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Applications of Algebraic Geometry in Contemporary Physics
Highlights in Science Engineering and TechnologyAlgebraic geometry, a branch of mathematics that studies solutions to polynomial equations, has found profound applications in physics, particularly in the context of addressing singularities with significant physical implications.
Yang Chen
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The Bell states in noncommutative algebraic geometry [PDF]
, 2013 We introduce new mathematical aspects of the Bell states using matrix factorizations, non-noetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial p consists of two matrices ϕ1, ϕ2 such that ϕ1ϕ2 = ϕ2ϕ1 = p id.
C. Beil
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Singularity analysis of 3T2R parallel mechanisms using Grassmann–Cayley algebra and Grassmann geometry [PDF]
Mechanism and Machine Theory, 2012 This paper deals with the singular configurations of symmetric 5-DOF parallel mechanisms performing three translational and two independent rotational DOFs. The screw theory approach is adopted in order to obtain the Jacobian matrices. The regularity of these matrices is examined using Grassmann-Cayley algebra and~Grassmann geometry.
Amine, Semaan +4 more
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Noncommutative Resolutions of Singularities
International Journal of Scientific Research in Science Engineering and TechnologySingularities in algebraic varieties present profound challenges in both classical and modern geometry. While Hironaka’s resolution of singularities provides a powerful tool in the commutative setting, many situations in higher-dimensional geometry and ...
Yashank Mittal, Dr. Narendra Swami
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Comput. Sci. J. Moldova, 1997 SINGULAR, as pointed out by the authors is a computer algebra system designed especially for commutative algebra, algebraic geometry and singular theory. For almost all computations SINGULAR requires a base ring. Computations are possible over such base rings as: polynomial rings, localizations of polynomial rings, quotient of such rings, exterior ...
G.-M. Greuel, G. Pfister, H. Schonemann
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Geometric selection rules for singularity formation in modified gravity
European Physical Journal C: Particles and FieldsWe argue that the polynomial degeneracies of curvature invariants can act as geometric selection rules for spacetime singularities in modified theories of gravity.
Soumya Chakrabarti
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An Analytically Derived Gauss–Legendre Quadrature for Axis-Aligned Ellipse–Ellipse Intersection
MathematicsAccurate and efficient evaluation of the intersection area between two axis-aligned ellipses is essential in applications where the coordinate system or underlying geometry naturally imposes alignment.
Mohamad Shatnawi, Péter Földesi
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Applications of Singularity Theory in Applied Algebraic Geometry and Algebraic Statistics
We survey recent applications of topology and singularity theory in the study of the algebraic complexity of concrete optimization problems in applied algebraic geometry and algebraic statistics.
Maxim, Laurentiu +2 more
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Worldline approach for spinor fields in manifolds with boundaries
Journal of High Energy PhysicsThe worldline formalism is a useful scheme in Quantum Field Theory which has also become a powerful tool for numerical computations. It is based on the first quantisation of a point-particle whose transition amplitudes correspond to the heat-kernel of ...
Lucas Manzo
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Approximate Fiber Products of Schemes and Their Étale Homotopical Invariants
MathematicsThe classical fiber product in algebraic geometry provides a powerful tool for studying loci where two morphisms to a base scheme, ϕ:X→S and ψ:Y→S, coincide exactly.
Dongfang Zhao
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