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Dynamic capabilities refer to an organization's ability to integrate, build, and reconfigure internal and external competencies to address rapidly changing environments. They are crucial for sustaining competitive advantage in volatile markets by enabling firms to adapt, innovate, and transform their strategies and operations effectively.
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An affine space is a geometric structure that generalizes the properties of Euclidean spaces, allowing for the definition of points and vectors without a fixed origin. It is characterized by the ability to perform vector addition and scalar multiplication while maintaining the concept of parallelism and affine transformations.
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Inner product preservation refers to the property of a transformation, typically a linear map or matrix, that maintains the Inner product (dot product) of vectors after transformation. This property is crucial in various fields such as quantum mechanics and computer graphics, ensuring that angles and lengths are preserved under the transformation, thus maintaining geometric integrity.
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Schur's Lemma is a fundamental result in representation theory stating that if a linear map between two irreducible representations of a group is an intertwiner, then it is either an isomorphism or the zero map. This lemma is crucial for understanding the structure of representations and has implications in the study of symmetry and quantum mechanics.
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Linear isomorphism is a bijective linear map between two vector spaces that preserves the operations of vector addition and scalar multiplication, effectively making the two spaces structurally identical. This concept is fundamental in linear algebra as it implies that isomorphic vector spaces have the same dimension and algebraic properties, allowing one to be transformed into the other without loss of information.
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An isometric *-isomorphism is a bijective linear map between two C*-algebras that preserves both the algebraic operations and the norm, ensuring that the structure and properties of the algebras are maintained. This concept is crucial in functional analysis as it allows for the classification of C*-algebras up to isometric *-isomorphism, which is central to understanding their representations and applications in quantum mechanics.
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In mathematics, particularly in the context of linear algebra and category theory, the kernel of a linear map is the set of elements that are mapped to zero, while the cokernel is the quotient of the codomain by the image of the map. These concepts are crucial for understanding the structure of linear transformations and for defining exact sequences in homological algebra.
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A multilinear map is a function that is linear in each of its arguments, meaning that if all but one argument is held constant, the map behaves as a linear transformation with respect to the remaining variable. These maps are fundamental in the study of tensor products and are crucial in fields like differential geometry, algebra, and functional analysis.
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A bilinear form is a mathematical function that takes two arguments from a vector space and returns a scalar, while being linear in each argument separately. It is a fundamental tool in linear algebra and geometry, often used to define inner products, study quadratic forms, and explore properties of vector spaces.
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A topological vector space is a vector space equipped with a topology that makes vector addition and scalar multiplication continuous operations, blending the structures of algebra and topology. This framework allows for the study of convergence, continuity, and compactness in infinite-dimensional spaces, crucial for functional analysis and related fields.
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In mathematics, the cokernel of a linear map between vector spaces is the quotient of the codomain by the image of the map, capturing the failure of the map to be surjective. It is a fundamental concept in homological algebra and is used to understand the structure of modules and abelian groups in terms of exact sequences.
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A bimodule is a generalization of the concept of a module, where an algebraic structure is defined with respect to two rings, allowing for multiplication from both the left and right. This structure is crucial in the study of ring theory and homological algebra, facilitating the interaction between different algebraic systems.
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A vector space homomorphism is a linear map between two vector spaces that preserves the operations of vector addition and scalar multiplication. It is a fundamental concept in linear algebra, ensuring that the structure of vector spaces is maintained under transformation.
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