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Cardiac function refers to the heart's ability to pump blood effectively throughout the body, ensuring adequate oxygen and nutrient delivery to tissues. It is influenced by factors such as heart rate, contractility, preload, and afterload, which together determine cardiac output and overall cardiovascular health.
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An orthogonal matrix is a square matrix whose rows and columns are orthogonal unit vectors, meaning it preserves the dot product and hence the length of vectors upon transformation. This property implies that the inverse of an orthogonal matrix is its transpose, making computations involving orthogonal matrices particularly efficient and stable in numerical analysis.
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Euclidean space is a mathematical construct that generalizes the properties of two-dimensional and three-dimensional spaces to any number of dimensions, characterized by the notions of distance and angle. It serves as the foundational setting for classical geometry and is defined by a coordinate system where the distance between points is given by the Euclidean distance formula.
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Matrix transpose involves flipping a matrix over its diagonal, effectively swapping the row and column indices of each element. It is a fundamental operation in linear algebra with applications in various fields such as computer graphics, data analysis, and machine learning.
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The matrix inverse is a fundamental concept in linear algebra, representing a matrix that, when multiplied by the original matrix, yields the identity matrix. Not all matrices have inverses, and a matrix must be square and have a non-zero determinant to be invertible.
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Group theory is a branch of abstract algebra that studies the algebraic structures known as groups, which are sets equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. It provides a unifying framework for understanding symmetry in mathematical objects and has applications across various fields including physics, chemistry, and computer science.
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Symmetry refers to a balanced and proportionate similarity found in two halves of an object, which can be divided by a specific plane, line, or point. It is a fundamental concept in various fields, including mathematics, physics, and art, where it helps to understand patterns, structures, and the natural order.
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Rotation refers to the circular movement of an object around a center or an axis. This fundamental concept is pivotal in various fields, including physics, engineering, and mathematics, where it describes phenomena ranging from the Earth's rotation to the angular momentum of particles.
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The determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties of the matrix, such as whether it is invertible. It is also used in various applications such as solving systems of linear equations, finding volumes in geometry, and analyzing linear transformations.
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Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces, focusing on the study of lines, planes, and subspaces. It is fundamental in various scientific fields, providing tools for solving systems of linear equations, performing transformations, and analyzing vector spaces and matrices.
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A Lie group is a mathematical structure that combines the properties of both groups and smooth manifolds, allowing for the study of continuous symmetries. They play a crucial role in various fields such as geometry, physics, and representation theory, providing a framework for analyzing the symmetry of differential equations and physical systems.
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The Lorentz group is a fundamental symmetry group in physics that describes how the laws of physics are invariant under transformations between inertial frames in special relativity. It includes rotations and boosts, which are transformations that relate different observers moving at constant velocities relative to each other.
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A classical group is a group of symmetries that preserves a certain bilinear form, such as orthogonal, unitary, or symplectic forms, and plays a fundamental role in various branches of mathematics and physics. These groups are central to the study of geometry, representation theory, and quantum mechanics due to their rich structure and symmetry properties.
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Classical groups are a set of groups that arise as symmetries of vector spaces equipped with additional structure, such as a bilinear form or a quadratic form. They play a fundamental role in various areas of mathematics, including geometry, algebra, and physics, due to their rich structure and symmetry properties.
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A rotation group is a mathematical concept that describes symmetries of objects under rotations in a given space, often represented by matrices that preserve geometric properties like distances and angles. These groups are foundational in fields like physics and engineering, as they allow for the study of rotational symmetries and conservation laws in physical systems.
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