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Euclidean geometry is a mathematical system attributed to the ancient Greek mathematician Euclid, which describes the properties and relations of points, lines, surfaces, and solids in two and three dimensions. It is based on five postulates, including the famous parallel postulate, which forms the foundation for much of classical geometry taught in schools today.
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A metric space is a set equipped with a metric, which is a function that defines a distance between any two elements in the set, satisfying properties like non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. metric spaces provide a framework for analyzing concepts of convergence, continuity, and compactness in a general setting, extending beyond the familiar Euclidean space.
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Inequality refers to the uneven distribution of resources, opportunities, and rights within a society, often leading to disparities in wealth, education, and power. Addressing inequality involves understanding its root causes, such as systemic discrimination and unequal access to resources, and implementing policies to promote equity and social justice.
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A normed vector space is a vector space equipped with a norm, which assigns a non-negative length or size to each vector in the space, facilitating the generalization of concepts like distance and convergence. This structure is foundational in functional analysis and provides a framework for studying the geometry of vector spaces and the behavior of linear transformations.
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A distance function is a mathematical construct used to quantify the similarity or dissimilarity between elements in a space, typically satisfying properties like non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. These functions are fundamental in various fields such as machine learning, optimization, and computational geometry, where they help in clustering, classification, and nearest neighbor searches.
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The Cauchy-Schwarz Inequality is a fundamental inequality in linear algebra and analysis, stating that for any vectors u and v in an inner product space, the absolute value of their inner product is less than or equal to the product of their magnitudes. This inequality underlies many mathematical proofs and is essential in fields such as statistics, quantum mechanics, and numerical analysis for establishing bounds and relationships between vector quantities.
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Convexity is a property of a set or function where, for any two points within the set or domain, the line segment connecting them lies entirely within the set or below the graph of the function. This concept is crucial in optimization, economics, and finance, as it often simplifies problem-solving and ensures the existence of optimal solutions.
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Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations such as stretching and bending, but not tearing or gluing. It provides a foundational framework for understanding concepts of convergence, continuity, and compactness in various mathematical contexts.
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A non-Archimedean norm is a type of norm on a field that satisfies the ultrametric inequality, which is stronger than the triangle inequality, implying that the distance between two points is dominated by the maximum of the distances to a third point. This property leads to unique topological and algebraic structures, such as totally disconnected spaces and the absence of small triangles, making non-Archimedean norms crucial in fields like p-adic number theory.
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Hyperbolic triangles are geometric figures formed in hyperbolic space, where the sum of the angles is less than 180 degrees, contrasting with Euclidean triangles. These triangles exhibit unique properties due to the curvature of hyperbolic space, influencing concepts like distance and parallelism.
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Entropy inequalities are mathematical expressions that provide bounds and relationships between different entropy measures, crucial for understanding information distribution and uncertainty in systems. They play a significant role in fields like information theory, thermodynamics, and quantum mechanics, helping to formalize the limits of information processing and transfer.
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The ultrametric inequality is a stronger form of the triangle inequality used in the context of ultrametric spaces, where the distance between any two points is always less than or equal to the maximum of the distances between each of them and a third point. This property leads to unique geometric structures where all triangles are isosceles with the two longer sides being equal in length, and often appears in fields like p-adic number theory and phylogenetics.
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A non-Archimedean valuation is a function that assigns a size or magnitude to elements of a field, satisfying a strong triangle inequality, which implies that the valuation of a sum is at most the maximum of the valuations of the summands. This property leads to the ultrametric inequality, which fundamentally alters the geometry of the field, making it a cornerstone in p-adic analysis and number theory.
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A geodesic triangle is a figure formed by three geodesics, which are the shortest paths between points on a curved surface, such as a sphere or a hyperbolic plane. These triangles help in understanding the geometry and intrinsic properties of the surface, differing significantly from Euclidean triangles in terms of angle sum and side relationships.
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Hyperbolic space is a non-Euclidean geometric structure where the parallel postulate of Euclidean geometry does not hold, allowing for multiple parallel lines through a given point. This space exhibits constant negative curvature, leading to unique properties such as exponential growth of area and volume with respect to radius, and is used in various fields including complex analysis, relativity, and network theory.
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A vector norm is a function that assigns a strictly positive length or size to each vector in a vector space, except for the zero vector, which is assigned a length of zero. It is a fundamental concept in linear algebra and functional analysis, providing a quantitative measure of vector magnitude and enabling the comparison of vector sizes and directions.
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Holder's Inequality is a fundamental inequality in measure theory and functional analysis, which generalizes the Cauchy-Schwarz inequality and provides a bound for the integral of the product of two functions. It is crucial in establishing convergence and integrability conditions in spaces known as Lp spaces, where it helps in proving the Minkowski inequality and the triangle inequality for integrals.
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Minkowski's Inequality is a fundamental result in mathematics that generalizes the triangle inequality to Lp spaces, providing a way to measure the 'distance' between functions in these spaces. It is essential in the study of functional analysis and is widely used in various fields such as probability theory and mathematical physics.
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Geometric inequalities are mathematical statements that compare different geometric quantities, such as lengths, areas, and volumes, under specific conditions. They play a crucial role in various fields of mathematics and physics by providing bounds and relationships that help in solving complex problems and proving theorems.
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Triangular geometry is a branch of geometry dealing with the properties and relations of triangles, which are fundamental shapes with three sides and three angles. It explores concepts like the Pythagorean theorem, trigonometric ratios, and the area and perimeter formulas, providing crucial insights into understanding other geometric shapes and principles.
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Distance matrix completion involves estimating missing values in a partially observed distance matrix, enabling the reconstruction of geometric or structural information essential for applications such as network topology and molecular conformation. The challenge lies in leveraging constraints like triangle inequalities and additional domain knowledge to accurately infer these missing distances.
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