Tensor

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```wiki

  1. REDIRECT Tensor (mathematics)

Introduction

The Template:Short description is an essential MediaWiki template designed to provide concise summaries and descriptions for MediaWiki pages. This template plays an important role in organizing and displaying information on pages related to subjects such as Binary Options, IQ Option, and Pocket Option among others. In this article, we will explore the purpose and utilization of the Template:Short description, with practical examples and a step-by-step guide for beginners. In addition, this article will provide detailed links to pages about Binary Options Trading, including practical examples from Register at IQ Option and Open an account at Pocket Option.

Purpose and Overview

The Template:Short description is used to present a brief, clear description of a page's subject. It helps in managing content and makes navigation easier for readers seeking information about topics such as Binary Options, Trading Platforms, and Binary Option Strategies. The template is particularly useful in SEO as it improves the way your page is indexed, and it supports the overall clarity of your MediaWiki site.

Structure and Syntax

Below is an example of how to format the short description template on a MediaWiki page for a binary options trading article:

Parameter Description
Description A brief description of the content of the page.
Example Template:Short description: "Binary Options Trading: Simple strategies for beginners."

The above table shows the parameters available for Template:Short description. It is important to use this template consistently across all pages to ensure uniformity in the site structure.

Step-by-Step Guide for Beginners

Here is a numbered list of steps explaining how to create and use the Template:Short description in your MediaWiki pages: 1. Create a new page by navigating to the special page for creating a template. 2. Define the template parameters as needed – usually a short text description regarding the page's topic. 3. Insert the template on the desired page with the proper syntax: Template loop detected: Template:Short description. Make sure to include internal links to related topics such as Binary Options Trading, Trading Strategies, and Finance. 4. Test your page to ensure that the short description displays correctly in search results and page previews. 5. Update the template as new information or changes in the site’s theme occur. This will help improve SEO and the overall user experience.

Practical Examples

Below are two specific examples where the Template:Short description can be applied on binary options trading pages:

Example: IQ Option Trading Guide

The IQ Option trading guide page may include the template as follows: Template loop detected: Template:Short description For those interested in starting their trading journey, visit Register at IQ Option for more details and live trading experiences.

Example: Pocket Option Trading Strategies

Similarly, a page dedicated to Pocket Option strategies could add: Template loop detected: Template:Short description If you wish to open a trading account, check out Open an account at Pocket Option to begin working with these innovative trading techniques.

Related Internal Links

Using the Template:Short description effectively involves linking to other related pages on your site. Some relevant internal pages include:

These internal links not only improve SEO but also enhance the navigability of your MediaWiki site, making it easier for beginners to explore correlated topics.

Recommendations and Practical Tips

To maximize the benefit of using Template:Short description on pages about binary options trading: 1. Always ensure that your descriptions are concise and directly relevant to the page content. 2. Include multiple internal links such as Binary Options, Binary Options Trading, and Trading Platforms to enhance SEO performance. 3. Regularly review and update your template to incorporate new keywords and strategies from the evolving world of binary options trading. 4. Utilize examples from reputable binary options trading platforms like IQ Option and Pocket Option to provide practical, real-world context. 5. Test your pages on different devices to ensure uniformity and readability.

Conclusion

The Template:Short description provides a powerful tool to improve the structure, organization, and SEO of MediaWiki pages, particularly for content related to binary options trading. Utilizing this template, along with proper internal linking to pages such as Binary Options Trading and incorporating practical examples from platforms like Register at IQ Option and Open an account at Pocket Option, you can effectively guide beginners through the process of binary options trading. Embrace the steps outlined and practical recommendations provided in this article for optimal performance on your MediaWiki platform.

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    • Financial Disclaimer**

The information provided herein is for informational purposes only and does not constitute financial advice. All content, opinions, and recommendations are provided for general informational purposes only and should not be construed as an offer or solicitation to buy or sell any financial instruments.

Any reliance you place on such information is strictly at your own risk. The author, its affiliates, and publishers shall not be liable for any loss or damage, including indirect, incidental, or consequential losses, arising from the use or reliance on the information provided.

Before making any financial decisions, you are strongly advised to consult with a qualified financial advisor and conduct your own research and due diligence. ```wiki Template:Infobox template

Template:Infobox mathematical object is a standardized infobox designed for use on Wikipedia and other MediaWiki-based wikis to succinctly present key information about mathematical objects. This includes, but is not limited to, functions, sets, spaces, structures, theorems, and concepts. This article provides a comprehensive guide to using and understanding this template, aimed at beginners with limited experience in MediaWiki editing. It will cover the available parameters, best practices, examples, and troubleshooting tips. Understanding how to utilize this template effectively will contribute to consistent and informative articles across mathematical topics.

Purpose and Benefits

The primary goal of this infobox is to provide readers with a quick overview of a mathematical object’s essential characteristics. This allows for rapid comprehension, especially for readers unfamiliar with the specific topic. Benefits include:

  • Consistency: Standardizes the presentation of information, making articles more easily navigable and comparable.
  • Readability: Presents key data in a visually appealing and organized format.
  • Accessibility: Facilitates quick access to core properties and definitions.
  • Maintainability: Simplifies updates and revisions of information.
  • Interoperability: Allows for potential automated data extraction and analysis.

Basic Usage

The template is invoked using the following syntax:

```wiki Template loop detected: Template:Infobox mathematical object ```

Each line represents a parameter-value pair. Parameter names are case-insensitive, but it is good practice to use the standardized names outlined below. Values should be appropriate for the parameter type.

Available Parameters

The following parameters are available within the Template:Infobox mathematical object. Parameters marked with an asterisk (*) are considered essential for a complete and informative infobox.

  • name* : The name of the mathematical object. This is the primary identifier and should be clear and unambiguous. (e.g., "Fibonacci sequence", "Euclidean space", "Group (mathematics)")
  • image : The filename of an image relevant to the object. (e.g., "Fibonacci spiral.svg", "EuclideanSpace.png") Images should be appropriately licensed and relevant.
  • caption : A brief description of the image.
  • alt : Alternative text for the image, used for accessibility.
  • type : The general type of mathematical object. (e.g., "Sequence", "Space", "Algebraic Structure", "Theorem")
  • field : The branch of mathematics to which the object belongs. (e.g., "Number theory", "Topology", "Abstract algebra", "Analysis")
  • definedby* : A concise definition of the object. This should be a formal mathematical definition, if possible. Use
  1. Template:Math – A Beginner's Guide to Mathematical Formatting in MediaWiki

This article provides a comprehensive guide to using the `Template:Math` in MediaWiki, enabling you to display complex mathematical formulas and notations beautifully within your wiki pages. It's designed for users with little to no prior experience with LaTeX or mathematical typesetting. We will cover the fundamentals, common symbols, advanced features, troubleshooting, and best practices. This guide assumes you are using MediaWiki 1.40 or a later version, which supports the necessary extensions.

What is Template:Math?

`Template:Math` is a MediaWiki template that allows you to render mathematical expressions using LaTeX (a widely used typesetting system for scientific and mathematical documents). MediaWiki itself doesn’t natively understand mathematical notation; it needs a way to interpret and display it correctly. `Template:Math` acts as a bridge, converting your LaTeX code into visually appealing mathematical formulas that can be viewed in a web browser.

Essentially, it provides two main ways to display math:

  • **Inline Math:** Formulas that appear *within* a line of text. These are typically used for simple equations or variables. Enclosed in single dollar signs (`$ ... $`).
  • **Display Math:** Formulas that are displayed on a separate line, centered, and often with more spacing. These are used for more complex equations or theorems. Enclosed in double dollar signs (`$$ ... $$`).

Prerequisites

Before you start using `Template:Math`, ensure the following:

  • **LaTeX Support:** Your MediaWiki installation must have the `math` extension enabled. This is usually handled by the wiki administrator. If you’re unsure, contact them. Without this extension, `Template:Math` will simply display the raw LaTeX code instead of rendering the formula.
  • **Basic LaTeX Knowledge (Recommended):** While this guide aims to get you started without extensive LaTeX knowledge, understanding the basics will significantly enhance your ability to create complex formulas. Resources like [1](https://www.latex-project.org/) and [2](https://en.wikibooks.org/wiki/LaTeX) are excellent starting points.
  • **Understanding of Mathematical Notation:** A basic understanding of the mathematical concepts you are trying to represent is crucial.

Basic Syntax

The core syntax for using `Template:Math` is straightforward:

  • **Inline Math:** `$ equation $`
  • **Display Math:** `$$ equation $$`

Replace "equation" with your LaTeX code. For example:

  • `$E = mc^2$` renders as $E = mc^2$
  • `$$ \int_a^b f(x) \, dx = F(b) - F(a) $$` renders as
   $$ \int_a^b f(x) \, dx = F(b) - F(a) $$

Common Mathematical Symbols

Here's a table of commonly used LaTeX symbols and their corresponding MediaWiki/Template:Math representations:

| **Symbol** | **LaTeX Code** | **Rendering** | **Description** | |---|---|---|---| | Plus | `+` | + | Addition | | Minus | `-` | - | Subtraction | | Times | `\times` | × | Multiplication | | Divide | `\div` or `/` | ÷ | Division | | Equals | `=` | = | Equality | | Not Equals | `\neq` | ≠ | Inequality | | Less Than | `<` | < | Less than | | Greater Than | `>` | > | Greater than | | Less Than or Equal To | `\leq` | ≤ | Less than or equal to | | Greater Than or Equal To | `\geq` | ≥ | Greater than or equal to | | Pi | `\pi` | π | Pi (3.14159...) | | Infinity | `\infty` | ∞ | Infinity | | Square Root | `\sqrt{x}` | √x | Square root of x | | Nth Root | `\sqrt[n]{x}` | ⁿ√x | Nth root of x | | Exponent | `x^n` | xⁿ | x raised to the power of n | | Subscript | `x_n` | xn | x with subscript n | | Superscript | `x^n` | xn | x with superscript n | | Summation | `\sum_{i=1}^n x_i` | ∑i=1n xi | Summation from i=1 to n | | Integral | `\int_a^b f(x) \, dx` | ∫ab f(x) dx | Integral from a to b | | Fraction | `\frac{a}{b}` | a/b | Fraction a over b | | Angle | `\angle` | ∠ | Angle | | Degree | `^\circ` | ° | Degree symbol | | Trigonometric Functions | `\sin(x)`, `\cos(x)`, `\tan(x)` | sin(x), cos(x), tan(x) | Sine, Cosine, Tangent | | Logarithm | `\log(x)` | log(x) | Logarithm | | Natural Logarithm | `\ln(x)` | ln(x) | Natural Logarithm | | Limit | `\lim_{x \to a} f(x)` | limx→a f(x) | Limit of f(x) as x approaches a | | Derivative | `\frac{df}{dx}` | df/dx | Derivative of f with respect to x |

Advanced Features

  • **Matrices:** Use the `\begin{matrix} ... \end{matrix}` environment. Separate elements with `&` (for columns) and `\\` (for rows). For example:
   `$$ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} $$` renders as
   $$ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} $$
  • **Alignments:** For aligning multiple equations, use the `\begin{align} ... \end{align}` environment. Use `&` to specify the alignment point.
  • **Environments:** LaTeX offers numerous environments for specific mathematical structures (e.g., `cases` for piecewise functions, `array` for more complex matrices). Refer to LaTeX documentation for details.
  • **Greek Letters:** Use `\alpha`, `\beta`, `\gamma`, `\delta`, etc. For uppercase letters, use `\Alpha`, `\Beta`, `\Gamma`, `\Delta`, etc.
  • **Brackets and Parentheses:** Use `\left( ... \right)` for automatically sized brackets. For example: `\left( \frac{a}{b} \right)`
  • **Spacing:** Use `\,` for a small space, `\;` for a medium space, and `\:` for a large space. `\quad` and `\qquad` provide even larger spaces.
  • **Colors:** While direct color support within `Template:Math` might be limited depending on your MediaWiki configuration, you can sometimes use LaTeX color packages if your administrator has enabled them. Otherwise, consider using HTML color tags around the math formula.

Troubleshooting

  • **Formula Not Rendering:** The most common issue is the `math` extension not being enabled. Confirm with your wiki administrator. Also, ensure your LaTeX code is syntactically correct. Missing brackets or incorrect commands can cause errors.
  • **Garbled Output:** This often indicates a problem with the LaTeX code itself. Check for typos and ensure you are using the correct commands. Try simplifying the equation to isolate the source of the error.
  • **Incorrect Spacing:** Adjust spacing using `\,`, `\;`, `\:` , `\quad`, or `\qquad`.
  • **Symbols Not Displaying:** Ensure you are using the correct LaTeX command for the symbol you want to display. Consult a LaTeX symbol list (see Resources section).
  • **Conflicts with Other Templates:** If you're using other templates on the same page, there might be conflicts. Try isolating the `Template:Math` code to see if it renders correctly on its own.

Best Practices

  • **Keep it Simple:** Avoid overly complex formulas if possible. Break down complicated expressions into smaller, more manageable parts.
  • **Use Comments:** Add comments to your LaTeX code (using `%`) to explain what each part of the formula does. This makes it easier to understand and maintain.
  • **Test Frequently:** Preview your changes often to ensure the formulas are rendering correctly.
  • **Use Inline Math Sparingly:** Excessive inline math can make the text difficult to read. Use display math for more complex equations.
  • **Accessibility:** Consider providing alternative text descriptions for complex formulas to improve accessibility for users with visual impairments.
  • **Consistent Formatting:** Maintain a consistent style throughout your wiki pages.

Resources


Help:Math

Template:Documentation

MediaWiki Help

LaTeX

MathJax

Extension:Math

Help:Formatting

Help:Wiki markup

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  1. Template:Math – A Beginner's Guide to Mathematical Formatting in MediaWiki

This article provides a comprehensive guide to using the `Template:Math` in MediaWiki, enabling you to display complex mathematical formulas and notations beautifully within your wiki pages. It's designed for users with little to no prior experience with LaTeX or mathematical typesetting. We will cover the fundamentals, common symbols, advanced features, troubleshooting, and best practices. This guide assumes you are using MediaWiki 1.40 or a later version, which supports the necessary extensions.

What is Template:Math?

`Template:Math` is a MediaWiki template that allows you to render mathematical expressions using LaTeX (a widely used typesetting system for scientific and mathematical documents). MediaWiki itself doesn’t natively understand mathematical notation; it needs a way to interpret and display it correctly. `Template:Math` acts as a bridge, converting your LaTeX code into visually appealing mathematical formulas that can be viewed in a web browser.

Essentially, it provides two main ways to display math:

  • **Inline Math:** Formulas that appear *within* a line of text. These are typically used for simple equations or variables. Enclosed in single dollar signs (`$ ... $`).
  • **Display Math:** Formulas that are displayed on a separate line, centered, and often with more spacing. These are used for more complex equations or theorems. Enclosed in double dollar signs (`$$ ... $$`).

Prerequisites

Before you start using `Template:Math`, ensure the following:

  • **LaTeX Support:** Your MediaWiki installation must have the `math` extension enabled. This is usually handled by the wiki administrator. If you’re unsure, contact them. Without this extension, `Template:Math` will simply display the raw LaTeX code instead of rendering the formula.
  • **Basic LaTeX Knowledge (Recommended):** While this guide aims to get you started without extensive LaTeX knowledge, understanding the basics will significantly enhance your ability to create complex formulas. Resources like [29](https://www.latex-project.org/) and [30](https://en.wikibooks.org/wiki/LaTeX) are excellent starting points.
  • **Understanding of Mathematical Notation:** A basic understanding of the mathematical concepts you are trying to represent is crucial.

Basic Syntax

The core syntax for using `Template:Math` is straightforward:

  • **Inline Math:** `$ equation $`
  • **Display Math:** `$$ equation $$`

Replace "equation" with your LaTeX code. For example:

  • `$E = mc^2$` renders as $E = mc^2$
  • `$$ \int_a^b f(x) \, dx = F(b) - F(a) $$` renders as
   $$ \int_a^b f(x) \, dx = F(b) - F(a) $$

Common Mathematical Symbols

Here's a table of commonly used LaTeX symbols and their corresponding MediaWiki/Template:Math representations:

| **Symbol** | **LaTeX Code** | **Rendering** | **Description** | |---|---|---|---| | Plus | `+` | + | Addition | | Minus | `-` | - | Subtraction | | Times | `\times` | × | Multiplication | | Divide | `\div` or `/` | ÷ | Division | | Equals | `=` | = | Equality | | Not Equals | `\neq` | ≠ | Inequality | | Less Than | `<` | < | Less than | | Greater Than | `>` | > | Greater than | | Less Than or Equal To | `\leq` | ≤ | Less than or equal to | | Greater Than or Equal To | `\geq` | ≥ | Greater than or equal to | | Pi | `\pi` | π | Pi (3.14159...) | | Infinity | `\infty` | ∞ | Infinity | | Square Root | `\sqrt{x}` | √x | Square root of x | | Nth Root | `\sqrt[n]{x}` | ⁿ√x | Nth root of x | | Exponent | `x^n` | xⁿ | x raised to the power of n | | Subscript | `x_n` | xn | x with subscript n | | Superscript | `x^n` | xn | x with superscript n | | Summation | `\sum_{i=1}^n x_i` | ∑i=1n xi | Summation from i=1 to n | | Integral | `\int_a^b f(x) \, dx` | ∫ab f(x) dx | Integral from a to b | | Fraction | `\frac{a}{b}` | a/b | Fraction a over b | | Angle | `\angle` | ∠ | Angle | | Degree | `^\circ` | ° | Degree symbol | | Trigonometric Functions | `\sin(x)`, `\cos(x)`, `\tan(x)` | sin(x), cos(x), tan(x) | Sine, Cosine, Tangent | | Logarithm | `\log(x)` | log(x) | Logarithm | | Natural Logarithm | `\ln(x)` | ln(x) | Natural Logarithm | | Limit | `\lim_{x \to a} f(x)` | limx→a f(x) | Limit of f(x) as x approaches a | | Derivative | `\frac{df}{dx}` | df/dx | Derivative of f with respect to x |

Advanced Features

  • **Matrices:** Use the `\begin{matrix} ... \end{matrix}` environment. Separate elements with `&` (for columns) and `\\` (for rows). For example:
   `$$ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} $$` renders as
   $$ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} $$
  • **Alignments:** For aligning multiple equations, use the `\begin{align} ... \end{align}` environment. Use `&` to specify the alignment point.
  • **Environments:** LaTeX offers numerous environments for specific mathematical structures (e.g., `cases` for piecewise functions, `array` for more complex matrices). Refer to LaTeX documentation for details.
  • **Greek Letters:** Use `\alpha`, `\beta`, `\gamma`, `\delta`, etc. For uppercase letters, use `\Alpha`, `\Beta`, `\Gamma`, `\Delta`, etc.
  • **Brackets and Parentheses:** Use `\left( ... \right)` for automatically sized brackets. For example: `\left( \frac{a}{b} \right)`
  • **Spacing:** Use `\,` for a small space, `\;` for a medium space, and `\:` for a large space. `\quad` and `\qquad` provide even larger spaces.
  • **Colors:** While direct color support within `Template:Math` might be limited depending on your MediaWiki configuration, you can sometimes use LaTeX color packages if your administrator has enabled them. Otherwise, consider using HTML color tags around the math formula.

Troubleshooting

  • **Formula Not Rendering:** The most common issue is the `math` extension not being enabled. Confirm with your wiki administrator. Also, ensure your LaTeX code is syntactically correct. Missing brackets or incorrect commands can cause errors.
  • **Garbled Output:** This often indicates a problem with the LaTeX code itself. Check for typos and ensure you are using the correct commands. Try simplifying the equation to isolate the source of the error.
  • **Incorrect Spacing:** Adjust spacing using `\,`, `\;`, `\:` , `\quad`, or `\qquad`.
  • **Symbols Not Displaying:** Ensure you are using the correct LaTeX command for the symbol you want to display. Consult a LaTeX symbol list (see Resources section).
  • **Conflicts with Other Templates:** If you're using other templates on the same page, there might be conflicts. Try isolating the `Template:Math` code to see if it renders correctly on its own.

Best Practices

  • **Keep it Simple:** Avoid overly complex formulas if possible. Break down complicated expressions into smaller, more manageable parts.
  • **Use Comments:** Add comments to your LaTeX code (using `%`) to explain what each part of the formula does. This makes it easier to understand and maintain.
  • **Test Frequently:** Preview your changes often to ensure the formulas are rendering correctly.
  • **Use Inline Math Sparingly:** Excessive inline math can make the text difficult to read. Use display math for more complex equations.
  • **Accessibility:** Consider providing alternative text descriptions for complex formulas to improve accessibility for users with visual impairments.
  • **Consistent Formatting:** Maintain a consistent style throughout your wiki pages.

Resources


Help:Math

Template:Documentation

MediaWiki Help

LaTeX

MathJax

Extension:Math

Help:Formatting

Help:Wiki markup

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  • notation : Common notation used to represent the object. (e.g., "ℝ", "ℤ", "lim")
  • firstappearance : The earliest known appearance of the object in mathematical literature. Use a year or a specific publication. (e.g., "1202", "Leonardo Pisano's *Liber Abaci*")
  • discoverer : The mathematician(s) credited with discovering or developing the object. (e.g., "Leonardo Pisano", "Euclid", "Bernhard Riemann")
  • properties : A list of key properties of the object, separated by semicolons (;). (e.g., "Recursive; Additive; Closed under addition")
  • relatedconcepts : Related mathematical concepts, separated by semicolons (;). Use internal links to other Wikipedia articles whenever possible. (e.g., "Golden ratio; Continued fraction; Lucas sequence")
  • applications : Real-world or theoretical applications of the object, separated by semicolons (;). (e.g., "Computer science; Finance; Nature")
  • parent : A broader mathematical object that encompasses this one. (e.g., "Field (mathematics)", "Topological space")
  • child : A more specific mathematical object derived from this one. (e.g., "Real number", "Metric space")
  • seealso : Additional related articles. Use internal links. (e.g., "Mathematical analysis; Discrete mathematics")
  • references : Citations to sources that support the information in the infobox. Use Template:Ref list to format the references.

Advanced Usage and Tips

  • LaTeX Rendering: Use the
  1. Template:Math – A Beginner's Guide to Mathematical Formatting in MediaWiki

This article provides a comprehensive guide to using the `Template:Math` in MediaWiki, enabling you to display complex mathematical formulas and notations beautifully within your wiki pages. It's designed for users with little to no prior experience with LaTeX or mathematical typesetting. We will cover the fundamentals, common symbols, advanced features, troubleshooting, and best practices. This guide assumes you are using MediaWiki 1.40 or a later version, which supports the necessary extensions.

What is Template:Math?

`Template:Math` is a MediaWiki template that allows you to render mathematical expressions using LaTeX (a widely used typesetting system for scientific and mathematical documents). MediaWiki itself doesn’t natively understand mathematical notation; it needs a way to interpret and display it correctly. `Template:Math` acts as a bridge, converting your LaTeX code into visually appealing mathematical formulas that can be viewed in a web browser.

Essentially, it provides two main ways to display math:

  • **Inline Math:** Formulas that appear *within* a line of text. These are typically used for simple equations or variables. Enclosed in single dollar signs (`$ ... $`).
  • **Display Math:** Formulas that are displayed on a separate line, centered, and often with more spacing. These are used for more complex equations or theorems. Enclosed in double dollar signs (`$$ ... $$`).

Prerequisites

Before you start using `Template:Math`, ensure the following:

  • **LaTeX Support:** Your MediaWiki installation must have the `math` extension enabled. This is usually handled by the wiki administrator. If you’re unsure, contact them. Without this extension, `Template:Math` will simply display the raw LaTeX code instead of rendering the formula.
  • **Basic LaTeX Knowledge (Recommended):** While this guide aims to get you started without extensive LaTeX knowledge, understanding the basics will significantly enhance your ability to create complex formulas. Resources like [57](https://www.latex-project.org/) and [58](https://en.wikibooks.org/wiki/LaTeX) are excellent starting points.
  • **Understanding of Mathematical Notation:** A basic understanding of the mathematical concepts you are trying to represent is crucial.

Basic Syntax

The core syntax for using `Template:Math` is straightforward:

  • **Inline Math:** `$ equation $`
  • **Display Math:** `$$ equation $$`

Replace "equation" with your LaTeX code. For example:

  • `$E = mc^2$` renders as $E = mc^2$
  • `$$ \int_a^b f(x) \, dx = F(b) - F(a) $$` renders as
   $$ \int_a^b f(x) \, dx = F(b) - F(a) $$

Common Mathematical Symbols

Here's a table of commonly used LaTeX symbols and their corresponding MediaWiki/Template:Math representations:

| **Symbol** | **LaTeX Code** | **Rendering** | **Description** | |---|---|---|---| | Plus | `+` | + | Addition | | Minus | `-` | - | Subtraction | | Times | `\times` | × | Multiplication | | Divide | `\div` or `/` | ÷ | Division | | Equals | `=` | = | Equality | | Not Equals | `\neq` | ≠ | Inequality | | Less Than | `<` | < | Less than | | Greater Than | `>` | > | Greater than | | Less Than or Equal To | `\leq` | ≤ | Less than or equal to | | Greater Than or Equal To | `\geq` | ≥ | Greater than or equal to | | Pi | `\pi` | π | Pi (3.14159...) | | Infinity | `\infty` | ∞ | Infinity | | Square Root | `\sqrt{x}` | √x | Square root of x | | Nth Root | `\sqrt[n]{x}` | ⁿ√x | Nth root of x | | Exponent | `x^n` | xⁿ | x raised to the power of n | | Subscript | `x_n` | xn | x with subscript n | | Superscript | `x^n` | xn | x with superscript n | | Summation | `\sum_{i=1}^n x_i` | ∑i=1n xi | Summation from i=1 to n | | Integral | `\int_a^b f(x) \, dx` | ∫ab f(x) dx | Integral from a to b | | Fraction | `\frac{a}{b}` | a/b | Fraction a over b | | Angle | `\angle` | ∠ | Angle | | Degree | `^\circ` | ° | Degree symbol | | Trigonometric Functions | `\sin(x)`, `\cos(x)`, `\tan(x)` | sin(x), cos(x), tan(x) | Sine, Cosine, Tangent | | Logarithm | `\log(x)` | log(x) | Logarithm | | Natural Logarithm | `\ln(x)` | ln(x) | Natural Logarithm | | Limit | `\lim_{x \to a} f(x)` | limx→a f(x) | Limit of f(x) as x approaches a | | Derivative | `\frac{df}{dx}` | df/dx | Derivative of f with respect to x |

Advanced Features

  • **Matrices:** Use the `\begin{matrix} ... \end{matrix}` environment. Separate elements with `&` (for columns) and `\\` (for rows). For example:
   `$$ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} $$` renders as
   $$ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} $$
  • **Alignments:** For aligning multiple equations, use the `\begin{align} ... \end{align}` environment. Use `&` to specify the alignment point.
  • **Environments:** LaTeX offers numerous environments for specific mathematical structures (e.g., `cases` for piecewise functions, `array` for more complex matrices). Refer to LaTeX documentation for details.
  • **Greek Letters:** Use `\alpha`, `\beta`, `\gamma`, `\delta`, etc. For uppercase letters, use `\Alpha`, `\Beta`, `\Gamma`, `\Delta`, etc.
  • **Brackets and Parentheses:** Use `\left( ... \right)` for automatically sized brackets. For example: `\left( \frac{a}{b} \right)`
  • **Spacing:** Use `\,` for a small space, `\;` for a medium space, and `\:` for a large space. `\quad` and `\qquad` provide even larger spaces.
  • **Colors:** While direct color support within `Template:Math` might be limited depending on your MediaWiki configuration, you can sometimes use LaTeX color packages if your administrator has enabled them. Otherwise, consider using HTML color tags around the math formula.

Troubleshooting

  • **Formula Not Rendering:** The most common issue is the `math` extension not being enabled. Confirm with your wiki administrator. Also, ensure your LaTeX code is syntactically correct. Missing brackets or incorrect commands can cause errors.
  • **Garbled Output:** This often indicates a problem with the LaTeX code itself. Check for typos and ensure you are using the correct commands. Try simplifying the equation to isolate the source of the error.
  • **Incorrect Spacing:** Adjust spacing using `\,`, `\;`, `\:` , `\quad`, or `\qquad`.
  • **Symbols Not Displaying:** Ensure you are using the correct LaTeX command for the symbol you want to display. Consult a LaTeX symbol list (see Resources section).
  • **Conflicts with Other Templates:** If you're using other templates on the same page, there might be conflicts. Try isolating the `Template:Math` code to see if it renders correctly on its own.

Best Practices

  • **Keep it Simple:** Avoid overly complex formulas if possible. Break down complicated expressions into smaller, more manageable parts.
  • **Use Comments:** Add comments to your LaTeX code (using `%`) to explain what each part of the formula does. This makes it easier to understand and maintain.
  • **Test Frequently:** Preview your changes often to ensure the formulas are rendering correctly.
  • **Use Inline Math Sparingly:** Excessive inline math can make the text difficult to read. Use display math for more complex equations.
  • **Accessibility:** Consider providing alternative text descriptions for complex formulas to improve accessibility for users with visual impairments.
  • **Consistent Formatting:** Maintain a consistent style throughout your wiki pages.

Resources


Help:Math

Template:Documentation

MediaWiki Help

LaTeX

MathJax

Extension:Math

Help:Formatting

Help:Wiki markup

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  1. Template:Math – A Beginner's Guide to Mathematical Formatting in MediaWiki

This article provides a comprehensive guide to using the `Template:Math` in MediaWiki, enabling you to display complex mathematical formulas and notations beautifully within your wiki pages. It's designed for users with little to no prior experience with LaTeX or mathematical typesetting. We will cover the fundamentals, common symbols, advanced features, troubleshooting, and best practices. This guide assumes you are using MediaWiki 1.40 or a later version, which supports the necessary extensions.

What is Template:Math?

`Template:Math` is a MediaWiki template that allows you to render mathematical expressions using LaTeX (a widely used typesetting system for scientific and mathematical documents). MediaWiki itself doesn’t natively understand mathematical notation; it needs a way to interpret and display it correctly. `Template:Math` acts as a bridge, converting your LaTeX code into visually appealing mathematical formulas that can be viewed in a web browser.

Essentially, it provides two main ways to display math:

  • **Inline Math:** Formulas that appear *within* a line of text. These are typically used for simple equations or variables. Enclosed in single dollar signs (`$ ... $`).
  • **Display Math:** Formulas that are displayed on a separate line, centered, and often with more spacing. These are used for more complex equations or theorems. Enclosed in double dollar signs (`$$ ... $$`).

Prerequisites

Before you start using `Template:Math`, ensure the following:

  • **LaTeX Support:** Your MediaWiki installation must have the `math` extension enabled. This is usually handled by the wiki administrator. If you’re unsure, contact them. Without this extension, `Template:Math` will simply display the raw LaTeX code instead of rendering the formula.
  • **Basic LaTeX Knowledge (Recommended):** While this guide aims to get you started without extensive LaTeX knowledge, understanding the basics will significantly enhance your ability to create complex formulas. Resources like [85](https://www.latex-project.org/) and [86](https://en.wikibooks.org/wiki/LaTeX) are excellent starting points.
  • **Understanding of Mathematical Notation:** A basic understanding of the mathematical concepts you are trying to represent is crucial.

Basic Syntax

The core syntax for using `Template:Math` is straightforward:

  • **Inline Math:** `$ equation $`
  • **Display Math:** `$$ equation $$`

Replace "equation" with your LaTeX code. For example:

  • `$E = mc^2$` renders as $E = mc^2$
  • `$$ \int_a^b f(x) \, dx = F(b) - F(a) $$` renders as
   $$ \int_a^b f(x) \, dx = F(b) - F(a) $$

Common Mathematical Symbols

Here's a table of commonly used LaTeX symbols and their corresponding MediaWiki/Template:Math representations:

| **Symbol** | **LaTeX Code** | **Rendering** | **Description** | |---|---|---|---| | Plus | `+` | + | Addition | | Minus | `-` | - | Subtraction | | Times | `\times` | × | Multiplication | | Divide | `\div` or `/` | ÷ | Division | | Equals | `=` | = | Equality | | Not Equals | `\neq` | ≠ | Inequality | | Less Than | `<` | < | Less than | | Greater Than | `>` | > | Greater than | | Less Than or Equal To | `\leq` | ≤ | Less than or equal to | | Greater Than or Equal To | `\geq` | ≥ | Greater than or equal to | | Pi | `\pi` | π | Pi (3.14159...) | | Infinity | `\infty` | ∞ | Infinity | | Square Root | `\sqrt{x}` | √x | Square root of x | | Nth Root | `\sqrt[n]{x}` | ⁿ√x | Nth root of x | | Exponent | `x^n` | xⁿ | x raised to the power of n | | Subscript | `x_n` | xn | x with subscript n | | Superscript | `x^n` | xn | x with superscript n | | Summation | `\sum_{i=1}^n x_i` | ∑i=1n xi | Summation from i=1 to n | | Integral | `\int_a^b f(x) \, dx` | ∫ab f(x) dx | Integral from a to b | | Fraction | `\frac{a}{b}` | a/b | Fraction a over b | | Angle | `\angle` | ∠ | Angle | | Degree | `^\circ` | ° | Degree symbol | | Trigonometric Functions | `\sin(x)`, `\cos(x)`, `\tan(x)` | sin(x), cos(x), tan(x) | Sine, Cosine, Tangent | | Logarithm | `\log(x)` | log(x) | Logarithm | | Natural Logarithm | `\ln(x)` | ln(x) | Natural Logarithm | | Limit | `\lim_{x \to a} f(x)` | limx→a f(x) | Limit of f(x) as x approaches a | | Derivative | `\frac{df}{dx}` | df/dx | Derivative of f with respect to x |

Advanced Features

  • **Matrices:** Use the `\begin{matrix} ... \end{matrix}` environment. Separate elements with `&` (for columns) and `\\` (for rows). For example:
   `$$ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} $$` renders as
   $$ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} $$
  • **Alignments:** For aligning multiple equations, use the `\begin{align} ... \end{align}` environment. Use `&` to specify the alignment point.
  • **Environments:** LaTeX offers numerous environments for specific mathematical structures (e.g., `cases` for piecewise functions, `array` for more complex matrices). Refer to LaTeX documentation for details.
  • **Greek Letters:** Use `\alpha`, `\beta`, `\gamma`, `\delta`, etc. For uppercase letters, use `\Alpha`, `\Beta`, `\Gamma`, `\Delta`, etc.
  • **Brackets and Parentheses:** Use `\left( ... \right)` for automatically sized brackets. For example: `\left( \frac{a}{b} \right)`
  • **Spacing:** Use `\,` for a small space, `\;` for a medium space, and `\:` for a large space. `\quad` and `\qquad` provide even larger spaces.
  • **Colors:** While direct color support within `Template:Math` might be limited depending on your MediaWiki configuration, you can sometimes use LaTeX color packages if your administrator has enabled them. Otherwise, consider using HTML color tags around the math formula.

Troubleshooting

  • **Formula Not Rendering:** The most common issue is the `math` extension not being enabled. Confirm with your wiki administrator. Also, ensure your LaTeX code is syntactically correct. Missing brackets or incorrect commands can cause errors.
  • **Garbled Output:** This often indicates a problem with the LaTeX code itself. Check for typos and ensure you are using the correct commands. Try simplifying the equation to isolate the source of the error.
  • **Incorrect Spacing:** Adjust spacing using `\,`, `\;`, `\:` , `\quad`, or `\qquad`.
  • **Symbols Not Displaying:** Ensure you are using the correct LaTeX command for the symbol you want to display. Consult a LaTeX symbol list (see Resources section).
  • **Conflicts with Other Templates:** If you're using other templates on the same page, there might be conflicts. Try isolating the `Template:Math` code to see if it renders correctly on its own.

Best Practices

  • **Keep it Simple:** Avoid overly complex formulas if possible. Break down complicated expressions into smaller, more manageable parts.
  • **Use Comments:** Add comments to your LaTeX code (using `%`) to explain what each part of the formula does. This makes it easier to understand and maintain.
  • **Test Frequently:** Preview your changes often to ensure the formulas are rendering correctly.
  • **Use Inline Math Sparingly:** Excessive inline math can make the text difficult to read. Use display math for more complex equations.
  • **Accessibility:** Consider providing alternative text descriptions for complex formulas to improve accessibility for users with visual impairments.
  • **Consistent Formatting:** Maintain a consistent style throughout your wiki pages.

Resources


Help:Math

Template:Documentation

MediaWiki Help

LaTeX

MathJax

Extension:Math

Help:Formatting

Help:Wiki markup

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Subscribe to our Telegram channel @strategybin to receive: ✓ Daily trading signals ✓ Exclusive strategy analysis ✓ Market trend alerts ✓ Educational materials for beginners` will display as $x^2 + y^2 = r^2$.

  • Internal Linking: Whenever possible, use internal links (link) to connect to other relevant Wikipedia articles. This improves navigability and provides readers with further information. Consider linking to concepts like Calculus, Linear Algebra, Probability theory, Differential equations, Geometry, Trigonometry, Statistics, Set theory, Mathematical logic, and Number theory.
  • Semicolon Separation: When listing multiple items in parameters like 'properties', 'relatedconcepts', or 'applications', separate them with semicolons (;).
  • Image Selection: Choose images that are clear, relevant, and appropriately licensed. Avoid using images that are too small or difficult to understand.
  • Conciseness: Keep the information in the infobox concise and to the point. Detailed explanations should be provided in the main article body.
  • Parameter Order: While the order of parameters doesn’t technically matter, it is good practice to follow the order presented above for consistency.
  • Empty Parameters: If a parameter is not applicable to the mathematical object, simply omit it from the infobox. Do not leave it blank.
  • Templates within Templates: You can utilize other templates *within* the infobox, provided they are appropriate and contribute to the clarity of the information.
  • Error Handling: If you encounter errors, double-check your syntax and parameter names. Consult the template documentation (Template:Infobox mathematical object/doc) for further assistance.

Examples

Example 1: Fibonacci Sequence

```wiki Template loop detected: Template:Infobox mathematical object ```

Example 2: Euclidean Space

```wiki Template loop detected: Template:Infobox mathematical object ```

Example 3: Group (mathematics)

```wiki Template loop detected: Template:Infobox mathematical object ```

Troubleshooting

  • Infobox Not Displaying: Ensure that the template name is spelled correctly (`Template loop detected: Template:Infobox mathematical object`). Check for syntax errors within the template code.
  • Incorrect LaTeX Rendering: Make sure you are using the
  1. Template:Math – A Beginner's Guide to Mathematical Formatting in MediaWiki

This article provides a comprehensive guide to using the `Template:Math` in MediaWiki, enabling you to display complex mathematical formulas and notations beautifully within your wiki pages. It's designed for users with little to no prior experience with LaTeX or mathematical typesetting. We will cover the fundamentals, common symbols, advanced features, troubleshooting, and best practices. This guide assumes you are using MediaWiki 1.40 or a later version, which supports the necessary extensions.

What is Template:Math?

`Template:Math` is a MediaWiki template that allows you to render mathematical expressions using LaTeX (a widely used typesetting system for scientific and mathematical documents). MediaWiki itself doesn’t natively understand mathematical notation; it needs a way to interpret and display it correctly. `Template:Math` acts as a bridge, converting your LaTeX code into visually appealing mathematical formulas that can be viewed in a web browser.

Essentially, it provides two main ways to display math:

  • **Inline Math:** Formulas that appear *within* a line of text. These are typically used for simple equations or variables. Enclosed in single dollar signs (`$ ... $`).
  • **Display Math:** Formulas that are displayed on a separate line, centered, and often with more spacing. These are used for more complex equations or theorems. Enclosed in double dollar signs (`$$ ... $$`).

Prerequisites

Before you start using `Template:Math`, ensure the following:

  • **LaTeX Support:** Your MediaWiki installation must have the `math` extension enabled. This is usually handled by the wiki administrator. If you’re unsure, contact them. Without this extension, `Template:Math` will simply display the raw LaTeX code instead of rendering the formula.
  • **Basic LaTeX Knowledge (Recommended):** While this guide aims to get you started without extensive LaTeX knowledge, understanding the basics will significantly enhance your ability to create complex formulas. Resources like [113](https://www.latex-project.org/) and [114](https://en.wikibooks.org/wiki/LaTeX) are excellent starting points.
  • **Understanding of Mathematical Notation:** A basic understanding of the mathematical concepts you are trying to represent is crucial.

Basic Syntax

The core syntax for using `Template:Math` is straightforward:

  • **Inline Math:** `$ equation $`
  • **Display Math:** `$$ equation $$`

Replace "equation" with your LaTeX code. For example:

  • `$E = mc^2$` renders as $E = mc^2$
  • `$$ \int_a^b f(x) \, dx = F(b) - F(a) $$` renders as
   $$ \int_a^b f(x) \, dx = F(b) - F(a) $$

Common Mathematical Symbols

Here's a table of commonly used LaTeX symbols and their corresponding MediaWiki/Template:Math representations:

| **Symbol** | **LaTeX Code** | **Rendering** | **Description** | |---|---|---|---| | Plus | `+` | + | Addition | | Minus | `-` | - | Subtraction | | Times | `\times` | × | Multiplication | | Divide | `\div` or `/` | ÷ | Division | | Equals | `=` | = | Equality | | Not Equals | `\neq` | ≠ | Inequality | | Less Than | `<` | < | Less than | | Greater Than | `>` | > | Greater than | | Less Than or Equal To | `\leq` | ≤ | Less than or equal to | | Greater Than or Equal To | `\geq` | ≥ | Greater than or equal to | | Pi | `\pi` | π | Pi (3.14159...) | | Infinity | `\infty` | ∞ | Infinity | | Square Root | `\sqrt{x}` | √x | Square root of x | | Nth Root | `\sqrt[n]{x}` | ⁿ√x | Nth root of x | | Exponent | `x^n` | xⁿ | x raised to the power of n | | Subscript | `x_n` | xn | x with subscript n | | Superscript | `x^n` | xn | x with superscript n | | Summation | `\sum_{i=1}^n x_i` | ∑i=1n xi | Summation from i=1 to n | | Integral | `\int_a^b f(x) \, dx` | ∫ab f(x) dx | Integral from a to b | | Fraction | `\frac{a}{b}` | a/b | Fraction a over b | | Angle | `\angle` | ∠ | Angle | | Degree | `^\circ` | ° | Degree symbol | | Trigonometric Functions | `\sin(x)`, `\cos(x)`, `\tan(x)` | sin(x), cos(x), tan(x) | Sine, Cosine, Tangent | | Logarithm | `\log(x)` | log(x) | Logarithm | | Natural Logarithm | `\ln(x)` | ln(x) | Natural Logarithm | | Limit | `\lim_{x \to a} f(x)` | limx→a f(x) | Limit of f(x) as x approaches a | | Derivative | `\frac{df}{dx}` | df/dx | Derivative of f with respect to x |

Advanced Features

  • **Matrices:** Use the `\begin{matrix} ... \end{matrix}` environment. Separate elements with `&` (for columns) and `\\` (for rows). For example:
   `$$ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} $$` renders as
   $$ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} $$
  • **Alignments:** For aligning multiple equations, use the `\begin{align} ... \end{align}` environment. Use `&` to specify the alignment point.
  • **Environments:** LaTeX offers numerous environments for specific mathematical structures (e.g., `cases` for piecewise functions, `array` for more complex matrices). Refer to LaTeX documentation for details.
  • **Greek Letters:** Use `\alpha`, `\beta`, `\gamma`, `\delta`, etc. For uppercase letters, use `\Alpha`, `\Beta`, `\Gamma`, `\Delta`, etc.
  • **Brackets and Parentheses:** Use `\left( ... \right)` for automatically sized brackets. For example: `\left( \frac{a}{b} \right)`
  • **Spacing:** Use `\,` for a small space, `\;` for a medium space, and `\:` for a large space. `\quad` and `\qquad` provide even larger spaces.
  • **Colors:** While direct color support within `Template:Math` might be limited depending on your MediaWiki configuration, you can sometimes use LaTeX color packages if your administrator has enabled them. Otherwise, consider using HTML color tags around the math formula.

Troubleshooting

  • **Formula Not Rendering:** The most common issue is the `math` extension not being enabled. Confirm with your wiki administrator. Also, ensure your LaTeX code is syntactically correct. Missing brackets or incorrect commands can cause errors.
  • **Garbled Output:** This often indicates a problem with the LaTeX code itself. Check for typos and ensure you are using the correct commands. Try simplifying the equation to isolate the source of the error.
  • **Incorrect Spacing:** Adjust spacing using `\,`, `\;`, `\:` , `\quad`, or `\qquad`.
  • **Symbols Not Displaying:** Ensure you are using the correct LaTeX command for the symbol you want to display. Consult a LaTeX symbol list (see Resources section).
  • **Conflicts with Other Templates:** If you're using other templates on the same page, there might be conflicts. Try isolating the `Template:Math` code to see if it renders correctly on its own.

Best Practices

  • **Keep it Simple:** Avoid overly complex formulas if possible. Break down complicated expressions into smaller, more manageable parts.
  • **Use Comments:** Add comments to your LaTeX code (using `%`) to explain what each part of the formula does. This makes it easier to understand and maintain.
  • **Test Frequently:** Preview your changes often to ensure the formulas are rendering correctly.
  • **Use Inline Math Sparingly:** Excessive inline math can make the text difficult to read. Use display math for more complex equations.
  • **Accessibility:** Consider providing alternative text descriptions for complex formulas to improve accessibility for users with visual impairments.
  • **Consistent Formatting:** Maintain a consistent style throughout your wiki pages.

Resources


Help:Math

Template:Documentation

MediaWiki Help

LaTeX

MathJax

Extension:Math

Help:Formatting

Help:Wiki markup

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Subscribe to our Telegram channel @strategybin to receive: ✓ Daily trading signals ✓ Exclusive strategy analysis ✓ Market trend alerts ✓ Educational materials for beginners template correctly to enclose mathematical expressions. Verify that the LaTeX code is valid.

  • Image Not Appearing: Confirm that the image filename is correct and that the image exists on Wikimedia Commons or the local wiki. Check the image license.
  • Parameters Not Working: Double-check the parameter names against the list above. Parameter names are case-insensitive, but using the standardized names is recommended.
  • Infobox Formatting Issues: Inspect the template code for any misplaced brackets or other syntax errors. Try clearing your browser cache.

Further Resources

Strategies, Technical Analysis, Indicators, and Trends

For those interested in applying mathematical concepts to real-world scenarios, particularly in finance, consider exploring these areas:

```

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Tensor: A Comprehensive Introduction for Beginners

A tensor is a mathematical object that generalizes scalars, vectors, and matrices to an arbitrary number of dimensions. While the term might sound intimidating, tensors are fundamental to many areas of science and engineering, including physics, machine learning, and data analysis. This article provides a beginner-friendly introduction to tensors, covering their definition, properties, and applications. We will build up the concept gradually, starting with familiar objects and extending to more complex ones. Understanding tensors is crucial for grasping advanced concepts in fields like Technical Analysis and Trend Following.

From Scalars to Tensors: A Hierarchy

To understand tensors, it's helpful to think of them as a hierarchy of data representations:

  • Scalar (Rank-0 Tensor): A single number. For example, the temperature is 25°C. It has no dimensions and is represented by a single value. In Candlestick Patterns, the close price of a stock is a scalar value.
  • Vector (Rank-1 Tensor): An ordered array of numbers. For example, the wind speed and direction can be represented as a vector (e.g., [5, 180]). It has one dimension. A time series of closing prices in Chart Patterns is essentially a vector.
  • Matrix (Rank-2 Tensor): A two-dimensional array of numbers, arranged in rows and columns. For example, a spreadsheet of sales data for different products in different regions. The Correlation Matrix used in financial analysis is a rank-2 tensor.
  • Rank-3 Tensor: A three-dimensional array of numbers. Imagine a cube of numbers. This could represent, for example, sales data for multiple products, across multiple regions, over multiple time periods.
  • Rank-n Tensor: An n-dimensional array of numbers. This generalization extends to any number of dimensions.

The rank (or order, or degree) of a tensor is the number of indices needed to specify a component of the tensor. So a scalar has rank 0, a vector has rank 1, a matrix has rank 2, and so on.

Formal Definition and Components

More formally, a tensor is a multilinear map from a set of vectors to a scalar. This means it takes vectors as input and produces a scalar output in a linear way. However, for a beginner, the array-based understanding is more practical.

Each element of a tensor is identified by a set of indices. For example, in a rank-3 tensor `T`, an element is specified by three indices: `Tijk`. The number of indices corresponds to the rank of the tensor.

The size or shape of a tensor is described by its dimensions. A matrix with `m` rows and `n` columns has a shape of `(m, n)`. A rank-3 tensor with dimensions `p`, `q`, and `r` has a shape of `(p, q, r)`.

Tensor Operations

Several operations can be performed on tensors:

  • Addition and Subtraction: Tensors of the *same shape* can be added or subtracted element-wise.
  • Scalar Multiplication: Each element of a tensor can be multiplied by a scalar.
  • Tensor Product (Outer Product): This operation combines two tensors to create a new tensor with a higher rank. For example, the tensor product of a vector and a matrix results in a rank-3 tensor. In Fibonacci Retracement, applying a scalar multiplier to a price level is a form of scalar multiplication.
  • Tensor Contraction: This operation reduces the rank of a tensor by summing over one or more pairs of indices. A common example is the trace of a matrix, which is the sum of its diagonal elements.
  • Dot Product: A special case of tensor contraction, commonly used with vectors and matrices. It's fundamental to many calculations in Moving Average Convergence Divergence.

Tensor Representation in Code (Python/NumPy Example)

The NumPy library in Python is widely used for working with tensors. Here's how you can represent and manipulate tensors using NumPy:

```python import numpy as np

  1. Scalar

scalar = np.array(5) print(f"Scalar: {scalar}, Shape: {scalar.shape}, Rank: {scalar.ndim}")

  1. Vector

vector = np.array([1, 2, 3]) print(f"Vector: {vector}, Shape: {vector.shape}, Rank: {vector.ndim}")

  1. Matrix

matrix = np.array([[1, 2], [3, 4]]) print(f"Matrix: {matrix}, Shape: {matrix.shape}, Rank: {matrix.ndim}")

  1. Rank-3 Tensor

tensor = np.array([[[1, 2], [3, 4]], [[5, 6], [7, 8]]]) print(f"Tensor: {tensor}, Shape: {tensor.shape}, Rank: {tensor.ndim}")

  1. Tensor operations

a = np.array([[1, 2], [3, 4]]) b = np.array([[5, 6], [7, 8]])

  1. Addition

addition = a + b print(f"Addition: \n{addition}")

  1. Scalar Multiplication

scalar_mult = 2 * a print(f"Scalar Multiplication: \n{scalar_mult}")

  1. Dot Product

dot_product = np.dot(a, b) print(f"Dot Product: \n{dot_product}") ```

This code demonstrates how to create tensors of different ranks and perform basic operations on them using NumPy. Understanding these operations is key to applying tensors in practical scenarios.

Applications of Tensors

Tensors are used extensively in various fields:

  • Physics: Tensors are used to describe physical quantities like stress, strain, and electromagnetic fields. The Elliott Wave Principle often involves analyzing wave patterns which can be mathematically represented using tensor-based techniques.
  • Engineering: Tensors are used in structural analysis, fluid dynamics, and materials science.
  • Machine Learning: Tensors are the fundamental data structure in deep learning frameworks like TensorFlow and PyTorch. Images, videos, and text data are often represented as tensors. Bollinger Bands calculations rely on tensors to represent price data and standard deviations.
  • Data Analysis: Tensors can be used to represent and analyze multi-dimensional data sets, such as customer behavior data or sensor data. Relative Strength Index calculations often utilize tensor operations for data manipulation.
  • Computer Graphics: Tensors can represent transformations, lighting, and material properties.
  • Financial Modeling: Tensors are increasingly used in financial modeling for portfolio optimization, risk management, and algorithmic trading. Ichimoku Cloud calculations can be efficiently implemented using tensor operations.
  • Image Processing: Images can be represented as 3D tensors (height, width, color channels). Convolutional Neural Networks (CNNs), a staple of image recognition, heavily rely on tensor operations.

Tensors in Deep Learning

In deep learning, tensors represent the weights and biases of neural networks, as well as the input and output data. The training process involves adjusting these tensor values to minimize a loss function.

  • Input Tensors: Represent the input data to the neural network.
  • Weight Tensors: Represent the parameters of the network that are learned during training.
  • Bias Tensors: Add a constant value to the output of each neuron.
  • Activation Tensors: Represent the output of each layer in the network.

The backpropagation algorithm uses tensor calculus to compute the gradients of the loss function with respect to the weights and biases, allowing the network to learn from its mistakes. The use of tensors is critical for the efficiency and scalability of deep learning models. Understanding Support Vector Machines often requires a grasp of how tensors represent data and decision boundaries.

Coordinate Systems and Tensor Transformations

The components of a tensor depend on the coordinate system used to represent it. When you change the coordinate system, the components of the tensor transform in a specific way to ensure that the tensor itself remains unchanged. These transformations are governed by the rules of tensor algebra. This concept is related to Head and Shoulders Patterns where identifying patterns is independent of the coordinate system of the chart.

Tensor Decomposition

Tensor decomposition techniques are used to approximate a tensor with a lower-rank tensor, which can reduce computational complexity and reveal underlying structure in the data. Common tensor decomposition methods include:

  • CP Decomposition (CANDECOMP/PARAFAC): Decomposes a tensor into a sum of rank-1 tensors.
  • Tucker Decomposition: Decomposes a tensor into a core tensor and a set of factor matrices.
  • Tensor Train Decomposition: Represents a tensor as a chain of smaller tensors.

These techniques are useful for dimensionality reduction, feature extraction, and data compression. Average True Range calculations can be optimized using tensor decomposition to analyze historical volatility data efficiently.

Advanced Concepts (Brief Overview)

  • Differential Forms: Tensors are closely related to differential forms, which are used in differential geometry and topology.
  • Riemannian Manifolds: Tensors are used to define the metric tensor on a Riemannian manifold, which measures distances and angles.
  • Kronecker Delta: A fundamental tensor used for indexing and summation conventions.
  • Einstein Summation Convention: A notation that simplifies tensor expressions by implicitly summing over repeated indices.

These advanced concepts are beyond the scope of this introductory article but provide a glimpse into the rich mathematical theory behind tensors. Exploring Japanese Candlesticks often requires understanding the underlying mathematical relationships that tensors can help represent.

Distinguishing Tensors from Matrices and Vectors

While matrices and vectors are special cases of tensors, there are key distinctions:

  • **Transformation Properties:** Tensors transform in a specific way under coordinate changes, ensuring their inherent properties remain invariant. Matrices and vectors don't necessarily follow this rule unless they are specifically defined as tensors.
  • **Generalization:** Tensors generalize the concept of matrices and vectors to higher dimensions. A matrix can be seen as a rank-2 tensor, and a vector as a rank-1 tensor.
  • **Multilinearity:** Tensors are multilinear maps, meaning they are linear in each of their arguments. This property is crucial for their applications in physics and engineering. Understanding Donchian Channels requires recognizing how price data (a vector) transforms over time, a concept related to tensor transformations.

Resources for Further Learning

Conclusion

Tensors are a powerful mathematical tool with applications in a wide range of fields. While the concept may seem abstract at first, understanding the basic principles of tensors is essential for anyone working with data analysis, machine learning, or physics. This article has provided a foundation for further exploration of this fascinating topic. By grasping the relationship between scalars, vectors, matrices, and tensors, and understanding the fundamental operations that can be performed on them, you will be well-equipped to tackle more advanced concepts and applications. Further study into Elliott Wave Trading will reveal how tensors can model cyclical patterns. Remember to practice with code examples to solidify your understanding and explore the vast possibilities of tensor manipulation.

Vector Space Matrix (mathematics) Multilinear Form Coordinate System Linear Transformation Eigenvalue Eigenvector Dimensionality Reduction Numerical Analysis Calculus

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