatives of the inverse trigonometric functions, which are respectively denoted as, ${\color{blue}{\arcsin x,\;}}\kern0pt \color{red}{\arccos x,\;}\kern0pt\color{darkgreen}{\arctan x,\;}\kern0pt\color{magenta}{\text{arccot }x,\;}\kern0pt\color{chocolate}{\text{arcsec }x,\;}\kern0pt\color{maroon}{\text{arccsc }x.\;}$. Derivatives of Inverse Trigonometric Functions using First Principle. Like before, we differentiate this implicitly with respect to $x$ to find, Solving for $d\theta/dx$ in terms of $\theta$ we quickly get, This is where we need to be careful. The Inverse Tangent Function. Examples: Find the derivatives of each given function. Since $\theta$ must be in the range of $\arcsin x$ (i.e., $[-\pi/2,\pi/2]$), we know $\cos \theta$ must be positive. Click or tap a problem to see the solution. To be a useful formula for the derivative of $\arccos x$ however, we would prefer that $\displaystyle{\frac{d\theta}{dx} = \frac{d}{dx} (\arccos x)}$ be expressed in terms of $x$, not $\theta$. The sine function (red) and inverse sine function (blue). Similarly, we can obtain an expression for the derivative of the inverse cosecant function: ${{\left( {\text{arccsc }x} \right)^\prime } = {\frac{1}{{{{\left( {\csc y} \right)}^\prime }}} }}= {-\frac{1}{{\cot y\csc y}} }= {-\frac{1}{{\csc y\sqrt {{{\csc }^2}y – 1} }} }= {-\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}.}$. Inverse Sine Function. $$\frac{d\theta}{dx} = \frac{-1}{\csc^2 \theta} = \frac{-1}{1+x^2}$$ Thus, Finally, plugging this into our formula for the derivative of $\arcsin x$, we find, Finding the Derivative of Inverse Cosine Function, $\displaystyle{\frac{d}{dx} (\arccos x)}$. $$\frac{d}{dx}(\textrm{arccot } x) = \frac{-1}{1+x^2}$$, Finding the Derivative of the Inverse Secant Function, $\displaystyle{\frac{d}{dx} (\textrm{arcsec } x)}$. This category only includes cookies that ensures basic functionalities and security features of the website. Derivatives of Inverse Trig Functions. }\], $\require{cancel}{y^\prime = \left( {\arcsin \left( {x – 1} \right)} \right)^\prime }={ \frac{1}{{\sqrt {1 – {{\left( {x – 1} \right)}^2}} }} }={ \frac{1}{{\sqrt {1 – \left( {{x^2} – 2x + 1} \right)} }} }={ \frac{1}{{\sqrt {\cancel{1} – {x^2} + 2x – \cancel{1}} }} }={ \frac{1}{{\sqrt {2x – {x^2}} }}. Coming to the question of what are trigonometric derivatives and what are they, the derivatives of trigonometric functions involve six numbers. Previously, derivatives of algebraic functions have proven to be algebraic functions and derivatives of trigonometric functions have been shown to be trigonometric functions. This lessons explains how to find the derivatives of inverse trigonometric functions. Review the derivatives of the inverse trigonometric functions: arcsin (x), arccos (x), and arctan (x). They are cosecant (cscx), secant (secx), cotangent (cotx), tangent (tanx), cosine (cosx), and sine (sinx). These cookies do not store any personal information. These functions are used to obtain angle for a given trigonometric value. In both, the product of \sec \theta \tan \theta must be positive. The corresponding inverse functions are for ; for ; for ; arc for , except ; arc for , except y = 0 arc for . Table 2.7.14. Arccotangent 5. Suppose \textrm{arccot } x = \theta. Differentiation of Inverse Trigonometric Functions Each of the six basic trigonometric functions have corresponding inverse functions when appropriate restrictions are placed on the domain of the original functions. g ( x) = arccos   ( 2 x) g (x)=\arccos\!\left (2x\right) g(x)= arccos(2x) g, left parenthesis, x, right parenthesis, … Arctangent 4. The inverse trigonometric functions actually perform the opposite operation of the trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. \dfrac {d} {dx}\arcsin (x)=\dfrac {1} {\sqrt {1-x^2}} dxd arcsin(x) = 1 − x2 To be a useful formula for the derivative of \arcsin x however, we would prefer that \displaystyle{\frac{d\theta}{dx} = \frac{d}{dx} (\arcsin x)} be expressed in terms of x, not \theta. However, since trigonometric functions are not one-to-one, meaning there are are infinitely many angles with , it is impossible to find a true inverse function for . Another method to find the derivative of inverse functions is also included and may be used. Then it must be the cases that, Implicitly differentiating the above with respect to x yields. Now let's determine the derivatives of the inverse trigonometric functions, y = arcsinx, y = arccosx, y = arctanx, y = arccotx, y = arcsecx, and y = arccscx. This implies. Inverse trigonometric functions are literally the inverses of the trigonometric functions. Derivative of Inverse Trigonometric Function as Implicit Function. Sec 3.8 Derivatives of Inverse Functions and Inverse Trigonometric Functions Ex 1 Let f x( )= x5 + 2x −1. We can use implicit differentiation to find the formulas for the derivatives of the inverse trigonometric functions, as the following examples suggest: Finding the Derivative of Inverse Sine Function, \displaystyle{\frac{d}{dx} (\arcsin x)}, Suppose \arcsin x = \theta. There are particularly six inverse trig functions for each trigonometry ratio. The inverse functions exist when appropriate restrictions are placed on the domain of the original functions. These six important functions are used to find the angle measure in a right triangle when two sides of the triangle measures are known. These cookies will be stored in your browser only with your consent. •Since the definition of an inverse function says that -f 1(x)=y => f(y)=x We have the inverse sine function, -sin 1x=y - π=> sin y=x and π/ 2 <=y<= / 2 The derivatives of inverse trigonometric functions are quite surprising in that their derivatives are actually algebraic functions. This website uses cookies to improve your experience while you navigate through the website. View Lesson 9-Differentiation of Inverse Trigonometric Functions.pdf from MATH 146 at Mapúa Institute of Technology. Domains and ranges of the trigonometric and inverse trigonometric functions Quick summary with Stories. }$, ${y^\prime = \left( {\text{arccot}\,{x^2}} \right)^\prime }={ – \frac{1}{{1 + {{\left( {{x^2}} \right)}^2}}} \cdot \left( {{x^2}} \right)^\prime }={ – \frac{{2x}}{{1 + {x^4}}}. Inverse Functions and Logarithms. Section 3-7 : Derivatives of Inverse Trig Functions. Nevertheless, it is useful to have something like an inverse to these functions, however imperfect. Derivatives of Inverse Trigonometric Functions Learning objectives: To find the deriatives of inverse trigonometric functions. Using this technique, we can find the derivatives of the other inverse trigonometric functions: \[{{\left( {\arccos x} \right)^\prime } }={ \frac{1}{{{{\left( {\cos y} \right)}^\prime }}} }= {\frac{1}{{\left( { – \sin y} \right)}} }= {- \frac{1}{{\sqrt {1 – {{\cos }^2}y} }} }= {- \frac{1}{{\sqrt {1 – {{\cos }^2}\left( {\arccos x} \right)} }} }= {- \frac{1}{{\sqrt {1 – {x^2}} }}\;\;}\kern-0.3pt{\left( { – 1 < x < 1} \right),}\qquad$, ${{\left( {\arctan x} \right)^\prime } }={ \frac{1}{{{{\left( {\tan y} \right)}^\prime }}} }= {\frac{1}{{\frac{1}{{{{\cos }^2}y}}}} }= {\frac{1}{{1 + {{\tan }^2}y}} }= {\frac{1}{{1 + {{\tan }^2}\left( {\arctan x} \right)}} }= {\frac{1}{{1 + {x^2}}},}$, ${\left( {\text{arccot }x} \right)^\prime } = {\frac{1}{{{{\left( {\cot y} \right)}^\prime }}}}= \frac{1}{{\left( { – \frac{1}{{{\sin^2}y}}} \right)}}= – \frac{1}{{1 + {{\cot }^2}y}}= – \frac{1}{{1 + {{\cot }^2}\left( {\text{arccot }x} \right)}}= – \frac{1}{{1 + {x^2}}},$, ${{\left( {\text{arcsec }x} \right)^\prime } = {\frac{1}{{{{\left( {\sec y} \right)}^\prime }}} }}= {\frac{1}{{\tan y\sec y}} }= {\frac{1}{{\sec y\sqrt {{{\sec }^2}y – 1} }} }= {\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}.}$. Here, we suppose $\textrm{arcsec } x = \theta$, which means $sec \theta = x$. Lesson 9 Differentiation of Inverse Trigonometric Functions OBJECTIVES • to 2 mins read. Note. Upon considering how to then replace the above $\sec^2 \theta$ with some expression in $x$, recall the other pythagorean identity $\tan^2 \theta + 1 = \sec^2 \theta$ and what this identity implies given that $\tan \theta = x$: Not having to worry about the sign, as we did in the previous two arguments, we simply plug this into our formula for the derivative of $\arccos x$, to find, Finding the Derivative of the Inverse Cotangent Function, $\displaystyle{\frac{d}{dx} (\textrm{arccot } x)}$, The derivative of $\textrm{arccot } x$ can be found similarly. Inverse trigonometric functions provide anti derivatives for a variety of functions that arise in engineering. which implies the following, upon realizing that $\cot \theta = x$ and the identity $\cot^2 \theta + 1 = \csc^2 \theta$ requires $\csc^2 \theta = 1 + x^2$, The process for finding the derivative of $\arctan x$ is slightly different, but the same overall strategy is used: Suppose $\arctan x = \theta$. The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. But opting out of some of these cookies may affect your browsing experience. The Inverse Cosine Function. Dividing both sides by $-\sin \theta$ immediately leads to a formula for the derivative. The formula for the derivative of y= sin 1 xcan be obtained using the fact that the derivative of the inverse function y= f 1(x) is the reciprocal of the derivative x= f(y). Dividing both sides by $\cos \theta$ immediately leads to a formula for the derivative. We then apply the same technique used to prove Theorem 3.3, “The Derivative Rule for Inverses,” to diﬀerentiate each inverse trigonometric function. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. Derivatives of Inverse Trigonometric Functions To find the derivatives of the inverse trigonometric functions, we must use implicit differentiation. Arcsine 2. Of course $|\sec \theta| = |x|$, and we can use $\tan^2 \theta + 1 = \sec^2 \theta$ to establish $|\tan \theta| = \sqrt{x^2 - 1}$. In modern mathematics, there are six basic trigonometric functions: sine, cosine, tangent, secant, cosecant, and cotangent. }\], {y’\left( x \right) }={ {\left( {\arctan \frac{{x + 1}}{{x – 1}}} \right)^\prime } }= {\frac{1}{{1 + {{\left( {\frac{{x + 1}}{{x – 1}}} \right)}^2}}} \cdot {\left( {\frac{{x + 1}}{{x – 1}}} \right)^\prime } }= {\frac{{1 \cdot \left( {x – 1} \right) – \left( {x + 1} \right) \cdot 1}}{{{{\left( {x – 1} \right)}^2} + {{\left( {x + 1} \right)}^2}}} }= {\frac{{\cancel{\color{blue}{x}} – \color{red}{1} – \cancel{\color{blue}{x}} – \color{red}{1}}}{{\color{maroon}{x^2} – \cancel{\color{green}{2x}} + \color{DarkViolet}{1} + \color{maroon}{x^2} + \cancel{\color{green}{2x}} + \color{DarkViolet}{1}}} }= {\frac{{ – \color{red}{2}}}{{\color{maroon}{2{x^2}} + \color{DarkViolet}{2}}} }= { – \frac{1}{{1 + {x^2}}}. Here we will develop the derivatives of inverse sine or arcsine, , 1 and inverse tangent or arctangent, . {\displaystyle {\begin{aligned}{\frac {d}{dz}}\arcsin(z)&{}={\frac {1}{\sqrt {1-z^{2}}}}\;;&z&{}\neq -1,+1\\{\frac {d}{dz}}\arccos(z)&{}=-{\frac {1}{\sqrt {1-z^{2}}}}\;;&z&{}\neq -1,+1\\{\frac {d}{dz}}\arctan(z)&{}={\frac {1}{1+z^{2}}}\;;&z&{}\neq -i,+i\\{\frac {d}{dz}}\operatorname {arccot}(z)&{}=-{\frac {1}{1+z^{2}}}\;;&z&{}\neq -i,+i\\{\frac {d}{dz}}\operatorname {arcsec}(z)&{}={\frac {1}{z^{2… Then it must be the case that. One example does not require the chain rule and one example requires the chain rule. Problem. The derivatives of the inverse trigonometric functions are given below. Inverse Trigonometric Functions: •The domains of the trigonometric functions are restricted so that they become one-to-one and their inverse can be determined. We also use third-party cookies that help us analyze and understand how you use this website. You also have the option to opt-out of these cookies. Dividing both sides by \sec^2 \theta immediately leads to a formula for the derivative. Trigonometric Functions (With Restricted Domains) and Their Inverses. Using similar techniques, we can find the derivatives of all the inverse trigonometric functions. The inverse of six important trigonometric functions are: 1. 1. Derivatives of the Inverse Trigonometric Functions. VIEW MORE. Inverse Trigonometric Functions Note. For example, the domain for $$\arcsin x$$ is from $$-1$$ to $$1.$$ The range, or output for $$\arcsin x$$ is all angles from $$– \large{\frac{\pi }{2}}\normalsize$$ to $$\large{\frac{\pi }{2}}\normalsize$$ radians. If f(x) is a one-to-one function (i.e. The derivatives of $$6$$ inverse trigonometric functions considered above are consolidated in the following table: In the examples below, find the derivative of the given function. Check out all of our online calculators here! Derivative of Inverse Trigonometric Functions using Chain Rule. Formula for the Derivative of Inverse Cosecant Function. -csc^2 \theta \cdot \frac{d\theta}{dx} = 1 Related Questions to study. In this section we are going to look at the derivatives of the inverse trig functions. The inverse of these functions is inverse sine, inverse cosine, inverse tangent, inverse secant, inverse cosecant, and inverse cotangent. We know that trig functions are especially applicable to the right angle triangle. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. Then it must be the case that. f(x) = 3sin-1 (x) g(x) = 4cos-1 (3x 2) Show Video Lesson. Email. Proofs of the formulas of the derivatives of inverse trigonometric functions are presented along with several other examples involving sums, products and quotients of functions. x = \varphi \left ( y \right) x = φ ( y) = \sin y = sin y. is the inverse function for. As such. Derivatives of Inverse Trigonometric Functions. }, ${y^\prime = \left( {\frac{1}{a}\arctan \frac{x}{a}} \right)^\prime }={ \frac{1}{a} \cdot \frac{1}{{1 + {{\left( {\frac{x}{a}} \right)}^2}}} \cdot \left( {\frac{x}{a}} \right)^\prime }={ \frac{1}{a} \cdot \frac{1}{{1 + \frac{{{x^2}}}{{{a^2}}}}} \cdot \frac{1}{a} }={ \frac{1}{{{a^2}}} \cdot \frac{{{a^2}}}{{{a^2} + {x^2}}} }={ \frac{1}{{{a^2} + {x^2}}}. You can think of them as opposites; In a way, the two functions “undo” each other. 3 mins read . Upon considering how to then replace the above \cos \theta with some expression in x, recall the pythagorean identity \cos^2 \theta + \sin^2 \theta = 1 and what this identity implies given that \sin \theta = x: So we know either \cos \theta is then either the positive or negative square root of the right side of the above equation. The basic trigonometric functions include the following $$6$$ functions: sine $$\left(\sin x\right),$$ cosine $$\left(\cos x\right),$$ tangent $$\left(\tan x\right),$$ cotangent $$\left(\cot x\right),$$ secant $$\left(\sec x\right)$$ and cosecant $$\left(\csc x\right).$$ All these functions are continuous and differentiable in their domains. a) c) b) d) 4 y = tan x y = sec x Definition [ ] 5 EX 2 Evaluate without a calculator. Example: Find the derivatives of y = sin-1 (cos x/(1+sinx)) Show Video Lesson. In Table 2.7.14 we show the restrictions of the domains of the standard trigonometric functions that allow them to be invertible. In order to derive the derivatives of inverse trig functions we’ll need the formula from the last section relating the derivatives of inverse functions. The derivatives of the inverse trigonometric functions can be obtained using the inverse function theorem. All the inverse trigonometric functions have derivatives, which are summarized as follows: 3 Definition notation EX 1 Evaluate these without a calculator. Derivatives of a Inverse Trigo function. This website uses cookies to improve your experience. Derivatives of inverse trigonometric functions. Then \cot \theta = x. Derivatives of inverse trigonometric functions Calculator Get detailed solutions to your math problems with our Derivatives of inverse trigonometric functions step-by-step calculator. }$, ${y^\prime = \left( {\text{arccot}\frac{1}{{{x^2}}}} \right)^\prime }={ – \frac{1}{{1 + {{\left( {\frac{1}{{{x^2}}}} \right)}^2}}} \cdot \left( {\frac{1}{{{x^2}}}} \right)^\prime }={ – \frac{1}{{1 + \frac{1}{{{x^4}}}}} \cdot \left( { – 2{x^{ – 3}}} \right) }={ \frac{{2{x^4}}}{{\left( {{x^4} + 1} \right){x^3}}} }={ \frac{{2x}}{{1 + {x^4}}}.}$. One way to do this that is particularly helpful in understanding how these derivatives are obtained is to use a combination of implicit differentiation and right triangles. In the last formula, the absolute value $$\left| x \right|$$ in the denominator appears due to the fact that the product $${\tan y\sec y}$$ should always be positive in the range of admissible values of $$y$$, where $$y \in \left( {0,{\large\frac{\pi }{2}\normalsize}} \right) \cup \left( {{\large\frac{\pi }{2}\normalsize},\pi } \right),$$ that is the derivative of the inverse secant is always positive. The deriatives of inverse functions and derivatives of inverse trigonometric functions values exactly once above-mentioned inverse trigonometric functions that them! These six important trigonometric functions provide anti derivatives for a given trigonometric value $\cos \theta,... Anti derivatives for a variety of functions that arise in engineering, geometry, navigation etc Lesson differentiation! 2X −1 and learn step by step with our math solver you use website., which means$ sec \theta = x $yields FUN‑3.E.2 ( EK ) Google Classroom Facebook Twitter use. The solution application in engineering learn step by step with our math solver our derivatives of inverse trigonometric follow. Prior to running these cookies on your website, FUN‑3.E ( LO ), FUN‑3.E LO! \Cos \theta$ must be positive line test, so it has inverse... Be invertible to be algebraic functions have been shown to be trigonometric functions like, inverse cosine, inverse! You inverse trigonometric functions derivatives ok with this, but you can think of them as opposites in. A given trigonometric value browsing experience these functions is also included and may be used differentiation... Require the chain rule and one example does not pass the horizontal test! Standard trigonometric functions: •The domains of the inverse trigonometric functions are literally Inverses. To see the solution at the derivatives of the inverse trigonometric functions ( with restricted domains ) and sine! 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Website to function properly use this website uses cookies to improve your experience while you inverse trigonometric functions derivatives through website. Has plenty of examples and worked-out practice problems ( cos x/ ( 1+sinx ) ) Show Video Lesson are! The above-listed functions is one-to-one, each has an inverse to these is... ( 1+sinx ) ) Show Video Lesson restricted appropriately, so it has of! Inverse secant, cosecant, and inverse sine, inverse secant, inverse secant,,. Functions have proven to be algebraic functions have been shown to be invertible Implicitly! ) Google Classroom Facebook Twitter ( 3x 2 ) Show Video Lesson that functions...


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# 29th RD50 Workshop (CERN)

Europe/Zurich
6/2-024 - BE Auditorium Meyrin (CERN)

### 6/2-024 - BE Auditorium Meyrin

#### CERN

6-2-024
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Description
29th RD50 Workshop on Radiation hard semiconductor devices for very high luminosity colliders
Participants
• Agnieszka Oblakowska-Mucha
• Alessandro La Rosa
• Alexander Zaluzhnyy
• Angel Merlos
• Anna Macchiolo
• Ashutosh Bhardwaj
• Carmen JIMENEZ RAMOS
• Daniel Muenstermann
• Donato Creanza
• Eckhart Fretwurst
• Eija Tuominen
• Elena Donegani
• Elena Verbitskaya
• Enrico Giulio Villani
• Esteban Curras
• Eva Vilella
• Francisco Rogelio Palomo Pinto
• Gianluigi Casse
• Giovanni Calderini
• Giulio Pellegrini
• Gregor Kramberger
• Hidetoshi Otono
• Igor Mandic
• Ioana Pintilie
• Isidre MATEU
• ivan vila
• Jaakko Härkönen
• Javier Garcia Lopez
• Joaquin Gonzalez
• Joern Lange
• Joern Schwandt
• John Cotterill
• Joost Vossebeld
• Julio Calvo
• Juozas Vaitkus
• Juozas Vysniauskas
• Kevin Lauer
• Laura Gonella
• Leena Diehl
• Marcela Mikestikova
• Marcin Bartosik
• Marco Bomben
• Marco Costa
• Marco Ferrero
• Marco Mandurrino
• Marcos Fernandez Garcia
• Maria Carna
• Maria del Mar Carulla Areste
• Marta Baselga Bacardit
• Matteo Centis Vignali
• Matteo Marone
• Maurizio Boscardin
• Michael Moll
• Michal Marcisovsky
• Mikko Juntunen
• Natascha Savic
• Oscar Alonso Casanovas
• Panja Luukka
• Pawel Kaminski
• Pedro Gonçalo Dias De Almeida
• Peter Kodys
• Ranjeet Dalal
• Roberta Arcidiacono
• Roberto Mulargia
• Sabina Ronchin
• Scott Edward Ely
• Sebastian White
• Sergey Rogozhkin
• Sofía Otero Ugobono
• Stefano Terzo
• Susanne Kuehn
• Sven Mägdefessel
• Sven Wonsak
• Tilman Rohe
• Ulrich Parzefall
• Veronique Wedlake
• Vuko Brigljevic
Support
• Monday, 21 November
• 08:30 09:30
RD50 CMOS project & working group meeting
• 08:30
RD50 CMOS project 1h
Speaker: Gianluigi Casse (University of Liverpool (GB))
• 09:30 10:15
Registration and Workshop opening
• 09:30
Coffee and Workshop registration 30m
• 10:00
Workshop opening 15m
Speaker: Michael Moll (CERN)
• 10:15 11:45
CMOS sensors
• 10:15
New results of measurements with irradiated CMOS detectors in Ljubljana 20m

New results of E-TCT and Sr90 measurements with CMOS detectors produced on substrates with different resistivities will be presented. With Edge-TCT method the thickness of depleted layer can be estimated and its dependence on irradiation flunece was studied. Collected charge deposited by MIPs from Sr90 source in passive CMOS detectors was measured with external amplifier. The dependence of collected charge on fluence will be presented and compared to E-TCT measurements.

Speaker: Igor Mandic (Jozef Stefan Institute (SI))
• 10:35
Two Photon edge-TCT measurements of a neutron irradiated HVCMOS 20m

First Two Photon edge-TCT measurements of an HVCMOSv3 (ams 180 nm) irradiated with neutrons to a fluence of 7e15 neq/cm2 will be presented. The superior spatial resolution of TPA allows to accurately calculate the depletion width and effective doping concentration of the bulk. We attempt to profile the electric field inside the detector, including its distribution inside the Deep Implant. We compare the results with those obtained from the non-irradiated device, already measured with this technique.

Speaker: Marcos Fernandez Garcia (Universidad de Cantabria (ES))
• 10:55
Edge-TCT studies of irradiated HV-CMOS sensors and first test beam results with monolithic H35demo chips 20m

The H35Demo is a large area High-Voltage CMOS demonstrator chip for tracking at LHC experiments. It is produced at the AMS foundry in 0.35 µm technology on wafers with resistivity ranging from 20 to 1000 Ohm cm.
Each chip includes two monolithic matrices using nMOS or CMOS transistors and 3x3 pixel test structures without electronics for sensor characterisations. H35demo samples of different resistivities have been irradiated with neutrons at the TRIGA reactor in Ljubljana up to a fluence of 2e15 neq/cm2. The depletion depth of the test structures was measured with the edge-TCT technique after each irradiation step. Results of the TCT measurements together with the very first test beam results of the monolithic CMOS matrices before irradiation will be presented.

Speaker: Stefano Terzo (IFAE Barcelona (ES))
• 11:15
Discussion session: CMOS & RD50 CMOS project 30m

Discussion of RD50 CMOS activity planing

Speaker: Gianluigi Casse (University of Liverpool (GB))
• 11:45 17:00
Defect and Material Characterization
• 11:45
Ion irradiation and semiconductor detector characterization at the Centro Nacional de Aceleradores [MONDAY] 20m

The National Accelerator Center (CNA) is a user’s facility dedicated to multidisciplinary applications of particle accelerators. In this talk, the infrastructure available at CNA for Ion Irradiation and Characterization of Materials, based on a 3 MV tandem accelerator and a compact cyclotron for 18 MeV protons, will be briefly described. Recent activities carried out at CNA with potential interest for RD50 participants will be presented. The examples include the strain response of proton-irradiated Fiber Bragg Gratings, soft errors produced in SRAM memories and applications of the Ion Beam Induced Current technique to study the transport properties of ion induced damaged Si and SiC diodes and the charge collection efficiency of silicon-3D and Low Gain Avalanche Detectors.

Speaker: Javier Garcia Lopez (University of Seville)
• 12:05
Studies of small-pitch CNM 3D detectors 20m

For the HL-LHC pixel detector upgrades, pixel sensors with small pixel size of 50x50 and 25x100 µm² are envisaged.
At CNM, a first run of small-pitch 3D sensors was produced, which are designed to be matched to existing front-end chips like the ATLAS FE-I4. Pixel detectors were bump-bonded and assembled at IFAE Barcelona and tested in the laboratory and beam tests at CERN before and after irradiation. Also strip and pad diodes were studied with IV characteristics and TCT. This talk will present an update of recently achieved results and give an overview on the plans for future developments.

Speaker: Joern Lange (IFAE Barcelona)
• 12:25
Lunch 1h 35m
• 14:00
Note on C-DLTS application for study of radiation damage produced by short range particles. V. Eremin, D. Mitina, E. Verbitskaya 20m

Short range particles like alphas, low energy protons, ions, etc. are cost effective approaches for the damage production in silicon. The physics of capacitance DLTS application for study of such structures is discussed. The approach is illustrated by the results on silicon detectors irradiated by heavy ions.

Speaker: Dr Vladimir Eremin (Ioffe Institute)
• 14:20
Proton energy dependent damage to thin Silicon pad diodes 20m

This work deals with the bulk damage due to 23 MeV, 188 MeV and 23 GeV protons
to 200$\mu$m silicon pad sensors (bulk materials: FTH, MC or dd-FZ). I-V, C-V-f and
TSC measurements were performed at subsequent annealing steps.
Two challenges in performing TSC measurements will be pointed out: the first one
concerns the application of a forward current in the order of 1mA at the filling
temperature T=10K; the second one is related to the impact of higher filling
temperature on the measured TSC spectra.
The TSC spectra are analyzed with a revisited SRH statistics, modified to account for
the cluster-related defect contributions.
A proton-energy dependent introduction of defects is found, except for cluster-related
defects. Moreover, shallow defects are present in different concentration according to
the material type. A correlation between the leakage current and the concentrations
of three defects (the V2, E5 and H(220K) defects) is notable. It is not excluded that the
changes in the space charge in p-type sensors are mainly due to the E(30K) and the
BiOi defects, and three deep acceptors (namely the H(116K), H(140K) and the
H(152K)) .

Speaker: Elena Donegani (University of Hamburg)
• 14:40
TSC Spectra – Point- versus Cluster-Defects 20m

TSC Spectra – Point- versus Cluster-Defects
E. Fretwurst, E. Donegani, E. Garutti, R. Klanner
Institute for Experimental Physics, University of Hamburg

Abstract:
The analysis of TSC spectra of silicon diodes after irradiation with GeV protons or neutrons rely on the knowledge of the induced defects which can be either point-like or cluster-related defects. It was found that TSC signals which are attributed to extended defects cannot be reproduced by defect parameter derived for point-like states. It is observed that the TSC signals of extended-defects exhibit much broader peak shapes compared to isolated point defects. A very strong broadening effect has been observed in DLTS spectra for dislocation loops (DLs) in CMOS devices and explained by A. Scheinemann and A Schenk1). The model is based on the change of the local potential due to the Coulomb energy of the charged defect states at the boundary of the DLs. This model has been applied to TSC spectra containing vacancy related cluster-defects. The results obtained for epitaxial diodes after irradiation with 23 GeV protons and neutrons will be presented and discussed.
1) A. Scheinemann and A Schenk, Phys. Status Solidi A 211, No. 1, 136-142 (2014)

Speaker: Eckhart Fretwurst (II. Institut fuer Experimentalphysik)
• 15:00
Magnetoresistance in the irradiated Si microstrip type samples 20m

The standard Si microstrip detector samples were irradiated to high fluence in TRIGA reactor.
The magnetoresistance was investigated at different temperature, and the preliminary results of mobility dependence was obtained. It was found a decrease of magnetoresistance mobility up to 640 cm2/sV @ room T in samples irradiated to 1e17 neutron/cm2 fluence. The mobility was near to the same in both FZ and MCZ Si samples but at the lower fluence the differences between the Si type was bigger. The details of the measurement technique and results needs to discuss.

Speaker: Juozas Vaitkus (Vilnius University)
• 15:20
Coffee Break 30m
• 15:50
Defect centers in nitrogen-enriched high-resistivity n-type silicon induced by high-energy protons 20m

Radiation damage in n-type high-resistivity FZ silicon wafers with a nitrogen concentration of ~1.5E15 cm^-3 exposed to 23-MeV protons has been studied by using high-resolution photoinduced transient spectroscopy (HRPITS), infrared absorption (FTIR) and photoluminescence (PL) and measurements. In order to determine the evolution of the radiation defect structure with increasing the proton fluence, the defect centers were produced by the irradiation with four proton fluences: 1E14, 5E14, 1E15, and 5E15 n(eq)/cm^2. The irradiation with each fluence resulted in the sharp increase of the material resistivity from ~2kΩcm to ~300 kΩcm. The HRPITS results show that 20 defect centers with activation energies ranging from 24 to 565 meV are formed during the irradiation and these centers can be involved in the charge compensation leading to the increase of the resistivity. The detected traps are tentatively assigned to thermal donors, interstitial carbon related complexes, small aggregates of self-interstitials, interstitial oxygen related complexes, and small aggregates of vacancies. According to the FTIR results, the introduction rate of VO complexes significantly decreases with increasing the proton fluence. For the fluences of 1E14 and 5E14 n(eq)/cm^2, the effect of the oxygen concentration in the wafer on the VO center introduction rate is also observed. With increasing the fluence from of 1E14 and 5E15 n(eq)/cm^2, the introduction rate of divacancies rises from ~0.15 to 0.3 cm-1. In the low-temperature PL spectra, the lines related to recombination of excitons bound to tri-interstitials, as well as to C(i)C(s), C(i)C(s)H and C(i)O(i) complexes are observed.

Speaker: Pawel Kaminski (Institute of Electronic Materials Technology (PL))
• 16:10
Impact of electron irradiation on N- and O-enriched FZ silicon p-in-n pad detectors 20m

Nitrogen and oxygen enriched FZ silicon was investigated with respect to the radiation hardness under electron irradiation. P-in-n pad detectors were fabricated on four groups of FZ silicon wafers (FZ, NFZ, DOFZ and DOPFZ). Group NFZ was enriched with nitrogen during crystal growth. The DOFZ and DOPFZ groups were enriched with oxygen by oxygen indiffusion during prolonged annealing. The DOPFZ group received an additional oxygen precipitation treatment. Nitrogen and oxygen concentrations were measured by low temperature FTIR spectroscopy. The p-in-n diodes were characterized by IV measurements prior and after irradiation. Additionally, the excess charge carrier lifetime was determined by measuring the open circuit voltage decay prior and after irradiation. It was found that for all types of enrichment the irradiation induced defect density is reduced in comparison to the reference FZ silicon group. Strongest reduction of the induced defect density was found for the DOPFZ group.

Speaker: Kevin Lauer (CIS Institut fuer Mikrosensorik GmbH (DE))
• 16:30
Discussion on Defects and Materials 30m
Speaker: Ioana Pintilie (NIMP Bucharest-Magurele, Romania)
• 17:00 19:00
Collaboration Board
• Tuesday, 22 November
• 09:00 14:00
• 09:00
Characterisation of HFS detectors 20m

Results from the study of HFS detectors from RMD will be presented. Some of these sensors have been subjected to neutron irradiation at the Jožef Stefan Institute (Ljubljana, Slovenia). The fluences to which these were exposed are 3E13, 6E13, 3E14 and 1E15 n/cm^2. TCT and CV/IV measurements were performed on irradiated and unirradiated samples. The results obtained from these studies will be shown in this presentation.

Speaker: Sofia Otero Ugobono (CERN/Universidade de Santiago de Compostela (ES))
• 09:20
Neutron Induced Radiation Damage of KETEK SiPMs 20m

Silicon photomultipliers (SiPMs), thanks to their excellent performance, are becoming the photodetectors of choice for many applications.
One major limitation, in particular for their use at high-luminosity colliders, is the radiation damage by hadrons.
In this work, SiPMs with 4384 pixels of $15 \times 15$ $\mu$m$^2$ size produced by KETEK have been irradiated by reactor neutrons to six fluences up to $\Phi_{eq} = 10^{12}$ cm$^{-2}$ (1 MeV equivalent neutrons).
Pulse-height, IV, and CV measurements with and without illumination by a LED for temperatures between $-30$ and $30^\circ$C have been performed.
In this paper results from the IV, and CV measurements are shown.
The fluence and the temperature dependence of the current and of the SiPM electrical parameters like pixel capacitance, quenching resistance and breakdown voltage allows to better understand the origin of the dark current and find ways to reduce the radiation-induced dark-count-rate.

Speaker: Matteo Centis Vignali (Hamburg University (DE))
• 09:40
Measurements on 50um thick LGAD from CNM 20m

For Ultra-fast Silicon Detectors, we are using thin Low-Gain Avalanche Diodes.
We report results from a series of measurements, including electrical characterization, charge collection and time resolution in beam tests.

Speaker: Hartmut Sadrozinski (University of California,Santa Cruz (US))
• 10:00
Radition damage studies in LGAD detectors from recent CNM and FBK runs 20m

The most important results from the lates LGAD runs at
CNM and FBK will be presented.

Speaker: Gregor Kramberger (Jozef Stefan Institute (SI))
• 10:20
Status of CNM technological developments on LGAD and future plans 20m

Last developments on LGAD fabrication, as well as future plans, will be presented.

Speaker: Maria del Mar Carulla Areste (Instituto de Microelectronica de Barcelona IMB-CNM)
• 10:45
Coffee Break 25m
• 11:10
Laboratory measurement and progress in Low Gain Avalanche Diodes 20m

We report on the status of the Ultra fast silicon Detector (USFD). UFSD are
silicon detectors based on the Low Gain Avalanche Diodes (LGAD); they LGAD
have internal moderate gain (~10 order of magnitude), they exhibit
fast and large signal and the signal is about 10 larger than the standard
silicon detector. Thanks to their properties they are good candidates
for time applications.
We will concentrate the discussion discuss the results on UFSD and LGAD
developed by the two international center manufacturers: the Institute of
Microelectronics of Barcelona (CNM) and the Fondazione Bruno Kessler of
Trento (FBK).

CNM has produced the first 50μm thin LGAD sensor with thickness = 50μm; in this contribution we focused the attention on sensors products for
the two CERN experiments TOTEM and CT-PPS.
In this presentation we will show the preliminary IV curves of these
detectors as measured in Turin.

FBK has produced many different LGAD structures with thickness = 300μm; we
will show the internal gain measurements performed in Turin and the its
comparison of these measurements with the gain simulated with the simulations carried over by the University of Trento.
We will also present results on the FBK production of LGAD MultiPad structures with readout segmentation either on n-side or on p-side segmentation; the difference
between n-side and p-side LGAD segmentation is the segmented internal gain layer for the first one and the single internal layer for the second one.
In this contribution we will show the preliminary results about the read-out signal for these two different LGAD structure.

Speaker: Mr Marco Ferrero (Universita e INFN Torino (IT))
• 11:30
Temperature dependence of the response of Ultra Fast Silicon Detectors 20m

The Ultra Fast Silicon Detectors (UFSD) are a novel concept of silicon detectors based on the Low Gain Avalanche Diode (LGAD) technology, which are able to obtain time resolution of the order of few tens of picoseconds. First prototypes with different geometries (pads/pixels/strips), thickness ($300$ and $50 \mu m$) and gain (between $5$ and $20$) have been recently designed and manufactured by CNM (Centro Nacional de Microelectrónica, Barcelona) and FBK (Fondazione Bruno Kessler, Trento).

Several measurements on these devices have been performed in laboratory and in beam test and a dependence of the gain on the temperature has been observed. Some of the first measurements will be shown (leakage current, breakdown voltage, gain and time resolution on the $300 \mu m$ from FBK and gain on the $50 \mu m$-thick sensor from CNM) and a comparison with the theoretically predicted trend will be discussed.

Speaker: Roberto Mulargia (Universita e INFN Torino (IT))
• 11:50
Test beam and TCT characterization results of i-LGAD and Strip LGAD sensors 20m

Results from the the test beam (at CERN's SPS) and TCT characterization of the first i-LGAD and Strip LGAD sensors will be presented.

Speaker: Ivan Vila Alvarez (Universidad de Cantabria (ES))
• 12:10
Studies of gain and time resolution of 50 µm LGADs before and after irradiation 20m

50 um LGADs from the recent CNM production were studied before and after irradiation with neutrons in Ljubljana to 3e14 and 1e15 neq/cm2. The gain was measured with TCT and Sr90 beta particles. The time resolution was measured with 120 GeV pions in two beam tests at CERN SPS in June/July and September 2016.

Speaker: Joern Lange (IFAE Barcelona)
• 12:30
Discussion: Detectors with Gain - Detectors for timing 30m
Speaker: Salvador Hidalgo Villena (Instituto de Microelectronica de Barcelona (ES))
• 13:00
Lunch Break 1h
• 14:00 19:00
TCT & Device simulations
• 14:00
Physical limits on the inter-strip resistance for planar strip sensors 20m

For conventional planar strip sensors the interstrip resistance was observed to have a significant variation with ionizing dose and operational temperature. In principle, it could become low enough to result in lateral charge spread across several strips leading to signal loss and deterioration of the positional resolution. There are two aspects of such evaluation that can be improved: 1) The resistance tests are typically done as static IV scans, whereas the realistic signal charge flow is a dymanic process. 2) The criteria used for sensor quality control sometimes use the bias resistor value as a metric for the inter-strip resistance. The bias resistor comparison has a large built-in safety margin since a more relevant metric is the readout amplifiers' input impedance. We sought to establish a more physical criteria for the interstrip resistance evaluation by using SPICE simulation of the sensor model with dynamic signal and a model of readout amplifier. A wide range of the inter-strip resistance values was simulated and its effect on the lateral signal spread was estimated.

Speaker: Vitaliy Fadeyev (University of California,Santa Cruz (US))
• 14:20
Calculation of the effective space charge profile of a detector using TRACS 20m

We present a new software that aims to calculate the effective space charge distribution of a silicon detector. The software uses TRACS (TRAnsient Current Simulator) to simulate the induced transients currents in a detector under edge-TCT illumination. Inside TRACS a Neff(z) profile is assumed. The parameters of the profile are extracted from a fit of measured transient currents to the simulation. Since this is a CPU demanding process, TRACS was parallelized and is ran in a multicore machine. This new TRACS version with fitting capabilities are available in the CERN-cloud service.

Speaker: Julio Calvo Pinto (CERN)
• 14:40

The extremely harsh radiation environment of the future trackers necessitates the upgrade of the existing silicon detector technologies. The LGAD detectors, based on their internal charge multiplication mechanism, have attracted a lot of interest in the silicon detector community. However, it has been reported that a rapid decrease in the LGAD gain with irradiation is limiting these devices as the possible detector candidates for future experiments. In this work, it is demonstrated, through the TCAD simulation results, that by tuning certain design parameters, LGAD can be made to sustain the high radiation environment.

Speaker: Ashutosh Bhardwaj (University of Delhi (IN))
• 15:00
Simulation of Low Gain Avalanche Detector characteristics based on the concept of negative feedback in irradiated silicon detectors with carrier impact ionization (Part II) 20m

Speaker: Dr Elena Verbitskaya (Ioffe Institute)
• 15:20
Simulation of Dynamic Characteristics of GaN p-i-n Avalanche Diode Type Particle Detector 20m

The future upgrading of detectors for the high luminosity colliders requests the detectors that are capable work in the conditions where fluence of neutrons exceeds 1e17 cm-2 (e.g., the forward callorimeter in CMS). We make an attempt to investigate the possibilities of GaN detectors that works at room and higher temperature.
The dynamic characteristics of the GaN p-i-n avalanche diodes have been simulated using a drift-diffusion model according to the software package Synopsys TCAD Sentaurus. Particle detection is emulated through photo-excitation of an excess carrier domain at different locations of the active volume of a diode. Shockley-Read-Hall (SRH), Auger and radiative recombination was taken into account. It has been shown that the pulse shape of a total current strongly depends on the excess carrier photo-injection location due to the considerable difference of the electron and hole saturation velocity and impact ionization coefficient in GaN. The internal gain due to charge multiplication ensures sufficient charge collection on electrodes of the relatively thin (5 micrometers) avalanche diodes. Our calculations show that the necessary charge collection exists even in the case of a very low carrier lifetime (about 10 ps).

Speaker: Juozas Vysniauskas (Vilnius University (LT))
• 15:40
Coffee Break 30m
• 16:10
TCAD simulations of p-bulk silicon sensors after a large range of fluences 20m

In this talk I will present recent results from TCAD simulations using Silvaco tools.
In particular I will focus on p-bulk sensors, both standard diodes and LGADs.
At the beginning I will present a comparison of two radiation damage models for the bulk (Perugia, New Delhi).
Then I will comment on the impact of radiation damage model parameters uncertainties on macroscopic observables.
Eventually I will present some results on simulated thin LGADs under laser red light illumination.

Speaker: Marco Bomben (Centre National de la Recherche Scientifique (FR))
• 16:30
Speaker: Francisco Rogelio Palomo Pinto (Universidad de Sevilla (ES))
• 16:50
Insight into the behaviour of futuristic low bulk resistivity Si sensors using device simulation. 20m

There has been a growing interest for the relatively higher bulk doping density (10^13 cm^(-3) to 2x10^14 cm^(-3)) Si sensors due to their better ability for internal charge multiplication and hence higher charge collection efficiency. The CMOS, APD and LGAD are some of the examples having high bulk doping density (or low resistivity) sensors. However, there has not been a detailed investigation of possible radiation damage effects, which these low resistivity devices may experience in the future collider experiments.
In the present work, we are reporting a systematic TCAD simulation study for the effect of proton fluence (up to 1x10^16 neqcm-2) on charge collection efficiency of low resistivity, p- and n-type Si diodes of thicknesses 100 micron, 200 micron and 300 micron. A two trap proton damage model (Delhi model) is used to implement the radiation damage within TCAD framework. As expected, the results show better radiation hardness behaviour for such devices.

• 17:10
Validation strategy for the simulation of highly irradiated silicon pixel sensors 20m

For the high-luminosity phase of the Large Hadron Collider (HL-LHC), at the expected position of the innermost pixel detector layer of the CMS and ATLAS experiment, the estimated equivalent neutron fluence after 3000 fb$^{-1}$ is 2$\cdot$10$^{16}$ n$_{eq}$/cm$^{-2}$, and the IEL (Ionizing Energy Loss) dose
in the SiO$_2$ is 5 MGy. The optimization of the pixel sensors and the understanding of their performance as a function of fluence and dose makes a radiation damage model for TCAD simulations, which describes the available experimental data, highly desirable. The currently available models are not able to describe simultaneously the measurements of dark current (I-V), capacitance-voltage (C-V) and charge collection efficiency (CCE) of pad diodes for fluences > 1$\cdot$10$^{15}$ n$_{eq}$/cm$^{-2}$. For the development and validation of an accurate radiation damage model of the silicon bulk we use I-V, C-V and CCE measurements on pad diodes available within the CMS-HPK campaign and data from samples irradiated recently with 24 GeV/c protons. To determine the parameter of a radiation damage model (energy levels, concentrations and cross sections) we use the “optimizer" of Synopsys TCAD to minimize the difference between the measured and simulated I-V, C-V and CCE. By this method we study in a systematic way the number and type of the defects, which are required for a consistent description of the bulk damage.

Speaker: Joern Schwandt (Hamburg University (DE))
• 17:30
Discussion: Device Simulation & RD50 Simulation Working Group 30m
• 19:30 22:30
Social Dinner 3h
• Wednesday, 23 November
• 09:00 13:00
Pixel and Strip sensors; LHC experiments
• 09:00
Study of prototype sensors for the LHCb Upstream Tracker Upgrade 20m

The UT working group is developing the Upstream Tracker for the LHCb experiment upgrade. Four testbeams were carried out in 2015 and 2016 to assess the performance of both full size and mini prototype silicon strip detectors. We present results on the performance of these sensors at various levels of sensor irradiation.

Speaker: Scott Edward Ely (Syracuse University (US))
• 09:20
Run II Radiation Damage Effects and Operation of the LHCb Vertex Locator 20m

The LHCb Vertex Locator (VELO) is a silicon micro-strip detector operating extremely close to the LHC proton beams. During nominal data-taking the innermost active strips are as close as ~8 mm to the beams. This proximity makes the LHCb VELO an ideal laboratory to study radiation damage effects in silicon detectors.
There are numerous challenges for VELO, both in proton and ion runs.
The VELO operation is monitored with a dedicated scans: IT, IV and CCE.
The CCE scans provide the best handle on to project the bias voltages needed to operate the VELO efficiently throughout LHC Run 2. Run 2 of the LHC exceeds the radiation damage requirements the LHCb VELO was originally designed for. The latest results from radiation damage studies and their impact on the operation of the LHCb VELO in LHC Run 2 will be presented.

Speaker: Agnieszka Oblakowska-Mucha (AGH University of Science and Technology (PL))
• 09:40
Charge collection and Lorentz angle measurement on ATLAS12 sensors 20m

As a result of the radiation damage expected at HL-LHC, the collected charge and the Lorentz angle on the silicon strip sensors present at the detectors is expected to change. Therefore, the collected charge and the Lorentz angle are measured on non-irradiated and highly irradiated future ATLAS silicon micro-strip sensors at the DESY II test beam. The results of the change in the collected charge and the Lorentz angle will be presented in this talk.

Speaker: Eda Yildirim (Johannes-Gutenberg-Universitaet Mainz (DE))
• 10:00
Long Term Annealing of ATLAS12 Sensors 20m

We will present a summary of long term annealing studies at Room Temperature and 60°C, using irradiated p-type sensors up to a fluence of 2e15 n_eq/cm^2. Measurements include the charge collection and leakage current behavior, and based on this estimate the scaling factor between the two temperatures and the behavior of the effective doping concentration.

Speaker: Ms Leena Diehl (University of Freiburg)
• 10:20
Near-unity quantum efficiency of broadband black silicon photodiodes with an induced junction 20m

Ideal photodiodes can detect all incoming photons independently of the wavelength, angle or intensity of the incident light. Present-day photodiodes notably suffer from optical losses and generated charge carriers are often lost via recombination. Here, we demonstrate a device with an external quantum efficiency above 96% over the wavelength range 250–950 nm. Instead of a conventional p–n junction, we use negatively charged alumina to form an inversion layer that generates a collecting junction extending to a depth of 30 µm in n-type silicon with bulk resistivity larger than 10 kΩ cm. We enhance the collection efficiency further by nanostructuring the photodiode surface, which results in higher effective charge density and increased charge-carrier concentration in the inversion layer. Additionally, nanostructuring and efficient surface passivation allow for a reliable device response with incident angles up to 70°. We expect the considered device to improve data quality, reduce the area of photodiodes as well as decrease the cost per pixel.

Speaker: Dr Mikko Juntunen (Helsinki Institute of Physics)
• 10:40
Coffee Break 30m
• 11:10
Characterization of irradiated thin n-in-planar pixel and active edge sensors 20m

Latest productions of thin n-in-p pixel sensors designed at MPP will be presented. Sensors of thicknesses of 50, 100 and 150 um have been produced at ADVACAM and CiS (100 and 150 um) and interconnected to FE-I4 chips. At ADVACAM SOI wafers were employed, while at CiS anisotropic KOH etching was carried out to create backside cavities in the wafer leaving thicker frames around each single structure.
To maximize the active area of the thin sensors, slim and active edges were implemented in the sensors of the ADVACAM production. The evaluation assemblies in the entire thickness range have been measured at beam tests and the results on charge collection and edge efficiency will be discussed for unirradiated and irradiated modules up to 3e15.
The performance of modules with a standard edge and with 150um thick sensors after irradiation to 1e16 will be discussed in terms of charge collection and hit efficiency. In addition, the charge collection properties and efficiencies at different depths inside the silicon bulk have been studied before and after irradiation in a thickness range of 50 to 200 um with the grazing angle technique.

Speaker: Natascha Savic (Max-Planck-Institut fur Physik (DE))
• 11:30
Proposal for the organization of an RD50 project aimed at the realization of an Edge-less Device 20m

The aim of the talk is to show at the RD50 community a proposal for the definition of a RD50 project with the goal of the realization of a edgless detector at FBK.
Respect to the technological approach that used up now so passivate the trench with doping and filling the trench , we suggest to use the idea proposed by Resarch Naval Insitute ( Fadeyev) to use the allumina as passivation layer.
We propose to extend this approach by inserting it directly into the process flow and not as a post process. The use of alumina will give a process simplification and also a great freedom in design achievable geometries.

Speakers: Maurizio Boscardin (Unknown), Maurizio Boscardin (FBK Trento)
• 11:50
Common project proposal: Thin-film based silicon sensors 20m

Recently, Fraunhofer ISE developed a technology with the potential to epitaxially produce very cost-efficient thin-films of 50-200 um thickness for the solar cell industry, aiming to lower the price for a 156mm by 156mm substrate to below the ~EUR level. Such thicknesses are precisely what is required for HL-LHC applications and the exitaxial growth might yield some additional benefits.

I would like to propose an RD50 common project to evaluate the suitability of such silicon for the production of sensors and - if successful - their radiation hardness. The presentation will provide details about the process and its limitations and a propose a roadmap for the procurement and processing of an initial batch of substrates.

Speaker: Daniel Muenstermann (Lancaster University (GB))
• 12:10
The RD50 database on measured data 20m
Speakers: Donato Creanza (Dipartimento Interateneo di Fisica & INFN - Bari), Donato Creanza (Universita e INFN, Bari (IT))
• 12:30
Discussion: LHC Experiments; New Technologies; Edgeless 30m

format: plain


text:
Online ISSN : 1881-8153
Print ISSN : 1340-4202
ISSN-L : 1340-4202
サイズが異なる構造物間における減衰特性の評価法および適用性

ジャーナル フリー

2009 年 74 巻 644 号 p. 1731-1739

An estimate method that used a parameter written Co is proposed in this paper. It's possible to estimate the damping performance by using the parameter Co between different size structures. Because the parameter Co is dimensionless, the size of the structure is irrelevant to estimate. The applicability of this method was proven to be high by using the observed data. In addition, this paper proposes a method to presume the parameter of the proportional damping equation considering the scale effect. And, the characteristics of the proposed equation and the relationship with Co were identified.


text:
# Why learn Mathematics?

I found a link to a blog posting on Wolf’s blog about the question: What is the point of math? It is quite an interesting read for me, but at some places very annoying.

Why do we learn the quadratic formula (also called the midnight formula in Bavaria because you are supposed be able to repeat it when someone awakens you ad midnight). After all, you may never have use this formula in your professional career. In fact, this is true even if you are mathematician working outside research. So what is the point of this formula then?

$$x_{1,2} = \dfrac{p}{2} \pm \sqrt{\dfrac{p^2}{4} – q}$$

You will be surprised to hear my answer as a math professor. Indeed, I agree, it is a useless fact to learn. You can absolutely do without that formula. If you asked me I’d cancel it from the syllabus altogether.

First of all, you can solve quadratic equations without the formula quite easily. The technique involved in this is called quadratic expansion. It is using algebraic tools to transform the equation into an equivalent equation which solves the problem. The underlying theory is the theory of fields. The technique can be applied in the field of complex numbers or in finite fields. So why learn a formula if you can derive it with just a few steps? Moreover, learning the algebra behind the calculations is far more useful than to remember the result.

Then, if you want to solve equations, why exactly quadratic equations? Just because you can solve an equation easily does not mean it is important. Admittedly, I can think of some applications besides elementary geometry. One is acceleration in physics or the ballistic curve in vacuum. But they are far to sparse to support a „midnight formula“.

Moreover, if you want to solve a formula, then start a computer. Install a software like my EMT, and it will print the answer in numerical and algebraic form (with the help of Maxima) in no time, even if the formula looks a bit more complicated than usually.

>&solve(a+x-b^2+a=(a^2+b^2)/x,x) | expand

4              2      2     2
sqrt(b  + (4 - 4 a) b  + 8 a )   b
[x = - ------------------------------ + -- - a,
2                  2
4              2      2     2
sqrt(b  + (4 - 4 a) b  + 8 a )   b
x = ------------------------------ + -- - a]
2                  2

Learning how to compute this by hand and practicing is a waste of time. We have computers to do this.

If I say such things some people understand that math should be abandoned altogether because it can be done by computers. This would be a complete misunderstanding. Besides the fact that you still must be able to do some simple computations in your head or with a pencil, there is more to math than computing.

So what is math all about? If you want to learn about this, but are not mathematically inclined enough to look at some research paper, visit one of the pages of the modern branch of school teachers that try to envision new ways to get students to be interested in math. Dan Meyer’s blog is an example. They start with real world activities. Their path is to have the students ask the questions by themselves if possible. Their conclusion is that math can be made to appear interesting or even necessary to answer the questions.

As a math professor I wished the students would understand the principles that prove the equivalences in the quadratic expansion and the logic behind the square root function, and not just memorize the quadratic formula. It is logic that students fight with the most. And that deficiency affects all branches of mathematics, and it also affects thinking in real life.

So please, teachers, give up on memorizing formulas or practicing boring examples. Have the courage to change the school book approach to something closer to real life and the things that really matter in math nowadays. I know that some school syllabuses have already changed. But the switch is not yet in the brains of most teachers. And it has not yet reached the public opinion about our subject.

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Torque calculation-help needed

1. Sep 2, 2009

pramura

Hello everyone!

I need a help from your side. I need the calculation for pulling torque needed to rotate a swing gate hinged at two points.Weight of the gate is 130kg, width 5ft(1.524m) and height is also 5ft. Upper hinge is 0.381m from top and lower hinge is 0.381m from bottom.
Do i need to include the total weight of the gate for this torque calculation, because the gate is supported by two hinges.

what is the suitable motor for this application?

2. Sep 2, 2009

Staff: Mentor

You will need to calculate the Moment of Inertia for the gate about the axis of rotation, and you will need to specify how quickly the gate should be accelerated in the move. Those numbers will lead you to the torque. Are you familiar with those terms? If not, www.wikipedia.org has reasonable introductions...

3. Sep 2, 2009

pramura

Thanks for your reply berkeman. Yes, for torque calculation we need moment of inertia and angular acceleration.I need to open the gate from 0 to 90 degrees in 15 secs. I calculated moment of inertia but for acceleration if I use constant speed motor to open the gate how can i calculate angular acceleration. I converted degrees into radians and i got 0.1047176 rad/s2. Is that correct.

4. Sep 5, 2009

nvn

Perhaps assume constant acceleration, alpha1, from 0 to theta1 radians. Then, using kinematics, compute final angular velocity, omega1, at theta1. Using statics, compute the frictional moment M, assuming a high mu value (rusty hinges). Motor torque T to overcome the total moment would be T = M + I*alpha. Perhaps turn off the motor at theta1 radians. Now T in the above equation becomes zero; therefore, solve for alpha2 to obtain the constant deceleration from theta1 to 90 deg. The initial angular velocity from theta1 radians to 90 deg is omega1. Using kinematics, compute the final (latching) angular velocity, omega2, at 90 deg. Using kinematics again, you can now compute the total swing time. Play around with this until you get what you consider to be an acceptable minimum (and maximum) latching angular velocity, and a total swing time of 15 s. You might want a damper or spring to handle the latching impact, to avoid damaging the structure (?). Maybe someone can let us know if the above approach sounds valid or incorrect.

5. Sep 5, 2009

nvn

pramura: Here is the key to solving the approach described in post 4. If you put all the above information together, it happens that T and t1 have a unique solution, as follows. (The motor is turned on at theta0 = 0 rad, when omega0 = 0 rad/s, which occurs at t0 = 0 s.)

t1 = (M*tt^2 + 2*I*omega2*tt - 2*I*thetat)/(M*tt + I*omega2),
T = (M*tt + I*omega2)/t1,

where T = motor torque (N*m),
t1 = time at which motor is turned off (s),
tt = gate total swing time (s),
thetat = gate total swing angle (rad),
M = frictional moment (N*m),
I = gate mass moment of inertia about hinge axis (kg*m^2),
omega2 = gate final (latching) angular velocity (rad/s).

After you compute T and t1, you can compute any other quantity, such as the following, using kinematics or kinetics.

alpha1 = gate constant angular acceleration while motor is running (rad/s^2),
alpha2 = gate constant angular acceleration while motor is not running (rad/s^2),
theta1 = swing angle at instant motor is turned off (rad),
theta2 = swing angle after motor is turned off (rad) = thetat - theta1,
t2 = gate swing time after motor is turned off (s) = tt - t1,
omega1 = gate angular velocity at theta1 (rad/s).

You can develop your own frictional moment M, but I arbitrarily used M = 2[0.5*Dp*muk*H + 0.5(1.25*Dp)*muk*V], where Dp = hinge pin diameter (m), muk = hinge kinetic coefficient of friction, H = horizontal reaction force applied to each hinge (N) = (m1*g)*0.5*(1.524 m)/(0.762 m), V = vertical reaction force applied to each hinge (N) = 0.5*m1*g, and m1 = gate mass (kg). Parameters omega2 and muk are fundamental input parameters you must specify to define your design.

It is interesting to see what happens to T as you decrease muk. Hint: To compute alpha1 and alpha2, see post 4.


text:
# When does the product automaton of two NFAs A and B not decide L(A) U L(B)?

We are given the following task:

Let $$\mathcal{A} = {}(Q_A,\Sigma, \delta_A, s_A, F_A)$$ and $$\mathcal{B} = {}(Q_B,\Sigma, \delta_B, s_B, F_B)$$ be two NFAs and let their product automaton be defined as:

$$\mathcal{A} \times \mathcal{B} = {}(Q_A \times Q_B,\Sigma, \delta, (s_A,s_B), F)$$ with

$$\delta = {}\{ ((p,q),\sigma,(p',q')) \mid (p,\sigma,p') \in \delta_A, (q,\sigma,q') \in \delta_B \}$$ and

$$F= {}(Q_A \times F_B) \space \cup \space (F_A \times Q_B)$$

(So the accepting (combined) states are the ones where at least one of the two individual states of $$\mathcal{A}$$ or $$\mathcal{B}$$ is accepting.)

Give an example for two NFAs $$\mathcal{A}$$ and $$\mathcal{B}$$ so that their product automaton $$\mathcal{A} \times \mathcal{B}$$ does not decide the language $$L(\mathcal{A}) \space \cup \space L(\mathcal{B})$$.

Now the one thing that is by far confusing me the most (and what I want to focus on in this question) is the following tip that is given as well:

Tip: There is an example in which $$\mathcal{A}$$ only has one state.

I can't see how that could be possible, with the following reasoning:

If $$\mathcal{A}$$ only has one state - let's call it $$q$$ - this means two things:

1. $$q$$ is either accepting or it is not accepting
2. All of the transitions of $$\mathcal{A}$$ start at and directly lead back to $$q$$.

Based on 1. we have two cases.

Let's first consider the case that $$q$$ is accepting ($$q \in F_\mathcal{A}$$). Because of 2. it follows that $$\mathcal{A}$$ accepts every word $$w \in \Sigma ^*$$ (so $$L(\mathcal{A}) = \{w \in \Sigma^*\}$$).

Let $$\mathcal{B}$$ be an arbitrary NFA. Based on the definition of $$F$$ and the fact that $$q$$ is accepting it is clear that all of the states of $$\mathcal{A} \times \mathcal{B}$$ will be accepting as well, which implies that the product automaton also accepts every $$w \in \Sigma ^*$$.

This means that $$\space L(\mathcal{A} \times \mathcal{B}) = \{w \in \Sigma^*\} = L(\mathcal{A}) \space \cup \space L(\mathcal{B})$$.

Now let $$q$$ be non-accepting ($$q \notin F_\mathcal{A})$$. Again, based on the definition of $$F$$ we can see that the accepting states of $$\mathcal{A} \times \mathcal{B}$$ are going exactly correspond to the accepting states of $$\mathcal{B}$$.

Point 2 implies that the transitions of $$\mathcal{A}$$ will "not have any impact on the transitions of $$\mathcal{A} \times \mathcal{B}$$", by which I mean that the product automaton is going to have the same transitions as $$\mathcal{B}$$ and therefore be isomorphic to $$\mathcal{B}$$, which again results in

$$L(\mathcal{A} \times \mathcal{B}) = L(\mathcal{B}) = L(\mathcal{B}) \space \cup \space \varnothing = L(\mathcal{B}) \space \cup \space L(\mathcal{A})$$

(since $$\mathcal{A}$$ doesn't accept any input, which means $$L(\mathcal{A}) = \varnothing$$)

This proof is trying to show that if $$\mathcal{A}$$ only has one state, then $$\mathcal{A} \times \mathcal{B}$$ will always decide $$L(\mathcal{B}) \space \cup \space L(\mathcal{A})$$.

Obviously, there has to be a logic fault in my thinking and in the proof above somewhere, which I can't seem to be able to figure out, so I'd really appreciate if anyone could point out what I'm missing.

The culprit is the fact that, in the presence of nondeterminism, automata need not have transitions to other states for the same input symbols.

Consider two NFA over the input alphabet $\{a,b\}$:

$$N_1 = (\{q_1\},\{a,b\},q_1,\delta_1,\{q_1\})$$

where $\delta_{1}(q_1,a) = \{ q_1 \}$ and $\delta_{1}(q_1,b) = \emptyset$

and

$$N_1 = (\{q_2\},\{a,b\},q_2,\delta_2,\{q_2\})$$

where $\delta_{2}(q_2,a) = \emptyset$ and $\delta_{2}(q_2,b) = \{ q_2 \}$

– that is, each of these automata has a single state, which is accepting, and a single transition, which is a loop on $a$ or $b$, respectively.

Clearly, $L(N_1)= \{ a^n \mid n \geq 0 \}$ and $L(N_2)= \{ b^n \mid n \geq 0 \}$. But the product construction will give us an automaton

$$N_{12} = (\{ (q_1,q_2 \}, \{ a,b \}, (q_1,q_2), \delta_{12}, \{ (q_1,q_2)\})$$

where $\delta_{12}((q_1,q_2),a) = \emptyset$ and $\delta_{12}((q_1,q_2),b) = \emptyset$, since there is no symbol $x$ such that both $N_1$ and $N_2$ admit an $x$-transition from their sole state.

Therefore $L(N_{12}) = \{ \varepsilon \} \neq L(N_1) \cup L(N_2)$.

• This is it! One of my attempts even included automata for the two languages in this answer, except that I used explicit second states instead of $\emptyset$, which then obviously turned the automata into DFAs, causing the example to fail. A quick sidenote though: Shouldn't it be $L(N_{12}) = \{\epsilon\}$ instead of $\emptyset$ at the end? (Since $N_{12}$ still accepts the empty word) May 12 '17 at 10:01
• Yes, indeed $L(N_{12}) = \{ \varepsilon \}$, since $(q_1,q_2)$ is an accepting state. May 12 '17 at 11:37


text:
How to search the internet for strings that consist mostly of math notation?

Is there a trick to search the internet for math notation? For example what if I want to search the internet for "X / ~" ? The default google search is useless for this and for similar searches where most of the search characters are not letters or numbers. Google code search allows this kind of search within source code, but this service will shut down in a few months and anyway it searches only source code not math.

-
There's (uni)quation, and there's $\LaTeX$ Search. –  Guess who it is. Oct 21 '11 at 16:40
By the way, since this probably came up in relation to my answer here, I was using the notation in the context of an equivalence relation and, in particular, it refers to the set of equivalence classes, a.k.a. the quotient set –  Zev Chonoles Oct 21 '11 at 16:43
I've retagged the question, since I believe that the tag searching was intended for questions about various search algorithms. If my impression is wrong, feel free to retag the question again. (If this is the case, it might be nice to correct the tag-wiki, too.) –  Martin Sleziak Aug 25 '12 at 13:20
I found this and this (But I am unable to search for the query in the latter, although according to description here is should be a search interface. –  Martin Sleziak Apr 8 '14 at 11:58


text:
# Solution of Potts-3 and Potts-∞ matrix models with the equations of motion method

@article{Bonnet1999SolutionOP,
title={Solution of Potts-3 and Potts-∞ matrix models with the equations of motion method},
author={G. Bonnet},
journal={Physics Letters B},
year={1999},
volume={459},
pages={575-581}
}
• G. Bonnet
• Published 8 April 1999
• Physics
• Physics Letters B
Abstract In this letter, we show how one can solve easily the Potts-3 + branching interactions and Potts-∞ matrix models, by the means of the equations of motion (loop equations). We give an algebraic equation for the resolvents of these models, and their scaling behaviour. This shows that the equations of motion can be a useful tool for solving such models.
5 Citations
Boundary States of the Potts Model on Random Planar Maps
• Physics, Mathematics
• 2016
We revisit the 3-states Potts model on random planar triangulations as a Hermitian matrix model. As a novelty, we obtain an algebraic curve which encodes the partition function on the disc with both
Yang-Lee zeros of the two- and three-state Potts model defined on phi3 Feynman diagrams.
• Mathematics, Medicine
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2003
For the Ising model, an argument based on a symmetry of the saddle point equations leads to a nonperturbative proof that the Yang-Lee zeros are located on the unit circle, although no circle theorem is known in this case of random graphs.
Sums of Random Matrices and the Potts Model on Random Planar Maps
• Mathematics, Physics
• 2015
We compute the partition function of the $q$-states Potts model on a random planar lattice with $p\leq q$ allowed, equally weighted colours on a connected boundary. To this end, we employ its matrix
Random Matrices, Boundaries and Branes
This thesis is devoted to the application of random matrix theory to the study of random surfaces, both discrete and continuous; special emphasis is placed on surface boundaries and the associated
Planar maps and random partitions
This habilitation thesis summarizes the research that I have carried out from 2005 to 2019. It is organized in four chapters. The first three deal with random planar maps. Chapter 1 is about their

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This book contains an edited comprehensive collection of reprints on the subject of the large N limit as applied to a wide spectrum of problems in quantum field theory and statistical mechanics. The
Simplicial Quantum Gravity and Random Lattices
Content: 1. Introduction 2. Regge calculus and dynamical triangulations Simplicial manifolds and piecewise linear spaces - dual complex and volume elements - curvature and Regge action -
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• 1995


text:
# Statistical Inference Formulas

Since I could not find a list of formulas anywhere, I thought I would create one.

### Notation

The notation used is fairly standard, but it is here for completeness.

Notation Meaning
$\mbox{df}$ Degrees of Freedom
$\mbox{SE}$ Standard Error
Confidence SE Confidence interval standard error
$n$ Sample Size
$s$ Sample standard deviation
$s^2$ Sample variance
$\bar{X}$ Sample mean
$N$ Population size
$\mu$ Population mean
$\sigma$ Population standard deviation
$\sigma^2$ Population variance
$s_p^2$ Pooled variance
$\alpha$ The significance level for a hypothesis test (or the probability of a type I errror)

### Hypothesis Testing

• $H_0$ and $H_a$ are the null and alternative hypotheses, respectively
• The p-value: the probability of returning a more extreme value than that of the test statistic given that the null hypothesis is true.
• Find the probability of a value being less than or equal to that of the test statistic under the chosen distribution (using it's cdf). Subtract the found value from 1, since we are looking for a more extreme value.
• For a one-sided test, this is the p-value.
• For a two-sided test, multiply by 2 (since we need the more extreme value for both sides)
• Rejection region: If the absolute value of the test statistic is larger than this value then the null hypothesis logically can't be true, since this directly corresponds to the significance level $\alpha$

### Single Population

Test df Test se Test Statistic Confidence se Confidence interval
$\mu$ when $\sigma$ known - $\mbox{SE} = \frac{\sigma}{\sqrt{n}}$ $z = \frac{\hat{\mu}}{\mbox{SE}}$ $\mbox{SE} = \frac{\sigma}{\sqrt{n}}$ $\hat{\mu} \pm z(1 - {\alpha \over 2}) \mbox{SE}$
$\mu$ when $\sigma$ unknown $\mbox{df} = n-1$ $\mbox{SE} = \frac{s}{\sqrt{n}}$ $t = \frac{\hat{\mu}}{\mbox{SE}}$ $\mbox{SE} = \frac{s}{\sqrt{n}}$ $\hat{\mu} \pm t_{n-1}(1 - \frac{\alpha}{2}) \mbox{SE}$
Population proportion $p$ - $\mbox{SE} = \sqrt{\frac{p_0 (1 - p_0)}{n}}$ $z = \frac{\hat{p} - p_0}{\mbox{SE}}$ $\mbox{SE} = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$ $\hat{p} \pm z(1 - {\alpha \over 2}) \mbox{SE}$

### Two Populations

Test df Test se Test Statistic Confidence se Confidence interval
two means equal variances $\mbox{df} = n_1 + n_2 - 2$ $\mbox{SE} = \sqrt{\frac{(n_1-1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$ $t = \frac{\mu_1 - \mu_2 - d_0}{\mbox{SE}}$ $\mbox{SE} = \sqrt{\frac{(n_1-1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$ $\hat{\mu} \pm t_{\mbox{df}}(1 - \frac{\alpha}{2}) \times \mbox{SE}$
two means unequal variances * $\mbox{df} \approx \frac{({s_{1}^2 \over n_1} + {s_2^2 \over n_2})^2}{{s_1^4 \over n_1^2 (n_1 - 1)} + {s_2^4 \over n_2^2 (n_2 - 1)}}$ $\mbox{SE} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ $t = \frac{\mu_1 - \mu_2-d_0}{\mbox{SE}}$ $\mbox{SE} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ $\hat{\mu} \pm t_{\mbox{df}}(1 - \frac{\alpha}{2}) \mbox{SE}$

Notes:

• *: If conservative $\mbox{df}$ is used, use the minimum of $n_1 - 1$ and $n_2 - 1$

### ANOVA

Source of Variance $\mbox{df}$ Sum of Square Mean square F statistic p-value
Treatment $I-1$ $\mbox{sst} = \sum_{i=1}^I n_i (\bar{X}_i - \bar{X})^2$ $\mbox{MST} = \frac{\mbox{sst}}{I-1}$ $f = \frac{\mbox{MST}}{\mbox{mse}}$ *
Error $n-I$ $\mbox{sse} = \sum_{i=1}^I s^2_i (n_i - 1)$ ** $\mbox{mse} = \frac{\mbox{sse}}{n-I}$
Total $n-1$ $\mbox{sst} + \mbox{sse}$

Notes:

• *: $f$ follows an F distribution with degrees of freedom equal to $I-1$ and $n-I$. $I-1$ corresponds to the numerator and $n-I$ the denominator.
• **: This is the pooled variance for all $I$ groups

### Contrasts and Linear Combinations of Group Means

• $\gamma = \sum_{i=1}^I c_i \mu_i$
• Estimated with $g = \sum_{i=1}^I c_i \bar{X}_i$
• If the coefficients sum to 0 then this linear combination is a contrast (if $\sum_{i=1}^I c_i = 0$)
• $\mbox{SE}(g) = s_p \sqrt{\sum_{i=1}^I \frac{c_i^2}{n_i}}$
• Use $s_p = \sqrt{\mbox{mse}}$ from the ANOVA table with the same degrees of freedom ($n-1$) This is because even if you're comparing less than $I$ groups in a contrast, those groups are still represented (just with 0 c coefficients)
• Confidence Interval: $g \pm t_{n-I}(1 - {\alpha \over 2}) \times \mbox{SE}(g)$
• Test Statistic: $t = \frac{g - \gamma}{\mbox{SE}\left(g\right)}$

### Post-ANOVA Comparison Methods

• Confidence intervals are constructed with $\mbox{Estimate} \pm \mbox{Multiplier} \times \mbox{SE}\left(\mbox{Estimate}\right)$
• Since pairs are really a contrast with coefficients (1, -1) and 0 for all other groups, the $\mbox{mse}$ from the ANOVA table is used for $s_p^2$.
Method Multiplier Notes
Least Significant Difference (LSD) $t_{n-I}(1 - {\alpha \over 2})$ No attempt to control the family wise error rate. If ANOVA is ran first, this is the f-protected LSD method
Tukey-Kramer $\frac{q_{I,n-I,1-\alpha}}{\sqrt{2}}$ Used for all pairwise comparisons, more conservative than the above two methods, uses the Studentized Range distribution for the multiplier
Bonferroni $t_{n-I}(1 - {\alpha \over 2k})$ Used for $k$ pairwise comparisons
Scheffe $\sqrt{(I-1)F_{I-1,n-I}(1 - \alpha)}$ Used for all contrasts, most conservative test

### Statistical Models

• Degrees of freedom for the model are the number of parameters in the model that vary.
• A reduced model is a model with fewer parameters.
• To determine lack of fit between two models:
• $H_0$: The reduced model adequately explains the data. $H_a$: The full model is required to adequately explain the data.
• Test statistic: $F = \frac{\frac{\text{ss}(error)_{red} - \text{ss}(error)_{full}}{\text{df}(error)_{red} - \text{df}(error)_{full}}}{\frac{\text{ss}(error)_{full}}{\text{df}(error)_{full}}}$
• F follows an f distribution with degrees of freedom equal to $\text{ss}(error)_{red}$ and $\text{ss}(error)_{full}$
• If F is less than the significance level $\alpha$, the full model is required to adequately explain the data.

### Regression

#### Simple Linear Regression and least squares regression

• $Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i$
• Estimated with $\hat{Y}_i = b_0 + b_1 X_i$
• Sums:
• $\mbox{ss}_{XY} = \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})$
• $ss_Y = \sum_{i=1}^n (Y_i - \bar{Y})^2$
• $ss_X = \sum_{i=1}^n (X_i - \bar{X})^2$
• The line of best fit must pass through the point $(\bar{X}, \bar{Y})$
• The slope ($\beta_1$): $\beta_1 = \frac{\mbox{ss}_{XY}}{\mbox{ss}_X}$
• The intercept ($\beta_0$): $\beta_0 = \bar{Y} - b_1 \bar{X}$
• $ss_e = \sum_{i=1}^n (Y_i - \hat{Y})^2$
• $ss_r = \sum_{i=1}^n (\hat{Y}_i - \bar{Y})^2$ This is the sum of squares explained by the model.
• $ss_t = \sum_{i=1}^n (Y_i - \bar{Y})^2$ same as $ss_Y$ above.
• The Coefficient of determination $R^2$
• Is the percentage of the sum of squares explained by the model to the sum of squares total
• $R^2 = \frac{ss_r}{ss_t}$
• It can also be rearranged as $R^2 = 1 - \frac{ss_e}{ss_t}$

#### Multiple Regression

• $Y_i = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + ... + \beta_I X_I$
• Predicted with b as per usual. Note the lack of $\varepsilon$


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Browse Questions

# Which of the following molecule is most suitable to disperse benzene in water?

(a)
(b)
(c)
(d)

Benzene (an oily substance) and water will form an emulsion in the presence of a micelle forming substance.
Out of the given four choices, only the choice:
gives anion with a polar heas and non-polar tail, which is suitable for micelle formation.


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# Why does the File Browser's 'filepath' property include the blend file name once the file has been saved?

I am puzzled by a weird behaviour of the Blender File Browser (v2.81) that I use to select the path to a folder containing the data my script processes:

• when it is invoked from a Blender project that was never saved, self.filepath returns the path selected e.g. my/selected/path/. This is the behaviour I am expecting.

• when it is invoked once the Blender project file as been saved (foo.blend), then self.filepath returns the path selected and appends the name of the project to it my/selected/path/foo.blend

I am getting around this by stripping foo.blend but I am puzzled by this.

import bpy
import os
from bpy_extras.io_utils import ImportHelper
from bpy.types import Operator

class OT_TestOpenFilebrowser(Operator, ImportHelper):
bl_idname = "test.open_filebrowser"
bl_label = "Select"
def execute(self, context):
filename, extension = os.path.splitext(self.filepath)
print('Selected:', self.filepath)
print('File name:', filename)
print('File extension:', extension)

return {'FINISHED'}

def register():
bpy.utils.register_class(OT_TestOpenFilebrowser)

def unregister():
bpy.utils.unregister_class(OT_TestOpenFilebrowser)

if __name__ == "__main__":
register()
bpy.ops.test.open_filebrowser('INVOKE_DEFAULT')

Run from a never saved Blender project:

Run again after saving Blender project:

Why is that? Am I missing a setting or parameter?

• self.filepath contains the concatenation of the current path and the filename. There is no check whether or not that file actually exist, how your script is currently implemented. You can enter any string you want in the filename field or set it to an empty string. Perhaps it's easier for us to help if we know what you're trying to do. Dec 1 '19 at 17:38
• IMO the filebrower is a tricky beast. What you are seeing is context.space_data.params.filename which you typed in when you saved the file. Other undocoed stuff like multi select files ie can define a files collection property of OperatorFileListElement see blender.stackexchange.com/a/39855/15543 (may be somewhat outdated) Suggest too checking out the code of bpy_extras.io_utils.py Dec 1 '19 at 18:39

Thanks to all for the input. I am answering my own question.

I wanted to have in my AddOn a way to choose a folder containing some input files. It turns out that it was very easy using bpy.props StringProperty as panel property:

import_path: StringProperty(
name = "Path",
description = "Path to the folder containing the files to import",
default = "",
subtype = 'DIR_PATH'
)

The DIR_PATH (or FILE_PATH, if expecting files) subtype is presenting a file system browser and retrieve the relative path, one just have to convert it to a absolute path in the operator:

bpy.path.abspath(self.import_path)

see here