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# Item 7: Prefer a nullable or Result result type when the lack of a result is possible

This is a chapter from the book Effective Kotlin. You can find it on LeanPub or Amazon.

Sometimes a function cannot produce its desired result. A few common examples are:

• We try to get data from a server, but there is a problem with the internet connection.
• We try to get the first element that matches some criteria, but there is no such element in our list.
• We try to parse an object from text, but this text is malformatted.

There are two main mechanisms to handle such situations:

• Return a null or Result.failure, thus indicating failure.
• Throw an exception.

There is an important difference between these two. Exceptions should not be used as a standard way to pass information. All exceptions indicate incorrect, special situations and should be treated this way. We should use exceptions only for exceptional conditions (Effective Java by Joshua Bloch). The main reasons for this are:

• The way exceptions propagate is not very readable for most programmers and might easily be missed in code.
• In Kotlin, all exceptions are unchecked. Users are not forced or even encouraged to handle them. They are often not well documented, and they are not very visible when we use an API.
• Because exceptions are designed for exceptional circumstances, there is little incentive for JVM implementers to make them as fast as explicit tests.
• Placing code inside a try-catch block inhibits certain optimizations that the compiler might otherwise perform.

On the other hand, null or Result.failure are both perfect for indicating an expected error. They are explicit, efficient, and can be handled in idiomatic ways. This is why the rule is that we should prefer to return null or Result.failure when an error is expected, and we should throw an exception when an error is not expected. Here are some examples:

inline fun <reified T> String.readObjectOrNull(): T? { //... if (incorrectSign) { return null } //... return result } inline fun <reified T> String.readObject(): Result<T> { //... if (incorrectSign) { return Result.failure(JsonParsingException()) } //... return Result.success(result) } class JsonParsingException : Exception()

Errors indicated like this are easier to handle and harder to miss. When we choose to use null, clients handling such a value can choose from a variety of null-safety-supporting features like a safe-call or the Elvis operator:

val age = userText.readObjectOrNull<Person>()?.age ?: -1

When we choose to return Result, the user of this function will be able to handle it using methods from the Result class:

userText.readObject<Person>() .onSuccess { showPersonAge(it) } .onFailure { showError(it) }

Using such error handling is not only more efficient than a try-catch block but is often also easier to use and more explicit. An exception can be missed and can stop our whole application; in contrast, a null value or a Result object needs to be explicitly handled, and it won’t interrupt the flow of the application.

The difference between a nullable result and the Result object is that we should prefer the latter when we need to pass additional information in the case of failure; otherwise, we should prefer null.

The Result class has a rich API of methods you can use to handle your result, including:

• isSuccess and isFailure properties, which we use to check if the result represents a success or a failure (isSuccess == !isFailure is always true).
• onSuccess and onFailure methods, which call their lambda expressions when the result is, respectively, a success or a failure.
• getOrNull method, which returns the value if the result is a success, or null otherwise.
• getOrThrow method, which returns the value if the result is a success, or throws the exception from the failure otherwise.
• getOrDefault method, which returns the value if the result is a success, or the default value provided as an argument if the result is a failure.
• getOrElse method, which returns the value if the result is a success, or calls its functional argument and returns its result.
• exceptionOrNull method, which returns the exception if the result is a failure, or null otherwise.
• map method for transforming the success value.
• recover method for transforming a throwable value into a success value.
• fold method for handling both success and failure in a single method.

To transform a function that throws exceptions into one that returns Result, use runCatching instead.

fun getA(): Result<T> = runCatching { getAThrowing() }

In important parts of an API, you might define two variants of functions: one that expects that failure can occur, and one that treats it as an unexpected situation. A good example is the fact that List has both:

• get, which is used when we expect an element to be at the given position; if it is not there, the function throws IndexOutOfBoundsException.
• getOrNull, which is used when we suppose that we might ask for an element that is out of range; if that happens, we’ll get null.

List also supports other options, like getOrDefault, which is useful in some cases but in general might be easily replaced with getOrNull and the Elvis operator ?:.

This is good practice because if developers know that an element must be at the specific position, they should not be forced to handle a nullable value; if they have any doubts, they should use getOrNull and properly handle the lack of a value.


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If to view examples, such short algorithm is correct for squared matrices 3x3 … Show Instructions. Adjoint of a Square Matrix Problems with Solutions 1. Here, M ij is the minor of a ij th element of the given matrix. Tags: adjoint matrix cofactor cofactor expansion determinant of a matrix how to find inverse matrix inverse matrix invertible matrix linear algebra minor matrix Next story Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$ The Adjoint of 3x3 Matrix block computes the adjoint matrix for the input matrix. The adjoint is the conjugate transpose of a matrix while the classical adjoint is another name for the adjugate matrix or cofactor transpose of a matrix. De inverse van een 3x3 matrix bepalen. In Part 2 we complete the process by finding the determinant of the matrix and its adjugate matrix. expand all. Finding inverse of matrix using adjoint Let’s learn how to find inverse of matrix using adjoint But first, let us define adjoint. Finding inverse of matrix using adjoint Let’s learn how to find inverse of matrix using adjoint But first, let us define adjoint. Learn more at BYJU'S. The adjugate has sometimes been Ports. Adjoint matrix is also referred as Adjunct matrix or Adjugate or classical adjoint matrix. Find the adjoint of the matrix: Solution: We will first evaluate the cofactor of every element, Therefore, (source: cliffnotes) The Relation between Adjoint and Inverse of a Matrix. ADJOINT OF A 3X3 MATRIX PDF - In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix. By using this website, you agree to our Cookie Policy. An adjoint matrix is also called an adjugate matrix. Adjoint (or Adjugate) of a matrix is the matrix obtained by taking transpose of the cofactor matrix of a given square matrix is called its Adjoint or Adjugate matrix. The calculator will find the adjoint (adjugate, adjunct) matrix of the given square matrix, with steps shown. Port_1 — Input matrix 3-by-3 matrix. You have one more cofactor to find. Also, the relation between inverse and adjoint are given along with their Physics. The Adjoint of 3x3 Matrix block computes the adjoint matrix for the input matrix. Ports. 3x3 identity matrices involves 3 rows and 3 columns. The Adjoint of 3x3 Matrix block computes the adjoint matrix for the input matrix. Video transcript . This is how we calculate adjoint of a 2x2 matrix. It is denoted by adj A . Adjoint And Inverse Of A Matrix: In this article, you will know how to find the adjoint of a matrix and its inverse along with solved example questions. For example, if a problem requires you to divide by a fraction, you can more easily multiply by its reciprocal. Your IP: 192.163.198.110 The matrix formed by taking the transpose of the cofactor matrix of a given original matrix. Port_1 — Output acceleration 3-by-3 matrix. Classical Adjoint The matrix formed by taking the transpose of the cofactor matrix of a given original matrix. Maths. Solution: Inverse of a Matrix A non-singular square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I n =BA . We'll assume you're ok with this, but you can opt-out if you wish. Port_1 — Output acceleration 3-by-3 matrix. Continue on with the rest of the matrix in kf fashion. Warnings Not all 3×3 matrices have inverses. $$2x+4y-10z=-2$$ $$3x+9y-21z=0$$ ... Stack Exchange Network Stack Exchange network consists of 176 … Press Enter, and the inverse matrix should appear on your screen. The Adjoint of 3×3 Matrix block computes the adjoint matrix for the input matrix. Elements of the matrix are the numbers which make up the matrix. ctranspose | det | inv | rank. See Also. Find more Mathematics widgets in Wolfram|Alpha. Data Types: double. The Adjoint of 3x3 Matrix block computes the adjoint matrix for the input matrix. Input. We chose element a 12, which is - on the sign chart. Not Helpful 3 Helpful Select a calculator with matrix capabilities. Biology. The adjoint of a matrix A is the transpose of the cofactor matrix of A . Input matrix, specified as a 3-by-3 matrix, in initial acceleration units. The name has changed to avoid ambiguity with a different defintition of the term adjoint. Free Matrix Adjoint calculator - find Matrix Adjoint step-by-step This website uses cookies to ensure you get the best experience. We strongly recommend you to refer below as a prerequisite of this. Find the adjoint of the matrix: Solution: We will first evaluate the cofactor of every element, Therefore, (source: cliffnotes) The Relation between Adjoint and Inverse of a Matrix To find the inverse of a matrix A, i.e A-1 we shall first Determinant of a Matrix. Let A be an n x n matrix… The adjugate has sometimes been called the . But opting out of some of these cookies may have an effect on your browsing experience. This website uses cookies to improve your experience. In the below Inverse Matrix calculator, enter the values for Matrix (A) and click calculate and calculator will provide you the Adjoint (adj A), Determinant (|A|) and Inverse of a 3x3 Matrix. For related equations, see Algorithms. Port_1 — Input matrix 3-by-3 matrix. Hermitian adjoint Adjugate matrix References a b a b a b c External links "Adjoint matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994] This page was last edited … It is mandatory to procure user consent prior to running these cookies on your website. The adjugate of matrix A is often written adj A. How to Find the Determinant of a 3X3 Matrix. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). A singular matrix is the one in which the determinant is not equal to zero. Matrices, when multiplied by its inverse will give a resultant identity matrix. This category only includes cookies that ensures basic functionalities and security features of the website. Check the determinant of the matrix. Are there any shortcuts for finding the inverse of a 3x3 matrix? We can calculate the Inverse of a Matrix by: Step 1: calculating the Matrix of Minors, Step Note: In the past, the term for adjugate used to be adjoint. Necessary cookies are absolutely essential for the website to function properly. Your calculator probably has a function that will automatically convert the decimals to fractions. Get the free "Adjoint of a 3x3 matrix" widget for your website, blog, Wordpress, Blogger, or iGoogle. The Adjoint of 3x3 Matrix block computes the adjoint matrix for the input matrix. By using this website, you agree to … It is estimated that the rate of change of production P with respect to the additional number of workers x is given by .If the firm employs 25 more workers, then the new level of production What is Adjoint? You clicked a link that corresponds to this MATLAB command: Run the command … We can calculate the Inverse of a Matrix by: Step 1: calculating the Matrix of Minors, Step 2: then turn that into the Matrix of Cofactors, Step 3: then the Adjugate, and; Step 4: multiply that by 1/Determinant. This is the currently selected item. The classical adjoint, or adjugate, of a square matrix A is the square matrix X, such that the (i,j)-th entry of X is the (j,i)-th cofactor of A. Then the IJ entry of adj r A is. Adjoint Matrix Calculator You can calculate the adjoint matrix, by taking the transpose of the calculated cofactor matrix. For every m×m square matrix there exist an inverse of it. Input. Select a Web Site Choose a web site to get translated content where available and see local events and offers. Simple 4-function calculators will not be able to help you directly find the inverse. Inverse of a Matrix using Minors, Cofactors and Adjugate (Note: also check out Matrix Inverse by Row Operations and the Matrix Calculator.) Matrices, when multiplied by its inverse will give a resultant identity matrix. Classical Adjoint. It’s the conjugate transpose. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The Adjoint of 3x3 Matrix block computes the adjoint matrix for the input matrix. Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are as essential for the working of basic functionalities of the website. Learn more Accept. Adjoint Matrix Calculator. It is estimated that the rate of change of production P with respect to the additional number of workers x is given by . For example one of the property is adj(AB)=adj(B).adj(A). expand all. At present, a firm is manufacturing 2000 items. Learn how to find the inverse of a matrix using different methods and formulas for 2x2 and 3x3 matrices with the solved examples. For related equations, see Algorithms. You could calculate the inverse matrix follow the steps below: Where a,b,c,d are numbers, The inverse is Description. A matrix whose determinant is 0 is said to be singular ; therefore, adoint matrix is invertible if and only if it is nonsingular. In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix. Adam Bede has been added to your Reading List! Calculate i for the third term in your reference row or column. We must change the sign of our answer: (-1)*(-120) = 120. In other words, adjoiint need to change the sign of alternate cells, like this:. The Adjoint of 3x3 Matrix block computes the adjoint matrix for the input matrix. expand all. Réduire la partie gauche de la matrice en forme échelon en appliquant les opérations élémentaires de lignes sur la matrice complète (incluant la partie droite). In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of the cofactor matrix. Inverting a 3x3 matrix using determinants Part 1: Matrix of minors and cofactor matrix. Next lesson. Collecting these determinants for the different possible i yields an equality of column vectors. You can also find the This is sometimes referred to as the adjoint matrix. This website uses cookies to ensure you get the best experience. In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix. Calculating the inverse of a 3×3 matrix by hand is a tedious job, but worth reviewing. NCERT DC Pandey Sunil Batra HC Verma Pradeep Errorless. NCERT NCERT Exemplar NCERT Fingertips Errorless Vol-1 Errorless Vol-2. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Data Types: double. You need to calculate the determinant of the matrix as an initial step. 8. Solve the following system using the adjoint matrix. To find the inverse of a 3×3 matrix, first calculate the determinant of the matrix. In the below Inverse Matrix … If the determinant is 0, then your work is finished, because the matrix has no inverse. You clicked a link that corresponds to this MATLAB command: Run the command … Dover Books on Mathematics. Adjoint Matrix | Self Adjoint Matrix | [email protected] Create a 3 x 3 matrix whose determinant is 1 and whose elements are all integers. Multiplying this system on the left by adj A and dividing by the determinant yields. For related equations, see Algorithms. expand all. Output . A 3 x 3 matrix has 3 rows and 3 columns. Calculating the inverse of a 3×3 matrix by hand is a tedious job, but worth reviewing. The adjointt screen will show a matrix. This video explains how to get ADJOINT of 3x3 matrix with examples. Your objective is to create the identity matrix on the left side of this newly augmented matrix. To find the inverse of a matrix A, i.e A-1 we shall first define the adjoint of a matrix. It is also occasionally known as adjunct matrix, though this nomenclature appears to have decreased in usage. MS Muhammad Saddique Jul 13, If A is invertible, this implies that adj A also commutes with B. Solution: 2. Inverse of 2x2 Matrix Formula. Cloudflare Ray ID: 5fb979d44e5a04ff please watch and enjoy For related equations, see Algorithms. Calculating the inverse of a 3×3 matrix by hand is a tedious job, but worth reviewing. 3x3 identity matrices involves 3 rows and 3 columns. The transpose of the matrix whose i, j entry is the a ij cofactor is mafrix the classical adjoint of A:. Get the free "Adjoint of a 3x3 matrix" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha. Port_1 — Output acceleration 3-by-3 matrix. 2018, zuletzt modifiziert: 18. Inverse of a matrix A is the reverse of it, represented as A-1. expand all. Given a square matrix, find adjoint and inverse of the matrix. Comme résultat vous aurez une inverse calculée à droite. But if the user enters 2 instead of 1 in ch, which means he wants to find the adjoint of a 3x3 matrix, then calculations will be like this. Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. adjoint of a 2x2 matrix, In linear algebra, When two matrix AB =BA = I n, B is the inverse matrix of A. To find the Inverse of a 3 by 3 Matrix is a little critical job but can be evaluated by following few steps. For related equations, see Algorithms. Conjugate transpose - Wikipedia The jth row becomes the jth column: Matrix Transpose Calculator - Symbolab Then each matrix element is converted to its complex conjugate (i → … Example: Find the adjoint of the matrix. Not Helpful 3 Helpful 3. Finding determinant of a 2x2 matrix Evalute determinant of a 3x3 matrix Area of triangle Equation of line using determinant Finding Minors and cofactors Evaluating determinant using minor and co-factor Find adjoint of a Now “Transpose” all elements of the previous matrix Did this summary help you?
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Measures DC and AC performance metrics of ADC output

• Library:
• Mixed-Signal Blockset / ADC / Measurements & Testbenches

## Description

The ADC Testbench block measures both DC and AC performance metrics. DC performance metrics include offset error and gain error. AC performance metrics include signal to noise ratio (SNR), signal to noise and distortion radio (SINAD), spurious free dynamic range (SFDR), effective number of bits (ENOB), noise floor, and conversion delay.

The ADC Testbench block generates the stimulus to drive the device under test (DUT) from the Stimulus tab. The setup parameters for validating the DUT are defined in the Setup tab and the target validation metrics are defined in the Target Metric tab.

You can use the ADC Testbench block to validate the ADC architectural models provided in Mixed-Signal Blockset™, or you can validate an ADC of your own implementation.

## Ports

### Input

expand all

Digital input signal from the ADC output, specified as a scalar.

Data Types: `fixed point` | `single` | `double` | `int8` | `int16` | `int32` | `uint8` | `uint16` | `uint32` | `Boolean`

Conversion ready signal from the ADC output, specified as a scalar. This signal indicates when the ADC conversion process is complete.

Data Types: `double`

### Output

expand all

Analog output stimulus signal for the ADC input, returned as a scalar.

Data Types: `double`

External start conversion clock for ADC, returned as a scalar. This signal starts the conversion process in ADC block.

Data Types: `double`

## Parameters

expand all

Select whether to measure static (DC) or dynamic (AC) performance metrics.

• Select `DC` to measure offset error and gain error.

• Select `AC` to measure SNR, SINAD, SFDR, ENOB, noise floor, and conversion delay.

Minimum time for which the simulation must run to obtain meaningful results, specified as a positive real scalar in seconds.

• To measure DC performance, the simulation must run so that ADC can sample each digital code 20 times, assuming a ramp input that traverses the full scale range of the ADC over the period of simulation. Based on this assumption, the analog input frequency (fanalog), generated by the ADC Testbench block for the sawtooth waveform is set as:

`${f}_{\text{analog}}=\frac{StartFreq}{{2}^{\left(Nbits+1\right)}·10}$`

where StartFreq is the frequency of the conversion start clock and Nbits is the resolution of the ADC.

So, the Recommended min. simulation stop time (s) T is calculated by using the formula: $T=\frac{1}{{f}_{\text{analog}}}$.

• To measure AC performance, the simulation must run so that the ADC can generate 6 spectral updates of the ADC output. The time to generate one spectral output based on Welch's method [1] is given by:

`$t=\frac{1.5·SamplingFrequency}{RBW}$`

where SamplingFrequency and RBW are the sampling frequency and resolution bandwidth of the spectrum estimator inside the ADC Testbench block.

This parameter is only reported by the testbench and is not editable.

Data Types: `double`

Click to automatically set the Recommended min. simulation stop time (s) as the stop time of the Simulink® model.

Measure the differential nonlinearity (DNL) error and integral nonlinearity (INL) error using the endpoint method. This method uses the end points of the actual transfer function to measure the DNL and INL error.

Measure the differential nonlinearity (DNL) error and integral nonlinearity (INL) error using the best fit method. This method uses a standard curve fitting technique to find the best fit to measure the DNL and INL error.

Click to plot DC analysis result for further analysis. To perform a complete DC analysis including integral nonlinearity (INL) and differential nonlinearity (DNL), use the ADC DC Measurement block.

#### Dependencies

This parameter is only available when Measurement option is set to `DC`.

Click to store detailed test results to a spreadsheet (XLS file) or as comma-separated values (CSV file) for further processing.

### Stimulus

Type of distortion the ADC Testbench block is set to measure, specified as `Harmonic` or `Intermodulation`.

#### Dependencies

To enable this parameter, set Measurement option as `AC`.

#### Programmatic Use

 Block parameter: `DistortionMeasurement` Type: character vector Values: `Harmonic` | `Intermodulation` Default: `Harmonic`

Frequency of the analog input signal to an ADC block, specified as a positive real scalar in hertz. Analog stimulus frequency must match the input frequency to the ADC device under test. By default, this parameter is calculated automatically. You can deselect Set automatically to customize the value.

• When the ADC Testbench is set to measure the `Harmonic` distortion, the default value of Analog stimulus frequency is `976.563`.

• When the ADC Testbench is set to measure the `Intermodulation` distortion, the default value of Analog stimulus frequency is `[878.907,1074.22]`.

Analog stimulus frequency needs to satisfy two requirements:

• All the output codes of the ADC must be activated.

• The Analog stimulus frequency must not share any common multiples other than 1 with the Start conversion frequency.

To satisfy both the conditions, use the equation ${f}_{\text{analog}}=\frac{J}{M}{f}_{\text{start}}$ [2],

where:

fanalog is the analog signal frequency,

fstart is the start conversion frequency,

$M>{2}^{Nbits}·\pi$, where Nbits is the number of bits of the ADC,

and J is an integer with no common factors with M.

#### Dependencies

To enable this parameter, set Measurement option as `AC`.

#### Programmatic Use

 Block parameter: `InputFrequency` Type: character vector Values: positive real scalar | positive real valued vector Default: `976.563`

Resolution bandwidth, specified as a positive real scalar in hertz. This parameter defines the smallest positive frequency that can be resolved. By default, this parameter is calculated automatically. You can deselect Set automatically to customize the value.

• When the ADC AC Measurement is set to measure the `Harmonic` distortion, the default value of Resolution bandwidth (Hz) is `97.6563`.

• When the ADC AC Measurement is set to measure the `Intermodulation` distortion, the default value of Resolution bandwidth (Hz) is `87.8907`.

#### Dependencies

To enable this parameter, set Measurement option as `AC`.

#### Programmatic Use

 Block parameter: `RBW` Type: character vector Values: positive real scalar Default: `97.6563`

Frequency of the start conversion clock of the ADC, specified as a positive real scalar in Hz. Start conversion frequency (Hz) must match the frequency of the start conversion clock of the ADC block.

#### Programmatic Use

 Block parameter: `StartFreq` Type: character vector Values: positive real scalar Default: `1e6`

Data Types: `double`

RMS aperture jitter to be added by the start conversion clock, specified as a positive real scalar in seconds.

#### Programmatic Use

 Block parameter: `RMSJitt` Type: character vector Values: positive real scalar Default: `40e-12`

Data Types: `double`

Maximum allowed difference in the amplitude of the successive samples of the analog input signal, specified as positive real scalar in least significant bit (LSB).

#### Dependencies

To enable this parameter, set Measurement option as `DC`.

Data Types: `double`

### Setup

Click to automatically propagate setup parameters from the ADC.

#### Dependencies

This parameter only works when the ADC is a Flash ADC or a SAR ADC from the Mixed-Signal Blockset.

Number of physical output bits, specified as a unitless positive real integer in the range `[1, 26]`. Number of bits determines the resolution of the ADC.

#### Programmatic Use

• Use `get_param(gcb,'NBits')` to view the current Number of bits.

• Use `set_param(gcb,'NBits',value)` to set Number of bits to a specific value.

Data Types: `double`

ADC dynamic range, specified as a 2-element row vector in volts.

#### Programmatic Use

• Use `get_param(gcb,'InputRange')` to view the current Input range (V).

• Use `set_param(gcb,'InputRange',value)` to set Input range (V) to a specific value.

Data Types: `double`

Delays measurement analysis to avoid corruption by transients, specified as a nonnegative real scalar in seconds.

#### Programmatic Use

 Block parameter: `HoldOffTime` Type: character vector Values: nonnegative real scalar Default: `0`

Data Types: `double`

Displays spectrum analyzer during simulation. By default, this option is deselected.

#### Dependencies

This parameter is only available when Measurement option is set to `AC`.

Select to enable increased buffer size during simulation. By default, this option is deselected.

Number of samples of the input buffering available during simulation, specified as a positive integer scalar.

Selecting different simulation solver or sampling strategies can change the number of input samples needed to produce an accurate output sample. Set the Buffer size to a large enough value that the input buffer contains all the input samples required.

#### Dependencies

This parameter is only available when Enable increased buffer size option is selected in the Configuration tab.

#### Programmatic Use

 Block parameter: `NBuffer` Type: character vector Values: positive integer scalar Default: `5`

Data Types: `double`

### Target Metric

Click to automatically propagate target metrics from ADC.

#### Dependencies

• To enable this parameter, set Measurement option to `DC`.

• This parameter only works when the ADC is a Flash ADC or a SAR ADC from the Mixed-Signal Blockset.

Shifts quantization steps by specific value, specified as a positive real scalar in %FS, FS, or LSB.

#### Dependencies

To enable this parameter, set Measurement option to `DC`.

#### Programmatic Use

 Block parameter: `TargetOffsetError` Type: character vector Values: real scalar Default: `1.5 LSB`

Data Types: `double`

Error on the slope of the straight line interpolating ADC transfer curve, specified as a positive real scalar in least significant bit %FS, FS, or LSB.

#### Dependencies

To enable this parameter, set Measurement option to `DC`.

#### Programmatic Use

 Block parameter: `TargetGainError` Type: character vector Values: real scalar Default: `1 LSB`

Data Types: `double`

## References

[2] IEEE Std 1241-2010. "IEEE Standard for Terminology and Test Methods for Analog-to-Digital Converters," pp. 29-30, 14 January 2011.


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# Two questions on Munkres -Topology: praoduct and subspace of separable spaces

I have two questions:

• If $X$ is a countable product of spaces having countable dense subsets then does $X$ have a countable dense subset?

Let $X$ $=\prod_{i=1}^\infty X_i$. Let $D_i$ denote the countable dense subset of the space $X_i$. Then $\scr D$ $=\prod_{i=1}^\infty D_i$ is a countable set as Cartesian product of countable sets is countable. Secondly $\overline {\prod _{i=1}^\infty D_i}=\prod _{i=1}^\infty \overline {D_i}=X$. Hence $\scr D$ is the required set. Is this correct?

• Is a subspace of a separable topological space separable?

$\Bbb R$ has $\Bbb Q$ as countable dense set, but $\Bbb Q^c$ does not have a countable dense subset. How should I prove this fact? Am I right?

Both of your arguments are wrong, I’m afraid. The Cartesian product of countably infinitely many countably infinite sets is uncountable, so $\prod_kD_k$ is not in general a countable subset of $X$, though it is a dense subset. It is nonetheless true that $X$ is separable. For each $k\in\Bbb N$ fix a point $p_k\in D_k$, and let

$$D=\left\{\langle x_k:k\in\Bbb N\rangle\in\prod_kD_k:\exists m\in\Bbb N\,\forall k\ge m(x_k=p_k)\right\}\;;$$

now show that this set $D$ is both countable and dense in $X$.

For the second question, $\Bbb R$ has a countable base, so every subspace of $\Bbb R$ has a countable base and therefore a countable dense subset. In particular, $\Bbb R\setminus\Bbb Q$ is separable. (A specific example of a countable dense subset of the irrationals is $\{q+\sqrt2:q\in\Bbb Q\}$.

It is true, however, that a subset of a separable space need not be separable. In fact, you can build an example by changing the topology of $\Bbb R$. Let each $x\in\Bbb Q$ be isolated. For $x\in\Bbb R\setminus\Bbb Q$ and $n\in\Bbb Z^+$ let $B_n(x)=\left(x-\frac1n,x+\frac1n\right)$, and let $\{B_n(x):n\in\Bbb Z^+\}$ be a local base at $x$.

• Verify that this yields a topology on $\Bbb R$.
• Show that $\Bbb R$ is separable in this topology.
• Show that $\Bbb R\setminus\Bbb Q$ is not separable in this topology.
• Sir two questions 1. I am really not getting how $D$ has been constructed or how does it's elements look like ;Is it possible to elaborate a bit 2.For specifying a topology on a set we need to give the collection of open sets but you did not do that .Can you please do that ?I am not getting the concept of local base here. – Learnmore Feb 6 '16 at 16:55
• @learnmore: Let $p=\langle p_k:k\in\Bbb N\rangle$; this is a point in $\prod_kD_k$. $D$ consists of all of the points in $\prod_kD_k$ that differ from $p$ on at most finitely many coordinates. If each $X_k$ were $\Bbb R$, for instance, each $D_k$ were $\Bbb Q$, and each $p_k=0$, $D$ would be the set of rational sequences with only finitely many non-zero coordinates. \\ A local base at a point is also called a neighborhood base at the point; you may know it under this name. It’s a fundamental concept that I’d expect to find in any topology text. At any rate, what I’ve defined is a ... – Brian M. Scott Feb 6 '16 at 17:32
• ... genuine system of local bases, and it generates a topology $\tau$ in the following way: a set $U\subseteq\Bbb R$ is in $\tau$ if and only if for each $x\in U$ there is a $B\in\mathscr{B}_x$ such that $B\subseteq U$, where $\mathscr{B}_x=\{B_n(x):n\in\Bbb Z^+\}$ if $x$ is irrational, and $\mathscr{B}_x=\big\{\{x\}\big\}$ if $x$ is rational. – Brian M. Scott Feb 6 '16 at 17:34
• My attempt: As constructed $D$ becomes the finite product of countably many dense sets and hence countable. Take any basic open set may be $\scr U=$ $U_1\times U_2\times... \times U_n\times X\times ...X$ Now as each $U_i$ intersects $D_i$ we have $x_i\in U_i\cap D_i$ for all $1\le i\le n$ then $(x_1,x_2,...,x_n,p,p...,p)\in \scr U\cap D$ – Learnmore Feb 7 '16 at 2:57
• Is the arguement correct? – Learnmore Feb 7 '16 at 2:58


text:
# Trouble finding inverse of a function

I have the following Mathematica code:

f[U_] = 1 - (Un/U)^3;

Un = 1;

L[Us_] := 3 NIntegrate[
Us^(5/2)/U^(3/2) Sqrt[f[Us]/f[U]] 1/
Sqrt[U^5 f[U] - Us^5 f[Us]], {U, Us, Infinity}]

UsvsL = Table[{Us, L[Us]}, {Us, 1.01, 2.01, 0.01}]

I am trying to find Us as a function of L or Us[L], for which I tried InverseSeries:

l[Us_] =
Integrate[
Series[Us^(5/2)/U^(3/2) Sqrt[f[Us]/f[U]] 1/
Sqrt[U^5 f[U] - Us^5 f[Us]], {U, 0, 4}], U]

Us1 = InverseSeries[l[Us], l] // PowerExpand // Simplify

but, to no avail. I need a table of LvsUs where I can vary the L values to obtain the corresponding Us values. Any help in this regard would be truly beneficial!

Look at your data:

ListLinePlot[UsvsL, AxesLabel -> {"U", "L"}]

Note that for some Ls there two Us. Therefore the inverse function is not single valued. We therefore create two inverse functions, One with the data from U= 1.01 .. 1.21 and a second one with data from U= 1.21 .. 2.01.

dat1 = Select[UsvsL, #[[1]] <= 1.21 &];
dat2 = Select[UsvsL, #[[1]] >= 1.21 &];

Now, how to create the inverse funtions? We already have a table of L[Us]. If we reverse the table entries, we have a table Us[L], but maybe not for the requested L. Toward this aim, we can calculate an interpolating function.

intpol1 = Interpolation[Reverse /@ dat1]
intpol2 = Interpolation[Reverse /@ dat2]

With this two functions you may get the Us for given Ls:

Plot[intpol1[l], {l, Min[dat1[[All, 2]]], Max[dat1[[All, 2]]]},
AxesLabel -> {"L", "U"}]
Plot[intpol2[l], {l, Min[dat2[[All, 2]]], Max[dat2[[All, 2]]]},
AxesLabel -> {"L", "U"}]

• Thank you for the answer! I just have one small doubt. How can I create the two tables from the above interpolating functions? Sep 26 at 13:31
• That is easy. If you want the U value to a certain L value you say e.g. : interpol1[L]. For a whole table: Table[interpol1[x],{x,xmin,xmax, xdelta}]. Note to take care that xmin and xmax are not outside the definition range, otherwise you may get arbitrary results. Sep 26 at 14:56


text:
# Introduction to eefAnalytics

library(eefAnalytics)

# DESCRIPTION

Package: eefAnalytics
Title: Robust Analytical Methods for Evaluating Educational Interventions using Randomised Controlled Trials Designs
Version: 1.1.0
Author: Germaine Uwimpuhwe, Akansha Singh, Dimitris Vallis, Steve Higgins, ZhiMin Xiao, Ewoud De Troyer and Adetayo Kasim
Maintainer: Dimitris Vallis <dimitris.vallis@durham.ac.uk>
Description: Analysing data from evaluations of educational interventions using a randomised controlled trial design. Various analytical tools to perform sensitivity analysis using different methods are supported (e.g. frequentist models with bootstrapping and permutations options, Bayesian models). The included commands can be used for simple randomised trials, cluster randomised trials and multisite trials. The methods can also be used more widely beyond education trials. This package can be used to evaluate other intervention designs using Frequentist and Bayesian multilevel models.
Imports: lme4, mvtnorm, graphics, stats, rstanarm, ggplot2, methods
Encoding: UTF-8
Roxygen: list(markdown = TRUE)
RoxygenNote: 7.2.0
Suggests: knitr, rmarkdown, testthat
VignetteBuilder: knitr
LazyData: true
URL: https://github.com/dimitris-90/eefanalytics
BugReports: https://github.com/dimitris-90/eefanalytics/issues

# ComparePlot: A plot function to compare different eefAnalytics S3 objects from the eefAnalytics package.

## Description

It generates bar plot that compares the effect size from eefAnalytics’ methods.

## Usage

ComparePlot(
eefAnalyticsList,
group,
Conditional = TRUE,
ES_Total = TRUE,
modelNames
)

## Arguments

Argument Description
eefAnalyticsList A list of eefAnalytics S3 objects from eefAnalytics package.
group a string/scalar value indicating which intervention to plot. This must be one of the values of intervention variable excluding the control group. For a two arm trial, the maximum number of values to consider is 1 and 2 for three arm trial.
Conditional a logical value to indicate whether to plot conditional effect size. The default is Conditional=TRUE, otherwise Conditional=FALSE should be specified for plot based on unconditional effect size. Conditional variance is total or residual variance a multilevel model with fixed effects, whilst unconditional variance is total variance or residual variance from a multilevel model with only intercept as fixed effect.
ES_Total A logical value indicating whether to plot the effect size based on total variance or within school variance. The default is ES_Total=TRUE, to plot effect size using total variance. ES_Total=FALSE should be specified for effect size based on within school or residuals variance.
modelNames a string factor containing the names of model to compare. See examples below.

## Details

ComparePlot produces a bar plot which compares the effect sizes and the associated confidence intervals from the different models. For a multilevel model, it shows the effect size based on residual variance and total variance.

## Value

Returns a bar plot to compare the different methods. The returned figure can be further modified as any ggplot

## Examples

data(mstData)
###############
##### SRT #####
###############

outputSRT <- srtFREQ(Posttest~ Intervention + Prettest,
intervention = "Intervention", data = mstData)

outputSRTBoot <- srtFREQ(Posttest~ Intervention + Prettest,
intervention = "Intervention",nBoot=1000, data = mstData)

###############
##### MST #####
###############

outputMST <- mstFREQ(Posttest~ Intervention + Prettest,
random = "School", intervention = "Intervention", data = mstData)

outputMSTBoot <- mstFREQ(Posttest~ Intervention + Prettest,
random = "School", intervention = "Intervention",
nBoot = 1000, data = mstData)

##################
#### Bayesian ####
##################

outputSRTbayes <- srtBayes(Posttest~ Intervention + Prettest,
intervention = "Intervention",
nSim = 2000, data = mstData)

## comparing different results

ComparePlot(list(outputSRT,outputSRTBoot,outputMST,outputMSTBoot,outputSRTbayes),
modelNames =c("ols", "olsBoot","MLM","MLMBoot","OLSBayes"),group=1)

# crtBayes: Bayesian analysis of cluster randomised education trials using Vague Priors.

## Description

crtBayes performs analysis of cluster randomised education trials using a multilevel model under a Bayesian setting, assuming vague priors.

## Usage

crtBayes(
formula,
random,
intervention,
baseln,
condopt,
uncopt,
nsim = 2000,
data,
threshold = 1:10/10,
...
)

## Arguments

Argument Description
formula the model to be analysed is of the form y ~ x1+x2+…. Where y is the outcome variable and Xs are the independent variables.
random a string variable specifying the “clustering variable” as contained in the data. See example below.
intervention a string variable specifying the “intervention variable” as appearing in the formula and the data. See example below.
baseln A string variable allowing the user to specify the reference category for intervention variable. When not specified, the first level will be used as a reference.
adaptD As this function uses rstanarm, this term provides the target average proposal acceptance probability during Stan’s adaptation period. Default is NULL.
nsim number of MCMC iterations per chain. Default is 2000.
condopt additional arguments of stan_lmer to be passed only to the conditional model specification (for example, defining priors only for the conditional model, etc.).
uncopt additional arguments of stan_lmer to be passed only to the unconditional model specification (for example, defining priors only for the unconditional model, etc.).
data data frame containing the data to be analysed.
threshold a scalar or vector of pre-specified threshold(s) for estimating Bayesian posterior probability such that the observed effect size is greater than or equal to the threshold(s).
... additional arguments of stan_lmer to be passed both to the conditional and unconditional model specifications.

## Value

S3 object; a list consisting of

• Beta : Estimates and credible intervals for variables specified in the model. Use summary.eefAnalytics to get Rhat and effective sample size for each estimate.

• ES : Conditional Hedges’ g effect size and its 95 % credible intervals.

• covParm : A vector of variance decomposition into between cluster variance (Schools) and within cluster variance (Pupils). It also contains intra-cluster correlation (ICC).

• SchEffects : A vector of the estimated deviation of each school from the intercept.

• ProbES : A matrix of Bayesian Posterior Probabilities such that the observed effect size is greater than or equal to a pre-specified threshold(s).

• Model : A stan_glm object used in ES computation, this object can be used for convergence diagnostic.

• Unconditional : A list of unconditional effect sizes, covParm and ProbES obtained based on between and within cluster variances from the unconditional model (model with only the intercept as a fixed effect).

## Examples

data(crtData)

########################################################
## Bayesian analysis of cluster randomised trials     ##
########################################################

output <- crtBayes(Posttest~ Intervention+Prettest,random="School",
intervention="Intervention",nsim=2000,data=crtData)

### Fixed effects
beta <- output$Beta beta ### Effect size ES1 <- output$ES
ES1

## Covariance matrix
covParm <- output$covParm covParm ### plot random effects for schools plot(output) ### plot posterior probability of an effect size to be bigger than a pre-specified threshold plot(output,group=1) ########################################################################################### ## Bayesian analysis of cluster randomised trials using informative priors for treatment ## ########################################################################################### ### define priors for explanatory variables my_prior <- normal(location = c(0,6), scale = c(10,1)) ### specify the priors for the conditional model only output2 <- crtBayes(Posttest~ Prettest+Intervention,random="School", intervention="Intervention",nsim=2000,data=crtData, condopt=list(prior=my_prior)) ### Fixed effects beta2 <- output2$Beta
beta2

### Effect size
ES2 <- output2$ES ES2  # crtData: Cluster Randomised Trial Data. ## Description A cluster randomised trial dataset containing 22 schools. The data contains a random sample of test data of pupils and not actual trial data. ## Format A data frame with 265 rows and 5 variables ## Details • Posttest: posttest scores • Prettest: prettest scores • Intervention: the indicator for intervention groups in a two arm trial, coded as 1 for intervention group and 0 for control group. • Intervention2: a simulated indicator for intervention groups in a three arm trial. • School: numeric school identifier # crtFREQ: Analysis of Cluster Randomised Education Trials using Multilevel Model under a Frequentist Setting. ## Description crtFREQ performs analysis of cluster randomised education trials using a multilevel model under a frequentist setting. ## Usage crtFREQ(formula, random, intervention, baseln, nPerm, nBoot, seed, data) ## Arguments Argument Description formula the model to be analysed is of the form y ~ x1+x2+…. Where y is the outcome variable and Xs are the independent variables. random a string variable specifying the “clustering variable” as contained in the data. See example below. intervention a string variable specifying the “intervention variable” as appearing in the formula and the data. See example below. baseln A string variable allowing the user to specify the reference category for intervention variable. When not specified, the first level will be used as a reference. nPerm number of permutations required to generate a permutated p-value. nBoot number of bootstraps required to generate bootstrap confidence intervals. type method of bootstrapping including case re-sampling at student level “case(1)”, case re-sampling at school level “case(2)”, case re-sampling at both levels “case(1,2)” and residual bootstrapping using “residual”. If not provided, default will be case re-sampling at student level. ci method for bootstrap confidence interval calculations; options are the Basic (Hall’s) confidence interval “basic” or the simple percentile confidence interval “percentile”. If not provided default will be percentile. seed seed required for bootstrapping and permutation procedure, if not provided default seed will be used. data data frame containing the data to be analysed. ## Value S3 object; a list consisting of • Beta : Estimates and confidence intervals for variables specified in the model. • ES : Conditional Hedges’ g effect size and its 95 % confidence intervals. If nBoot is not specified, 95% confidence intervals are based on standard errors. If nBoot is specified, they are non-parametric bootstrapped confidence intervals. • covParm : A vector of variance decomposition into between cluster variance (Schools) and within cluster variance (Pupils). It also contains intra-cluster correlation (ICC). • SchEffects : A vector of the estimated deviation of each school from the intercept. • Perm : A “nPerm x 2w” matrix containing permutated effect sizes using residual variance and total variance. “w” denotes number of intervention. “w=1” for two arm trial and “w=2” for three arm trial excluding the control group. It is produced only when nPerm is specified. • Bootstrap : A “nBoot x 2w” matrix containing the bootstrapped effect sizes using residual variance (Within) and total variance (Total), where “w” denotes number of interventions excluding the control group. For example, “w=1” for a two arm trial and “w=2” for a three arm trial excluding the control group. It is only produced when nBoot is specified. • Unconditional : A list of unconditional effect sizes, covParm, Perm and Bootstrap obtained based on variances from the unconditional model (model with only the intercept as a fixed effect). ## Examples  data(crtData) ######################################################## ## MLM analysis of cluster randomised trials + 1.96SE ## ######################################################## output1 <- crtFREQ(Posttest~ Intervention+Prettest,random="School", intervention="Intervention",data=crtData) ### Fixed effects beta <- output1$Beta
beta

### Effect size
ES1 <- output1$ES ES1 ## Covariance matrix covParm <- output1$covParm
covParm

### plot random effects for schools

plot(output1)

##################################################
## MLM analysis of cluster randomised trials    ##
## with residual bootstrap confidence intervals ##
##################################################

output2 <- crtFREQ(Posttest~ Intervention+Prettest,random="School",
intervention="Intervention",nBoot=1000,type="residual",data=crtData)

### Effect size

ES2 <- output2$ES ES2 ### plot bootstrapped values plot(output2, group=1) ####################################################################### ## MLM analysis of cluster randomised trials with permutation p-value## ####################################################################### output3 <- crtFREQ(Posttest~ Intervention+Prettest,random="School", intervention="Intervention",nPerm=1000,data=crtData) ### Effect size ES3 <- output3$ES
ES3

### plot permutated values

plot(output3, group=1)

# eefAnalytics-defunct: Defunct functions in eefAnalytics

## Description

These functions are marked as defunct and have been removed from eefAnalytics.

These functions are marked as defunct and have been removed from eefAnalytics.

These functions are marked as defunct and have been removed from eefAnalytics.

These functions are marked as defunct and have been removed from eefAnalytics.

## Usage

mlmbayes(...)
caceMSTBoot(...)
caceCRTBoot(...)
caceSRTBoot(...)

# mstBayes: Bayesian analysis of Multisite Randomised Education Trials using Vague Priors.

## Description

mstBayes performs analysis of multisite randomised education trials using a multilevel model under a Bayesian setting assuming vague priors.

## Usage

mstBayes(
formula,
random,
intervention,
baseln,
nsim = 2000,
condopt,
uncopt,
data,
threshold = 1:10/10,
...
)

## Arguments

Argument Description
formula the model to be analysed is of the form y ~ x1+x2+…. Where y is the outcome variable and Xs are the independent variables.
random a string variable specifying the “clustering variable” as contained in the data. See example below.
intervention a string variable specifying the “intervention variable” as appearing in the formula and the data. See example below.
baseln A string variable allowing the user to specify the reference category for intervention variable. When not specified, the first level will be used as a reference.
adaptD As this function uses rstanarm, this term provides the target average proposal acceptance probability during Stan’s adaptation period. Default is NULL.
nsim number of MCMC iterations per chain. Default is 2000.
condopt additional arguments of stan_lmer to be passed only to the conditional model specification (for example, defining priors only for the conditional model, etc.).
uncopt additional arguments of stan_lmer to be passed only to the unconditional model specification (for example, defining priors only for the unconditional model, etc.).
data data frame containing the data to be analysed.
threshold a scalar or vector of pre-specified threshold(s) for estimating Bayesian posterior probability that the observed effect size is greater than or equal to the threshold(s).
... additional arguments of stan_lmer to be passed both to the conditional and unconditional model specifications.

## Value

S3 object; a list consisting of

• Beta : Estimates and credible intervals for variables specified in the model. Use summary.eefAnalytics to get Rhat and effective sample size for each estimate.

• ES : Conditional Hedges’ g effect size and its 95 % credible intervals.

• covParm : A list of variance decomposition into between cluster variance-covariance matrix (schools and school by intervention) and within cluster variance (Pupils). It also contains intra-cluster correlation (ICC).

• SchEffects : A vector of the estimated deviation of each school from the intercept and intervention slope.

• ProbES : A matrix of Bayesian posterior probabilities such that the observed effect size is greater than or equal to a pre-specified threshold(s).

• Model : A stan_glm object used in ES computation, this object can be used for convergence diagnostic.

• Unconditional : A list of unconditional effect sizes, covParm and ProbES obtained based on between and within cluster variances from the unconditional model (model with only the intercept as a fixed effect).

## Examples

data(mstData)

########################################################
## Bayesian analysis of multisite randomised trials   ##
########################################################

output <- mstBayes(Posttest~ Intervention+Prettest,random="School",
intervention="Intervention",nsim=2000,data=mstData)

### Fixed effects
beta <- output$Beta beta ### Effect size ES1 <- output$ES
ES1

## Covariance matrix
covParm <- output$covParm covParm ### plot random effects for schools plot(output) ### plot posterior probability of an effect size to be bigger than a pre-specified threshold plot(output,group=1) ############################################################################################# ## Bayesian analysis of multisite randomised trials using informative priors for treatment ## ############################################################################################# ### define priors for explanatory variables my_prior <- normal(location = c(0,6), scale = c(10,1)) ### specify the priors for the conditional model only output2 <- mstBayes(Posttest~ Prettest+Intervention,random="School", intervention="Intervention",nsim=2000,data=mstData, condopt=list(prior=my_prior)) ### Fixed effects beta2 <- output2$Beta
beta2

### Effect size
ES2 <- output2$ES ES2  # mstData: Multisite Trial Data. ## Description A multisite trial dataset containing 54 schools. This data contains a random sample of test data of pupils and not actual trial data. ## Format A data frame with 210 rows and 5 variables ## Details • Posttest: posttest scores • Prettest: prettest scores • Intervention: the indicator for the intervention groups in a two arm trial, coded as 1 for intervention group and 0 for control group. • Intervention2: a simulated indicator for intervention groups in a three arm trial. • School: numeric school identifier # mstFREQ: Analysis of Multisite Randomised Education Trials using Multilevel Model under a Frequentist Setting. ## Description mstFREQ performs analysis of multisite randomised education trials using a multilevel model under a frequentist setting. ## Usage mstFREQ(formula, random, intervention, baseln, nPerm, data, seed, nBoot) ## Arguments Argument Description formula the model to be analysed is of the form y ~ x1+x2+…. Where y is the outcome variable and Xs are the independent variables. random a string variable specifying the “clustering variable” as contained in the data. See example below. intervention a string variable specifying the “intervention variable” as appearing in the formula and the data. See example below. baseln A string variable allowing the user to specify the reference category for intervention variable. When not specified, the first level will be used as a reference. nPerm number of permutations required to generate permutated p-value. data data frame containing the data to be analysed. seed seed required for bootstrapping and permutation procedure, if not provided default seed will be used. nBoot number of bootstraps required to generate bootstrap confidence intervals. type method of bootstrapping including case re-sampling at student level “case(1)”, case re-sampling at school level “case(2)”, case re-sampling at both levels “case(1,2)” and residual bootstrapping using “residual”. If not provided, default will be case re-sampling at student level. ci method for bootstrap confidence interval calculations; options are the Basic (Hall’s) confidence interval “basic” or the simple percentile confidence interval “percentile”. If not provided default will be percentile. ## Value S3 object; a list consisting of • Beta : Estimates and confidence intervals for variables specified in the model. • ES : Conditional Hedge’s g effect size (ES) and its 95 % confidence intervals. If nBoot is not specified, 95% confidence intervals are based on standard errors. If nBoot is specified, they are non-parametric bootstrapped confidence intervals. • covParm : A list of variance decomposition into between cluster variance-covariance matrix (schools and school by intervention) and within cluster variance (Pupils). It also contains intra-cluster correlation (ICC). • SchEffects : A vector of the estimated deviation of each school from the intercept and intervention slope. • Perm : A “nPerm x 2w” matrix containing permutated effect sizes using residual variance and total variance. “w” denotes number of intervention. “w=1” for two arm trial and “w=2” for three arm trial excluding the control group. It is produced only when nPerm is specified. • Bootstrap : A “nBoot x 2w” matrix containing the bootstrapped effect sizes using residual variance (Within) and total variance (Total), where “w” denotes number of interventions excluding the control group. For example, “w=1” for a two arm trial and “w=2” for a three arm trial excluding the control group. It is only produced when nBoot is specified. • Unconditional : A list of unconditional effect sizes, covParm, Perm and Bootstrap obtained based on variances from the unconditional model (model with only the intercept as a fixed effect). ## Examples  data(mstData) ############################################### ## MLM analysis of multisite trials + 1.96SE ## ############################################### output1 <- mstFREQ(Posttest~ Intervention+Prettest,random="School", intervention="Intervention",data=mstData) ### Fixed effects beta <- output1$Beta
beta

### Effect size
ES1 <- output1$ES ES1 ## Covariance matrix covParm <- output1$covParm
covParm

### plot random effects for schools

plot(output1)

###############################################
## MLM analysis of multisite trials          ##
## with bootstrap confidence intervals       ##
###############################################

output2 <- mstFREQ(Posttest~ Intervention+Prettest,random="School",
intervention="Intervention",nBoot=1000,data=mstData)

tp <- output2$Bootstrap ### Effect size ES2 <- output2$ES
ES2

### plot bootstrapped values

plot(output2, group=1)

################################################################
## MLM analysis of mutltisite trials with permutation p-value ##
################################################################

output3 <- mstFREQ(Posttest~ Intervention+Prettest,random="School",
intervention="Intervention",nPerm=1000,data=mstData)

ES3 <- output3$ES ES3 #### plot permutated values plot(output3, group=1)  # plot.eefAnalytics: A plot method for an eefAnalytics S3 object obtained from the eefAnalytics package. ## Description Plots different figures based on output from eefAnalytics package. ## Usage list(list("plot"), list("eefAnalytics"))(x, group, Conditional = TRUE, ES_Total = TRUE, slope = FALSE, ...) ## Arguments Argument Description x an output object from the eefAnalytics package. group a string/scalar value indicating which intervention to plot. This must be one of the values of intervention variable excluding the control group. For a two arm trial, the maximum number of values to consider is 1 and 2 for three arm trial. Conditional a logical value to indicate whether to plot the conditional effect size. The default is Conditional=TRUE, otherwise Conditional=FALSE should be specified for plot based on the unconditional effect size. Conditional variance is total or residual variance from a multilevel model with fixed effects, whilst unconditional variance is total variance or residual variance from a multilevel model with only intercept as fixed effect. ES_Total A logical value indicating whether to plot the effect size based on total variance or within school variance. The default is ES_Total=TRUE, to plot the effect size using total variance. ES_Total=FALSE should be specified for the effect size based on within school or residuals variance. slope A logical value indicating whether to return the plot of random intercept (default is slope=FALSE). return other school-by-intervention interaction random slope (s) is slope=TRUE. This argument is suitable only for mstBayes and mstFREQ functions. ... arguments passed to plot.default ## Details Plot produces a graphical visualisation depending on which model is fitted: • For srtFREQ() , plot can only be used when nBoot or nPerm is specified to visualise the distribution of bootstrapped or permutated values. • For crtFREQ() or mstFREQ() , plot shows the distribution of random intercepts when group=NULL . It produces histogram of permutated or bootstrapped values when group is specified and either nBoot or nPerm is also specified. ## Value Returns relevant plots for each model. ## Examples  #### read data data(mstData) data(crtData) ############### ##### SRT ##### ############### ##### Bootstrapped outputSRTBoot <- srtFREQ(Posttest~ Intervention + Prettest, intervention = "Intervention",nBoot=1000, data = mstData) plot(outputSRTBoot,group=1) ##### Permutation outputSRTPerm <- srtFREQ(Posttest~ Intervention + Prettest, intervention = "Intervention",nPerm=1000, data = mstData) plot(outputSRTPerm,group=1) ############### ##### MST ##### ############### #### Random intercepts outputMST <- mstFREQ(Posttest~ Intervention + Prettest, random = "School", intervention = "Intervention", data = mstData) plot(outputMST) #### Bootstrapped outputMSTBoot <- mstFREQ(Posttest~ Intervention + Prettest, random = "School", intervention = "Intervention", nBoot = 1000, data = mstData) plot(outputMSTBoot) plot(outputMSTBoot,group=1) #### Permutation outputMSTPerm <- mstFREQ(Posttest~ Intervention + Prettest, random = "School", intervention = "Intervention", nPerm = 1000, data = mstData) plot(outputMSTPerm) plot(outputMSTPerm,group=1) ############### ##### CRT ##### ############### #### Random intercepts outputCRT <- crtFREQ(Posttest~ Intervention + Prettest, random = "School", intervention = "Intervention", data = crtData) plot(outputCRT) ## Bootstrapped outputCRTBoot <- crtFREQ(Posttest~ Intervention + Prettest, random = "School", intervention = "Intervention", nBoot = 1000, data = crtData) plot(outputCRTBoot,group=1) ##Permutation outputCRTPerm <- crtFREQ(Posttest~ Intervention + Prettest, random = "School", intervention = "Intervention", nPerm = 1000, data = crtData) plot(outputCRTPerm,group=1)  # print.eefAnalytics: Print for a fitted model represented by an eefAnalytics object. ## Description Print for a fitted model represented by an eefAnalytics object. ## Usage list(list("print"), list("eefAnalytics"))(x, ...) ## Arguments Argument Description x Object of class eefAnalytics ... Additional arguments of print ## Value Print conditional and unconditional effect sizes. # srtBayes: Analysis of Simple Randomised Education Trials using Bayesian Linear Regression Model with Vague Priors. ## Description srtBayes performs analysis of educational trials under the assumption of independent errors among pupils using Bayesian framework with Stan. This can also be used with schools as fixed effects. ## Usage srtBayes( formula, intervention, baseln, adaptD, nsim = 2000, condopt, uncopt, data, threshold = 1:10/10, ... ) ## Arguments Argument Description formula The model to be analysed is of the form y~x1+x2+…. Where y is the outcome variable and Xs are the independent variables. intervention A string variable specifying the “intervention variable” as appearing in the formula and the data. See example below. baseln A string variable allowing the user to specify the reference category for intervention variable. When not specified, the first level will be used as a reference. adaptD As this function uses rstanarm, this term provides the target average proposal acceptance probability during Stan’s adaptation period. Default is NULL. nsim A number of MCMC iterations per chain. Default is 2000. condopt additional arguments of stan_lmer to be passed only to the conditional model specification (for example, defining priors only for the conditional model, etc.). uncopt additional arguments of stan_lmer to be passed only to the unconditional model specification (for example, defining priors only for the unconditional model, etc.). data Data frame containing the data to be analysed. threshold a scalar or vector of pre-specified threshold(s) for estimating Bayesian posterior probability such that the observed effect size is greater than or equal to the threshold(s). ... Additional arguments of stan_lmer to be passed both to the conditional and unconditional model specifications. ## Value S3 object; a list consisting of • Beta : Estimates and credible intervals for the variables specified in the model. Use summary.eefAnalytics to get Rhat and effective sample size for each estimate. • ES : Conditional Hedges’ g effect size and its 95 % credible intervals. • sigma2 : Residual variance. • ProbES : A matrix of Bayesian posterior probabilities such that the observed effect size is greater than or equal to a pre-specified threshold(s). • Model : A stan_glm object used in ES computation, this object can be used for convergence diagnostic. • Unconditional : A list of unconditional effect sizes, sigma2 and ProbES obtained based on residual variance from the unconditional model (model with only the intercept as a fixed effect). ## Examples  data(mstData) ######################################################## ## Bayesian analysis of simple randomised trials ## ######################################################## output <- srtBayes(Posttest~ Intervention+Prettest, intervention="Intervention",nsim=2000,data=mstData) ### Fixed effects beta <- output$Beta
beta

### Effect size
ES1 <- output$ES ES1 ## Covariance matrix covParm <- output$covParm
covParm

### plot random effects for schools

plot(output)

### plot posterior probability of an effect size to be bigger than a pre-specified threshold

plot(output,group=1)

###########################################################################################
## Bayesian analysis of simple randomised trials using informative priors for treatment  ##
###########################################################################################

### define priors for explanatory variables

my_prior <- normal(location = c(0,6), scale = c(10,1))

### specify the priors for the conditional model only

output2 <- srtBayes(Posttest~ Prettest+Intervention,
intervention="Intervention",
nsim=2000,data=mstData,
condopt=list(prior=my_prior))

### Fixed effects
beta2 <- output2$Beta beta2 ### Effect size ES2 <- output2$ES
ES2

# srtFREQ: Analysis of Simple Randomised Education Trial using Linear Regression Model.

## Description

srtFREQ performs analysis of educational trials under the assumption of independent errors among pupils. This can also be used with schools as fixed effects.

## Usage

srtFREQ(formula, intervention, baseln, nBoot, nPerm, seed, data)

## Arguments

Argument Description
formula the model to be analysed is of the form y~x1+x2+…. Where y is the outcome variable and Xs are the independent variables.
intervention a string variable specifying the “intervention variable” as appearing in the formula and the data. See example below.
baseln A string variable allowing the user to specify the reference category for intervention variable. When not specified, the first level will be used as a reference.
nBoot number of bootstraps required to generate bootstrap confidence intervals.
nPerm number of permutations required to generate permutated p-value.
seed seed required for bootstrapping and permutation procedure, if not provided default seed will be used.
ci method for bootstrap confidence interval calculations; options are the Basic (Hall’s) confidence interval “basic” or the simple percentile confidence interval “percentile”. If not provided default will be percentile.
data data frame containing the data to be analysed.

## Value

S3 object; a list consisting of

• Beta : Estimates and confidence intervals for the variables specified in the model.

• ES : Conditional Hedges’g effect size and its 95 % confidence intervals. If nBoot is not specified, 95% confidence intervals are based on standard errors. If nBoot is specified, they are non-parametric bootstrapped confidence intervals.

• sigma2 : Residual variance.

• Perm : A “nPerm x w” matrix containing permutated effect sizes using residual variance. “w” denotes number of intervention. “w=1” for two arm trial and “w=2” for three arm trial excluding the control group. It is produced only if nPerm is specified.

• Bootstrap : A “nBoot x w” matrix containing the bootstrapped effect sizes using residual variance. “w” denotes number of intervention. “w=1” for two arm trial and “w=2” for three arm trial excluding the control group. It is produced only if nBoot is specified.

• Unconditional : A list of unconditional effect size, sigma2, Perm and Bootstrap obtained based on variances from the unconditional model (model with only intercept as fixed effect).

## Examples

data(mstData)

###################################################################
## Analysis of simple randomised trials using Hedges Effect Size ##
###################################################################

output1 <- srtFREQ(Posttest~ Intervention+Prettest,
intervention="Intervention",data=mstData )
ES1 <- output1$ES ES1 ################################################################### ## Analysis of simple randomised trials using Hedges Effect Size ## ## with Permutation p-value ## ################################################################### output2 <- srtFREQ(Posttest~ Intervention+Prettest, intervention="Intervention",nPerm=1000,data=mstData ) ES2 <- output2$ES
ES2

#### plot permutated values

plot(output2, group=1)

###################################################################
## Analysis of simple randomised trials using Hedges Effect Size ##
## with non-parametric Basic bootstrap confidence intervals      ##
###################################################################

output3 <- srtFREQ(Posttest~ Intervention+Prettest,
intervention="Intervention",nBoot=1000,ci="basic",data=mstData)

ES3 <- output3$ES ES3 ### plot bootstrapped values plot(output3, group=1) #################################################################### ## Analysis of simple randomised trials using Hedges' effect size ## ## with schools as fixed effects ## #################################################################### output4 <- srtFREQ(Posttest~ Intervention+Prettest+as.factor(School), intervention="Intervention",data=mstData ) ES4 <- output4$ES
ES4

####################################################################
## Analysis of simple randomised trials using Hedges' effect size ##
## with schools as fixed effects and with permutation p-value     ##
####################################################################

output5 <- srtFREQ(Posttest~ Intervention+Prettest+as.factor(School),
intervention="Intervention",nPerm=1000,data=mstData )

ES5 <- output5$ES ES5 #### plot permutated values plot(output5, group=1) #################################################################### ## Analysis of simple randomised trials using Hedges' effect size ## ## with schools as fixed effects and with permutation p-value ## #################################################################### output6 <- srtFREQ(Posttest~ Intervention+Prettest+as.factor(School), intervention="Intervention",nBoot=1000,data=mstData) ES6 <- output6$ES
ES6

### plot bootstrapped values

plot(output6, group=1)

# summary.eefAnalytics: Summary for a fitted model represented by an eefAnalytics object.

## Description

Summary for a fitted model represented by an eefAnalytics object.

## Usage

list(list("summary"), list("eefAnalytics"))(object, ...)

## Arguments

Argument Description
object Object of class eefAnalytics
... Additional arguments of summary

## Value

Returns relevant summary including Rhat and effective sample sizes.


text:
Overall Objectives
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography
 PDF e-Pub

## Section: New Results

### Hardware and FPGA arithmetic

#### Reconfiguring arithmetic

With B. Pasca (Altera), F. de Dinechin contributed a book chapter about of the opportunities and challenges of computer arithmetic for reconfigurable/FPGA computing [32] . The main point of this chapter is to look beyond the heritage of processor arithmetic. Using many examples from the FloPoCo project and others, it shows the benefits of merging and fusing standard operators, it introduces an open-ended space of non-standard operators, and illustrates the power of machine-generation of such arithmetic cores.

#### The bit heap framework for fixed-point arithmetic

N. Brunie, F. de Dinechin, and M. Istoan, with students G. Sergent, K. Illyes, and B. Popa, extended FloPoCo with a versatile framework for manipulating sums of weighted bits [28] , [18] . Such bit heaps may be used to express and optimize at the bit level a wide range of operators (from adders and multipliers to polynomials, filters, and other coarse arithmetic cores). A single piece of code can then be used to generate an architecture for any of these operators.

#### Elementary functions

F. de Dinechin, with P. Echeverria and M. Lopez-Vallejo (U. Madrid) and B. Pasca (Altera), published a hardware architecture for the floating-point pow and powr functions of the IEEE-754-2008 standard [3] . These functions compute ${x}^{y}$, and differ only in the specification of special cases. The implementation, distributed in FloPoCo, is parameterized in exponent and significand size. It combines suitably modified exponential and logarithm units.

F. de Dinechin and M. Istoan, with student G. Sergent, compared several hardware algorithms for the implementation of sine, cosine, and combined sine/cosine [21] : unrolled CORDIC in two variants with several minor improvements, polynomial approximation, and an ad-hoc architecture based on trigonometric identities. A surprising result is that the ad-hoc architecture betters CORDIC even when its multipliers and tables are synthesized as logic.

#### Contributions to processor architecture

S. Collange (ALF team) and N. Brunie with G. Diamos (Nvidia) suggested improvements for the architecture of general-purpose graphical processing units [11] . As threads take different paths across the control-flow graph, SIMD lockstep execution is partially lost, and must be regained whenever possible in order to maximize the occupancy of SIMD units. Two techniques are described to handle SIMT control divergence and identify reconvergence points. The most advanced one operates in constant space and handles indirect jumps and recursion. In terms of performance, this solution is at least as efficient as state-of-the-art techniques in use in current GPUs.

N. Brunie and F. de Dinechin studied with B. de Dinechin (Kalray) the integration of a tightly coupled reconfigurable accelerator in a massively parallel multiprocessor [27] . For this purpose, they described an architecture exploration framework that produces an architecture along with the relevant compilation software. This framework was demonstrated on AES, SHA2, and a FIR filter.


text:
3

# Question 222 ptsNow; pretend You actually ran the study You came Up with: Make up the results (the values; means and test statistic SDs) and write Up the results aS...

## Question

###### Question 222 ptsNow; pretend You actually ran the study You came Up with: Make up the results (the values; means and test statistic SDs) and write Up the results aS you would see them in APA style research paperEdit Viewy Insert Format Tools Table12ptParagraph

Question 22 2 pts Now; pretend You actually ran the study You came Up with: Make up the results (the values; means and test statistic SDs) and write Up the results aS you would see them in APA style research paper Edit Viewy Insert Format Tools Table 12pt Paragraph

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# Is every group homomorphism of the rationals an isomorphism

Is it true that every non trivial group homomorphism from $\mathbb Q$ to $\mathbb Q$ is a group isomorphism. The trivial homomorphism being the map that sends every rational to $0$.

-
What have you tried? –  Chris Eagle Oct 27 '12 at 14:37
HINT: Show that if $h:\Bbb Q\to\Bbb Q$ is a group homomorphism, then $h(q)=qh(1)$ for every $q\in\Bbb Q$. You might want to begin by showing it for $q\in\Bbb Z^+$.
so if $h(1)\not = 0$ then $h$ is obviously a group isomorphism. Does this generalize to every group homomorphism $\mathbb Q\oplus \mathbb Q\rightarrow \mathbb Q\oplus \mathbb Q$? –  palio Oct 27 '12 at 14:55
@palio: Such a homomorphism is completely determined by $h(\langle 0,1\rangle)$ and $h(\langle 1,0\rangle)$, but it can collapse the range to a copy of $\Bbb Q$: consider $h(\langle p,q\rangle)=\langle p,0\rangle$. –  Brian M. Scott Oct 27 '12 at 15:00


text:
# Propagation constant

(Redirected from Phase coefficient)

The propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The quantity being measured can be the voltage or current in a circuit or a field vector such as electric field strength or flux density. The propagation constant itself measures change per metre but is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

The propagation constant is expressed logarithmically, almost universally to the base e, rather than the more usual base 10 used in telecommunications in other situations. The quantity measured, such as voltage, is expressed as a sinusoidal phasor. The phase of the sinusoid varies with distance which results in the propagation constant being a complex number, the imaginary part being caused by the phase change.

## Alternative names

The term propagation constant is somewhat of a misnomer as it usually varies strongly with ω. It is probably the most widely used term but there are a large variety of alternative names used by various authors for this quantity. These include, transmission parameter, transmission function, propagation parameter, propagation coefficient and transmission constant. In plural, it is usually implied that α and β are being referenced separately but collectively as in transmission parameters, propagation parameters, propagation coefficients, transmission constants and secondary coefficients. This last occurs in transmission line theory, the term secondary being used to contrast to the primary line coefficients. The primary coefficients being the physical properties of the line; R,C,L and G, from which the secondary coefficients may be derived using the telegrapher's equation. Note that, at least in the field of transmission lines, the term transmission coefficient has a different meaning despite the similarity of name. Here it is the corollary of reflection coefficient.

## Definition

The propagation constant, symbol γ, for a given system is defined by the ratio of the amplitude at the source of the wave to the amplitude at some distance x, such that,

$\frac{A_0}{A_x}=e^{\gamma x}$

Since the propagation constant is a complex quantity we can write:

$\gamma = \alpha +i \beta \,$

where

α, the real part, is called the attenuation constant
β, the imaginary part, is called the phase constant

That β does indeed represent phase can be seen from Euler's formula;

$e^{i\theta}=\cos{\theta}+i\sin{\theta}\,\!$

which is a sinusoid which varies in phase as θ varies but does not vary in amplitude because;

$\left|e^{i\theta}\right|=\sqrt{\cos^2{\theta}+\sin^2{\theta}}=1$

The reason for the use of base e is also now made clear. The imaginary phase constant, iβ, can be added directly to the attenuation constant, α, to form a single complex number that can be handled in one mathematical operation provided they are to the same base. Angles measured in radians require base e, so the attenuation is likewise in base e.

The propagation constant for copper (or any other conductor) lines can be calculated from the primary line coefficients by means of the relationship;

$\gamma=\sqrt{ZY}$

where;

$Z=R+i\omega L\,\!$, the series impedance of the line per metre and,
$Y=G+i\omega C\,\!$, the shunt admittance of the line per metre.

## Attenuation constant

In telecommunications, the term attenuation constant, also called attenuation parameter or attenuation coefficient, is the attenuation of an electromagnetic wave propagating through a medium per unit distance from the source. It is the real part of the propagation constant and is measured in nepers per metre. A neper is approximately 8.7dB. Attenuation constant can be defined by the amplitude ratio;

$\left|\frac{A_0}{A_x}\right|=e^{\alpha x}$

The propagation constant per unit length is defined as the natural logarithmic of ratio of the sending end current or voltage to the receiving end current or voltage.

### Copper lines

The attenuation constant for copper lines (or ones made of any other conductor) can be calculated from the primary line coefficients as shown above. For a line meeting the distortionless condition, with a conductance G in the insulator, the attenuation constant is given by;

$\alpha=\sqrt{RG}\,\!$

however, a real line is unlikely to meet this condition without the addition of loading coils and, furthermore, there are some frequency dependent effects operating on the primary "constants" which cause a frequency dependence of the loss. There are two main components to these losses, the metal loss and the dielectric loss.

The loss of most transmission lines are dominated by the metal loss, which causes a frequency dependency due to finite conductivity of metals, and the skin effect inside a conductor. The skin effect causes R along the conductor to be approximately dependent on frequency according to;

$R \propto \sqrt{\omega}$

Losses in the dielectric depend on the loss tangent (${\tan \delta}$) of the material, which depends inversely on the wavelength of the signal and is directly proportional to the frequency.

$\alpha_d={{\pi}\sqrt{\varepsilon_r}\over{\lambda}}{\tan \delta}$

### Optical fibre

The attenuation constant for a particular propagation mode in an optical fiber, the real part of the axial propagation constant.

## Phase constant

In electromagnetic theory, the phase constant, also called phase change constant, parameter or coefficient is the imaginary component of the propagation constant for a plane wave. It represents the change in phase per metre along the path travelled by the wave at any instant and is equal to real part of the angular wavenumber of the wave. It is represented by the symbol β and is measured in units of radians per metre.

From the definition of (angular) wavenumber;

$k = \frac{2\pi}{\lambda} = \beta$

For a transmission line, the Heaviside condition of the telegrapher's equation tells us that the wavenumber must be proportional to frequency for the transmission of the wave to be undistorted in the time domain. This includes, but is not limited to, the ideal case of a lossless line. The reason for this condition can be seen by considering that a useful signal is composed of many different wavelengths in the frequency domain. For there to be no distortion of the waveform, all these waves must travel at the same velocity so that they arrive at the far end of the line at the same time as a group. Since wave phase velocity is given by;

$v_p=\frac{\lambda}{T}=\frac{f}{\tilde{\nu}}=\frac{\omega}{\beta}$

it is proved that β is required to be proportional to ω. In terms of primary coefficients of the line, this yields from the telegrapher's equation for a distortionless line the condition;

$\beta = \omega \sqrt{LC}$

However, practical lines can only be expected to approximately meet this condition over a limited frequency band.

## Filters and two-port networks

The term propagation constant or propagation function is applied to filters and other two-port networks used for signal processing. In these cases, however, the attenuation and phase coefficients are expressed in terms of nepers and radians per network section rather than per metre. Some authors[1] make a distinction between per metre measures (for which "constant" is used) and per section measures (for which "function" is used).

The propagation constant is a useful concept in filter design which invariably uses a cascaded section topology. In a cascaded topology, the propagation constant, attenuation constant and phase constant of individual sections may be simply added to find the total propagation constant etc.

Three networks with arbitrary propagation constants and impedances connected in cascade. The Zi terms represent image impedance and it is assumed that connections are between matching image impedances.

The ratio of output to input voltage for each network is given by,[2]

$\frac{V_1}{V_2}=\sqrt{\frac{Z_{I1}}{Z_{I2}}}e^{\gamma_1}$

$\frac{V_2}{V_3}=\sqrt{\frac{Z_{I2}}{Z_{I3}}}e^{\gamma_2}$

$\frac{V_3}{V_4}=\sqrt{\frac{Z_{I3}}{Z_{I4}}}e^{\gamma_3}$

The terms $\sqrt{\frac{Z_{In}}{Z_{Im}}}$ are impedance scaling terms[3] and their use is explained in the image impedance article.

The overall voltage ratio is given by,

$\frac{V_1}{V_4}=\frac{V_1}{V_2}\cdot\frac{V_2}{V_3}\cdot\frac{V_3}{V_4}=\sqrt{\frac{Z_{I1}}{Z_{I4}}}e^{\gamma_1+\gamma_2+\gamma_3}$

Thus for n cascaded sections all having matching impedances facing each other, the overall propagation constant is given by,

$\gamma_{Tot}=\gamma_1 + \gamma_2 + \gamma_3 + \cdots + \gamma_n$


text:
How do you see the genus of a curve, just looking at its function field?

Yuhao asked in the 20-questions seminar:

The genus of a curve is a birational invariant; the function field of a curve determines it up to birational equivelance.

How do you see the genus directly from the field?

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Let $k$ be the ground field.

The Kahler differentials $\Omega_{K/k}$ are the $K$ vector space generated by formal symbols $dg$ subject to $d(f+g)=df+dg$, $d(fg) = f dg + g df$ and $da=0$ for $a \in k$. This is a one dimensional $K$ vector space.

Let $\omega$ be a differential. For any valuation $v$ on $K$, let $t$ be such that $v(t)=1$. We say that $\omega$ is regular at v if $v(\omega/dt) \geq 0$. We say that $\omega$ is regular if it is regular at every valuation of $K/k$. Then the space of regular differentials is a $k$ vector space of dimension $g$.

FURTHER THOUGHTS

Both Ben's answer and mine used the set of valuations of $K/k$. This essentially means that we used the ground field $k$. A valuation of $K/k$ is defined as a valuation of $K$ which is trivial on $k$; conversely, the ground field can be recovered from the valuations that respect it by the formula $k = \bigcap_{v} v^{-1}(\mathbb{R}_{\geq 0})$.

Here is a cautionary example to show that there can not be any solution which only uses properties of the field $K$, without reference to $k$. Let $C$ and $D$ be two irreducible curves of different genuses. Let $K$ be the field of meromorphic functions on $C \times D$, let $k$ and $\ell$ be the fields of functions on $C$ and $D$. Then $K$ is a transcendence degree 1 extension of both $k$ and $\ell$, but has different genuses when considered in these two ways.

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Assuming you work over $\mathbb C$, you can see it in the structure of the Galois group. The points of the curve can be seen as non-Archimedean valuations on the field, and you can see whether a map of curves is ramified by looking at valuations (each prime in the smaller field should have valuation 1 or 0 for any valuation of the bigger field). Basically, this means that around any point in the domain and its target, the completed local rings look like $\mathbb{C}[[t]]$ and $\mathbb{C}[[s]]$, and the map should always be $s\mapsto a_1t+a_2t^2\cdots$, which $a_1\neq 0$ (so the map on completions is an isomorphism).

There's a maximal unramified field extension, and the Galois group of this extension is the profinite completion of $\pi_1(X)$ in the topological sense, so the genus can be obtained by looking at the abelianization.

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But one should be sure first that the field of constants is algebraically closed. –  JSE Oct 16 '09 at 3:25
Hmm, it's becoming more and more clear that in my world, there are only two kinds of fields: algebraically closed and finite. Oddly enough, not everyone seems to agree. –  Ben Webster Oct 17 '09 at 4:05

Danny Calegari addresses the question of how to see the genus of an affine curve. To obtain the genus of an algebraic curve from the function field, take two generic elements in the field (giving a map to ℂ2), and then take a minimal polynomial relation between them. Then compute the Newton polygon of this polynomial, and count the number of lattice points inside as described in Danny's blog entry. Maybe this is as unsatisfying an answer though as taking a map to ℂℙ1 and use Riemann-Hurwitz....

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The Riemann-Roch theorem (or even a fragment of it for a sequence of divisors of increasing degree) can be stated in terms of the field $K$, its constant field $k$ (as David Speyer pointed out this is very important) and the valuations of $K/k$. The genus is the unique number making the Riemann-Roch theorem true. More specifically, for divisors $D$ of sufficiently large degree, $g = \deg D - l(D) + 1$.

@shenghao If you use a similar definition for number fields, using $l(D)$ for the logarithm of $|L(D)|$ (instead of its dimension, which is not defined) you get that the logarithm of the absolute discriminant is the analogue of the genus.

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Chapter 3 in Neukirch's Algebraic Number Theory develops a geometric language for number fields which is analogous (this is THE analogy) to that which one has at one's disposal when studying function fields. This includes some notion of the genus of a number field. He proves a Riemann Roch type theorem and I think there is also some type of Hurwitz formula.

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Thanks for the pointer to Neukirch's book! That's useful. Have added a pointer to it on the nLab here: ncatlab.org/nlab/show/function+field+analogy#Neukirch92 –  Urs Schreiber Jul 19 at 14:24

See C. Chevalley, Introduction to the theory of Algebraic functions of one variable. That book develops the whole theory of algebraic curves using just its function field.

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Here is a review by A. Weil of this book: ams.org/journals/bull/2000-37-01/S0273-0979-99-00834-4/… –  Martin Brandenburg Apr 18 '11 at 11:01
the full review is here: ams.org/journals/bull/1951-57-05/S0002-9904-1951-09522-1/… –  roy smith Apr 22 '11 at 21:03

I saw a definition for the genus of a global field (like number field) somewhere, in the spirit of Hurwitz formula. The title of that book is something like "zeta functions and L functions"; I forget.

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text:
Hopf-Lax type formula for sub- and supersolutions

Abstract

We study the continuous as well as discontinuous viscosity solutions of a certain Hamilton-Jacobi equation, $u_t + H(u, D u)=0$ in $\mathbb R^{\,n} \times \mathbb R_+$ with $u(x,0)=u_0(x)$. We obtain explicit formulas for continuous as well as for the sub- and supersolutions. In the latter case, furthermore, if $H(u,p) > 0$ for $|p| \neq 0$ then the supersolution becomes a solution.

Article information

Source
Adv. Differential Equations Volume 5, Number 1-3 (2000), 97-119.

Dates
First available in Project Euclid: 27 December 2012


text:
MathSciNet bibliographic data MR1362937 11R20 (11R29) Miyada, Ichiro On imaginary abelian number fields of type \$(2,2,\cdots,2)\$$(2,2,\cdots,2)$ with one class in each genus. Manuscripta Math. 88 (1995), no. 4, 535–540. Article

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