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# Vector (mathematics and physics)

(Redirected from Vector (mathematics))

In mathematics and physics, a vector is an element of a vector space.

For many specific vector spaces, the vectors have received specific names, which are listed below.

Historically, vectors were introduced in geometry and physics (typically in mechanics) before the formalization of the concept of vector space. Therefore, one talks often of vectors without specifying the vector space to which they belong. Specifically, in a Euclidean space, one consider spatial vectors, also called Euclidean vectors which are used to represent quantities that have both magnitude and direction, and may be added and scaled (that is multiplied by a real number) for forming a vector space.

## Vectors in Euclidean geometry

In classical Euclidean geometry (that is in synthetic geometry), vectors were introduced (during 19th century) as equivalence classes, under equipollence, of ordered pairs of points; two pairs (A, B) and (C, D) being equipollent if the points A, B, D, C, in this order, form a parallelogram. Such an equivalence class is called a vector, more precisely, a Euclidean vector.[1] The equivalence class of (A, B) is often denoted ${\displaystyle {\overrightarrow {AB}}.}$

A Euclidean vector, is thus an entity endowed with a magnitude (the length of the line segment (A, B)) and a direction (the direction from A to B). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars, which have no direction. For example, velocity, forces and acceleration are represented by vectors.

In modern geometry, Euclidean spaces are often defined from linear algebra. More precisely, a Euclidean space E is defined as a set to which is associated a inner product space of finite dimension over the reals ${\displaystyle {\overrightarrow {E}},}$  and a group action of the additive group of ${\displaystyle {\overrightarrow {E}},}$  which is free and transitive (See Affine space for details of this construction). The elements of ${\displaystyle {\overrightarrow {E}}}$  are called translations.

It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations.

Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of ${\displaystyle \mathbb {R} ^{n}}$  equipped with the dot product. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, ${\displaystyle \mathbb {R} ^{n}}$  is a Euclidean space, with itself as an associated vector space, and the dot product as a inner product.

The Euclidean space ${\displaystyle \mathbb {R} ^{n}}$  is often presented as the Euclidean space of dimension n. This is motivated by the fact that every Euclidean space of dimension n is isomorphic to the Euclidean space ${\displaystyle \mathbb {R} ^{n}.}$  More precisely, given such a Euclidean space, one may choose any point O as an origin. By Gram–Schmidt process, one may also find an orthonormal basis of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point P of the space, as the coordinates on this basis of the vector ${\displaystyle {\overrightarrow {OP}}.}$  These choices define an isomorphism of the given Euclidean space onto ${\displaystyle \mathbb {R} ^{n},}$  by mapping any point to the n-uple of its Cartesian coordinates, and every vector to its coordinate vector.

## Vectors in specific vector spaces

• Column vector, a matrix with only one column. The column vectors with a fixed number of rows form a vector space.
• Row vector, a matrix with only one row. The row vectors with a fixed number of columns form a vector space.
• Coordinate vector, the n-tuple of the coordinates of a vector on a basis of n elements. For a vector space over a field F, these n-uples form the vector space ${\displaystyle F^{n}}$  (where the operation are pointwise addition and scalar multiplication).
• Displacement vector, a vector that specifies the change in position of a point relative to a previous position. Displacement vectors belong to the vector space of translations.
• Position vector of a point, the displacement vector from a reference point (called the origin) to the point. A position vector represents the position of a point in a Euclidean space or an affine space.
• Velocity vector, the derivative, with respect of the time, of the position vector. It does not depend of the choice of the origin, and, thus belongs to the vector space of translations.
• Pseudovector, also called axial vector, an element of the dual of a vector space. In a inner product space, the inner product defines an isomorphism between the space and its dual, which may make difficult to distinguish a pseudo vector from a vector. The distinction becomes apparent when one changes coordinates: the matrix used for a change of coordinates of pseudovectors is the transpose of that of vectors.
• Tangent vector, an element of the tangent space of a curve, a surface or, more generally, a differential manifold at a given point (these tangent spaces are naturally endowed with a structure of vector space)
• Normal vector or simply normal, in a Euclidean space or, more generally, in an inner product space, a vector that is perpendicular to a tangent space at a point. Normals are pseudovectors that belong to the dual of the tangent space.
• Gradient, the coordinates vector of the partial derivatives of a function of several real variables. In a Euclidean space the gradient gives the magnitude and direction of maximum increase of a scalar field. The gradient is a pseudo vector that is normal to a level curve.
• Four-vector, in the theory of relativity, a vector in a four-dimensional real vector space called Minkowski space

## Tuples that are not really vectors

The set ${\displaystyle \mathbb {R} ^{n}}$  of tuples of n real numbers has a natural structure of vector space defined by component-wise addition and scalar multiplication. When such tuples are used for representing some data, it is common to call them vectors even if the vector addition does not mean anything for these data, which may make the terminology confusing. Similarly, some physical phenomena involve a direction and a magnitude. They are often represented by vectors, even if operations of vector spaces do not apply to them.

## Vectors in algebras

Every algebra over a field is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called vectors, mainly for historical reasons.

### Vector fields

A vector field is a vector-valued function that, generally, has a domain of the same dimension (as a manifold) as its codomain,

### Miscellaneous

• Ricci calculus
• Vector Analysis, a textbook on vector calculus by Wilson, first published in 1901, which did much to standardize the notation and vocabulary of three-dimensional linear algebra and vector calculus
• Vector bundle, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space
• Vector calculus, a branch of mathematics concerned with differentiation and integration of vector fields
• Vector differential, or del, a vector differential operator represented by the nabla symbol ${\displaystyle \nabla }$
• Vector Laplacian, the vector Laplace operator, denoted by ${\displaystyle \nabla ^{2}}$ , is a differential operator defined over a vector field
• Vector notation, common notation used when working with vectors
• Vector operator, a type of differential operator used in vector calculus
• Vector product, or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector
• Vector projection, also known as vector resolute or vector component, a linear mapping producing a vector parallel to a second vector
• Vector-valued function, a function that has a vector space as a codomain
• Vectorization (mathematics), a linear transformation that converts a matrix into a column vector
• Vector autoregression, an econometric model used to capture the evolution and the interdependencies between multiple time series
• Vector boson, a boson with the spin quantum number equal to 1
• Vector measure, a function defined on a family of sets and taking vector values satisfying certain properties
• Vector meson, a meson with total spin 1 and odd parity
• Vector quantization, a quantization technique used in signal processing
• Vector soliton, a solitary wave with multiple components coupled together that maintains its shape during propagation
• Vector synthesis, a type of audio synthesis

## Notes

1. ^ In some old texts, the pair (A, B) is called a bound vector, and its equivalence class is called a free vector.


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## Data setup

This demonstration will use the example3 dataset included in the causalRisk package, which was generated using the coxed package. This dataset of 10,000 observations includes a treatment variable (Statin), an outcome variable (Death), a censoring variable (EndofEnrollment), and two covariates (X1 and X2). The data are setup such that X1 is a confounder and effect measure modifier and X2 is a predictor of the outcome, and censoring is non-informative.

Statin X1 X2 EndofEnrollment Death
Control -0.5633082 0.2089505 NA NA
Control -0.3456137 1.7601274 NA 2.650168
Treat 0.3637345 -0.9931112 NA NA
Control 0.0530444 -0.2091482 NA 19.365315
Control 0.2136881 -1.0809493 NA 60.591595
Control 0.5691088 0.3711670 NA NA

## Analysis without missing data

To begin, we conduct an analysis without missing data. The crude risk is 17.1% in the control group and 9.0% in the treated group. The adjusted risk is 15.2% in the control group and 9.8% in the treated group. The crude and adjusted risk differences are -8.1% and -5.4%, respectively.

  #No missingness, for comparisons
models_nomiss_crude <- specify_models(identify_treatment(Statin),
identify_censoring(EndofEnrollment),
identify_outcome(Death))

fit_nomiss_crude <- estimate_ipwrisk(data,
models_nomiss_crude,
times = seq(0,20,.1),
labels = c("No missingness - Crude"))

models_nomiss <- specify_models(identify_treatment(Statin, ~X1),
identify_censoring(EndofEnrollment),
identify_outcome(Death))

fit_nomiss <- estimate_ipwrisk(data,
models_nomiss,
times = seq(0,20,.1),
labels = c("No missingness"))

plot(fit_nomiss_crude, fit_nomiss, ncol=2, scales="fixed")

  make_table2(fit_nomiss_crude, fit_nomiss, risk_time=20)

## Analysis with data missing completely at random (MCAR)

Next, we induce missingness that is MCAR, meaning that missingness is unconditionally independent of the outcome. We compare adjusted analyses of the full data, of the complete cases, and using inverse probability of missingness weights. Because the data are MCAR and the missingness model does not contain covariates, the complete case and IPMW analyses are essentially the same. In each analysis, the adjusted risk in the control group is 15.2%, in the treated is 10.1%, yielding a risk difference of -5.2%.

#Induce MCAR
data_mcar <- data
pmiss <- .10
miss <- rbinom(nrow(data_mcar),1,pmiss)
data_mcar$X1 <- ifelse(miss, NA, data_mcar$X1)

#Analyze
models_mcar <- specify_models(identify_treatment(Statin, ~X1),
identify_censoring(EndofEnrollment),
identify_outcome(Death),
identify_missing(~1))

fit_mcar <- estimate_ipwrisk(data_mcar,
models_mcar,
times = seq(0,20,.1),
labels = c("IPMW - MCAR"))

models_mcar_cc <- specify_models(identify_treatment(Statin, ~X1),
identify_censoring(EndofEnrollment),
identify_outcome(Death))

fit_mcar_cc <- estimate_ipwrisk(data_mcar,
models_mcar_cc,
times = seq(0,20,.1),
labels = c("Complete case - MCAR"))

plot(fit_nomiss, fit_mcar_cc, fit_mcar, ncol=3, scales="fixed")

make_table2(fit_nomiss, fit_mcar_cc, fit_mcar, risk_time=20)

## Analysis with data missing at random (MAR)

Finally, we induce missingness that is MAR, meaning that missingness is independent of the outcome conditional on some variables, in this case X2. Again, we compare adjusted analyses of the full data, of the complete cases, and using inverse probability of missingness weights. Because of the missing data structure, we expect the estimates from the complete case analysis to be biased, but for the IPMW to be unbiased. Indeed, the risks from the complete case analysis are 12.3% and 7.6% for the control and treatment groups, respectively, yielding a risk difference of -4.7%. The risks from the IPMW analysis are 15.3% and 9.1% for the control and treatment groups, respectively, for a risk difference of -6.2%.

#Induce MAR
data_mar <- data
data_mar$pmiss <- 1/(1+exp(-(log(.9)+log(5)*data_mar$X2)))
data_mar$miss <- apply(as.data.frame(data_mar$pmiss), 1, function(df) rbinom(1,1,df))
data_mar$X1 <- ifelse(data_mar$miss==1, NA, data_mar\$X1)

#Analyze
models_mar <- specify_models(identify_treatment(Statin, ~X1),
identify_censoring(EndofEnrollment),
identify_outcome(Death),
identify_missing(~X2))

fit_mar <- estimate_ipwrisk(data_mar,
models_mar,
times = seq(0,20,.1),
labels = c("IPMW - MAR"))

models_mar_cc <- specify_models(identify_treatment(Statin, ~X1),
identify_censoring(EndofEnrollment),
identify_outcome(Death))

fit_mar_cc <- estimate_ipwrisk(data_mar,
models_mar_cc,
times = seq(0,20,.1),
labels = c("Complete case - MAR"))

plot(fit_nomiss, fit_mar_cc, fit_mar, scales="fixed", ncol=3)

make_table2(fit_nomiss, fit_mar_cc, fit_mar, risk_time=20)

format: plain


text:
Installation

Toubles compiling? See FIXING VS CODE

Visual Studio Code (or VS Code) is a modern, open-source* code editor with extension support, and is developed by Microsoft, together with a community. There is an extension which adds the LaTeX-specific features, meaning we use this beautiful editor for writing LaTeX code!

See the image below for an impression. Clicking it opens the image in a new tab.

Visual Studio Code is the editor: it's mostly concerned with presenting the code in a nice manner. To produce a .pdf file from a .tex file, we need another program too. This is the compiler, which is part of a distribution. Refer to the following snippet taken from a-es2.nl/texnicie to see which distribution you need to install:

The distributions don't differ much from each other. On Windows, MiKTeX can install packages automatically, TeX Live can't. TeX Live rather chooses to install all or most packages, which takes a couple of gigabytes of storage. TeX Live is however better integrated with the command line, if you plan on using that.

warning

LaTeX Workshop (see below) compiles your .tex by default with latexmk (a program shipped by each distribution), which needs Perl installed on your computer. On Windows, Perl isn't installed by default, and only TeX Live comes bundled with it. If you want to use MiKTeX, you can configure VS Code to use pdflatex instead of latexmk (latexmk adds only marginal features on top of the original pdflatex). Instructions for this are at FIXING VS CODE. Or you can install Perl from https://strawberryperl.com.

For more on latexmk: https://mg.readthedocs.io/latexmk.html

Encountering installation problems? Restarting your computer and trying again might help, else ask for help. For help outside a LaTeX course, send a mail to

Not available.

The installation can take some time. In the meantime you can already have a look inside Visual Studio Code. Some of the steps below should be doable while the distribution isn't installed yet. However, if you want to play on the safe side, wait for the distribution to be installed before proceeding.

In Visual Studio Code you need to install the extension 'LaTeX Workshop'. In the menu on the left (called 'the Activity bar'), select 'Extensions'. Alternatively, use the shortcut Ctrl+Shift+X to open the menu. Search and install the extension. This mainly adds code syntax highlighting for LaTeX to VS Code, as well as a set of tools necessary for compiling the code into a PDF.

Select File > Open Folder and choose a new directory on your computer. Next, create a file (File > New File) and paste the following rudimentary document:

\documentclass[a4paper]{article}

\title{De titel}
\begin{document}
\maketitle

Hallo!
\end{document}

Save (Ctrl+S) the file with name document.tex (or something else ending in .tex). In the left menu (the Activity bar), an icon appears with the text 'TeX'. In this menu, click on Build LaTeX project, or expand the menu and select a specific Recipe. Wait till compilation is done.

If you don't get a success, have a look at FIXING VS CODE. If you get the success checkmark, click the View LaTeX project button.

Now change something in your .tex and press Ctrl+S or Command+S to save. With the default settings, LaTeX workshop should automatically compile your file again, and the PDF viewer should refresh automatically after that compilation.

You see the project directory now contains a document.pdf alongside the document.tex. There are a bunch of other files too, like .aux, .log, .synctex.gz and possibly more files. These are auxiliary files and you can ignore them. If you want, you can harmlessly remove them by selecting Clean up auxiliary files in the Build LaTeX project menu.

The installation is now complete.


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# [OS X TeX] Wish list

Matthew Hills hills_tex at tina.stanford.edu
Sat Feb 12 15:19:46 EST 2005

On Feb 12, 2005, at 11:41 AM, Alexandre Enkerli wrote:
>> (1) Command Select to select discontinuous text to which to apply
>> typeface formatting from the LaTeX panel.
> Personally, a keyboard shortcut for, say, \textbf{} (cmd-b would be
> nice) and \emph{} (cmd-i?) would be much more help than the panel...
> By any chance, is this there already? It seems so obvious...

Take a look at the Macro menu in TeXShop.
There are short macros that will do most of these styling commands.
If you select "Open Macro Editor..." you can then assign keyboard

How often do you apply boldface?
If there is a class of things in bold, then I'll usually define a new
command for that group of things
that applies the bold (this way I can change the definition later)

I use italics more commonly in writing, and even then I'll often define
a new command for a word
that I use really often...

Matt

--------------------- Info ---------------------
Mac-TeX Website: http://www.esm.psu.edu/mac-tex/
& FAQ: http://latex.yauh.de/faq/
TeX FAQ: http://www.tex.ac.uk/faq
List Post: <mailto:MacOSX-TeX at email.esm.psu.edu>


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The Power Distribution Field Technician is an experienced role supporting our Field Services Group in performing installation, troubleshooting, repair and preventative maintenance on electrical distribution equipment. Use the Help and type in the function name to get more information. Practice Problems 2. 5 is called the failure rate of the terminal. _____ Problem 5. Once you understand the idea, you will see how easy it is. I want to create a formula in Excel to solve for M in the equation below. please help me with this problem of double exponential distribution Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Exponential Smoothing is the best technique to do this in excel. SupertutorTV 129,731 views. The exponential distribution is used to calculate the probability of occurrence of an event that is the result of a continuous decaying or declining process such as the time until a radioactive particle decays. Everyday low prices and free delivery on eligible orders. to generate a sample of size 25 which follows a normal distribution with mean 60 and standard deviation 20, you simply use the formula =NORMINV(RAND(),60,20) 25 times. If you buy a new car and. as we did in class. Homework Help: Bus simulation in excel, exponential distribution departure. Several different sources of data are normally distributed. Example : In a factory, the machines break down and require service according to a Poisson distribution at the average of four per day. The mean inter-arrival time and standard deviation of this distribution are both 1/ (λ) where, (λ) is the arrival rate. Definition 1: The exponential distribution has probability density function (pdf) given by Excel Function: Excel provides the following function for the exponential distribution: EXPONDIST ( x, λ, cum) where λ is the parameter in Definition 1 and cum = TRUE or FALSE EXPONDIST( x, λ,. To see this, think of an exponential random variable in the sense of tossing a lot of coins until observing the first heads. It is a continuous analog of the geometric distribution. Yjy It is worth distinguishing between the distribution of jy, which represents our uncertainty about the average number of children in these two groups, and the distribution of the actual number of children in the two groups We can derive this distribution as well, however; letting Y. The rate of population growth at any given time can be written:. " It can also be shown (do you want to show that one too?) that if X is exponentially distributed with mean θ, then: $$P(X>k)=e^{-k/\theta}$$. 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However, the use of distribution fitting is connected with complex calculations which require special knowledge in the field of statistics and/or programming skills. Given the memoryless property of the exponential distribution, we can write the Laplace transform of the conditional distribution of given as (32) Laplace transform of a nonnegative continuous random variable is defined as [3], page 196: (33). Interpret Functions in Context HSF-LE. Find the probability that the sample maximum is greater than 4. The property owner insured the property against damage by purchasing an insurance policy with a deductible of 5. 1 De nitions: The goals of this unit are to introduce notation, discuss ways of probabilisti-cally describing the distribution of a 'survival time' random variable, apply these to several common parametric families, and discuss how observations of survival times can be right. The models and characterization investigated in this paper lead to the multivariate distribution with exponential marginals, which in the. I want to sampling from a truncated exponential distribution with lambda, and the truncate values are 40 for lower bound, and 10000 for upper bound. Exponential regression equation is the form y=a*e^(bx). Solution: Note that 1 6 = 6 1 and 36 = 62. org I just found there are two implementations of 'cumulative distribution function' for Exponential Distribution and there. Another way of expressing this would be an exponential random variable X conditioned on a <= X <= b. Create Custom Headers and Footers 7. Exponential Distribution The exponential distribution is often used to model the reliability of electronic systems, which do not typically experience wearout type failures. He has also proposed a revised method for multi-choice goal programming model which does not. The most important property of the exponential distribution is known as the memoryless property, which says that if the time to wait for an event to occur has. The 7 Excel Tricks Every Office Worker Should Know [Infographic]. When there is a damage to the property, the amount of damage (in thousands) has an exponential distribution with mean 20. For example, we might measure the number of miles traveled by a given car before its transmission ceases to function. Evaluate alternative actions in a relative manner Exponential. It looks something like that. Exponential regression equation is the form y=a*e^(bx). Sample Exponential and Logarithm Problems 1 Exponential Problems Example 1. The waiting time to observe the occurrence of an event in a Poisson process with intensity is a random variable that follows the exponential distribution with parameter (which must be positive). You have observed that the number of hits to your web site follow a Poisson distribution at a rate of 2 per day. The exponential value falls very close to the empirical distribution at. , with scale parameter) form of the function. There are two methods for solving exponential equations. Problem statement Consider a vector of N values that are the results of an experiment. Austin is naturally correct. { If ris a positive integer, the distribution is called an Erlang distribution. In this simulation, you will investigate the distribution of averages of 40 exponential(0. I am trying to calculate safety stock using an exponential distribution need to calculate the safety factor for a desired service level is there a function like NORMSINV for exponential distribution - ExcelBanter. Because there are an infinite number of possible constants θ, there are an infinite number of possible exponential distributions. Practice Problems 2. DIST function in Microsoft Excel. You have observed that the number of hits to your web site follow a Poisson distribution at a rate of 2 per day. DIST function in Microsoft Excel. This article describes the formula syntax and usage of the HYPGEOM. DIST function syntax has the following arguments:. If an input is given then it can easily show the result for the given number. The time for washing and cleaning each car varies but is found to follow an exponential distribution with mean 10 minutes per car. If this waiting time is unknown it can be considered a random variable, x, with an exponential distribution. Further, if the data aren't exponential, that adjustment may be badly impacted by large outliers. Remember that we learned about using the Sign Chart or Sign Pattern method for inequalities here in the Quadratic Inequalities section, and also we have the domain restriction that the argument of a log has to be $$>0$$. (a) leasyl Derive the Exponential from the Geometric r. Lilliefors, H. To generate a hyperexponential distribution, let be independent exponential random variables with rates parameters , and with weights , respectively. For example, if you flip a coin, you either get heads or tails. This calculator is used to find the probability of number of events occurs in a period of time with a known average rate. : Multi-choice Stochastic Transportation Problem with Exponential Distribution [2]. Byju's Exponential Distribution Calculator is a tool which makes calculations very simple and interesting. Many people think of Excel as a mere calculator or place to keep data. The main properties of this new distribution is studied, with special emphasis on its median, mode and moments function and some characteristics related to reliability. In compartmental modelling, the Exponential distribution plays a role as the probability distribution underlying the sojourn time in a compartment. The number of hooligans is geometrically distributed with mean 10. 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Log-normal distributions can model a random variable X , where log( X ) is. Exponential distribution is a continuous probability distribution with probability density function given by P(x)=ae-ax where a is the parameter of the distribution and x is a random variable Code to add this calci to your website. Insert a Second Table into a Worksheet 5. DIST() function to create your data set for the chart, e. The paper proposes the original procedures for solving some exponential and Poisson distribution inverse problems by recently developed special trans function theory. Too expensive to directly estimate Nhat every year 2. The following is the density function of the lifetime. x and mu can be vectors, matrices, or multidimensional arrays that all have the same size. It is faster and simpler than the original, and will produce, for example, normal or exponential variates at the rate of 15 million per second with a C version on a 400MHz PC. Volodin; translated by Margaret Stein and edited by Ingram Olkin. Binomial distribution definition and formula. Excel has an Exponential Distribution function called EXPONDIST(X, Lamda,Cumulative). In example 1, the lifetime of a certain computer part has the exponential distribution with a mean of ten years (X ~ Exp(0. The practice problems in this post are additional practice problems in addition to Practice Problem Set 7 and Practice Problem Set 8. …The POISSON distribution lets you estimate…the number of customers who will come into a store…during a given time period such as. For reasons that we will not cover here, the best estimate of the population variance will equal the sample variance times n/(n-1), where n is the. As we have seen earlier, M/M/1 refers to negative exponential arrivals and service times with a single server. Exponential growth occurs when a function's rate of change is proportional to the function's current value. Fiverr freelancer will provide Data Analysis & Reports services and help with ms excel problems and data analysis including Include Source Code within 2 days. Note that. KEATINGE Abstract Finding a parametric model that fits loss data well is often difficult. • The exponential function, Y=c*EXP(b*x), is useful for fitting some non-linear single-bulge data patterns. So the Excel command includes "INV" e. Buy Characterization Problems Associated with the Exponential Distribution Softcover reprint of the original 1st ed. that show how to apply the techniques to real-world. , The Annals of Statistics, 1978; Alternative Estimators for the Scale Parameter of the Exponential Distribution with Unknown Location Brewster, J. Exponential Population Growth Why model? 1. Here we discuss how to calculate the Probability of X using Poisson distribution formula in excel with examples and downloadable excel template. Excel Solver - Define and Solve a Problem Excel Solver - Understanding Solver Results messages Users who solve problems with integer constraints occasionally report that “Solver claims it found an optimal solution, but I manually found an even better solution. You will enter some of your answers to the questions that follow in this sample Excel spreadsheet: P(x< 5) P(24 0. In this simulation, you will investigate the distribution of averages of 40 exponential(0. Exponential Distribution. You can learn more about financial modeling from the following articles – Poisson Distribution in Excel; What is the Break-Even Chart?. Work Example 1 again by assuming that the lifetime of the type of computers in questions follows the exponential distribution with mean 4 years. In Excel 2010, 2013 and 2016 there is the additional function EXPON. { If ris a positive integer, the distribution is called an Erlang distribution. DIST function in Microsoft Excel. What you can do is create your range for the x-values. What's a Good SAT Score? 2019 EDITION UPDATED! Test Score Ranges! Charts! College Admission Tips! - Duration: 12:38. Posted on January 16, 2019 Updated on March 4, 2019. No Data Sets for Homework Problems for this chapter. Uniform Probability Distribution (15:04) Excel 2013 Exponential Probability Distribution & EXPON. Posts about Exponential Distribution written by Dan Ma. It is the continuous counterpart of the geometric distribution, which is instead discrete. This is the first study that integrates ACO with DES for identification and optimisation of maritime traffic. In fact, this is fairly easy to do using conditional chart formatting techniques I describe on my web site (see Simple Conditional Charts). We do not offer that service, and that is not the purpose of this contact form. The memoryless property says that knowledge of what has occurred in the past has no effect on future probabilities. Fiverr freelancer will provide Data Analysis & Reports services and help with ms excel problems and data analysis including Include Source Code within 2 days. The other is to use the following formula for the skewness coefficient for beta distribution. In this article we will focus on M/M/1 queueing system. Any problems email [email protected] Pris: 959 kr. Exponential Growth and Decay. Returns the hypergeometric distribution. The exponential distribution is also used to model the waiting times between rare events. Exponential Excel function in excel is also known as the EXP function in excel which is used to calculate the exponent raised to the power of any number we provide, in this function the exponent is constant and is also known as the base of the natural algorithm, this is an inbuilt function in excel. DIST to determine the probability. The most important of these properties is that the exponential distribution is memoryless. , with scale parameter) form of the function. For daily posts on probability, follow @ProbFact on Twitter. Exponential random variables (sometimes) give good models for the time to failure of mechanical devices. Just copy and paste the below code to your webpage where you want to display this calculator. Exponential smoothing is used to smooth out irregularities (peaks and valleys) to easily recognize trends. p = expcdf(x,mu) computes the exponential cdf at each of the values in x using the corresponding mean parameter mu. First, the Bayes test rule of one-sided test is derived in the case of independent and identically distributed random variables under weighted linear loss function. MODELING LOSSES WITH THE MIXED EXPONENTIAL DISTRIBUTION CLIVE L. In example 1, the lifetime of a certain computer part has the exponential distribution with a mean of ten years (X ~ Exp(0. The property owner insured the property against damage by purchasing an insurance policy with a deductible of 5. August 2014. distribution. Generate 200 samples of 30 observations each by generating 200 rows of data and storing the results in columns C1-C30. The buses depart from the terminal station A with headways that follow the negative exponentioal distributiona with parameter 10 trips per hour. Exponential Distribution of jyvs. If you have a probability distribution of a discrete random variable and you want to find the. Byju's Exponential Distribution Calculator is a tool which makes calculations very simple and interesting. 1 month) such that for a FF $\theta < \theta_0$. The exponential distribution is used to model data with a constant failure rate (indicated by the hazard plot which is simply equal to a constant).


text:
# tricky memoization

Let's say I have the NDSolve example for documentation involving splitting 2nd order into set of 1st order ODEs:

NDSolve[{x'[t] == y[t], y'[t] == -x[t], x[0] == 1, y[0] == 0}, {x,
y}, {t, 0, 10}]

I need to solve my second order system like this for various reasons.

My ODE is more complicated, and depends on some other parameters call them $(a,b)$, and so when I solved it as a second order system I used memoization in the form below, so that if a solution for a given parameter set had already been computed, Mathematica wouldn't redo it all again. I would like to do this, or something like it for the first order system and where NDSolve is giving the two variables $x[a,b],y[a,b]$.

x[a_,b_]:=
x[a,b] = x /.NDSolve[{eq[a,b],x[0]== init[a,b],x'[0]==dinit[a,b],x,{r, 0, 10}][[1]];

-
Have you, by any chance, seen the NDSolveStateData tutorial? – J. M. Sep 30 '12 at 1:38

## 1 Answer

When I see this right, you have one single function call (NDSolve) depending on two parameters which results in a vector (the interp. functions). Here is an equivalent minimal example function for this call

f[a_, b_] := {b, a}

The underlying issue is now, that you always get both answers from f (x and y) even if you need only x for a specific choice of a and b. You can solve this by memorizing both, x and y, no matter which function you call.

x[a_, b_] := {x[a, b], y[a, b]} = f[a, b];
y[a_, b_] := {x[a, b], y[a, b]} = f[a, b];

A quit check shows you that you memorize the values for x and y even if you compute only x

Table[x[a, b], {a, 0, 2}, {b, 0, 2}];
?x
?y

Lets try this with your NDSolve example. I only wrapped Module around to localize x and y.

ClearAll[x, y, f];
f[a_, b_] := Module[{x, y}, {x, y} /.
First@NDSolve[{x'[t] == y[t], y'[t] == -x[t], x[0] == a, y[0] == b},
{x, y}, {t, 0, 10}]];

x[a_, b_] := {x[a, b], y[a, b]} = f[a, b];
y[a_, b_] := {x[a, b], y[a, b]} = f[a, b];

Table[x[a, b], {a, 0, 2}, {b, 0, 2}];

?y

Finally, the repetition of the exact same function definition for x and y looks kind of like we are not experts. Lets improve this. The only thing which is different is the head of the left side in the :=. So we write the line one time and substitute x and y resp.

SetDelayed[#[a_, b_], {x[a, b], y[a, b]} = f[a, b]] & /@ {x, y}
`
-
one small problem with this is on the initial call, if I do say: x[1,2][3.0] instead of getting a numeric answer back I get two interpolating functions. The second call works fine, and gives the result desired. This is in contrast to standard memoization. I think it can be solved by just changing to x[a_, b_] := ({x[a, b], y[a, b]} = f[a, b]; x[a,b]) etc – fpghost Sep 30 '12 at 12:14
@fpghost Yes, I haven't addressed that. I only worked out the problem with the memoization. I'll have a look at it later. – halirutan Sep 30 '12 at 14:08


text:
# Finding the Jordan Normal Form for a General Linear Transformation

Hey everyone here's the problem:

Let V be a vector space with dim(V)=n

For a particular linear transformation,f, we are given that there are two distinct eigenvalues, λ1 and λ2, with corresponding eigenspaces, E(λ1) and E(λ2).

dim(E(λ1))=m, dim(E(λ2))=n-m

I'm just struggling to figure out why the Jordan normal form of this linear transformation is diagonal and what it looks like. Would anyone be able to give me a reason that the Jordan normal form of this linear transformation is strictly diagonal?

Thanks!

The Jordan form is diagonal if the matrix has a basis of eigenvectors. If you have a basis $\{ x_1,x_2,\cdots x_m \}$ of $E(\lambda_1)$ and a basis $\{ y_1,y_2,\cdots,y_{n-m}\}$ of $E(\lambda_2)$, then the union of these two bases is a basis of the full space because, if $$(\alpha_1 y_1 + \alpha_2 y_2 + \cdots+\alpha_mx_m)+(\alpha_{m+1}y_1+\alpha_{m+2}y_2+\cdots+\alpha_ny_{n-m}) = 0,$$ then you can apply $A-\lambda_2 I$ to the above in order to conclude $$(\lambda_1-\lambda_2)(\alpha_1 y_1 + \alpha_2 y_2 + \cdots+\alpha_mx_m) = 0.$$ Because $\lambda_1-\lambda_2 \ne 0$ then $\alpha_1 y_1+\alpha_2 y_2+\cdots+\alpha_m y_m=0$, which means $\alpha_1=\alpha_2=\cdots=\alpha_m = 0$ because $\{ y_j \}$ is a linearly independent set of vectors. The same argument gives $\alpha_{m+1}=\alpha_{m+2}=\cdots=\alpha_{n}=0$. Once you have a basis of eigenvectors for $A$, then $A$ is similar to a diagonal matrix. So the Jordan form must also be diagonal.


text:
### Cosines Rule

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

### Squ-areas

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more triangular areas are enclosed. What is the area of this convex hexagon?

### Far Horizon

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

# Sine and Cosine

##### Stage: 4 Challenge Level:

The Flash "movie" allows students to see the flow of function values as the dot circles the centre point or origin. Students can be encouraged to see the "spot" values provided by a calculator as snap-shots of this flow.

Check that students grasp that an angle of zero is when the dot is at (1,0), that the red line corresponds to the Sine value and the blue line to the Cosine value.

There may be a need to guide students into noticing that there is a the change of sign, and noticing when it occurs.


text:
# View sign-in history

A few days ago I recall finding a visual calendar of my sign-in history on Math Stack Exchange. It looked like an ordinary monthly calendar, where the days were white if I hadn't signed in, and green if I had. (I found it when I wondering how close I've been in the past to getting the "fanatic" badge.)

I thought it might have been a feature of Stack Exchange as a whole, but I was unable to find an answer on Meta Stack Exchange.

A brief answer is that you should go to your profile and click on the number of consecutive days. (Remember that you'll find this in the profile tab, not in the activity tab.)

You can find more details here:

(Originally I considered that this question could be closed as a duplicate of the first one in this list. But maybe they are slightly different after all - the other one asks also whether it is possible to see this information for other users, not just your own visits. That's why I decided to convert my comments to an answer.)

When you click your own profile, the default tab is the activity tab. You should go to the "Profile" tab, where you can see the "visited $x$ days, $y$ consecutive" text. Click this text to show the calendar.

• Wow, $x = y$. I wonder who else on math.SE has this level of attendance. – achille hui Oct 29 '17 at 0:15
• @achillehui I think on math.SE this is unmatched. There are two on SO with x=y and 3k+ days though. – quid Oct 29 '17 at 20:26
• @quid: Give it a year... ;p – Asaf Karagila Oct 29 '17 at 23:06
• Sure. Those two just had a head-start. I believe to recall there is somebody on tex.se too with a comparable record. But I cannot locate the meta.se post where it came up. something about a display bug maybe. – quid Oct 29 '17 at 23:31
• Well, I'm sure there are others like me who check the website every day, and have been doing that for a long enough period. It's not hard, just requires basic dedication... :) – Asaf Karagila Oct 29 '17 at 23:32


text:
Story posted as a reason to

# Tiësto in Kosovo

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text:
quadrupole moment and higher for simple current loop

I'm working on a field configuration that needs to die off rapidly and I got to a $1/r^5$ dependence with canceling the dipole moment of the system cleverly, but to go get better arrangements I need to figure out the higher order moments of a simple loop as well.

Griffiths (page 243, 3rd ed) refers to this integral $\oint (r')^{n}P_n(cos \theta ') d L'$, with $P_n$ as the Legendre polynomials but I am unclear on how to go about evaluating it for higher moments because for the dipole term Griffiths does a clever trick that I'm not super sure extrapolates. I feel like the current loop is such a simple, tidy system it's higher moments would be tabulated somewhere just as an example of how higher orders are calculated, but I'm unable to find them after a few a days of googling and librarying.

Whenever $|\mathbf{r}|$ lies far outside the loop, we can expand the vector potential $\mathbf{A}(\mathbf{r})$ in an irregular solid harmonic series $$\mathbf{A}(\mathbf{r})=\frac{\mu_0}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r} '|}dV=\frac{\mu_0}{4\pi}\sum_{L=0}^\infty\sum_{m=-L}^L I_{Lm}(\mathbf{r})\langle R_{Lm},\mathbf{J}\rangle$$ where $R_{Lm},I_{Lm}$ are the regular and irregular real solid harmonic functions and $\langle R_{Lm},\mathbf{J}\rangle$ is the integral inner product between the two (aka, the $Lm$ multipole moment).

Letting the loop with radius $\rho$ lie in the $xy$-plane with $z$-axis height $h$, the interior moments $\langle R_{Lm},\mathbf{J}\rangle$ can be computed as $$\langle R_{Lm},\mathbf{J}\rangle=\int_0^{2\pi}R_{Lm}\left(\sqrt{\rho^2+h^2},\arctan(h,\rho),\phi\right)\mathbf{J}(\phi)\,\rho d\phi.$$ Letting the loop current flow counterclockwise, in Cartesian coordinates we have $$\mathbf{J}(\phi)= \left( \begin{array}{cc} -J_0\sin(\phi)\\ J_0\cos(\phi)\\0 \end{array} \right)$$ and we obtain the following moments up to octupole order (copy paste into Mathematica and hit Cell > Convert To > StandardForm to properly format it):

SphericalHarmonicYr[L_, m_, \[Theta]_, \[Phi]_] := (-1)^m*
Sqrt[((2*L + 1)*(L - Abs[m])!)/((2*Pi)*(L + Abs[m])!)]*
LegendreP[L, Abs[m], Cos[\[Theta]]]*
Piecewise[{{Sin[Abs[m]*\[Phi]], m < 0}, {Sqrt[1/2],
m == 0}, {Cos[Abs[m]*\[Phi]], m > 0}}];
SolidHarmonicIr[l_, m_,
r_, \[Theta]_, \[Phi]_] := (Sqrt[(4*Pi)/(2*l + 1)]*
SphericalHarmonicYr[l, m, \[Theta], \[Phi]])/r^(l + 1);
SolidHarmonicRr[l_, m_, r_, \[Theta]_, \[Phi]_] :=
Sqrt[(4*Pi)/(2*l + 1)]*r^l*
SphericalHarmonicYr[l, m, \[Theta], \[Phi]];
J[\[Phi]_] := J0*{-Sin[\[Phi]], Cos[\[Phi]], 0};
Grid[Simplify[
Table[MatrixForm[
Integrate[\[Rho]*J[\[Phi]]*
SolidHarmonicRr[L, m, Sqrt[\[Rho]^2 + h^2],
ArcTan[h, \[Rho]], \[Phi]], {\[Phi], 0, 2*Pi}]],
{L, 0, 3}, {m, -L, L}],
Assumptions -> {\[Rho] > 0, Element[h, Reals]}], Frame -> All]

The code above computes the moments out to $L=3$, but you can change 3 to whatever multipole order is necessary. Here is a numerical verification that the irregular solid harmonic series $\frac{\mu_0}{4\pi}\sum_{L=0}^\infty\sum_{m=-L}^L I_{Lm}(\mathbf{r})\langle R_{Lm},\mathbf{J}\rangle$ and the integral $\frac{\mu_0}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r} '|}dV$ are actually the same thing (again, hit Cell > Convert To > StandardForm to properly format the appearance):

Subscript[Q, L_, m_] :=
Integrate[\[Rho]*J[\[Phi]]*
SolidHarmonicRr[L, m, Sqrt[\[Rho]^2 + h^2],
ArcTan[h, \[Rho]], \[Phi]], {\[Phi], 0, 2*Pi}];
r = N[{5, 6, 7}];
rSph = CoordinateTransform["Cartesian" -> "Spherical", r];
h = 2;
\[Rho] = 1;
J0 = 2;
NIntegrate[(J[\[Phi]]/
Norm[r - {\[Rho]*Cos[\[Phi]], \[Rho]*Sin[\[Phi]],
h}])*\[Rho], {\[Phi], 0, 2*Pi}]
Sum[(Subscript[Q, L, m]*SolidHarmonicIr[L, m, ##1] & ) @@ rSph, {L,
0, 5}, {m, -L, L}]
Clear[h, \[Rho], J0]

{-0.0471743, 0.039312, 0.}

{-0.0471859, 0.0393216, 0.}

Alternately, as an additional simplification, note that $\mathbf{J}$ can be expressed in spherical vector coordinates as $$\mathbf{J}(\phi)=\left( \begin{array}{cc} 0\\ 0\\ J_0 \end{array} \right)$$ giving (this time up to $L=4$):

J[\[Phi]_] := {0, 0, J0};
Grid[Simplify[
Table[MatrixForm[
Integrate[\[Rho]*J[\[Phi]]*
SolidHarmonicRr[L, m, Sqrt[\[Rho]^2 + h^2],
ArcTan[h, \[Rho]], \[Phi]], {\[Phi], 0, 2*Pi}]],
{L, 0, 4}, {m, -L, L}],
Assumptions -> {\[Rho] > 0, Element[h, Reals]}], Frame -> All]

although for this coordinate choice I'm a bit confused as to why the $L=0$ coefficient is not zero (will have to check that later).

The magnetic field can then be obtained from the above coefficients $\langle R_{Lm},\mathbf{J}\rangle$ by $$\mathbf{B}(\mathbf{r})=\nabla\times\mathbf{A}(\mathbf{r})=\frac{\mu_0}{4\pi}\sum_{L=0}^\infty\sum_{m=-L}^L \nabla\times I_{Lm}(\mathbf{r})\langle R_{Lm},\mathbf{J}\rangle.$$


text:
# How to detect if the first non-space/non-tab character in a string is '-' (minus)?

I have a macro \myNumericString that expands to a string of numbers, with the possibility to present a minus sign as a first non-space character. How can I construct a macro:

\isNegative{\myNumericString}{ true }{ false }

in order to make decisions on how to treat \myNumericString in successive typesetting steps? Here's a simple case of usage:

$4 \times \isNegative{\myNumericString}{ ( \myNumericString ) }{ \myNumericString }$

-
\ifnum\myNumericString<0? –  egreg Feb 4 '12 at 10:59

If you are sure that \myNumericString is what TeX expects as a <number>, then

\makeatletter
\newcommand{\isNegative}[1]{%
\ifnum#1<0
\expandafter\@firstoftwo
\else
\expandafter\@secondoftwo
\fi}
\makeatother

If \myNumericString possibly has a decimal part, you can use

\ifdim#1pt<0pt

instead (the integer part cannot exceed 16383).

A better definition that only examines the first token in the expansion of \myNumericString is

\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand{\isNegative}{m}
{
\exp_args:Nx \str_if_eq:nnTF { \tl_head:f { #1 } } { - }
}
\ExplSyntaxOff

Here's an example document

\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand{\isNegative}{m}
{
\exp_args:Nx \str_if_eq:nnTF { \tl_head:f { #1 } } { - }
}
\ExplSyntaxOff

\begin{document}
\def\myNumericString{123.456}
$4\times\isNegative{\myNumericString}{(\myNumericString)}{\myNumericString}$

\def\myNumericString{-123.456}
$4\times\isNegative{\myNumericString}{(\myNumericString)}{\myNumericString}$
\end{document}

-

if you need it for all arguments:

\documentclass{article}
\makeatletter
\newcommand\isNegative[1]{\expandafter\is@Negative#1\@nil}
\def\is@Negative{\@ifnextchar-{\@negative}{\@positive}}
\def\@negative-#1\@nil{#1 starts with a minus}
\def\@positive#1\@nil{#1 starts not with a minus}
\makeatother
\begin{document}
\def\myNumericString{123.456}
\isNegative{\myNumericString}

\def\myNumericString{-123.456}
\isNegative{\myNumericString}

\isNegative{     -foo}
\end{document}

-
+1 This is a nice solution as well, but here \isNegative doesn't fit with the macro proposed in the answer. –  agodemar Feb 5 '12 at 9:30


text:
# Application and projection¶

## Iterables¶

An iterable is a list, dictionary,process handle, or function. It is a mapping from its argument/s to its result.

• A list is a mapping from its indexes to its items.
• A dictionary is a mapping from its keys to its values.
• A matrix is a mapping from its row indexes to its rows; or a mapping from its row and column indexes to its items.
• A table is a mapping from its row indexes to its tuples; or a mapping from its column names to its columns; or a mapping from its row indexes and column names to its items.

A function is a mapping from its domain/s (all its possible valid arguments) to its range – all its possible results.

Operators, keywords and lambdas are all functions.

## Application¶

To apply an iterable means

• to evaluate a function on its arguments
• to select items from a list or dictionary

There are several ways to do it.

All iterables can be applied using either bracket notation or the Apply operator.

Functions can also be applied prefix, infix, postfix, or using the Apply At operator.

rank
of f
argument
notation
Apply Apply At other note
0 f[] f . enlist(::) f@(::)
1 f[x] f . enlist x f@x f x, xf prefix, postfix
2 f[x;y] f . (x;y) x f y infix
>2 f[x;y;z;…] f . (x;y;z;…)

Binary operators and many binary keywords can be applied infix.

q)2+2                         / binary operator
4
q)3 in 0 1 2 3 4              / binary keyword
1b

Unary iterables (keywords, lambdas, dictionaries, and lists – but not iterators) can be applied prefix.

q)count "zero"                / unary keyword
4
q){x*x}4                      / unary lambda
16
q)d:TomDickHarry!42 97 35  / dictionary
q)d HarryTom
35 42
q)m:3 4#"abcdefghijkl"        / a matrix (binary iterable)
q)m 1 3                       / is also a list (unary iterable)
"def"
"jkl"

Iterators can be applied postfix, and usually are.

q)subtots:sum'                / '[sum]
q)subtots 3 4#til 12
3 12 21 30

## Long right scope¶

Iterables applied prefix or infix, have long right scope. In other words:

When a unary iterable is applied prefix, its argument is everything to its right.

q)sqrt count "It's about time!"
4

When a binary iterable is applied infix, its right argument is everything to its right.

q)7 * 2 + 4
42

Republic of iterables

There is no precedence among iterables. In 7*2+4 the right argument of * is the result of evaluating the expression on its right.

This rule applies without exception.

## Iterators¶

The iterators are almost invariably applied postfix.

q)+/[17 13 12]
42

In the above, the Over iterators / is applied postfix to its single argument + to derive the function +/ (sum).

An iterator applied postfix has short left scope. That is, its argument is the object immediately to its left. For the Case iterator that object is an int vector; for all other iterators, an iterable. But note that an iterator’s argument may itself be a deerived function.

q)txt:(("Now";"is";"the";"time");("for";"all";"good";"folk"))
q)txt
"Now" "is"  "the"  "time"
"for" "all" "good" "folk"
q)count[txt]
2
q)count'[txt]
4 4
q)count''[txt]
3 2 3 4
3 3 4 4

In the last example, the derived function count' is the argument of the second ' (Each).

Only iterators can be applied postfix.

Apply/Index and Apply/Index At for how to apply functions and index lists

## Rank and syntax¶

The rank of an iterable is the number of

• arguments it evaluates, if it is a function
• indexes required to select an atom, if it is a list

An iterable is variadic if it can be used with more than one rank. All matrixes and some derived functions are variadic.

q)+/[til 5]           / unary
10
q)+/[1000000;til 5]   / binary
1000010

Rank is a semantic property, and is independent of syntax. This is a ripe source of confusion.

The syntax of aderived function is determined by the application that produced it.

Postfix application produces an infix.

The derived function +/ is variadic buthas infix syntax. Applying it infix is straightforward.

q)1000000+/til 5
1000010

How then to apply it as a unary? Bracket notation ‘overrides’ infix syntax.

q)+/[til 5]           / unary
10
q)+/[1000000;til 5]   / binary
1000010

Or isolate it with parentheses. It remains variadic.

q)(+/)til 5           / unary
10
q)(+/)[1000000;til 5] / binary
1000010

The potential for confusion is even greater when the argument of a unary operator is a unary function. Here the derived function is unary – but it is still an infix! Only parentheses or brackets can save us.

q)count'[txt]
4 4
q)(count')txt
4 4

Or the each keyword.

q)count each txt
4 4

Conversely, if the unary operator is applied, not postfix, but with bracket notation (unusual and not recommended) the derived function is not an infix. But it is still variadic.

q)/[+]til 5               / oops, a comment
q);/[+]til 5              / unary, prefix
10
q);/[+][til 5]            / unary, bracket notation
10
q);/[+][10000;til 5]      / binary, bracket notation
10010
q)100000/[+]til 5         / but not infix
'type
[0]  100000/[+]til 5
^
q)'[count]txt             / unary, prefix
4 4

Applying a unary operator with bracket notation is unusual and not recommended.

## Projection¶

When an iterable of rank $n$ is applied to $m$ arguments and $m, the result is a projection of the iterable onto the supplied arguments (indexes), now known as the projected arguments or indexes.

In the projection, the values of projected arguments (or indexes) are fixed.

The rank of the projection is $n-m$.

q)double:2*
q)double 5                         / unary
10
q)halve:%[;2]
q)halve[10]                        / unary
5
q)f:{x+y*z}                        / ternary
q)f[2;3;4]
14
q)g:f[2;;4]
q)g 3                              / unary
14
q)(f . 2 3) 4
14
q)l:("Buddy can you spare";;"?")
q)l "a dime"                       / unary
"Buddy can you spare"
"a dime"
"?"
q)m:("The";;;"fox")
q)m["quick";"brown"]               / binary
"The"
"quick"
"brown"
"fox"

Make projections explicit

When projecting a function onto an argument list, make the argument list full-length. This is not always necessary but it is good style, because it makes it clear the iterable is being projected, not applied.

q)foo:{x+y+z}
q)goo:foo[2]    / discouraged
q)goo:foo[2;;]  / recommended

You could make an exception for operators and keywords, where the rank is well known.

q)f:?["brown"]
q)f "fox"
5 2 5
q)g:like["brown"]
q)g "*ow*"
1b
`

When projecting a variadic function the argument list must always be full-length.


text:
# An Inertial Newton Algorithm for Deep Learning

1 IRIT-SC - Signal et Communications
IRIT - Institut de recherche en informatique de Toulouse
6 IRIT-ADRIA - Argumentation, Décision, Raisonnement, Incertitude et Apprentissage
IRIT - Institut de recherche en informatique de Toulouse
Abstract : We introduce a new second-order inertial optimization method for machine learning called INNA. It exploits the geometry of the loss function while only requiring stochastic approximations of the function values and the generalized gradients. This makes INNA fully implementable and adapted to large-scale optimization problems such as the training of deep neural networks. The algorithm combines both gradient-descent and Newton-like behaviors as well as inertia. We prove the convergence of INNA for most deep learning problems. To do so, we provide a well-suited framework to analyze deep learning loss functions involving tame optimization in which we study a continuous dynamical system together with its discrete stochastic approximations. We prove sublinear convergence for the continuous-time differential inclusion which underlies our algorithm. Additionally, we also show how standard optimization mini-batch methods applied to non-smooth non-convex problems can yield a certain type of spurious stationary points never discussed before. We address this issue by providing a theoretical framework around the new idea of $D$-criticality; we then give a simple asymptotic analysis of INNA. Our algorithm allows for using an aggressive learning rate of $o(1/\log k)$. From an empirical viewpoint, we show that INNA returns competitive results with respect to state of the art (stochastic gradient descent, ADAGRAD, ADAM) on popular deep learning benchmark problems.
Keywords :
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-02140748
Contributor : Camille Castera <>
Submitted on : Friday, July 2, 2021 - 12:00:56 PM
Last modification on : Sunday, August 22, 2021 - 3:24:01 AM

### File

1905.12278.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-02140748, version 5
• ARXIV : 1905.12278

### Citation

Camille Castera, Jérôme Bolte, Cédric Févotte, Edouard Pauwels. An Inertial Newton Algorithm for Deep Learning. 2021. ⟨hal-02140748v5⟩

Record views


text:
edit

# Grove - Oxygen Sensor Pro(GGC2330-O2)

Grove - Oxygen Sensor Pro(GGC2330-O2) is a kind of sensor to test the oxygen concentration in air, which is based on the principle of the electrochemical cell to the original work.When it starts to work, it will produce a current proportional to the concentration of oxygen, then you can determine the concentration of oxygen by measuring the Current size.

Grove - Oxygen Sensor Pro(GGC2330-O2) is an enhanced version of Grove - Oxygen Sensor(GG2020-O2), Compared to the Grove - Oxygen Sensor(GG2020-O2), It has a built-in temperature sensor to make temperature compensation, which can correct the measurement error caused by the temperature difference and make the measurement more accurate.On top of that, it is equipped with a high-performance microprocessor, which allows it to have digital output and analog voltage output capabilities at the same time, making the sensor easier to use and calibrate and shorten the development period.

Tip

We've released the Seeed Gas Sensor Selection Guide, it will help you choose the gas sensor that best suits your needs.

## Feature¶

• Pre-calibration(With build-in MCU and temperature sensor)
• High precision and resolution
• Long life, low power consumption
• Strong anti-interference ability
• Good stability
• UART signal output

Tip

## Specification¶

Items Parameter
Target Gas O2
Resolution 0.1%VOL
Measurement Range 0-25%VOL
ReOperating Voltage DC 5±0.1V DC
Operating Current ＜ 10 mA
Output Output UART(TTL electrical Level, 3V)Analog voltage(refer table 2 for sensor original amplifying signal)
Operating Conditions -20～25℃/15～90%RH(no condensation)
Storage Conditions -20～50℃/15～90%RH(no condensation)
Size ø23.5mm*24.5mm
Life Expectancy 2 years(in air)

Note

The aging time of power-on is not less than 5min; Avoid long-term contact with organic volatile solvents; The use or storage environment cannot be an acid-base environment.

## Applications¶

• Portable gas detector
• Fixed gas alarm and detector

## Platforms Supported¶

Arduino Raspberry Pi

Caution

The platforms mentioned above as supported is/are an indication of the module's software or theoritical compatibility. We only provide software library or code examples for Arduino platform in most cases. It is not possible to provide software library / demo code for all possible MCU platforms. Hence, users have to write their own software library.

## Getting Started¶

Note

This chapter is based on Win10 and Arduino IDE 1.7.9

This new Grove - Oxygen Sensor Pro(GGC2330-O2) usage method is is not used in the same way as the Grove - Oxygen Sensor(GG2020-O2) GG2020-O2.

Here we will show you how this Grove - Oxygen Sensor Pro(GGC2330-O2) works via a simple demo. First of all, you need to prepare the below stuffs:

Seeeduino Cortex-M0+ Grove - Oxygen Sensor Pro(GGC2330-O2) Base Shield
Get ONE Now Get ONE Now Get ONE Now

### Connection¶

Thanks to the benefit of Grove series modules, you don't need to make soldering or bread board, what you need to do is connect the modules to the right port of Base Shield. For this demo, we have only one Grove module.

• Grove - Oxygen Sensor Pro(GGC2330-O2) is an analog input module, we connect it to UART in this demo.

### Upload the code to Arduino¶

Copy the below code to Arduino IDE.

void setup() {
Serial.begin(9600);
Serial2.begin(9600);
}
void loop() {
if (Serial2.available()){
delay(10);
delay(10);
delay(10);
delay(10);
delay(10);
delay(10);
delay(10);
delay(10);
delay(10);

if(begin_code == 255 && state_code == 134){

float O2_val = ((high_code * 256) + low_code) * 0.1 ;
Serial.print("O2: ");
Serial.print(O2_val);
Serial.println(" %");
}
}
}

Then choose the right Board and COM port, and then click on the Upload button, this process take few seconds.

### Get data¶

Open serial monitor of your Arduino IDE, and you will get the data now.

Warning

It need about 20~30 minutes to preheat the sensor, or you will get a larger value.

## Projects¶

LoRa IoTea: An automatic information collection system applied to tea plantation. It is part of intelligent agricultural information collection.

A Plant Box with Lighting and Raining You never seen such a way to water you plant.

## Tech Support¶

Please submit any technical issue into our forum.


text:
## Sparse Plus Low Rank Matrix Decomposition: A Discrete Optimization Approach

We study the Sparse Plus Low Rank decomposition problem (SLR), which is the problem of decomposing a corrupted data matrix D into a sparse matrix Y containing the perturbations plus a low rank matrix X. SLR is a fundamental problem in Operations Research and Machine Learning arising in many applications such as data compression, latent … Read more

## Efficient Joint Object Matching via Linear Programming

Joint object matching, also known as multi-image matching, namely, the problem of finding consistent partial maps among all pairs of objects within a collection, is a crucial task in many areas of computer vision. This problem subsumes bipartite graph matching and graph partitioning as special cases and is NP-hard, in general. We develop scalable linear … Read more

## On Piecewise Linear Approximations of Bilinear Terms: Structural Comparison of Univariate and Bivariate Mixed-Integer Programming Formulations

Bilinear terms naturally appear in many optimization problems. Their inherent nonconvexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of … Read more

## An Alternative Perspective on Copositive and Convex Relaxations of Nonconvex Quadratic Programs

We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family … Read more

## A Polynomial-time Algorithm with Tight Error Bounds for Single-period Unit Commitment Problem

This paper proposes a Lagrangian dual based polynomial-time approximation algorithm for solving the single-period unit commitment problem, which can be formulated as a mixed integer quadratic programming problem and proven to be NP-hard. Tight theoretical bounds for the absolute errors and relative errors of the approximate solutions generated by the proposed algorithm are provided. Computational … Read more

## Dual-density-based reweighted $\ell_{1}hBcalgorithms for a class of$\ell_{0}hBcminimization problems

The optimization problem with sparsity arises in many areas of science and engineering such as compressed sensing, image processing, statistical learning and data sparse approximation. In this paper, we study the dual-density-based reweighted $\ell_{1}$-algorithms for a class of $\ell_{0}$-minimization models which can be used to model a wide range of practical problems. This class of … Read more

## Solving Heated Oil Pipeline Problems Via Mixed Integer Nonlinear Programming Approach

It is a crucial problem how to heat oil and save running cost for crude oil transport. This paper strictly formulates such a heated oil pipeline problem as a mixed integer nonlinear programming model. Nonconvex and convex continuous relaxations of the model are proposed, which are proved to be equivalent under some suitable conditions. Meanwhile, … Read more

## A convex relaxation to compute the nearest structured rank deficient matrix

Given an affine space of matrices L and a matrix \theta in L, consider the problem of finding the closest rank deficient matrix to \theta on L with respect to the Frobenius norm. This is a nonconvex problem with several applications in estimation problems. We introduce a novel semidefinite programming (SDP) relaxation, and we show … Read more

## Tightening McCormick Relaxations Toward Global Solution of the ACOPF Problem

We show that a strong upper bound on the objective of the alternating current optimal power flow (ACOPF) problem can significantly improve the effectiveness of optimization-based bounds tightening (OBBT) on a number of relaxations. We additionally compare the performance of relaxations of the ACOPF problem, including the rectangular form without reference bus constraints, the rectangular … Read more

## Approximate Positively Correlated Distributions and Approximation Algorithms for D-optimal Design

Experimental design is a classical problem in statistics and has also found new applications in machine learning. In the experimental design problem, the aim is to estimate an unknown vector x in m-dimensions from linear measurements where a Gaussian noise is introduced in each measurement. The goal is to pick k out of the given … Read more


text:
# Question #ea570

Jan 15, 2018

Suppose, you had a rifle of mass $M$ and bullet of $m$ in it,now after firing the bullet goes away from the rifle with a velocity of $u$,if you consider that the velocity of rifle due to this process will be $v$ then we can use the law of conservation of momentum as no external force is acting on the system.

so,we can write,
$M \cdot 0 + m \cdot 0 = M \cdot v + m \cdot u$

or, $v = - \left(\frac{m}{M}\right) \cdot u$

negative sign means the rifle will move in the opposite direction of the bullet, and as $m$ is << than $M$ so from this equation we can say, $v$ will also be << than $u$ i.e the velocity of rifle will be very small than that of bullet.

so,that's how their mass affected their velocities.

But as the bullet will go through air before hitting something it's retardation will occur based on air resistance.

Whereas the rifle will move very little and almost no acceleration because it's so heavy.


text:
• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Constructing the "analytical" formula for tetration. mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 01/28/2011, 09:12 PM (This post was last modified: 01/29/2011, 12:25 AM by mike3.) Well, here goes... this is LONG! I'm going to also give an explanation of the derivation, since you were asking for more "insight" into the problem. OK, so you have the recurrence I gave. $a_1 = r_{1, 1}$ $a_n = \sum_{m=1}^{n-1} r_{n, m} a_m$. If we expand a few terms, we get $a_2 = r_{2, 1} r_{1, 1}$ $a_3 = r_{3, 1} r_{1, 1} + r_{3, 2} r_{2, 1} r_{1, 1}$ $a_4 = r_{4, 1} r_{1, 1} + r_{4, 2} r_{2, 1} r_{1, 1} + r_{4, 3} r_{3, 1} r_{1, 1} + r_{4, 3} r_{3, 2} r_{2, 1} r_{1, 1}$ $a_5 = r_{5, 1} r_{1, 1} + r_{5, 2} r_{2, 1} r_{1, 1} + r_{5, 3} r_{3, 1} r_{1, 1} + r_{5, 3} r_{3, 2} r_{2, 1} r_{1, 1} + r_{5, 4} r_{4, 1} r_{1, 1} + r_{5, 4} r_{4, 2} r_{2, 1} r_{1, 1} + r_{5, 4} r_{4, 3} r_{3, 1} r_{1, 1} + r_{5, 4} r_{4, 3} r_{3, 2} r_{2, 1} r_{1, 1}$ ... Looking at this, we see that the number of products in each $a_n$ is 1, 1, 2, 4, 8, ... , or $2^{n-2}$ for $n > 1$. It is possible to prove this by considering how the above process works: each sum adds together the previous terms, which are themselves sums, thus the number of products in that sum should equal the sum of the numbers of products in the previous terms. That is, $PN_1 = 1$ $PN_n = \sum_{m=1}^{n-1} PN_m$ Taking the forward difference $\Delta$ of that last equation gives $\Delta PN_n = \sum_{m=1}^{n} PN_m - \sum_{m=1}^{n-1} PN_m = PN_n$. This kind of difference equation has as solutions $PN_n = r 2^n + s$. We see that for the given initial conditions, i.e. $PN_2 = 1$ and $PN_3 = 2$, $PN_n = 2^{n-2}$ for $n > 1$. Now, we move on to consider the number of factors in each product. Counting them, we see that n = 1: 1 n = 2: 2 n = 3: 2, 3 n = 4: 2, 3, 3, 4 n = 5: 2, 3, 3, 4, 3, 4, 4, 5 ... Looking at the formula, we see the nth term should be the concatenation (due to the sum) of the previous terms with their elements each incremented by 1 (due to the multiplication of a new $r_{n, m}$ factor.). That is, $PFN_n = \mathrm{incr}(PFN_1) + \mathrm{incr}(PFN_2) + ... + \mathrm{incr}(PFN_{n-1}) = \sum_{m=1}^{n-1} \mathrm{incr}(PFN_m)$. (where the "+" are actually string concatenations. Formally, $PFN_n \in \mathbb{N}*$.) We can "refactor" this process by noting that the "increment" "distributes" over the concatenation, so $PFN_n = \sum_{m=1}^{n-1} \mathrm{incr}(PFN_m) = \mathrm{incr}\left(\sum_{m=1}^{n-1} PFN_m\right)$. Now, we know that the previous term is also an "incremented concatenation" of the terms before it, thus if we decremented that term and then concatenated itself to that, we would have the concatenation of all terms before $PFN_n$, and thus $PFN_n = \mathrm{incr}(\mathrm{decr}(PFN_{n-1}) + PFN_{n-1}) = PFN_{n-1} + \mathrm{incr}(PFN_{n-1})$ i.e. the sequence can be formed by taking the previous term, incrementing all its elements, and then concatenating it to the unincremented version, which is also easily observable from the pattern, but this gives a kind of "proof". Anyway, by looking at the dictionary of integer sequences at oeis.org, I found that these strings appear within the sequence of "binary weights of n", the number of 1 bits in an integer. Looking at it, it appears element j (first is j = 1) of $PFN_n$ is given by the number of 1 bits in $2^{n-1} + 2j - 1$. This can be written explicitly. Note that $\lfloor\frac{N}{2^k}\rfloor$ is a "lossy right shift" of $N$ (i.e. shift right k bits, knock off everything past the binary point, like the ">>" operation in the C programming language.). Then, we can get the kth bit (0th is the LSB) of $N$ as $\lfloor\frac{N}{2^k}\rfloor - 2\lfloor\frac{N}{2^{k+1}}\rfloor$. The second term is a lossy right shift by k+1 bits followed by a left shift by 1 bit, which is like the right shift by k bits but with its LSB cleared. So we can add up all the bits in $N$ (thus giving the number of 1 bits) by $\sum_{i=0}^{\lfloor\log_2(N)\rfloor} \lfloor\frac{N}{2^i}\rfloor - 2\lfloor\frac{N}{2^{i+1}}\rfloor$. Taking $N = 2^{n-1} + 2j - 1$ and noting that the first and last bit are always 1 (by virtue of $2^{n-1}$ and note also that this is always odd as $2^{n-1}$ is even and $2j - 1$ is odd, and even + odd = odd), we get ${(PFN_n)}_j = 2 + \sum_{i=0}^{\lfloor\log_2(j)\rfloor} \lfloor\frac{j}{2^i}\rfloor - 2\lfloor\frac{j}{2^{i+1}}\rfloor$. These connections to binary made me wonder about whether or not this whole thing could be interpreted in terms of binary. I noticed that the $r_{n,m}$ are present at most only once in a product, i.e. there are no higher-degree $r_{n,m}$ factors. And the indices, for the nth term of our original sequence, range from 1 to n. So I thought that each product could be represented as a binary nxn matrix with 1-bit elements, and we could look for patterns there. This led me to see part of, but not all of, the pattern. However, the most fruitful way of viewing it, I found, was to arrange the terms and factors like this: Code:r11 r21 r11 r31 r32     r21 r11 r11 r41 r42 r43 r43         r31 r32     r21     r21 r11 r11 r11 r11 r51 r52 r53 r53 r54 r54 r54 r54                 r41 r42 r43 r43         r31 r32         r31 r32     r21     r21     r21     r21 r11 r11 r11 r11 r11 r11 r11 r11 where the value of each term can be found by multiplying the columns' elements together and then adding up the results. We can now see an interesting alternating pattern... which looks like binary counting! We see that the factors appear and disappear as the bits of $2^{n-1} + 2j - 1$, which also explains the "number of 1 bits of $2^{n-1} + 2j - 1$" observation. The left indices of the $r_{n, m}$ in a product, then, are generated by acquiring the position of each one bit in $2^{n-1} + 2j - 1$, with the LSB counted as position 1, and putting them in numerical order for convenience. That is, $\mathrm{Left\ index\ of\ }k\mathrm{th\ factor\ of\ jth\ product\ in\ }n\mathrm{th\ term} = \mathrm{position\ of\ }k\mathrm{th\ 1\ bit\ in\ }2^{n-1} + 2j - 1\mathrm{,\ with\ the\ LSB\ counted\ as\ position\ 1}$. Now for the right indices. If we look at them, they look like (here, for the 5th term) Code:11 211 311 3211 411 4211 4311 43211 The same "bit index" interpretation works here as well. Ignoring the last "1", these terms give the positions of 1 bits in $2j - 1$. So, we have $\mathrm{Right\ index\ of\ }k\mathrm{th\ factor\ of\ jth\ product\ in\ }n\mathrm{th\ term} = \mathrm{if\ }k-1 > 0\mathrm{,\ position\ of\ }k-1\mathrm{th\ 1\ bit\ in\ }2j - 1\mathrm{,\ with\ the\ LSB\ counted\ as\ position\ 1,\ otherwise\ 1}$. Then, the not-quite-symbolic but still "explicit" and "non-recursive" formula is $a_n = \sum_{j=1}^{2^{n-2}} \prod_{k=1}^{(\mathrm{number\ of\ 1\ bits\ in\ }2^{n-1} + 2j - 1)} r_{(\mathrm{position\ of\ }k\mathrm{th\ 1\ bit\ in\ }2^{n-1} + 2j - 1\mathrm{,\ with\ the\ LSB\ counted\ as\ position\ 1}),(\mathrm{if\ }k-1 > 0\mathrm{,\ position\ of\ }k-1\mathrm{th\ 1\ bit\ in\ }2j - 1\mathrm{,\ with\ the\ LSB\ counted\ as\ position\ 1,\ otherwise\ 1})}$. « Next Oldest | Next Newest »

 Messages In This Thread Constructing the "analytical" formula for tetration. - by mike3 - 01/17/2011, 01:05 PM RE: Constructing the "analytical" formula for tetration. - by Gottfried - 01/17/2011, 10:10 PM RE: Constructing the "analytical" formula for tetration. - by mike3 - 01/22/2011, 04:00 AM RE: Constructing the "analytical" formula for tetration. - by Gottfried - 01/24/2011, 08:56 AM RE: Constructing the "analytical" formula for tetration. - by mike3 - 01/28/2011, 03:12 AM RE: Constructing the "analytical" formula for tetration. - by Gottfried - 01/28/2011, 02:49 PM RE: Constructing the "analytical" formula for tetration. - by mike3 - 01/28/2011, 09:12 PM RE: Constructing the "analytical" formula for tetration. - by Gottfried - 01/28/2011, 10:42 PM RE: Constructing the "analytical" formula for tetration. - by mike3 - 01/29/2011, 12:10 AM RE: Constructing the "analytical" formula for tetration. - by mike3 - 02/10/2011, 04:20 AM RE: Constructing the "analytical" formula for tetration. - by sheldonison - 02/10/2011, 05:59 AM RE: Constructing the "analytical" formula for tetration. - by mike3 - 02/10/2011, 07:35 AM RE: Constructing the "analytical" formula for tetration. - by tommy1729 - 01/23/2011, 10:59 PM RE: Constructing the "analytical" formula for tetration. - by mike3 - 01/24/2011, 04:34 AM

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