Some fundamental algebraic formulas and identities Part 3

 

101. Euler's Rotation Theorem:

• Euler's rotation theorem states that any displacement of a rigid body in three-dimensional space is equivalent to a rotation about a fixed axis.

102. Schröder-Bernstein Theorem:

• The Schröder-Bernstein theorem states that if there exist injective functions $�:�\to �$ and $�:�\to �$, then there exists a bijective function $ℎ:�\to �$.

103. Girsanov's Theorem:

• Girsanov's theorem is a result in stochastic calculus that provides a way to change the probability measure for a given stochastic process.

104. Möbius Strip:

• The Möbius strip is a non-orientable surface with only one side and one boundary component.

105. Hausdorff Dimension:

• The Hausdorff dimension is a measure of the "roughness" or "fractality" of a set, generalizing the concept of dimension.

106. Monotone Convergence Theorem:

• The Monotone Convergence Theorem is a fundamental result in measure theory stating that if a sequence of measurable functions is monotone and bounded, then it converges almost everywhere.

107. Radon-Nikodym Theorem:

• The Radon-Nikodym theorem is a result in measure theory that provides conditions under which one measure is absolutely continuous with respect to another measure.

108. Ricci Flow:

• Ricci flow is a process in differential geometry that evolves the metric of a Riemannian manifold, used in the study of geometric structures.

109. Prime Number Theorem for Arithmetic Progressions:

• This theorem extends the Prime Number Theorem to predict the distribution of prime numbers in arithmetic progressions.

110. Haar Measure:

• Haar measure is a way of assigning "volume" to subsets of locally compact topological groups, important in harmonic analysis and representation theory.

111. Gaussian Integral:

• The Gaussian integral is the integral of the Gaussian function, widely used in probability theory and quantum mechanics.

112. Archimedean Property:

• The Archimedean property states that, for any real numbers $�$ and $�$ where $�>0$, there exists a positive integer $�$ such that $��>�$.

113. Ideal Gas Law:

• The ideal gas law, $��=���$, relates the pressure ($�$), volume ($�$), amount of substance ($�$), and temperature ($�$) of an ideal gas.

114. De Bruijn Sequence:

• A De Bruijn sequence is a cyclic sequence in which every possible subsequence of a certain length appears exactly once.

115. Wiener Process:

• The Wiener process, also known as Brownian motion, is a continuous-time stochastic process that models the random motion of particles.

116. Cofactor Expansion:

• Cofactor expansion is a method used to find the determinant of a matrix by expanding along a row or column.

117. Minkowski-Bouligand Dimension:

• The Minkowski-Bouligand dimension, or box-counting dimension, is a method for measuring the fractal dimension of a set.

118. P-NP Problem:

• The P-NP problem is a major open question in computer science and mathematics, asking whether every problem for which a solution can be verified quickly (in polynomial time) can also be solved quickly (in polynomial time).

119. Sturm's Theorem:

• Sturm's theorem is a result in real algebraic geometry that provides a method for counting the number of real roots of a polynomial within a given interval.

120. Tropical Geometry:

• Tropical geometry is an algebraic geometric framework that uses min-max algebra to study certain algebraic varieties.

121. Burnside's Lemma:

• Burnside's Lemma is a result in group theory used to count the number of orbits of a group action under certain conditions.

122. Sylow Theorems:

• The Sylow theorems are a set of results in group theory that provide information about the number of subgroups of a certain order in a finite group.

123. Cauchy-Riemann Equations:

• The Cauchy-Riemann equations are a system of partial differential equations that characterize complex differentiability.

124. Box-Muller Transform:

• The Box-Muller transform is a method for generating pairs of independent, standard normally distributed random numbers from pairs of independent, uniformly distributed random numbers.

125. Matroid Theory:

• Matroid theory is a branch of discrete mathematics that generalizes concepts from linear algebra, graph theory, and combinatorics.

126. Affine Transformation:

• An affine transformation is a combination of linear transformations and translations, preserving points, straight lines, and planes.

127. Maschke's Theorem:

• Maschke's theorem is a result in the representation theory of finite groups, stating that every representation of a finite group can be decomposed into a direct sum of irreducible representations.

128. Haar Wavelet:

• The Haar wavelet is a mathematical function used in signal and image processing for data compression and analysis.

129. Frobenius Automorphism:

• In the context of finite fields, the Frobenius automorphism is a field automorphism that raises each element to a fixed power, usually the prime characteristic of the field.

130. Levi-Civita Symbol:

• The Levi-Civita symbol is a mathematical notation used in vector calculus and differential geometry to represent the permutation of indices.

131. Pascal's Law:

• Pascal's Law, also known as Pascal's Principle, states that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.

132. Pumping Lemma for Regular Languages:

• The Pumping Lemma is a property of regular languages in formal language theory, stating that any sufficiently long string in a regular language can be "pumped" to generate more strings in the language.

133. Rational Root Theorem:

• The Rational Root Theorem helps identify potential rational roots (or zeros) of a polynomial equation with integer coefficients.

134. Karnaugh Map (K-map):

• A Karnaugh map is a graphical representation of truth tables used for simplifying Boolean algebra expressions and digital circuits.

135. Stochastic Calculus:

• Stochastic calculus is a branch of mathematics that extends traditional calculus to deal with random processes and continuous-time stochastic models.

136. Euler's Number (e):

• Euler's number ($�$) is an irrational mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and has widespread applications in mathematics and science.

137. Hausdorff Distance:

• Hausdorff distance measures how far two sets of points in a metric space are from each other. It is used in image analysis and pattern recognition.

138. Residue Theorem:

• The Residue Theorem is a powerful tool in complex analysis, used to evaluate contour integrals by summing the residues of singular points inside a closed curve.

139. Divisibility Rules:

• Divisibility rules are shortcuts for determining whether a given integer is divisible by another integer without performing the actual division.

140. Linearity of Expectation:

• In probability theory, the Linearity of Expectation states that the expected value of the sum of random variables is equal to the sum of their individual expected values.

141. Cauchy-Schwarz-Minkowski Inequality:

• This inequality generalizes the Cauchy-Schwarz Inequality and the Triangle Inequality, providing bounds for various mathematical quantities.

142. Sheaf Theory:

• Sheaf theory is a branch of mathematics that generalizes the concept of locally defined data, often used in algebraic geometry and topology.

143. Poisson's Equation:

• Poisson's equation is a partial differential equation that arises in physics, particularly in the study of electrostatics and heat conduction.

144. Spectral Theorem:

• The Spectral Theorem relates linear operators on a finite-dimensional, complex inner product space to the spectral decomposition of the associated matrix.

145. Jordan Canonical Form:

• The Jordan Canonical Form is a way of representing matrices as a sum of a diagonal matrix and a nilpotent matrix.

146. Hodge Theory:

• Hodge theory is a branch of algebraic geometry and topology that studies harmonic forms on smooth manifolds.

147. Catalan Numbers:

• Catalan numbers are a sequence of natural numbers that appear in various counting problems, such as the number of ways to parenthesize expressions.

148. Sierpinski Triangle:

• The Sierpinski Triangle is a fractal named after Wacław Sierpiński, obtained by recursively removing triangles from a larger equilateral triangle.

149. Sieve of Eratosthenes:

• The Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to a given limit.

150. Random Walk:

• A random walk is a mathematical concept that describes a path formed by a series of random steps. It has applications in various fields, including physics, finance, and computer science.