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# Sum, Add, Subtract, Sum of Elements

Math Operations

## Description

The Sum block performs addition or subtraction on its inputs. This block can add or subtract scalar, vector, or matrix inputs. It can also collapse the elements of a signal.

You specify the operations of the block with the List of signs parameter. Plus (`+`), minus (`-`), and spacer (`|`) characters indicate the operations to be performed on the inputs:

• If there are two or more inputs, then the number of `+` and `-` characters must equal the number of inputs. For example, "`+-+`" requires three inputs and configures the block to subtract the second (middle) input from the first (top) input, and then add the third (bottom) input.

• All nonscalar inputs must have the same dimensions. Scalar inputs will be expanded to have the same dimensions as the other inputs.

• A spacer character creates extra space between ports on the block's icon.

• For a round Sum block, the first input port is the port closest to the 12 o'clock position going in a counterclockwise direction around the block. Similarly, other input ports appear in counterclockwise order around the block.

• If only addition of all inputs is required, then a numeric parameter value equal to the number of inputs can be supplied instead of "`+`" characters.

• If only one input port is required, a single "`+`" or "`-`" collapses the element via the specified operation.

The Sum block first converts the input data type(s) to its accumulator data type, then performs the specified operations. The block converts the result to its output data type using the specified rounding and overflow modes.

### Calculation of Block Output

Output calculation for the Sum block depends on the number of block inputs and the sign of input ports:

If the Sum block has...And...The formula for output calculation is...Where...

One input port

The input port sign is +

y = e[0] + e[1] + e[2] ... + e[m]

`e[i]` is the ith element of input u

The input port sign is –

y = 0.0 – e[0] – e[1] – e[2] ... – e[m]

Two or more input ports

All input port signs are –

y = 0.0 – u[0] – u[1] – u[2] ... – u[n]

`u[i]` is the input to the ith input port

The kth input port is the first port where the sign is +

y = u[k] – u[0] – u[1] – u[2] – u[k–1] (+/–) u[k+1] ... (+/–) u[n]

## Data Type Support

The Sum block accepts real or complex signals of the following data types:

• Floating point

• Built-in integer

• Fixed point

• Boolean

The inputs can be of different data types, unless you select the Require all inputs to have the same data type parameter. For more information, see Data Types Supported by Simulink in the Simulink® documentation.

## Parameters and Dialog Box

The Main pane of the Sum block dialog box appears as follows:

The Signal Attributes pane of the Sum block dialog box appears as follows:

### Show data type assistant

Display the Data Type Assistant.

#### Settings

The Data Type Assistant helps you set the Output data type parameter.

### Icon shape

Designate the icon shape of the block.

#### Settings

Default: `round`

`rectangular`

Designate the icon shape of the block as rectangular.

`round`

Designate the icon shape of the block as round.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### List of signs

Enter plus (`+`) and minus (`-`) characters.

#### Settings

Default: ` |++`

• Addition is the default operation, so if you only want to add the inputs, enter the number of input ports.

• For a single vector input, "`+`" or "`-`" will collapse the vector using the specified operation.

• Enter as many plus (`+`) and minus (`-`) characters as there are inputs.

#### Tips

You can manipulate the positions of the input ports on the block by inserting spacers (`|`) between the signs in the List of signs parameter. For example, "`++|--`" creates an extra space between the second and third input ports.

#### Dependencies

Entering only one element enables the Sum over parameter.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Sum over

Select dimension over which to perform the sum over operation.

#### Settings

Default: `All dimensions`

`All dimensions`

Sum all input elements, yielding a scalar.

`Specified dimension`

Display the Dimension parameter, where you specify the dimension over which to perform the operation.

#### Dependencies

Selecting `Specified dimension` enables the Dimension parameter.

List of signs (when it has only one element) enables this parameter.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Dimension

Specify the dimension over which to perform the operation.

#### Settings

Default: `1`

The block follows the same summation rules as the MATLAB® `sum` function.

Suppose that you have a 2-by-3 matrix U.

• Setting Dimension to `1` results in the output Y being computed as:

`$Y={\sum }_{i=1}^{2}U\left(i,j\right)$`
• Setting Dimension to `2` results in the output Y being computed as:

`$Y={\sum }_{j=1}^{3}U\left(i,j\right)$`

If the specified dimension is greater than the dimension of the input, an error message appears.

#### Dependencies

Setting Sum over to ```Specified dimension``` enables this parameter.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Sample time

 Note:   This parameter is not visible in the block dialog box unless it is explicitly set to a value other than `-1`. To learn more, see Blocks for Which Sample Time Is Not Recommended.

### Require all inputs to have the same data type

Require that all inputs have the same data type.

#### Settings

Default: Off

On

Require that all inputs have the same data type.

Off

Do not require that all inputs have the same data type.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Lock data type settings against changes by the fixed-point tools

Select to lock data type settings of this block against changes by the Fixed-Point Tool and the Fixed-Point Advisor.

#### Settings

Default: Off

On

Locks all data type settings for this block.

Off

Allows the Fixed-Point Tool and the Fixed-Point Advisor to change data type settings for this block.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Integer rounding mode

Specify the rounding mode for fixed-point operations.

#### Settings

Default: `Floor`

`Ceiling`

Rounds both positive and negative numbers toward positive infinity. Equivalent to the MATLAB `ceil` function.

`Convergent`

Rounds number to the nearest representable value. If a tie occurs, rounds to the nearest even integer. Equivalent to the Fixed-Point Designer™ `convergent` function.

`Floor`

Rounds both positive and negative numbers toward negative infinity. Equivalent to the MATLAB `floor` function.

`Nearest`

Rounds number to the nearest representable value. If a tie occurs, rounds toward positive infinity. Equivalent to the Fixed-Point Designer `nearest` function.

`Round`

Rounds number to the nearest representable value. If a tie occurs, rounds positive numbers toward positive infinity and rounds negative numbers toward negative infinity. Equivalent to the Fixed-Point Designer `round` function.

`Simplest`

Automatically chooses between round toward floor and round toward zero to generate rounding code that is as efficient as possible.

`Zero`

Rounds number toward zero. Equivalent to the MATLAB `fix` function.

#### Command-Line Information

 Parameter: `RndMeth` Type: string Value: `'Ceiling'` | `'Convergent'` | `'Floor'` | `'Nearest'` | `'Round'` | `'Simplest'` | `'Zero'` Default: `'Floor'`

### Saturate on integer overflow

Specify whether overflows saturate.

#### Settings

Default: Off

On

Overflows saturate to either the minimum or maximum value that the data type can represent.

For example, an overflow associated with a signed 8-bit integer can saturate to -128 or 127.

Off

Overflows wrap to the appropriate value that the data type can represent.

For example, the number 130 does not fit in a signed 8-bit integer and wraps to -126.

#### Tips

• Consider selecting this check box when your model has a possible overflow and you want explicit saturation protection in the generated code.

• Consider clearing this check box when you want to optimize efficiency of your generated code.

Clearing this check box also helps you to avoid overspecifying how a block handles out-of-range signals. For more information, see Checking for Signal Range Errors.

• When you select this check box, saturation applies to every internal operation on the block, not just the output or result.

• In general, the code generation process can detect when overflow is not possible. In this case, the code generator does not produce saturation code.

#### Command-Line Information

 Parameter: `SaturateOnIntegerOverflow` Type: string Value: `'off'` | `'on'` Default: `'off'`

### Accumulator data type

Specify the accumulator data type.

#### Settings

Default: ```Inherit: Inherit via internal rule```

`Inherit: Inherit via internal rule`

Use internal rule to determine accumulator data type.

`Inherit: Same as first input`

Use data type of first input signal.

`double`

Accumulator data type is `double`.

`single`

Accumulator data type is `single`.

`int8`

Accumulator data type is `int8`.

`uint8`

Accumulator data type is `uint8`.

`int16`

Accumulator data type is `int16`.

`uint16`

Accumulator data type is `uint16`.

`int32`

Accumulator data type is `int32`.

`uint32`

Accumulator data type is `uint32`.

`fixdt(1,16,0)`

Accumulator data type is fixed point `fixdt(1,16,0)`.

`fixdt(1,16,2^0,0)`

Accumulator data type is fixed point `fixdt(1,16,2^0,0)`.

`<data type expression>`

The name of a data type object, for example `Simulink.NumericType`

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Mode

Select the category of accumulator data to specify

#### Settings

Default: `Inherit`

`Inherit`

Specifies inheritance rules for data types. Selecting `Inherit` enables a list of possible values:

• `Inherit via internal rule` (default)

• `Same as first input`

`Built in`

Specifies built-in data types. Selecting ```Built in``` enables a list of possible values:

• `double` (default)

• `single`

• `int8`

• `uint8`

• `int16`

• `uint16`

• `int32`

• `uint32`

`Fixed point`

Specifies fixed-point data types.

`Expression`

Specifies expressions that evaluate to data types. Selecting `Expression` enables you to enter an expression.

#### Dependency

Clicking the Show data type assistant button for the accumulator data type enables this parameter.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Data type override

Specify data type override mode for this signal.

#### Settings

Default: `Inherit`

`Inherit`

Inherits the data type override setting from its context, that is, from the block, `Simulink.Signal` object or Stateflow® chart in Simulink that is using the signal.

`Off`

Ignores the data type override setting of its context and uses the fixed-point data type specified for the signal.

#### Tip

The ability to turn off data type override for an individual data type provides greater control over the data types in your model when you apply data type override. For example, you can use this option to ensure that data types meet the requirements of downstream blocks regardless of the data type override setting.

#### Dependency

This parameter appears only when the Mode is ```Built in``` or `Fixed point`.

### Signedness

Specify whether you want the fixed-point data to be signed or unsigned.

#### Settings

Default: `Signed`

`Signed`

Specify the fixed-point data to be signed.

`Unsigned`

Specify the fixed-point data to be unsigned.

#### Dependencies

Selecting Mode > ```Fixed point``` for the accumulator data type enables this parameter.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Word length

Specify the bit size of the word that will hold the quantized integer.

#### Settings

Default: `16`

Minimum: `0`

Maximum: `32`

Large word sizes represent large values with greater precision than small word sizes.

#### Dependencies

Selecting Mode > ```Fixed point``` for the accumulator data type enables this parameter.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Scaling

Specify the method for scaling your fixed-point data to avoid overflow conditions and minimize quantization errors.

#### Settings

Default: ```Binary point```

`Binary point`

Specify binary point location.

`Slope and bias`

Enter slope and bias.

#### Dependencies

Selecting Mode > ```Fixed point``` for the accumulator data type enables this parameter.

Selecting `Binary point` enables:

• Fraction length

Selecting `Slope and bias` enables:

• Slope

• Bias

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Fraction length

Specify fraction length for fixed-point data type.

#### Settings

Default: `0`

Binary points can be positive or negative integers.

#### Dependencies

Selecting Scaling > ```Binary point``` for the accumulator data type enables this parameter.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Slope

Specify slope for the fixed-point data type.

#### Settings

Default: `2^0`

Specify any positive real number.

#### Dependencies

Selecting Scaling > ```Slope and bias``` for the accumulator data type enables this parameter.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Bias

Specify bias for the fixed-point data type.

#### Settings

Default: `0`

Specify any real number.

#### Dependencies

Selecting Scaling > ```Slope and bias``` for the accumulator data type enables this parameter.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Output minimum

Lower value of the output range that Simulink checks.

#### Settings

Default: `[]` (unspecified)

Specify this number as a finite, real, double, scalar value.

Simulink uses the minimum to perform:

 Note:   Output minimum does not saturate or clip the actual output signal. Use the Saturation block instead.

#### Command-Line Information

 Parameter: `OutMin` Type: string Value: `'[ ]'` Default: `'[ ]'`

### Output maximum

Upper value of the output range that Simulink checks.

#### Settings

Default: `[]` (unspecified)

Specify this number as a finite, real, double, scalar value.

Simulink uses the maximum value to perform:

 Note:   Output maximum does not saturate or clip the actual output signal. Use the Saturation block instead.

#### Command-Line Information

 Parameter: `OutMax` Type: string Value: `'[ ]'` Default: `'[ ]'`

### Output data type

Specify the output data type.

#### Settings

Default: ```Inherit: Inherit via internal rule```

`Inherit: Inherit via internal rule`

Simulink chooses a data type to balance numerical accuracy, performance, and generated code size, while taking into account the properties of the embedded target hardware. If you change the embedded target settings, the data type selected by the internal rule might change. It is not always possible for the software to optimize code efficiency and numerical accuracy at the same time. If the internal rule doesn't meet your specific needs for numerical accuracy or performance, use one of the following options:

• Specify the output data type explicitly.

• Use the simple choice of ```Inherit: Same as first input```.

• Explicitly specify a default data type such as `fixdt(1,32,16)` and then use the Fixed-Point Tool to propose data types for your model. For more information, see `fxptdlg`.

• To specify your own inheritance rule, use ```Inherit: Inherit via back propagation``` and then use a Data Type Propagation block. Examples of how to use this block are available in the Signal Attributes library Data Type Propagation Examples block.

 Note:   The accumulator internal rule favors greater numerical accuracy, possibly at the cost of less efficient generated code. To get the same accuracy for the output, set the output data type to ```Inherit: Inherit same as accumulator```.
`Inherit: Inherit via back propagation`

Use data type of the driving block.

`Inherit: Same as first input`

Use data type of first input signal.

`Inherit: Same as accumulator`

Output data type is the same as accumulator data type.

`double`

Output data type is `double`.

`single`

Output data type is `single`.

`int8`

Output data type is `int8`.

`uint8`

Output data type is `uint8`.

`int16`

Output data type is `int16`.

`uint16`

Output data type is `uint16`.

`int32`

Output data type is `int32`.

`uint32`

Output data type is `uint32`.

`fixdt(1,16,0)`

Output data type is fixed point `fixdt(1,16,0)`.

`fixdt(1,16,2^0,0)`

Output data type is fixed point `fixdt(1,16,2^0,0)`.

`<data type expression>`

Use a data type object, for example, `Simulink.NumericType`.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Mode

Select the category of data to specify.

#### Settings

Default: `Inherit`

`Inherit`

Inheritance rules for data types. Selecting `Inherit` enables a second menu/text box to the right. Select one of the following choices:

• `Inherit via internal rule` (default)

• `Inherit via back propagation`

• `Same as first input`

• `Same as accumulator`

`Built in`

Built-in data types. Selecting `Built in` enables a second menu/text box to the right. Select one of the following choices:

• `double` (default)

• `single`

• `int8`

• `uint8`

• `int16`

• `uint16`

• `int32`

• `uint32`

`Fixed point`

Fixed-point data types.

`Expression`

Expressions that evaluate to data types. Selecting `Expression` enables a second menu/text box to the right, where you can enter the expression.

#### Dependency

Clicking the Show data type assistant button enables this parameter.

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Signedness

Specify whether you want the fixed-point data as signed or unsigned.

#### Settings

Default: `Signed`

`Signed`

Specify the fixed-point data as signed.

`Unsigned`

Specify the fixed-point data as unsigned.

#### Dependencies

Selecting Mode > ```Fixed point``` enables this parameter.

### Word length

Specify the bit size of the word that holds the quantized integer.

#### Settings

Default: `16`

Minimum: `0`

Maximum: `32`

#### Dependencies

Selecting Mode > ```Fixed point``` enables this parameter.

### Scaling

Specify the method for scaling your fixed-point data to avoid overflow conditions and minimize quantization errors.

#### Settings

Default: ```Binary point```

`Binary point`

Specify binary point location.

`Slope and bias`

Enter slope and bias.

#### Dependencies

Selecting Mode > ```Fixed point``` enables this parameter.

Selecting `Binary point` enables:

• Fraction length

• Calculate Best-Precision Scaling

Selecting `Slope and bias` enables:

• Slope

• Bias

• Calculate Best-Precision Scaling

#### Command-Line Information

See Block-Specific Parameters for the command-line information.

### Fraction length

Specify fraction length for fixed-point data type.

#### Settings

Default: `0`

Binary points can be positive or negative integers.

#### Dependencies

Selecting Scaling > ```Binary point``` enables this parameter.

### Slope

Specify slope for the fixed-point data type.

#### Settings

Default: `2^0`

Specify any positive real number.

#### Dependencies

Selecting Scaling > ```Slope and bias``` enables this parameter.

### Bias

Specify bias for the fixed-point data type.

#### Settings

Default: `0`

Specify any real number.

#### Dependencies

Selecting Scaling > ```Slope and bias``` enables this parameter.

## Examples

### How the Sum Block Reorders Inputs

If you use `-` on the first input port, the Sum block reorders the inputs so that, if possible, the first input uses a `+` operation. For example, in the expression ```output = -a-b+c```, the Sum block reorders the inputs so that ```output = c-a-b```. To initialize the accumulator, the Sum block uses the first `+` input port.

The block avoids performing a unary minus operation on the first operand `a` because doing so can change the value of `a` for fixed-point data types. In that case, the output value differs from the result of accumulating the values for `a`, `b`, and `c`.

 Tip   To explicitly specify a unary minus operation for ```output = -a-b+c```, you can use the Unary Minus block in the Math Operations library.

Suppose that you have the following model:

The following block parameters apply:

• Both Constant blocks, `Input1` and ```Input 2```, use `int8` for the Output data type.

• The Sum block uses `int8` for both Accumulator data type and Output data type.

• The Sum block has Saturate on integer overflow turned `on`.

The Sum block reorders the inputs so that the following operations occur and you get the ideal result of 127.

StepBlock Operation
1

Reorders inputs from `(–Input1 + Input2)` to ```(Input2 – Input1)```.

2

Initializes the accumulator by using the first `+` input port:

`Accumulator = int8(-1) = -1`

3

Continues to accumulate values:

```Accumulator = Accumulator – int8(-128) = 127```

4

Calculates the block output:

```Output = int8(127) = 127```

If the Sum block does not reorder the inputs, the following operations occur instead and you get the nonideal result of 126.

StepBlock Operation
1

Initializes the accumulator by using the first input port:

`Accumulator = int8(-(-128)) = 127`

Because saturation is on, the initial value of the accumulator saturates at 127 and does not wrap.

2

Continues to accumulate values:

```Accumulator = Accumulator + int8(-1) = 126```

3

Calculates the block output:

```Output = int8(126) = 126```

## Characteristics

 Data Types Double | Single | Boolean | Base Integer | Fixed-Point Sample Time Inherited from driving block Direct Feedthrough Yes Multidimensional Signals Yes Variable-Size Signals Yes Zero-Crossing Detection No Code Generation Yes


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Indico search will be reestablished in the next version upgrade of the software: https://getindico.io/roadmap/

# Neutrino 2020

June 22, 2020 to July 2, 2020
US/Central timezone

## Background model of the CUPID-Mo 0$\nu\beta\beta$ experiment

Not scheduled
10m
Poster

### Speaker

Dr Pia Loaiza (IJCLab, CNRS, Université Paris Saclay)

### Description

CUPID-Mo is searching for neutrinoless double beta decay with enriched Li$_{2}$$^{100}$MoO$_4$ scintillating crystals, operating at the Modane underground laboratory. During the first physics runs, started at the beginning of 2019, CUPID-Mo collected an exposure of more than 2 kg.y. The data validates the high level of radiopurity of the crystals and the excellent alpha-particle rejection.
In this work we present a model to reconstruct the CUPID-Mo energy spectra. The model is built from the Monte Carlo simulations of the background sources and exploits their distinctive signatures.
The investigation of the background sources in CUPID-Mo is precious to finely tune the construction choices of the next-generation experiment CUPID.

### Mini-abstract

We present a background model to reconstruct the CUPID-Mo energy spectra.

Experiment/Collaboration CUPID-Mo

### Primary author

Dr Pia Loaiza (IJCLab, CNRS, Université Paris Saclay)

format: plain


text:
## Functiones et Approximatio Commentarii Mathematici

### On spectral large sieve inequalities

Matti Jutila

#### Abstract

The spectral large sieve inequality due to H. Iwaniec, that is an estimate for the mean square over a spectral interval of a linear form in the Fourier coefficients of Maass wave forms is reported by use of a formula of Y. Motohashi and the “hybrid” large sieve inequality. Then the result is generalized to the case, where the coefficients of the linear form may depend on the respective eigenvalue of the hyperbolic Laplacian, and also on a well-spaced set of points.

#### Note

Dedicated to Włodzimierz Staś on the occasion of his 75th birthday

#### Article information

Source
Funct. Approx. Comment. Math., Volume 28 (2000), 7-18.

Dates
First available in Project Euclid: 29 September 2018

https://projecteuclid.org/euclid.facm/1538186680

Digital Object Identifier
doi:10.7169/facm/1538186680

Mathematical Reviews number (MathSciNet)
MR1823989

Zentralblatt MATH identifier
1007.11027

Subjects
Primary: llF30
Secondary: l1F72

Keywords
spectral theory large sieve

#### Citation

Jutila, Matti. On spectral large sieve inequalities. Funct. Approx. Comment. Math. 28 (2000), 7--18. doi:10.7169/facm/1538186680. https://projecteuclid.org/euclid.facm/1538186680


text:
# Create Song Books in XeLaTeX

Recently, I decided to digitise my songbook (lyrics with chords annotated) using the TeX typesetting system. TeX provides amazing packages for typesetting songbooks. Since my collection had mostly Malayalam songs, XeLaTeX was my choice of TeX engine which has better support for non-Roman scripts – like Malayalam, Tamil etc.

There are several TeX packages available for typesetting song lyrics and chord books. I am using the songbook package which is fairly advanced in its features. The following gist is a sample source code for generating lyrics of a famous Malayalam Melody – “തേരിറങ്ങും മുകിലെ”, annotated with the chords.

Running the command xelatex songbook.tex produces a PDF that uses Rachana font released by Swathanthra Malayalam Computing.


text:
• ### A Review of Statistical Methods in Imaging Genetics(1707.07332)

March 6, 2018 stat.ME
With the rapid growth of modern technology, many large-scale biomedical studies have been/are being/will be conducted to collect massive datasets with large volumes of multi-modality imaging, genetic, neurocognitive, and clinical information from increasingly large cohorts. Simultaneously extracting and integrating rich and diverse heterogeneous information in neuroimaging and/or genomics from these big datasets could transform our understanding of how genetic variants impact brain structure and function, cognitive function, and brain-related disease risk across the lifespan. Such understanding is critical for diagnosis, prevention, and treatment of numerous complex brain-related disorders (e.g., schizophrenia and Alzheimer). However, the development of analytical methods for the joint analysis of both high-dimensional imaging phenotypes and high-dimensional genetic data, called big data squared (BD$^2$), presents major computational and theoretical challenges for existing analytical methods. Besides the high-dimensional nature of BD$^2$, various neuroimaging measures often exhibit strong spatial smoothness and dependence and genetic markers may have a natural dependence structure arising from linkage disequilibrium. We review some recent developments of various statistical techniques for the joint analysis of BD$^2$, including massive univariate and voxel-wise approaches, reduced rank regression, mixture models, and group sparse multi-task regression. By doing so, we hope that this review may encourage others in the statistical community to enter into this new and exciting field of research.
• ### CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data(1708.03749)

Aug. 12, 2017 math.PR
This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form $\mathbf{B}_n=n^{-1}\sum_{j=1}^{n}\mathbf{Q}\mathbf{x}_j\mathbf{x}_j^{*}\mathbf{Q}^{*}$ where $\mathbf{Q}$ is a nonrandom matrix of dimension $p\times k$, and $\{\mathbf{x}_j\}$ is a sequence of independent $k$-dimensional random vector with independent entries, under the assumption that $p/n\to y>0$. A key novelty here is that the dimension $k\ge p$ can be arbitrary, possibly infinity. This new model of sample covariance matrices $\mathbf{B}_n$ covers most of the known models as its special cases. For example, standard sample covariance matrices are obtained with $k=p$ and $\mathbf{Q}=\mathbf{T}_n^{1/2}$ for some positive definite Hermitian matrix $\mathbf{T}_n$. Also with $k=\infty$ our model covers the case of repeated linear processes considered in recent high-dimensional time series literature. The CLT found in this paper substantially generalizes the seminal CLT in Bai and Silverstein (2004). Applications of this new CLT are proposed for testing the structure of a high-dimensional covariance matrix. The derived tests are then used to analyse a large fMRI data set regarding its temporary correlation structure.
• ### Bayesian analysis of ambulatory blood pressure dynamics with application to irregularly spaced sparse data(1511.05372)

Jan. 10, 2017 stat.AP
Ambulatory cardiovascular (CV) measurements provide valuable insights into individuals' health conditions in "real-life," everyday settings. Current methods of modeling ambulatory CV data do not consider the dynamic characteristics of the full data set and their relationships with covariates such as caffeine use and stress. We propose a stochastic differential equation (SDE) in the form of a dual nonlinear Ornstein--Uhlenbeck (OU) model with person-specific covariates to capture the morning surge and nighttime dipping dynamics of ambulatory CV data. To circumvent the data analytic constraint that empirical measurements are typically collected at irregular and much larger time intervals than those evaluated in simulation studies of SDEs, we adopt a Bayesian approach with a regularized Brownian Bridge sampler (RBBS) and an efficient multiresolution (MR) algorithm to fit the proposed SDE. The MR algorithm can produce more efficient MCMC samples that is crucial for valid parameter estimation and inference. Using this model and algorithm to data from the Duke Behavioral Investigation of Hypertension Study, results indicate that age, caffeine intake, gender and race have effects on distinct dynamic characteristics of the participants' CV trajectories.
• ### Bayesian spatial transformation models with applications in neuroimaging data(1607.07664)

July 26, 2016 stat.AP
The aim of this paper is to develop a class of spatial transformation models (STM) to spatially model the varying association between imaging measures in a three-dimensional (3D) volume (or 2D surface) and a set of covariates. Our STMs include a varying Box-Cox transformation model for dealing with the issue of non-Gaussian distributed imaging data and a Gaussian Markov Random Field model for incorporating spatial smoothness of the imaging data. Posterior computation proceeds via an efficient Markov chain Monte Carlo algorithm. Simulations and real data analysis demonstrate that the STM significantly outperforms the voxel-wise linear model with Gaussian noise in recovering meaningful geometric patterns. Our STM is able to reveal important brain regions with morphological changes in children with attention deficit hyperactivity disorder.
• ### Local Region Sparse Learning for Image-on-Scalar Regression(1605.08501)

May 27, 2016 cs.LG, stat.ML
Identification of regions of interest (ROI) associated with certain disease has a great impact on public health. Imposing sparsity of pixel values and extracting active regions simultaneously greatly complicate the image analysis. We address these challenges by introducing a novel region-selection penalty in the framework of image-on-scalar regression. Our penalty combines the Smoothly Clipped Absolute Deviation (SCAD) regularization, enforcing sparsity, and the SCAD of total variation (TV) regularization, enforcing spatial contiguity, into one group, which segments contiguous spatial regions against zero-valued background. Efficient algorithm is based on the alternative direction method of multipliers (ADMM) which decomposes the non-convex problem into two iterative optimization problems with explicit solutions. Another virtue of the proposed method is that a divide and conquer learning algorithm is developed, thereby allowing scaling to large images. Several examples are presented and the experimental results are compared with other state-of-the-art approaches.
• ### Extrinsic local regression on manifold-valued data(1508.02201)

Aug. 10, 2015 math.ST, stat.TH
We propose an extrinsic regression framework for modeling data with manifold valued responses and Euclidean predictors. Regression with manifold responses has wide applications in shape analysis, neuroscience, medical imaging and many other areas. Our approach embeds the manifold where the responses lie onto a higher dimensional Euclidean space, obtains a local regression estimate in that space, and then projects this estimate back onto the image of the manifold. Outside the regression setting both intrinsic and extrinsic approaches have been proposed for modeling i.i.d manifold-valued data. However, to our knowledge our work is the first to take an extrinsic approach to the regression problem. The proposed extrinsic regression framework is general, computationally efficient and theoretically appealing. Asymptotic distributions and convergence rates of the extrinsic regression estimates are derived and a large class of examples are considered indicating the wide applicability of our approach.
• ### TPRM: Tensor partition regression models with applications in imaging biomarker detection(1505.05482)

May 20, 2015 stat.AP
Many neuroimaging studies have collected ultra-high dimensional imaging data in order to identify imaging biomarkers that are related to normal biological processes, diseases, and the response to treatment, among many others. These imaging data are often represented in the form of a multi-dimensional array, called a tensor. Existing statistical methods are insufficient for analysis of these tensor data due to their ultra-high dimensionality as well as their complex structure. The aim of this paper is to develop a tensor partition regression modeling (TPRM) framework to establish an association between low-dimensional clinical outcomes and ultra-high dimensional tensor covariates. Our TPRM is a hierarchical model and efficiently integrates four components: (i) a partition model; (ii) a canonical polyadic decomposition model; (iii) a factor model; and (iv) a generalized linear model. Under this framework, ultra-high dimensionality is not only reduced to a manageable level, resulting inefficient estimation, but also prediction accuracy is optimized to search for informative sub-tensors. Posterior computation proceeds via an efficient Markov chain Monte Carlo algorithm. Simulation shows that TPRM outperforms several other competing methods. We apply TPRM to predict disease status (Alzheimer versus control) by using structural magnetic resonance imaging data obtained from Alzheimer's Disease Neuroimaging Initiative (ADNI) study.
• ### Multiscale adaptive smoothing models for the hemodynamic response function in fMRI(1312.5859)

Dec. 20, 2013 stat.AP
In the event-related functional magnetic resonance imaging (fMRI) data analysis, there is an extensive interest in accurately and robustly estimating the hemodynamic response function (HRF) and its associated statistics (e.g., the magnitude and duration of the activation). Most methods to date are developed in the time domain and they have utilized almost exclusively the temporal information of fMRI data without accounting for the spatial information. The aim of this paper is to develop a multiscale adaptive smoothing model (MASM) in the frequency domain by integrating the spatial and frequency information to adaptively and accurately estimate HRFs pertaining to each stimulus sequence across all voxels in a three-dimensional (3D) volume. We use two sets of simulation studies and a real data set to examine the finite sample performance of MASM in estimating HRFs. Our real and simulated data analyses confirm that MASM outperforms several other state-of-the-art methods, such as the smooth finite impulse response (sFIR) model.
• ### Spatially Varying Coefficient Model for Neuroimaging Data with Jump Discontinuities(1310.1183)

Oct. 22, 2013 stat.ME
Motivated by recent work on studying massive imaging data in various neuroimaging studies, we propose a novel spatially varying coefficient model (SVCM) to spatially model the varying association between imaging measures in a three-dimensional (3D) volume (or 2D surface) with a set of covariates. Two key features of most neuorimaging data are the presence of multiple piecewise smooth regions with unknown edges and jumps and substantial spatial correlations. To specifically account for these two features, SVCM includes a measurement model with multiple varying coefficient functions, a jumping surface model for each varying coefficient function, and a functional principal component model. We develop a three-stage estimation procedure to simultaneously estimate the varying coefficient functions and the spatial correlations. The estimation procedure includes a fast multiscale adaptive estimation and testing procedure to independently estimate each varying coefficient function, while preserving its edges among different piecewise-smooth regions. We systematically investigate the asymptotic properties (e.g., consistency and asymptotic normality) of the multiscale adaptive parameter estimates. We also establish the uniform convergence rate of the estimated spatial covariance function and its associated eigenvalue and eigenfunctions. Our Monte Carlo simulation and real data analysis have confirmed the excellent performance of SVCM.
• ### Varying coefficient model for modeling diffusion tensors along white matter tracts(1304.4439)

April 16, 2013 stat.AP
Diffusion tensor imaging provides important information on tissue structure and orientation of fiber tracts in brain white matter in vivo. It results in diffusion tensors, which are $3\times3$ symmetric positive definite (SPD) matrices, along fiber bundles. This paper develops a functional data analysis framework to model diffusion tensors along fiber tracts as functional data in a Riemannian manifold with a set of covariates of interest, such as age and gender. We propose a statistical model with varying coefficient functions to characterize the dynamic association between functional SPD matrix-valued responses and covariates. We calculate weighted least squares estimators of the varying coefficient functions for the log-Euclidean metric in the space of SPD matrices. We also develop a global test statistic to test specific hypotheses about these coefficient functions and construct their simultaneous confidence bands. Simulated data are further used to examine the finite sample performance of the estimated varying coefficient functions. We apply our model to study potential gender differences and find a statistically significant aspect of the development of diffusion tensors along the right internal capsule tract in a clinical study of neurodevelopment.
• ### Surprising Asymptotic Conical Structure in Critical Sample Eigen-Directions(1303.6171)

March 25, 2013 math.ST, stat.TH
The aim of this paper is to establish several deep theoretical properties of principal component analysis for multiple-component spike covariance models. Our new results reveal a surprising asymptotic conical structure in critical sample eigendirections under the spike models with distinguishable (or indistinguishable) eigenvalues, when the sample size and/or the number of variables (or dimension) tend to infinity. The consistency of the sample eigenvectors relative to their population counterparts is determined by the ratio between the dimension and the product of the sample size with the spike size. When this ratio converges to a nonzero constant, the sample eigenvector converges to a cone, with a certain angle to its corresponding population eigenvector.In the High Dimension, Low Sample Size case, the angle between the sample eigenvector and its population counterpart converges to a limiting distribution.Several generalizations of the multi-spike covariance models are also explored, and additional theoretical results are presented.
• ### Multivariate varying coefficient model for functional responses(1302.4211)

Feb. 18, 2013 math.ST, stat.TH
Motivated by recent work studying massive imaging data in the neuroimaging literature, we propose multivariate varying coefficient models (MVCM) for modeling the relation between multiple functional responses and a set of covariates. We develop several statistical inference procedures for MVCM and systematically study their theoretical properties. We first establish the weak convergence of the local linear estimate of coefficient functions, as well as its asymptotic bias and variance, and then we derive asymptotic bias and mean integrated squared error of smoothed individual functions and their uniform convergence rate. We establish the uniform convergence rate of the estimated covariance function of the individual functions and its associated eigenvalue and eigenfunctions. We propose a global test for linear hypotheses of varying coefficient functions, and derive its asymptotic distribution under the null hypothesis. We also propose a simultaneous confidence band for each individual effect curve. We conduct Monte Carlo simulation to examine the finite-sample performance of the proposed procedures. We apply MVCM to investigate the development of white matter diffusivities along the genu tract of the corpus callosum in a clinical study of neurodevelopment.
• ### High Dimensional Principal Component Scores and Data Visualization(1211.2679)

Nov. 20, 2012 stat.AP
Principal component analysis is a useful dimension reduction and data visualization method. However, in high dimension, low sample size asymptotic contexts, where the sample size is fixed and the dimension goes to infinity,a paradox has arisen. In particular, despite the useful real data insights commonly obtained from principal component score visualization, these scores are not consistent even when the sample eigen-vectors are consistent. This paradox is resolved by asymptotic study of the ratio between the sample and population principal component scores. In particular, it is seen that this proportion converges to a non-degenerate random variable. The realization is the same for each data point, i.e. there is a common random rescaling, which appears for each eigen-direction. This then gives inconsistent axis labels for the standard scores plot, yet the relative positions of the points (typically the main visual content) are consistent. This paradox disappears when the sample size goes to infinity.
• ### Perturbation and scaled Cook's distance(1204.0064)

June 7, 2012 math.ST, stat.TH, stat.ME
Cook's distance [Technometrics 19 (1977) 15-18] is one of the most important diagnostic tools for detecting influential individual or subsets of observations in linear regression for cross-sectional data. However, for many complex data structures (e.g., longitudinal data), no rigorous approach has been developed to address a fundamental issue: deleting subsets with different numbers of observations introduces different degrees of perturbation to the current model fitted to the data, and the magnitude of Cook's distance is associated with the degree of the perturbation. The aim of this paper is to address this issue in general parametric models with complex data structures. We propose a new quantity for measuring the degree of the perturbation introduced by deleting a subset. We use stochastic ordering to quantify the stochastic relationship between the degree of the perturbation and the magnitude of Cook's distance. We develop several scaled Cook's distances to resolve the comparison of Cook's distance for different subset deletions. Theoretical and numerical examples are examined to highlight the broad spectrum of applications of these scaled Cook's distances in a formal influence analysis.
• ### Tensor Regression with Applications in Neuroimaging Data Analysis(1203.3209)

March 14, 2012 stat.ME
Classical regression methods treat covariates as a vector and estimate a corresponding vector of regression coefficients. Modern applications in medical imaging generate covariates of more complex form such as multidimensional arrays (tensors). Traditional statistical and computational methods are proving insufficient for analysis of these high-throughput data due to their ultrahigh dimensionality as well as complex structure. In this article, we propose a new family of tensor regression models that efficiently exploit the special structure of tensor covariates. Under this framework, ultrahigh dimensionality is reduced to a manageable level, resulting in efficient estimation and prediction. A fast and highly scalable estimation algorithm is proposed for maximum likelihood estimation and its associated asymptotic properties are studied. Effectiveness of the new methods is demonstrated on both synthetic and real MRI imaging data.
• ### Two-stage empirical likelihood for longitudinal neuroimaging data(1108.2334)

Aug. 11, 2011 stat.AP
Longitudinal imaging studies are essential to understanding the neural development of neuropsychiatric disorders, substance use disorders, and the normal brain. The main objective of this paper is to develop a two-stage adjusted exponentially tilted empirical likelihood (TETEL) for the spatial analysis of neuroimaging data from longitudinal studies. The TETEL method as a frequentist approach allows us to efficiently analyze longitudinal data without modeling temporal correlation and to classify different time-dependent covariate types. To account for spatial dependence, the TETEL method developed here specifically combines all the data in the closest neighborhood of each voxel (or pixel) on a 3-dimensional (3D) volume (or 2D surface) with appropriate weights to calculate adaptive parameter estimates and adaptive test statistics. Simulation studies are used to examine the finite sample performance of the adjusted exponential tilted likelihood ratio statistic and TETEL. We demonstrate the application of our statistical methods to the detection of the difference in the morphological changes of the hippocampus across time between schizophrenia patients and healthy subjects in a longitudinal schizophrenia study.
• ### Perturbation selection and influence measures in local influence analysis(0803.2986)

March 20, 2008 math.ST, stat.TH
Cook's [J. Roy. Statist. Soc. Ser. B 48 (1986) 133--169] local influence approach based on normal curvature is an important diagnostic tool for assessing local influence of minor perturbations to a statistical model. However, no rigorous approach has been developed to address two fundamental issues: the selection of an appropriate perturbation and the development of influence measures for objective functions at a point with a nonzero first derivative. The aim of this paper is to develop a differential--geometrical framework of a perturbation model (called the perturbation manifold) and utilize associated metric tensor and affine curvatures to resolve these issues. We will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model. Moreover, we will introduce new influence measures that are applicable to objective functions at any point. Examples including linear regression models and linear mixed models are examined to demonstrate the effectiveness of using new influence measures for the identification of influential observations.
• ### Generalized score test of homogeneity for mixed effects models(math/0608028)

Aug. 1, 2006 math.ST, stat.TH
Many important problems in psychology and biomedical studies require testing for overdispersion, correlation and heterogeneity in mixed effects and latent variable models, and score tests are particularly useful for this purpose. But the existing testing procedures depend on restrictive assumptions. In this paper we propose a class of test statistics based on a general mixed effects model to test the homogeneity hypothesis that all of the variance components are zero. Under some mild conditions, not only do we derive asymptotic distributions of the test statistics, but also propose a resampling procedure for approximating their asymptotic distributions conditional on the observed data. To overcome the technical challenge, we establish an invariance principle for random quadratic forms indexed by a parameter. A simulation study is conducted to investigate the empirical performance of the test statistics. A real data set is analyzed to illustrate the application of our theoretical results.


text:
# How to solve for the roots of a 4th degree polynomial with complex coefficients?

Given the following equation

$c_4Z^4+c_3Z^3+c_2Z^2+(c_1+\frac{i}{\beta})Z^1-\frac{i}{\beta} =0$

where $Z$ lies in the complex domain. Each of the coefficients, $c_n$ for $n=1:4$, are real and can be either positive or negative. They are known, but left as variables here for the sake of generality. My objective is to determine expressions for each of the 4 roots and plot them versus $\beta$.

I have done this numerically using Matlab's roots function and got the correct results, but in order to plot each of the roots as a function of $\beta$ in a meaningful way, I have to sort them to appear as continuous curves or plot them as points in a scatter plot. Either way, it takes way too long to explore the full complex domain for $\beta$. This is a whole other issue, so don't lose focus here.

That being said, I'm wondering if I can solve for each of the roots in the form of an analytic expression as a function of $\beta$? If so, I could program each of the expressions as anonymous functions and save a bunch of time exploring the complex domain for $\beta$.

Unfortunately, my subpar math skills have left me scratching my head on how to proceed. I've looked at a bunch of different examples/tricks on determining the roots for 4th degree polynomials, but they all seem to apply to real polynomials with real coefficients. In the polynomial I've expressed above, $\beta$ can be complex. So again, my question is, can I solve for each of the roots in the form of an analytic expression as a function of $\beta$ and if so, how do I proceed given the complex nature of the polynomial?

• both α and β can be complex Where is $\alpha$? – dxiv Oct 13 '16 at 6:08
• You can, but it's going to be very, very messy and tedious. Ferrari's method for solving the general quartic applies with complex coefficients too. You might want to look into using a Computerised Algebra System, although I'm unsure if it can do this. – Deepak Oct 13 '16 at 6:10
• @Deepak Writing the roots as analytic expressions is not necessarily the same as solving by radicals. I imagine OP is looking for something along the lines of Puiseux series although I don't know enough about that to answer – user376902 Oct 13 '16 at 6:13
• @dxiv $Z=e^{i\alpha}$ – ThatsRightJack Oct 13 '16 at 6:18
• Ah, thanks, missed that somehow. Though I don't see how $Z=e^{i \alpha} \;\;|\;\; \alpha \in \mathbb{C}$ adds anything vs. simply $Z \in \mathbb{C}$ in the context of this quartic. – dxiv Oct 13 '16 at 6:22

Here is a solution to the general quartic compactified. Given,

$$Ax^4+Bx^3+Cx^2+Dx+E = 0$$

divide by $A$ to get the simpler,

$$x^4+ax^3+bx^2+cx+d=0$$

Then the four roots are,

$$x_{1,2} = -\frac{a}{4}+\frac{\color{red}\pm\sqrt{u}}{2}\color{blue}+\frac{1}{4}\sqrt{3a^2-8b-4u+\frac{-a^3+4ab-8c}{\color{red}\pm\sqrt{u}}}\tag1$$

$$x_{3,4} = -\frac{a}{4}+\frac{\color{red}\pm\sqrt{u}}{2}\color{blue}-\frac{1}{4}\sqrt{3a^2-8b-4u+\frac{-a^3+4ab-8c}{\color{red}\pm\sqrt{u}}}\tag2$$

where,

$$u = \frac{a^2}{4}-\frac{2b}{3} +\frac{1}{3}\left(v_1^{1/3}\zeta_3+\frac{b^2 - 3 a c + 12 d}{v_1^{1/3}\zeta_3}\right)$$

with $v_1$ any non-zero root of the quadratic,

$$v^2 + (-2 b^3 + 9 a b c - 27 c^2 - 27 a^2 d + 72 b d)v + (b^2 - 3 a c + 12 d)^3 = 0$$

and a chosen cube root of unity $\zeta_3^3 = 1$ such that $u$ is also non-zero. (Normally, just use $\zeta_3=1$, but not when $a^3-4ab+8c = 0$.)

P.S. This is essentially the method used by Mathematica, though much simplified for aesthetics.

• Do you already have this programmed and readily available in Mathematica? I tried to plot it in Matlab (using symbolic math) and did not receive the same solution as I did before using their internal "roots" function. One of us made an error copying and I just want to give a word of caution. – ThatsRightJack Oct 17 '16 at 23:58
• @ThatsRightJack: I re-implemented it by copying from scratch the solution above and got a correct result. What's the $a,b,c,d$ of the test quartic you used and what did you get for $u,v,x$? – Tito Piezas III Oct 18 '16 at 2:16
• @ThatsRightJack: For example, using $x^4 - 2 x^3 + 5x^2 - 11x + 19=0$, I got $u\approx4.7769,\;v\approx-1130.5-2293.7246i,\,x_1\approx1.5928+1.1588i$. What values do you get? – Tito Piezas III Oct 18 '16 at 2:22
• Hmmm....now I'm really scratching my head. I got the same answer you did for your test case. Let me take a closer look at it again tomorrow and follow up with more details regarding the known coeffs and solution for comparison purposes. – ThatsRightJack Oct 18 '16 at 3:55
• Just to be clear, this general solution does hold for complex coeffs? If you notice in my question, D and E (or after dividing through by A, c and d) are complex. Just wanted to be sure....again, I'll look closer tomorrow and get back to you. – ThatsRightJack Oct 18 '16 at 4:13

First depress the equation by translating the variable $$x\to x-\dfrac a4$$, giving

$$x^4+px^2+qx+r=0.$$

Now factor this in two quadratic polynomials in such a way that the cubic term is missing (mind the reuse of the coefficient names)

$$(x^2+ax+b)(x^2-ax+c)=x^4+(-a^2+b+c)x^2+a(c-b)x+bc.$$

By identification,

$$\begin{cases}-a^2+b+c=p\implies ab+ac=a^3+pa,\\ac-ab=q,\\bc=r\end{cases}.$$

From the first two equations, you draw $$ab,ac$$ in terms of $$a$$ and plug in the third.

$$\left(a^3+pa-q\right)\left(a^3+pa+q\right)=4ra^2,$$

giving by magic a cubic equation in $$a^2$$,

$$(a^2)^3+2p(a^2)^2+(p^2-4r)a^2-q^2=0.$$

Once solved for $$a$$, you get $$b,c$$ and solve the two quadrics.

For the cubic $$x^3+ax^2+bx+c$$, first depress by the change $$x\to x-\dfrac a3$$ to the form

$$x^3+px+q=0.$$

Then set

$$x=u-\frac{p}{3u}$$ so that by substitution

$$u^3+q-\frac{p^3}{27u^3}=0$$ or

$$(u^3)^2+qu^3-\frac{p^3}{27}=0.$$

Solve the quadratic for $$u^3$$ and you are... done.


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anonymous 4 years ago Find the distance between the points (2, -3) and (5, -4)

• This Question is Closed
1. hartnn

Distance between points (x1,y1) and (x2,y2) is $$\huge d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$

2. anonymous

show work please

3. hartnn

work needs to be done by you....

4. mayankdevnani

|dw:1353937644917:dw|

5. anonymous

i need help and if u arent going to help then bye and be rude.

6. TranceNova

Hi Rebecca - the aim of openstudy is to teach, which means people aren't going to give you answers, and you need to interact to find the answer. hartnn provided a formula to find the answer - if you don't understand it, please ask for clarification.

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## Logistic Regression

Preliminaries ‘An Introduction to Probabilistic Generative Models for Linear Classification’ Idea of logistic regression1 Logistic sigmoid function(logistic function for short) had been introduced in post ‘An Introduction to Probabilistic Generative Models for Linear Classification’. It has an elegant form: $\delta(a)=\frac{1}{1+e^{-a}}\tag{1}$ and when $$a=0$$, $$\delta(a)=\frac{1}{2}$$ and this is just the half of the range of logistic function. This gives us a strong implication that we can set $$a$$ equals to some functions $$y(\mathbf{x})$$, and then...

February 20, 2020 · (Last Modification: April 28, 2022) · Anthony Tan

## An Introduction to Probabilistic Generative Models

Preliminaries Probability Bayesian Formular Calculus Probabilistic Generative Models1 The generative model used for making decisions contains an inference step and a decision step: Inference step is to calculate $$\Pr(\mathcal{C}_k|\mathbf{x})$$ which means the probability of $$\mathbf{x}$$ belonging to the class $$\mathcal{C}_k$$ given $$\mathbf{x}$$ Decision step is to make a decision based on $$\Pr(\mathcal{C}_k|\mathbf{x})$$ which was calculated in step 1 In this post, we just give an introduction and a framework for the probabilistic generative model in classification....

February 20, 2020 · (Last Modification: April 28, 2022) · Anthony Tan


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# GRE Math : Proportion / Ratio / Rate

## Example Questions

← Previous 1 3 4 5 6 7

### Example Question #1 : Proportion / Ratio / Rate

A solution made up of  alcohol by volume is mixed with  liters of solution that is  alcohol by volume. How much, in liters, of the  alcoholic solution is needed to make a mixture that is  alcohol by volume?

Explanation:

Let  represent the number of liters of the 40% solution. Then it follows that  liters of the 40% solution plus 4 liters of the 10% solution will equal (x+4) liters of a 25% solution. This can be represented by the following equation:

Now solve for x:

You will need 4 liters of the 40% solution in order to make a mixture that is 25% alcohol by volume

### Example Question #1 : Proportion / Ratio / Rate

A solution is  parts water,  parts wine, and  part honey.  If a container of this solution contains  gallons of water, how much total solution is there in it?

gallons

gallons

gallons

gallons

gallons

gallons

Explanation:

To begin, notice that there is a  ratio between the water in your container and the water specified by the mix of the components.  Given that there are  total parts in your solution, this means that you can set up this equation:

Multiplying both sides by 8, you get:

There are  total gallons of solution.

### Example Question #3 : Proportion / Ratio / Rate

A solution is made up of  parts water,  parts orange juice, and  parts apple juice.  If you wanted the percentage of orange juice to be % of the solution, how many parts would you need to add?

Explanation:

To begin, you know that the basic form of the solution has a total of , or  parts.  Now, we know that we are going to have to add  parts of orange juice.  This means that the new solution will have  parts orange juice and  total parts (since we are adding to the original).  Since we want this to be % orange juice, we really want the following equation to be true:

Solve the following equation, therefore:

Multipy by :

Now, isolate :

### Example Question #1 : Proportion / Ratio / Rate

Flour, eggs, sugar, and chocolate chips are mixed by weight in the ratio of 12 : 5 : 3 : 5, respectively. How many pounds of chocolate chips are there in 75 pounds of the mixture?

15

25

18

5

15

Explanation:

First, add up the four parts of the ratio. This equals 25 parts. These 25 parts make up the 75 pound mixture, which means the 75 pound mixture is composed of 3 times the 25 parts (25x = 75 so x = 3).

This allows you to know that the number of pounds of chocolate chips is 3 times the ratio, i.e. 3 * 5 = 15. The mixture includes 15 pounds of chocolate chips. The answer is 15.

### Example Question #1 : Proportion / Ratio / Rate

If there are 120 men and women on a committee, and the ratio of men to women is 5 : 1, how many more men are on the committee than women?

80

72

100

96

80

Explanation:

The best approach is to add the numerator and denominator of the ratio (5 + 1 = 6), and then divide the total by that sum (120/6 = 20). This gives you the value for 1 part when the total is divided into 6 parts and, luckily for us, there is only 1 part women for every 5 parts men. Now set up equal ratios: men/women = 5/1 = x/20. Solve for the number of men by cross multiplying (5 * 20 = 1x). So there are 100 men and 20 women, which makes 80 more men than women.

### Example Question #2 : Proportion / Ratio / Rate

You have a rope of some length, but 2/3rds of it is cut off and thrown away.  1/4th of the remaining rope is cut off and thrown away. What proportion of the original rope remains?

1/4

1/6

1/10

1/3

The answer cannot be determined from the given information.

1/4

Explanation:

If 2/3 is cut off and thrown away, that means 1/3 of the original length remains. Of this, 1/4 gets cut off and thrown away, meaning 3/4 of 1/3 still remains. Multiplying 3/4 * 1/3, we get 1/4.

### Example Question #1 : Proportion / Ratio / Rate

If the sum of a, b and c is 400, and a is 1/3 b and b is  1/4 c, what is the value of a?

25

20

300

66

33

25

Explanation:

Explanation: For this type of problem, build an equation that represents the relationships between the quantities and solve for the quantitiy you need. The problem states that b=1/4c and a = 1/3b. Thus, a = 1/3(1/4)c, or 1/12 of c. Now put everything in terms of c, thus 1/12c + 1/4c +c = 400. Now comes the tricky step--combine like terms and create the improper fraction (1/12c + 3/12c + 12/12c = 16/12c). Reduce the fraction to 4/3. So 400 is 4/3 of c. Thus c is 3/4 of 400, or 300. a is 1/12 of 300, or 25.

### Example Question #1 : Proportion / Ratio / Rate

Erin went to the movies with her friends.  She spent 1/4 of her allowance on movie tickets and 3/5 of the remaining money on popcorn.  If her allowance is $10, how much money remains? Possible Answers:$5.00

$8.50$1.50

$3.00$7.00

$3.00 Explanation: Since she spent 1/4 on the ticket, 3/5 of the remaining 3/4 of the money was spent on popcorn: 3/5 x 3/4 = 9/20. This means 9/20 of the money was spent on popcorn so in total: 1/4 + 9/20 = 14/20 = 7/10 of her money was spent. This leaves 3/10 of her money behind: 3/10x 10 = 3.00. ### Example Question #2 : How To Find Proportion Fudge sells at$18.50 for 5 pounds.  What is the cost for 2 pounds?

$5.50$6.75

$3.70$9.25

$7.40 Correct answer:$7.40

Explanation:

Set up a proportion: 18.50/5 = x/2.  Cross multiply and solve for x: 37 = 5x.... x = 7.40.

### Example Question #1 : Proportion / Ratio / Rate

3 men can paint 3 rooms in 3 hours. How long would it take 1 man to paint 1 room?

2.5 hours

3 hours

1.5 hours

2 hours

1 hour


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## Introduction to Algorithms(Asymptotic Analysis)-Part 3

·  ☕ 3 min read  ·  🤖 Arjit Sharma

# Practise problems on Asymptotic Analysis

## Finding complexity of code

1. Statements : Statements have constant complexity.

 if(n>0){ # c1
printf("Something"); # c2
}else{ # c3
printf("Something else"); # c4
}

$T(n)=c1+c2+c3+c4=O(c)=O(1)$

2. Loops : Check number of times the loop is running

 for(i=1;i≤n;++i){
//Statment; # c1
}

$T(n)=c1*n = O(n)$

3. Nested Loops : Look at how many times each loop execute and multiply them.

 for(i=0;i≤m;++i){ # Runs m times
for(j=0;j≤n;++j){ # Runs n times
//Statment; # c1
}
}

$T(n)=O(m*n)$

4. Log Complexity :

 for(i=1;i≤n;i*=2){
Statement;
}

i=2,4,8,16…k
$2^k=n$
Taking log both side
$log 2^k = log n$ ⇒ $k=log n$

$T(n)=O(log n)$

5. Recursive functions :

 int fibo(int n){
if(n==0)
return 0;

else if(n==1)
return 1;

else
return fibo(n-1)+fibo(n-2)
}

$T(n)=T(n-1)+T(n-2)+c$
for larger value of n
T(n) = 2T(n-1) + O(1)
Using Master Theorem for Subtract and Conquer ⇒ Case 3 satisfies

$T(n) = O(n^02^n)=O(2^n)$

## Practise problems on Code complexity

Master Theorem questions can be done directly. Following are the question that use some sort of trick.

### Ques 1 :

    void func(int n){
int i=1,s=1;
while(s≤n){
i++;
s+=i;
printf("Hello World");
}
}

Soln →

$s_i=s_\left( i-1\right) + i$ where i increments 1,2,3…k
it takes k iteration to get out of loop
1+2+3….+k > n
$\left(k(k+1) \right)/2 > n$

$T(n)=O(\sqrt{n})$

### Ques 2 :

    void func(int n){
int i,count=0;
for(i=1;i*i≤n;++i){
count++;
}
}

Soln →

Loop exits if $i^2>n$ ⇒ $T(n)=O(\sqrt{n})$

### Ques 3 :

    func(int n){
if(n==1) return;
for(i=1;i≤n;++i) // n times
for(j=1;j≤n;++j) // n times
printf("Hello World");

func(n-3); // Subtract and conquer ?
}

Soln →

We can see in the problem that there is recursive call so there is a possibility it can be solved by Master Theorem.
We make recursive eqn by following 2 steps :
step 1: Time taken by each recursive call ⇒ $n^2$
step 2: How much Input size was reduced ⇒ T(n) to T(n-3)

Recursive Eqn = $T(n)=T(n-3)+cn^2$

Now using Master theorem Subtract and Conquer Case 2
$T(n) = O(n^3)$

### Ques 4:

    for (i=1;i≤n;++i){ // n times
for(j=1;j≤n;kj+=i){ // n/i times
//Statment
}
}

Soln→

$n(\sum n/i) = O(nlogn)$

### Ques 5:

    func(n){
if(n≤1) return; // c1
for(i=1;i≤3;++i){ // Divided into 3 subproblems
func(n/3);  //problem Input size reduced by n/3
}
}

Soln→

T(n)=3T(n/3) + O(1)
Master Theorem Case 1 satisfies
$T(n)=O(n^\left( log_33 \right)))=O(n)$

### Ques 6:

    func(n){
if(n≤1) return;
for(i=1;i≤3;++i){ // Divided into 3 subproblems
func(n-1); // Input size reduced by 1
}
}

Soln→

T(n)=3T(n-1)+O(1)

Master Theroem Subtract and Conquer Case 3 satisfies

$O(n^03^n)=O(3^n)$

### Ques 7:

Find complexity of $T(n)=2T(\sqrt{n})+logn$ .

Soln →

It is not in form of Master Theorem
These type of problems uses iteratively subsituting a value to reach complexity
Assume $n=2^m$

Taking log both sides we get ⇒ log n = m log2 ⇒ m=log n
$T(n) = 2T(2\left( m/2 \right)) + logn$

To simplify let a function $S(m)=T(2^m)$
Then $S(m/2)=T(m/2)$
$S(m) = S(m/2)+ m$

Using Master Theroem ⇒ $S(m) = O(mlogm)$

We know m=log n
$T(n)=O(log n log logn)$

Summary of what we did →

step 1: Transform n in form of m (because in form of n it was not solvable with Master Theorem)
step 2: Take a func S(m) and relate to T(m)
step 3: Find complexity of function S(m)
step 4: Subsitute value of m with n

### Ques 8:

Find complexity of $T(n)=T(\sqrt{n})+1$ .

Soln →

It is not in form of Master Theorem
These type of problems uses iteratively subsituting a value to reach complexity
Assume $n=2^m$

Taking log both sides we get ⇒ log n = m log2 ⇒ m=log n
$T(n) = T(2\left( m/2 \right)) + 1$

To simplify let a function $S(m)=T(2^m)$
Then $S(m/2)=T(m/2)$
$S(m) = S(m/2)+ 1$

Using Master Theroem ⇒
$S(m) = O(mlog_21logm)$ ⇒ $S(m) = O(m^0logm)$ ⇒ $S(m) = O(logm)$

We know m=log n
$T(n)=O(log logn)$

Referrences

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WRITTEN BY
Arjit Sharma
Yo, I am a CS enthusiast or am I ? Just kidding.


text:
# Inspired by Li Hang

Geometry Level 5

There are 1 circle and 2 semicircles inscribed in a square with side length 2. If the radius of the smaller semicircle is $$r$$, find $$r$$.

×


text:
# All Factors Divide Integer then Whole Divides Integer

## Theorem

Let $S = \set {a_1, a_2, \ldots, a_r} \subseteq \Z$ be a finite subset of the integers.

Let $S$ be pairwise coprime.

Let:

$\forall j \in \set {1, 2, \ldots, r}: a_r \divides b$

where $\divides$ denotes divisibility.

Then:

$\displaystyle \prod_{j \mathop = 1}^r a_j \divides b$

## Proof

Proof by induction:

In the following, it is assumed at all times that $S = \set {a_1, a_2, \ldots, a_r} \subseteq \Z$ is pairwise coprime.

For all $r \in \N_{> 1}$, let $\map P r$ be the proposition:

$\displaystyle \prod_{j \mathop = 1}^r a_j \divides b$

### Basis for the Induction

$\map P 2$ is the case:

$a_1 a_2 \divides b$

By definition of divisibility:

$b = a_1 q_1 = a_2 q_2$

where $q_1, q_2 \in \Z$.

$\exists x, y \in \Z: a_1 x + a_2 y = 1$

So:

 $\displaystyle b$ $=$ $\displaystyle a_2 q_2 \cdot a_1 x + a_1 q_1 \cdot a_2 y$ $\displaystyle$ $=$ $\displaystyle a_1 a_2 \paren {q_2 x + q_1 y}$

and so $a_1 a_2 \divides b$.

This is our basis for the induction.

### Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:

$\displaystyle \prod_{j \mathop = 1}^k a_j \divides b$

Then we need to show:

$\displaystyle \prod_{j \mathop = 1}^{k + 1} a_j \divides b$

### Induction Step

This is our induction step:

$a_{k + 1} \divides b$
$\displaystyle a_{k + 1} \perp \prod_{j \mathop = 1}^k a_j$

By the induction hypothesis:

$\displaystyle \prod_{j \mathop = 1}^k a_j \divides b$

So by the basis for the induction:

$\displaystyle \prod_{j \mathop = 1}^{k + 1} a_j \divides b$

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore: $\displaystyle r \in \N_{> 1}: \prod_{j \mathop = 1}^r a_j \divides b$

$\blacksquare$


text:
# Fisher-Snedecor distribution

The Fisher-Snedecor distribution (also known as F-distribution) is a probability distribution defined on the range $[0, \infty]$. The Fisher-Snedecor distribution with parameters $p_{1}$  and $p_{2}$  arises as the ratio of two Chi-square variables with resp. $p_{1}$  and $p_{2}$  degrees of freedom.

$\begin{matrix} &f(x) = \dfrac{\sqrt{\dfrac{(p_1\,x)^{p_1}\,\,p_2^{p_2}} {(p_1\,x+p_2)^{p_1+p_2}}}} {x\,\mathrm{B}\!\left(\dfrac{p_1}{2},\dfrac{p_2}{2}\right)}\ \\ &F(x)=I_{\dfrac{p_1 x}{p_1 x + p_2}}(p_1/2, p_2/2) \\ \end{matrix}$

where $I$  is the regularised incomplete beta function.

$\mu = \frac{p_2}{p_2-2}$ for $p_2 > 2$

$\sigma = \sqrt{\dfrac{2\,p_2^2\,(p_1+p_2-2)}{p_1 (p_2-2)^2 (p_2-4)}}$ for $p_2 > 4$


text:
# Relation between inlet and outlet pressure in a classical capillary tube

I'm looking into capillary flow in a tube as explained in this link. The system consists of two liquid phases, one wetting and one non-wetting, inside a tube,

What is the relation between the inlet pressure $p_i$ and the outlet pressure $p_r$? I would personally expect that $p_i>p_r$ due to the capillary forces that establish a pressure gradient.

EDIT: In the figure below the point B denotes the tube above and the point A the reservoir that the tube is "attached" to. Say that the velocity in reservoir A is nonzero -- then by Bernoulli's law, should the pressure in reservoir A be lower than that in tube B?

The purpose of the experiment is to find the unknown capillary pressure $p_c=p_w-p_{nw}$ from measured quantities : pressure difference $\Delta p=p_l-p_r$, flow rate $u$, viscosity $\mu$ and length $d$ and radius $r$ of the tube. The pressures $p_l$ and $p_r$ are variables which you control in the experiment.
$p_r$ would usually be atmospheric pressure $P_0$. $p_l$ is the pressure applied to the liquid by a pump or a constant 'head' of liquid - ie $\rho gh+P_0$.
• But say that this is a pore inside a piece of rock or chalk and that the outlet is, e.g., a reservoir. Then what would the relation be? I'd expect $p_I>p_R$ still in the case of a solid boundary that is water-wet, since that is the only way for the capillary forces to "push" the non-wetting fluid out – BillyJean Nov 30 '16 at 14:00
• $p_l$ and $p_r$ are the fluid pressures at the left and right sides of the tube. I do not think it would be easy to apply this experiment to a pore inside a rock. $p_l$ is easy : it is as given in the last sentence of my answer, where $h$ is the depth of the reservoir. But if the air or gas on the other side of the pore is not open to the air, $p_r$ is not atmospheric pressure $P_0$. I think it would be difficult to know what $p_r$ is in that case. If you knew $p_c$ and all other variables, you could deduce $p_r$. But what use would that be? – sammy gerbil Nov 30 '16 at 14:09
• If the liquid flows left to right then $p_l>p_r$. But you could have $p_l<p_r$. Then the liquid flows right to left and the interface is concave instead of convex. – sammy gerbil Nov 30 '16 at 14:11
• I see, that is a good point. As a thought experiment, I now consider there to be a velocity in the reservoir attached to the outlet (see the OP) - then by Bernoulli's law, can we now state that $p_l>p_r$? – BillyJean Nov 30 '16 at 14:53
• Sorry I don't understand how your 2nd figure relates to the 1st. If you know that the liquid is flowing left to right, then it follows that $p_l > p_r$. You don't need Bernoulli's Law to tell you this. – sammy gerbil Nov 30 '16 at 15:22


text:
Conference item

### Time−Bounded Verification of CTMCs Against Real−Time Specifications

Abstract:

Abstract. In this paper we study time-bounded verification of a finite continuous-timeMarkov chain (CTMC) C against a real-time specification, provided either as a metric temporal logic (MTL) property 'φ', or as a timed automaton (TA) A. The key question is: what is the probability of the set of timed paths of C that satisfy 'φ' (or are accepted by A) over a time interval of fixed, bounded length? We provide approximation algorithms to solve these problems. We first derive a bound N such that...

### Authors

Publisher:
Springer
Publication date:
2011-01-01
URN:
Local pid:
cs:5085


text:
arrow_back Find the negatives of (a) $u=(3,-5,-8,4)$, (b) $v=(-4,2 \pi, 0)$, (c) $0=(0,0,0,0)$.

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Find the negatives of

(a) $u=(3,-5,-8,4)$,
(b) $v=(-4,2 \pi, 0)$,
(c) $0=(0,0,0,0)$.

Take the negative of each component of the vector.

$(a)-u=(-3,-(-5),-(-8),-4)=(-3,5,8,-4) ;$

(b) $-v=(4,-2 \pi, 0) ;$

(c) $-0=(-0,-0,-0,-0)=(0,0,0,0)=\boldsymbol{\theta}$.
by Bronze Status
(5,384 points)

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text:
# Tetration Forum

Full Version: Universal uniqueness criterion?
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Pages: 1 2 3 4 5 6
As I now see, even if the uniqueness criterion is valid, the precondition:
$\lim_{n\to\infty} \exp_b^{\circ n}(z)=\infty$ for each $z\in f(S)$
can not be true for any $f$ for $b=e$ (and probably for each $b>e^{1/e}$)!

Michał Misiurewicz [1] showed 1981 that the Julia set of $\exp$ is the whole $\mathbb{C}$ (that is not directly visible from the $e^z$ fractal). The Julia set is the boundary of the set $K$ of all $z$ such that $\lim_{n\to\infty} \exp^{\circ n}(z)=\infty$. That means that the set $K$ and its complement $\mathbb{C}\setminus K$ is dense in $\mathbb{C}$. In every neighborhood of any complex number $z_0$ there is a complex number $z$ such that $\lim_{n\to\infty} \exp^{\circ n}(z)=\infty$ and also a $w$ such that $\lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty$!

And that implies that $f(S)$, which contains an open non-empty set, always contains points $w$ such that $\lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty$.

[1] Michał Misiurewicz (1981). On iterates of e^z. Ergodic Theory and Dynamical Systems, 1 , pp 103-106, doi:10.1017/S014338570000119X
bo198214 Wrote:...
(U2) $\lim_{n\to\infty}\exp_b^{\circ n}(z)= \infty$ for all $z\in f(S)$ ...

$f(\delta(z))=\exp_b^{{\delta_Z(z)}} (f({\delta_S(z)})$
...
$
\infty=\lim_{m\to\infty} \exp_b^{\circ \delta_Z(w_m)}( c)
...$

Do you make any difference between $\exp_b^{\circ p}(q)$ and $\exp_b^{p}(q)$ ?
bo198214 Wrote:...
And that implies that $f(S)$, which contains an open non-empty set, always contains points $w$ such that $\lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty$.
...
Why do you think that this set is open? Wiki says:
In metric topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U. In other words, the distance between any point x in U and the edge of U is always greater than zero.
http://en.wikipedia.org/wiki/Open_set

The small deviation from a point where the limit is finite allows to get (in the limit) so high walues as you want, because the right hand side of the plot of tetration is pretty scratched. I would replace "open non-empty set" to "non-empty set".
Uniqueness of analytic tetration, scheme of the proof.

Notations.
Let $\mathbb{C}$ be set of complex numbers.
Let $\mathbb{R}$ be set of real numbers.
Let $\mathbb{N}$ be set of integer numbers.
Let $\mathbb{C}^{\prime}=\mathbb{C} \backslash \{ z| z \in \mathbb{R}, z\le 2 \}$
Let $b>1$ be base of tetration.
Assume that there exist analytic tetration $F$ on base $b$, id est,
(0) $F$ is analytic at $\mathbb{C}^{\prime}$
(1) for all $z\in \mathbb{C}^{\prime}$, the relation $F(z+1)=\exp_b(F(z))$ holds
(2) $F(0)=1$
(3) $F$ is real increasing function at $\{z \in \mathbb{R}| z>-2 \}$.

Properties.
From assumption (1) and (2) it follows, that function $F$ has singularity at -2, at -3 and so on.

Consider following Assumption:
There exist entire 1-periodic function $h$ such that
$G(z)=F(z+h(z))$ is also analytic tetration on base $b$.

Then, function $I(z)=z+h(z)$ is not allowed to take values -2, -3, ..
being evaluated at elements of $\mathbb{C}^{\prime}$.

This means that function
$J(z)=I(z) \frac{z+2}{I(z)+2} =(z+h(z)) \frac{z+2}{z+h(z)+2}$
is entire function.

(Weak statement which seems to be true)
Function $J$ cannot grow faster than linear function at infinity in any direction.

Therefore, it is linear function. Therefore, $h=0$

Therefore, there exist only one analytic tetration.
Kouznetsov Wrote:
bo198214 Wrote:...
And that implies that $f(S)$, which contains an open non-empty set, always contains points $w$ such that $\lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty$.
...
Why do you think that this set is open?

Just take a point $w$ of the interior of $S$ such that $f'(w)\neq 0$ (if such a point does not exist then $f$ is constant), then there is a neighborhood $W$ of $w$ such that $f$ is bijective on $W$. That means $f$ and $f^{-1}$ are continuous (because analytic). And the preimage of an open set is open under a continuous map. So the preimage $f(W)$ of $W$ under $f^{-1}$ is open and so we find arbitrary many points $z$ in $f(W)\subseteq f(S)$ such that $\exp^{\circ n}(z)\not\to\infty$.

Kouznetsov Wrote:Do you make any difference between $\exp_b^{\circ p}(q)$ and $\exp_b^{p}(q)$ ?

Oh no its both the same here.
Kouznetsov Wrote:Consider following Assumption:
There exist entire 1-periodic function $h$ such that
$G(z)=F(z+h(z))$ is also analytic tetration on base $b$.

Then, function $I(z)=z+h(z)$ is not allowed to take values -2, -3, ..
being evaluated at elements of $\mathbb{C}^{\prime}$.

But $I(z)$ does not need to be entire, it can also omit the values -2, -3 , ... as arguments.
For an entire function we would know that it can omit at most 1 value (Little Picard).

Edit: Oh I see now $I(z)$ needs not to be entire, but can be continued being entire by $I(z\pm n)=I(z)\pm n$ if $I=f^{-1}\circ g$ can be/is defined on the strip $S$.
Ok, I give it another try:

Proposition.
Let $S=\{z:0<\Re(z)\le 1\}$, $S_\epsilon=\{z:0<\Re(z)<1+\epsilon\}$, $\epsilon>0$ and $D\supseteq S_\epsilon$ being a domain (open and connected) of definition and $H\supseteq S_\epsilon$ being a domain (open and connected) of values for a holomorphic function.
Let $f$ be a holomorphic function on $D$, $G:=f(S_\epsilon)\subseteq H$, such that
(0) $H\subseteq f(D)$.
(1) $f(0)=1$
(2) $f(z+1)=F(f(z))$
(U) $f^{-1}(G)$ has bounded real part.
Then $g=f$ for every other on $D$ holomorphic $g$ satisfying (0), (1), (2), (U).

Proof.
$h(z):=g^{-1}(z)-f^{-1}(z)$ has bounded real part on $G$. We consider $f^{-1}$ and $g^{-1}$ and so $h$ to be holomorphic on the same Riemann surface $G$. $\delta(z):=h(f(z))=g^{-1}(f(z))-z$ is a 1-periodic function, holomorphic on $S_\epsilon$. As $S_\epsilon\supset S$ it can be continued to an entire function, so it has to take on every complex value with at most one exception already on the strip $S$ otherwise it is a constant. Now $\delta(S)=\delta(S_\epsilon)=h(G)$ has bounded real part and hence can not take on every value, so $h(z)=0$ and $g=f$.
bo198214 Wrote:Ok, I give it another try:
..
Bo, Congratulations! You seem to do well. I copypast your theorem into the latex source, but I shall not work on it until I collect all the pieces into the single file.
(I have not yet type the uniqueness of the "Children's factorial" you suggested). While, you generate results faster than I describe them. The draft becomes longer, but does not yet approach to the finish. If you want to work with the text, let me know; I send you the last version. Keep in touch, K.
(10/05/2008, 12:22 AM)Kouznetsov Wrote: [ -> ]Bo, Congratulations! You seem to do well...
The preprint is here: http://www.ils.uec.ac.jp/~dima/PAPERS/2009matan.pdf
Any comments and ctyrics is welcommed.
"We call a region H an initial region of F iff F(z) {is not in} H for all z {in} H..."
does this mean $\exists z : F(z) \notin H$ (there is SOME z such that F(z) is NOT in H) or $\forall z (F(z) \notin H$) (there is NO z such that F(z) IS in H)?
Pages: 1 2 3 4 5 6


text:
# Meaning of $\geqslant$, $\leqslant$, $\eqslantgtr$, $\eqslantless$

What do slanted inequality signs mean? Specifically, these are $\geqslant$, $\leqslant$; and the variation: $\eqslantgtr$, $\eqslantless$.

Is there any place I can look this up? I've searched Wikipedia and the web and can't find anything about them. The last two were found when looking up the first two.

$\geqslant$ is an alternative to $\geq$ and means the same: $a \geqslant b$ = "$a$ is greater than or equal to $b$".

Likewise, $\leqslant$ is an alternative notation for $\leq$, with $a \leqslant b$ = "$a$ is less than or equal to $b$".

I haven't encountered $\eqslantgtr$ or $\eqslantless$, but given that the former is formatted eqslantgtr and the latter eqslantless, I would venture to guess that they likewise denote "greater than or equal to" and "less than or equal to", respectively. Perhaps with these symbols, where the emphasis appears to be on the "equals" component, they are read as "equal to or greater than" and "equal to or less than", respectively.

• Thanks. The context in which I saw $\geqslant$ and $\leqslant$ did make it look like they meant what their unslanted counterparts do, but I couldn't tell for sure.
– anon
Nov 30 '12 at 1:36

In Russia $\geqslant$ and $\leqslant$ are used instead of $\geq$ and $\leq$.
We don't use $\geq$ and $\leq$ at all.
But I've never seen $\eqslantgtr$ and $\eqslantless$ in Russian math texts.

P.S.
1. There are more interesting distinctions between Russian and English math notations.
For example, we occasionally using $\ n\vdots d\$ instead of $\ d|n$.
2. There are a lot of strange math symbols exists: see here and here.

• in Vietnam we use ⩾ and ⋮ too, so it confuses me occasionally when seeing d | n. I think it's due to the influence of France. ≥ is only seen when typing Jul 30 '18 at 2:50

I haven't seen $\eqslantless$ or $\eqslantgtr$ used. A Google search for "eqslantless" turned up http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=3775&option_lang=eng but the actual paper (both the Russian original and the English translation) used $\leqslant$.

• I have: here. In particular the definitions of minimizing and maximizing orders. Of course I was forced by sheer shortage of still available symbols... ;-)
– WimC
Aug 28 '14 at 17:19

In France too, we use $\geqslant$ and $\leqslant$, at least in high school teaching.

The $\geq$ and $\leq$ signs are understandable though, and used by pocket calculators.

• In Vietnam ⩾ and ⩽ are also used. In fact most of the Vietnamese mathematics and physics are heavily influenced by France (like U for voltage, tg for tangent, sin for waveforms) and Russia (as old Communist allies). In the first half of the 20th century, most students have to learn French, and in the next quarter century they have to learn Russian. However recently the new textbooks changed to the more US-centric style (tg → tan, sin → cos for wave function) Jul 30 '18 at 3:01

Since people agreed with $\leq$ vs $\leqslant$ is a style issue. I checked a list of books from various disciplines,

the "$\leq$" group:

R. Shankar, Principle of Quantum Mechanics.

O.C. Zienkiewicz, et al. The Finite Element Method.

J. Munkres, Topology.

S. Boyd, et al. Convex Optimization.

S.M. Carroll, Spacetime and Geometry

R.W. Hamming, Numerical Methods for Scientists and Engineers

T. Sauer, Numerical Analysis.

R.A. Horn, et al. Matrix Analysis

G.H. Golub, et al. Matrix Computations

W. Fulton, et al. Representation Theory

T.W. Hungerford, Algebra

N. Jacobson, Basic Algebra

I. Goodfellow, Deep Learning

the "$\leqslant$" group:

D.V. Hutton, Fundamentals of Finite Element Analysis.

S.J. Farlow, Partial Differential Equations for Scientistes and Engineers

R.L. Graham, Concrete Mathematics

C.M. Bishop, Pattern Recognition and Machine Learning

It seems that most gurus preferred "$\leq$".

• it depends on the gurus' language. It seems the slanted version is preferred in European countries and their colonies Jul 30 '18 at 2:53
• I encountered the slanted version in The C Programming Language second edition by B. Kernighan and D. Ritchie, which led me to research and end up here. Aug 10 '18 at 20:46

As other answers have remarked, $$\eqslantless$$ and $$\eqslantgtr$$ are rarely seen, and the difference between $$\leqslant$$ and $$\leq$$, and similarly between $$\geqslant$$ and $$\geq$$, is just a matter of style. I would like to add that $$\leqslant$$ and $$\geqslant$$ were the preferred standard of many publishers in the UK and the US until recent decades when the direct setting of authors' LaTeX files took over in books and journals. In LaTeX, the default characters are $$\leq$$ and $$\geq$$, the slanted versions requiring an extra five keystrokes each for the author to type.

Personally, I prefer the aesthetics of the slanted version, to the extent that I am willing to type the extra letters.


text:
## Journal of Symbolic Logic

### The Church-Rosser property in dual combinatory logic

Katalin Bimbó

#### Abstract

Dual combinators emerge from the aim of assigning formulas containing $\leftarrow$ as types to combinators. This paper investigates formally some of the properties of combinatory systems that include both combinators and dual combinators. Although the addition of dual combinators to a combinatory system does not affect the unique decomposition of terms, it turns out that some terms might be redexes in two ways (with a combinator as its head, and with a dual combinator as its head). We prove a general theorem stating that no dual combinatory system possesses the Church-Rosser property. Although the lack of confluence might be problematic in some cases, it is not a problem per se. In particular, we show that no damage is inflicted upon the structurally free logics, the system in which dual combinators first appeared.

#### Article information

Source
J. Symbolic Logic, Volume 68, Issue 1 (2003), 132-152.

Dates
First available in Project Euclid: 21 February 2003

https://projecteuclid.org/euclid.jsl/1045861508

Digital Object Identifier
doi:10.2178/jsl/1045861508

Mathematical Reviews number (MathSciNet)
MR1959314

Zentralblatt MATH identifier
1045.03017

#### Citation

Bimbó, Katalin. The Church-Rosser property in dual combinatory logic. J. Symbolic Logic 68 (2003), no. 1, 132--152. doi:10.2178/jsl/1045861508. https://projecteuclid.org/euclid.jsl/1045861508

#### References

• K. Bimbó Investigation into combinatory systems with dual combinators, Studia Logica, vol. 66 (2000), pp. 285--296.
• K. Bimbó and J. M. Dunn Two extensions of the structurally free logic $LC$, Logic Journal of IGPL, vol. 6 (1998), pp. 403--424.
• A. Church The calculi of lambda-conversion, 1st ed., Princeton University Press, Princeton,1941.
• H. B. Curry and R. Feys Combinatory logic, 1st ed., vol. I, North-Holland, Amsterdam,1958.
• H. B. Curry, J. R. Hindley, and J. P. Seldin Combinatory logic, vol. II, North-Holland, Amsterdam,1972.
• J. M. Dunn and R. K. Meyer Combinatory logic and structurally free logic, Logic Journal of IGPL, vol. 5 (1997), pp. 505--537.
• J. R. Hindley An abstract form of the Church-Rosser theorem, I, Journal of Symbolic Logic, vol. 34 (1969), pp. 545--560.
• J. R. Hindley and J. P. Seldin Introduction to combinators and $\lambda$-calculus, Cambridge University Press, Cambridge (UK),1986.
• S. C. Kleene Proof by cases in formal logic, Annals of Mathematics, vol. 35 (1934), pp. 529--544.
• J. Lambek From categorial grammar to bilinear logic, Substructural logics (K. Došen and Schroeder-Heister P., editors), Clarendon and Oxford University Press, Oxford (UK),1993, pp. 207--237.
• R. K. Meyer, K. Bimbó, and J. M. Dunn Dual combinators bite the dust, (abstract), Bulletin of Symbolic Logic, vol. 4 (1998), pp. 463--464.
• B. K. Rosen Tree-manipulating systems and Church-Rosser theorems, Journal of the Association for Computing Machinery, vol. 20 (1973), pp. 160--187.
• J. B. Rosser A mathematical logic without variables, Annals of Mathematics, vol. 2 (1936), pp. 127--150.
• J. Staples Church-Rosser theorems for replacement systems, Algebra and logic (J. N. Crossley, editor), Lecture Notes in Mathematics, vol. 450, Springer, Berlin,1975, pp. 291--307.


text:
# Start and Pause in a Streaming Plot using plotly

plotly has a very nice example about how to making a streaming plot in R. When I was working on the arduino package, I need this feature because I want to create a plotting panel where data will be plotted once collected. So I used that example and quickly created a streaming app.

Here is the core part from the example

p <- plot_ly(
y = c(rand(),rand(),rand()),
type = 'scatter',
mode = 'lines',
line = list(
color = '#25FEFD',
width = 3
)
) %>%
layout(yaxis = list(range = c(0,10)))

output$plot <- renderPlotly(p) observeEvent(input$button, {
while(TRUE){
Sys.sleep(1)
plotlyProxy("plot", session) %>%
plotlyProxyInvoke("extendTraces", list(y=list(list(rand()))), list(0))
}
})

(For the complete example, check out the original link.)

So basically this is done by creating a shiny app and use plotlyProxyInvoke to send additional data through the extendTraces track to a named plotly plot called “plot”. I’m very satisfied with the result.

However, I quickly run into a problem that I don’t know how to pause this streaming without stopping this app. Below is my first attempt.

observeEvent(input$start, { rv$state <- 1 - rv$state while(rv$state == 1){
Sys.sleep(1)
plotlyProxy("plot", session) %>%
plotlyProxyInvoke("extendTraces", list(y=list(list(rand()))), list(0))
}
})

Here, I created a reactiveValue in shiny and was hoping it could help me to controll the while loop. I mean, in theory, this mechanism looks like it will work, right? However, it turned out that I was underestimating the power of while. When we use to while or repeat to get the streaming feature, this iteration process completely blocks the current R thread, which was also used by the shiny app. In this case, there is no way for the shiny app to move around during the looping. One obvious solution is to create a child thread here but do we really need to get that complicated?

All of a sudden, I recalled a magical command in Shiny that Joe talked about during useR! 2016: invalidateLater(). This command “schedules the current reactive context to be invalidated in the given number of milliseconds.”. What does it mean? It will force a reactive object to be re-evaluated after a certain period of time. This is exactly what I need! So I made some modifacations to the previous code.

observeEvent(input$start, { rv$state <- 1 - rv$state }) observe({ invalidateLater(1) if (rv$state == 1) {
Sys.sleep(1)
plotlyProxy("plot", session) %>%
plotlyProxyInvoke("extendTraces", list(y=list(list(rand()))), list(0))
}
})

So, in the real example, I got

I can’t expect more!

You can find the real example in the arduino package.


text:
Volume 320 - 12th International Workshop on High-pT Physics in the RHIC/LHC Era (High-pT2017) - Direct photons, heavy flavor, quarkonia
Impact of the initial high gluon density on the prompt photon yield and $v2$ in heavy-ion collisions with magnetic fields
A. Ayala,* J. David Castaño-Yepes, C. Dominguez, L. Alberto Hernández, S. Hernández-Ortíz, M. Elena Tejeda-Yeomans
*corresponding author
Full text: pdf
Pre-published on: 2019 June 20
Published on: 2019 October 09
Abstract
We compute prompt photon production from gluon fusion in the presence of a magnetic field in semi-central relativistic heavy-ion collisions. The main ingredient is the treatment of the high gluon density at early stages of the collision where intense magnetic fields are also present. The magnetic field opens new channels for photon production that are forbidden otherwise. The elliptic flow coefficient $v_2$ is also computed. The calculation takes into account the saturation scale and phenomenological factors. Overall, the treatment gives a good description of the excess photon yield and $v_2$, particularly at low values of photon transverse momentum.
DOI: https://doi.org/10.22323/1.320.0024
Open Access


text:
# Electric field from a charged disk

1. Aug 17, 2013

### Bipolarity

This is rather strange and has been bugging me, from chapter 22 (Electric fields) of Resnick & Halliday:

So suppose you have a charged disk of radius $R$ (the disk has no thickness and charge is uniformly spread on the surface of the disk). If you are at a distance $z$ away from the center of the disk (and orthogonal to the disk), the net electric field you experience is:

$E = \frac{\displaystyle zσ}{\displaystyle 4ε_{0}}\int^{R}_{0}(z^{2}+r^{2})^{-3/2}(2r)dr$

Now if $z = 0$, then it is clear that the field will be 0. This makes sense, because if you are at the center of the disk, then symmetry will ensure that the field induced by all charge is cancelled out by the charge diametrically opposite to it on the disk.

But if you actually evaluate the integral, you get
$E = \frac{\displaystyle σ}{\displaystyle 2ε_{0}}(1- \frac{\displaystyle z}{\displaystyle \sqrt{z^{2}+R^{2}}} )$

Now the problem with this is that if you set z=0 here, you end up something nonzero! What is the issue here? Thanks!

BiP

2. Aug 17, 2013

### Astrum

The equation is only valid for $z > 0$

The electric field (and magnetic field of a surface current) due to a surface charge is discontinuous when you cross the surface.

3. Aug 17, 2013

### voko

Why do you think the integral is non-zero?

4. Aug 17, 2013

### Bipolarity

Plugging in z=0 for the second expression yields a nonzero value.

BiP

5. Aug 17, 2013

### WannabeNewton

Astrum answered the question, it isn't valid for $z = 0$.

6. Aug 17, 2013

### voko

When you integrate, you get $1/z - 1/\sqrt {R^2 + z^2}$. This is clearly only valid when $z > 0$. You can then talk about the limit of that at $z = 0$, but the limit of a function does not have to be equal to the value of the function.

In this case, the function is not continuous at $z = 0$. Physically this is because we are dealing with an idealized situation, where the charge is spread over a 2D domain. Obviously you can expect that the idealization works reasonably wellwhen you are far enough from the surface, so that its thickness (or non-thickness) can be neglected, but when you get very close you need a better physical model, so puzzling over why the mathematical device breaks down is rather pointless.