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Equations describing magnetic monopoles
mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation F A = ⋆ d A Φ , {\displaystyle F_{A}=\star d_{A}\Phi
Bogomolny_equations
Study of vector bundles, principal bundles, and fibre bundles
the equations be invariant under a group of translational or other symmetries. Through this process the Yang–Mills equations lead to the Bogomolny equations
Gauge_theory_(mathematics)
configuration of the scalar and gauge fields which satisfies the Bogomolny equations and has finite action. Due to the presence of a scalar field, this
Monopole_(mathematics)
Partial differential equations whose solutions are instantons
differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal
Yang–Mills_equations
Lower energy limit in string theory
a simplified set of first-order partial differential equations known as the Bogomolny equations. Classical solutions that saturate the BPS bound are called
Bogomol'nyi–Prasad–Sommerfield bound
Bogomol'nyi–Prasad–Sommerfield_bound
moduli space is a space parametrizing monopoles (solutions of the Bogomolny equations). Atiyah and Hitchin (1988) studied the moduli space for 2 monopoles
Monopole_moduli_space
differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the
Nahm_equations
Hypothetical particle with one magnetic pole
Giacomelli (2000) and by S. Balestra (2011) in the Bibliography section. Bogomolny equations Dirac string Dyon Felix Ehrenhaft Flatness problem Gauss's law for
Magnetic_monopole
Relation between sides of a right triangle
a mathematician nor a scientist, remains the consensus. Bogomolny (2016), Proof #4. Bogomolny (2016), Proof #3. Loomis 1940. Maor (2007), p. 39. Schroeder
Pythagorean_theorem
Model of mesons in the massless quark limit
term. It also has a formulation which is formally identical to the Bogomolny equations but with Lorentz signature. The relation between these formulations
Chiral_model
Mathematical puzzle
to solve the equations is the condition that the solutions be integers. Any solution must satisfy all equations. Some Diophantine equations have no solution
The_monkey_and_the_coconuts
Mathematical polynomial factorization
Divisibility and Primality, Carnegie Institute of Washington, p. 382 Bogomolny, Alexander, "Sophie Germain's identity", Cut-the-Knot, retrieved 2023-06-19
Sophie_Germain's_identity
3 intersections of any triangle's adjacent angle trisectors form an equilateral triangle
Substituting equations (2) and (5) in the β {\displaystyle \beta } equation and equations (3) and (6) in the γ {\displaystyle \gamma } equation gives h =
Morley's_trisector_theorem
Infinite sequence of differential equations
sequence of mutually compatible nonlinear evolution equations containing the Korteweg–de Vries equation as its first nontrivial member. It is one of the
Korteweg–De_Vries_hierarchy
Line which touches a circle at exactly one point
circle with a straightedge". Stack Exchange. August 15, 2015. Alexander Bogomolny "When A Quadrilateral Is Inscriptible?" at Cut-the-knot Paul Kunkel. "Tangent
Tangent_lines_to_circles
Number, approximately 1.618
related to Golden ratio. Weisstein, Eric W. "Golden Ratio". MathWorld. Bogomolny, Alexander (2018). "Golden Ratio in Geometry". Cut-the-Knot. Knott, Ron
Golden_ratio
Mathematical concept
Elements Book X Proposition 9". Clark University. Retrieved 2008-10-29. Bogomolny, Alexander. "Square root of 2 is irrational". Interactive Mathematics
Quadratic_irrational_number
Self-similar growth curve
2021-05-01 at the Wayback Machine, University of Georgia (1999) Alexander Bogomolny, Spira Mirabilis - Wonderful Spiral, at cut-the-knot Wikimedia Commons
Logarithmic_spiral
Illustration of the Pythagorean theorem
{1}{2}}ab)=(a^{2}-2ab+b^{2})+(2ab)=a^{2}+b^{2}.} Hsuan thu Alex Bogomolny. "Bride's Chair". www.cut-the-knot.org. Retrieved 28 November 2023. David
Bride's_Chair
Used to count, measure, and label
"Transcendental Numbers" (PDF). Stanford University. Retrieved 22 October 2025. Bogomolny, A. "What's a number?". Interactive Mathematics Miscellany and Puzzles
Number
Generalization of Pythagorean theorem
Clark University. Alexander Bogomolny credits this proof to teacher John Molokach (2011), but it may be older. Bogomolny, Alexander. "The Law of Cosines
Law_of_cosines
Line constructed from a triangle
quasi-Euler line of a quadrilateral and a hexagon at Dynamic Geometry Sketches Bogomolny, Alexander, "Altitudes and the Euler Line" and "Euler Line and 9-Point
Euler_line
Problem of constructing equal-area shapes
Wikisource has original text related to this article: Squaring the circle Bogomolny, Alexander. "Squaring the Circle". cut-the-knot. Grime, James (25 March
Squaring_the_circle
Increasing sequence of reduced fractions
"Symmetries of period-doubling maps" (PDF). Bogomolny, Alexander. "Farey series". Cut-the-Knot. Bogomolny, Alexander. "Stern-Brocot Tree". Cut-the-Knot
Farey_sequence
In mathematics, a Riemann surface
(but omitting 4, 24, 48, 72, 140, and various higher values) (Aurich, Bogomolny & Steiner 1991) and where m {\displaystyle m} is the unique odd integer
Bolza_surface
Expression in mathematical analysis
Mathematics Magazine. 50 (1): 41–42. doi:10.2307/2689754. JSTOR 2689754. Bogomolny, Alexander (2018). "Undefined vs Indeterminate in Mathematics". Cut The
Indeterminate_form
Fixed number that has received a name
of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection Bogomolny, Alexander. "Square root of 2 is irrational". Aubrey J. Kempner (Oct 1916)
Mathematical_constant
Perpendicular line segment from a triangle's side to opposite vertex
Geometry: Theorems and Constructions, Prentice Hall, ISBN 0-13-087121-4 Bogomolny, Alexander. "Existence of the Orthocenter". Cut the Knot. Retrieved 2022-12-17
Altitude_(triangle)
Convex 4-sided polygon whose sidelines are all tangent to an outside circle
also a chordal one", Mathematical Communications, 12 (2007) pp. 33–52. Bogomolny, Alexander, "Inscriptible and Exscriptible Quadrilaterals", Interactive
Ex-tangential_quadrilateral
Fractal composed of tangent circles
fractals Weisstein, Eric W., "Apollonian Gasket", MathWorld Alexander Bogomolny, Apollonian Gasket, cut-the-knot An interactive Apollonian gasket running
Apollonian_gasket
Theorem on an equilateral triangle constructed from three equilateral triangles
Revisited, pages 60-63. "Napoleon's Theorem". MathPages.com. Alexander Bogomolny. "Proof #2 (an argument by symmetrization)". Cut-the-knot.org. Retrieved
Napoleon's_theorem
Game variations and descriptions of intransitive dice and their behaviour
Eric W. "Efron's Dice". Wolfram MathWorld. Retrieved 12 January 2021. Bogomolny, Alexander. "Non-transitive Dice". Cut the Knot. Archived from the original
Intransitive_dice
Quadrilateral whose vertices lie on a circle
1929). Inequalities proposed in "Crux Mathematicorum", 2007, [1]. A. Bogomolny, An Identity in (Cyclic) Quadrilaterals, Interactive Mathematics Miscellany
Cyclic_quadrilateral
Form of entertainment in mathematics
following: The Breaking Math Podcast - Episode 001 Cut-the-Knot by Alexander Bogomolny Futility Closet by Greg Ross Mathologer by Burkard Polster Numberphile
Recreational_mathematics
Integers formed by rounding down the integer multiples of a positive irrational number
4153/CMB-1976-071-6. MR 0444558. Includes many references. Weisstein, Eric W. "Beatty Sequence". MathWorld. Alexander Bogomolny, Beatty Sequences, Cut-the-knot
Beatty_sequence
Theorem concerning ratios of line segments
related to Intercept theorem. Intercept Theorem at PlanetMath Alexander Bogomolny: Thales' Theorems and in particular Thales' Theorem at Cut-the-Knot intercept
Intercept_theorem
American mathematician
of Viewpoints: Mathematical Perspective and Fractal Geometry in Art: Bogomolny, Alexander (September 2011), "Review", MAA Reviews Mellor, Blake (December
Annalisa_Crannell
Fractal sets in complex dynamics of mathematics
one complex variable: Introductory lectures". arXiv:math.DS/9201272. Bogomolny, Alexander. "Mandelbrot Set and Indexing of Julia Sets". cut-the-knot
Julia_set
Phenomenon in quantum systems
insight on scarring was acquired with a real-space approach by E. B. Bogomolny and a phase-space alternative by Michael V. Berry complementing the wave-packet
Quantum_scar
Birth of Modern Geology. New York: HarperCollins. ISBN 0-14-028039-1. Bogomolny, Alexander. "Simson Line: What is it?". Cut The Knot: Interactive Mathematics
1799_in_science
Application of K-theory in string theory
such branes can decay, whereas no superposition of branes that satisfy a Bogomolny bound may ever decay. However the charge of such branes is conserved modulo
K-theory_(physics)
Convex, 4-sided shape with an incircle and a circumcircle
Euclidean and Non-Euclidean Geometry a metric approach, [8], pp. 153–158. Bogomolny, Alex, Collinearity in Bicentric Quadrilaterals [9], 2004. L. V. Nagarajan
Bicentric_quadrilateral
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