diff --git a/tech_docs/math.md b/tech_docs/math.md new file mode 100644 index 0000000..1721cc5 --- /dev/null +++ b/tech_docs/math.md @@ -0,0 +1,81 @@ +Here's the summary of the mathematical concepts using LaTeX syntax: + +1. High-dimensional Vectors: + - Definition: In mathematics and computer science, a vector is an ordered list of numbers (elements). The number of elements in a vector determines its dimension. + - High-dimensional Vectors: Vectors with a large number of dimensions, often in the hundreds, thousands, or even millions. + - Applications: In machine learning and data science, high-dimensional vectors are commonly used to represent complex data objects, such as text documents, images, or user preferences. + +2. First-order Equations: + - Definition: In mathematics, a first-order equation is an equation that involves only first derivatives of one or more variables. + - Types: First-order equations can be linear or nonlinear, depending on the form of the equation. + - Examples: + - First-order linear differential equation: $\frac{dy}{dx} + P(x)y = Q(x)$ + - First-order nonlinear differential equation: $\frac{dy}{dx} = f(x, y)$ + - Applications: First-order equations are used to model various phenomena in physics, engineering, and other fields. + +3. Tensors: + - Definition: Tensors are mathematical objects that generalize vectors and matrices to higher dimensions. + - Dimensions: + - Scalar (a single number): 0-dimensional tensor + - Vector: 1-dimensional tensor + - Matrix: 2-dimensional tensor + - Higher-order Tensors: Tensors with three or more dimensions + - Applications: + - Physics and Engineering: Tensors describe physical quantities with magnitude and direction in multiple dimensions, such as stress, strain, and electromagnetic fields. + - Machine Learning and Deep Learning: Tensors represent multi-dimensional data arrays and the parameters of neural networks. + +4. Scalars: + - Definition: A scalar is a single numerical value, a quantity that has magnitude but no direction. + - Properties: Scalars are zero-dimensional entities, meaning they have no spatial extent or orientation. + - Examples: Temperature, mass, density, and electric charge. + - Applications: In linear algebra, scalars are often used to scale (multiply) a vector or matrix. + +5. Relationship between Scalars and First-order Equations: + - Similarity: Both scalars and first-order equations can involve a single variable. + - Difference: + - Scalars are single numerical values, while first-order equations describe the relationship between a function and its first derivative. + - Scalars have no dimension, while first-order equations can involve functions of one or more variables (usually one variable for first-order equations). + +In summary, high-dimensional vectors, tensors, and scalars are mathematical objects used to represent and manipulate multi-dimensional data, while first-order equations describe the relationship between a function and its first derivative. These concepts find applications in various fields, including machine learning, data science, physics, and engineering. + +--- + +Here's a summary of the mathematical concepts we discussed in our conversation: + +1. High-dimensional Vectors: + - Definition: In mathematics and computer science, a vector is an ordered list of numbers (elements). The number of elements in a vector determines its dimension. + - High-dimensional Vectors: Vectors with a large number of dimensions, often in the hundreds, thousands, or even millions. + - Applications: In machine learning and data science, high-dimensional vectors are commonly used to represent complex data objects, such as text documents, images, or user preferences. + +2. First-order Equations: + - Definition: In mathematics, a first-order equation is an equation that involves only first derivatives of one or more variables. + - Types: First-order equations can be linear or nonlinear, depending on the form of the equation. + - Examples: + - First-order linear differential equation: y' + P(x)y = Q(x) + - First-order nonlinear differential equation: y' = f(x, y) + - Applications: First-order equations are used to model various phenomena in physics, engineering, and other fields. + +3. Tensors: + - Definition: Tensors are mathematical objects that generalize vectors and matrices to higher dimensions. + - Dimensions: + - Scalar (a single number): 0-dimensional tensor + - Vector: 1-dimensional tensor + - Matrix: 2-dimensional tensor + - Higher-order Tensors: Tensors with three or more dimensions + - Applications: + - Physics and Engineering: Tensors describe physical quantities with magnitude and direction in multiple dimensions, such as stress, strain, and electromagnetic fields. + - Machine Learning and Deep Learning: Tensors represent multi-dimensional data arrays and the parameters of neural networks. + +4. Scalars: + - Definition: A scalar is a single numerical value, a quantity that has magnitude but no direction. + - Properties: Scalars are zero-dimensional entities, meaning they have no spatial extent or orientation. + - Examples: Temperature, mass, density, and electric charge. + - Applications: In linear algebra, scalars are often used to scale (multiply) a vector or matrix. + +5. Relationship between Scalars and First-order Equations: + - Similarity: Both scalars and first-order equations can involve a single variable. + - Difference: + - Scalars are single numerical values, while first-order equations describe the relationship between a function and its first derivative. + - Scalars have no dimension, while first-order equations can involve functions of one or more variables (usually one variable for first-order equations). + +In summary, high-dimensional vectors, tensors, and scalars are mathematical objects used to represent and manipulate multi-dimensional data, while first-order equations describe the relationship between a function and its first derivative. These concepts find applications in various fields, including machine learning, data science, physics, and engineering.