The Einstein summation convention removes the need to write summation symbols: when an index appears twice, the sum is implied. A simple idea introduced by Einstein that keeps tensor and physics formulas compact and readable. Have you used it before?
#Math #MathType #Physics
How do you combine two matrices into a much larger structure while preserving their internal relationships? The Kronecker product does exactly that.
Have you ever worked with the Kronecker product?
#Math #Mathematics #LinearAlgebra #MathType #STEM
The kernel (null space) of a linear map is the set of vectors mapped to zero. It tells you what the transformation collapses and whether it’s injective. Do you check the kernel first?
#LinearAlgebra #Mathematics #STEM
The I–V equation describes how a solar cell converts light into electricity and defines key parameters such as short-circuit current and open-circuit voltage. A compact model with powerful insights. #SolarEnergy #Physics #STEM
Jordan’s Lemma bounds integrals over large semicircles in the complex plane, ensuring exponential terms decay and making residue calculus and Fourier methods work smoothly. Do you remember using it?
#ComplexAnalysis #Mathematics #STEM #MathEducation #MathType
In mathematics, the length (or magnitude) of a vector is defined as the square root of the sum of the squares of its components, giving us a precise way to measure distance in space. What other “simple” ideas are more powerful than they appear?
#math #mathematics #mathematical
Prime ideals are one of the central building blocks of modern algebra.
They extend the idea of prime numbers from the integers to arbitrary rings, preserving the key property: if a product belongs to the ideal, then at least one of the factors must belong to it.
When is a complex function truly differentiable? The Cauchy–Riemann equations give us the answer. By linking the partial derivatives of the real and imaginary parts of a function, they establish the precise condition for complex differentiability.
#ComplexAnalysis #Math
In mathematics, the Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the mathematician Ludwig Otto Hesse.
Nuclear fission occurs when a heavy nucleus, such as uranium-235, absorbs a neutron, becomes unstable, and splits into two lighter nuclei, releasing energy due to a small loss of mass and emitting additional neutrons that can trigger a chain reaction.
📐 Introduced by Jean le Rond d’Alembert, the ratio test became a practical and reliable method for determining whether an infinite series converges or diverges by comparing consecutive terms. How often do you use the ratio test when teaching or solving series problems?
This fundamental result is known as Euclid’s Lemma. Introduced in Euclid’s Elements, this simple yet powerful idea underpins the unique factorization of integers into prime numbers.
#math #mathematics #mathematical
This elegant result is known as the Four Colour Theorem, a landmark theorem in mathematics stating that any planar map can be coloured using no more than four colours so that no two adjacent regions share the same colour.
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Absolute continuity has important implications in calculus and analysis, particularly in the exploration of integration and differentiation. Functions that possess absolute continuity exhibit a range of desirable properties. which ones you know?
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A quick logic challenge 🧠
Place 3, 4, 5, 6, and 7 in the grid so the vertical and horizontal sums match.
Simple question, tricky thinking:
How many values can go in the center?
#WirisQuizzes #mathquiz #mathproblem #mathexercise #problem
An elegant example of the conservation of angular momentum, the rotational equivalent of linear momentum, is when a figure skater does a pirouette. Their rotational speed increases as their moment of inertia decreases by drawing in their arms and legs.
#MathType #math #mathfacts
How many integer solutions do you think this equation has? 👀
Take a moment, choose your answer, and let the math do the talking.
#WirisQuizzes #mathquiz #mathproblem #mathexercise #problem
The central limit theorem states that as the sample size of a random variable increases, the distribution of sample means becomes more and more normal, regardless of the underlying distribution of the population. It is fundamental in probability theory and #statistics. #MathType
Twin primes are pairs of primes that differ only by 2. It remains an open problem to prove whether there exists an infinite number of them. What are your thoughts?
#MathType #NumberTheory #math #mathematics #mathfacts
Rolle's theorem states that for a differentiable function with real values, if it takes the same value at two distinct points, then there exists at least one point between them where the first derivative is zero.
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Wolstenholme's Theorem states a property that all primes greater than 3 hold. The converse (if a number satisfies the theorem, then it must be prime) is an open problem that could give a new necessary and sufficient condition for primality.
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Was it Andrew Wiles?
My issue with his theorem is its complexity. Surely such a simple problem should have an equally simple proof. I realise this rational does not necessarily follow, but in my perfectly balanced mathematical universe it does.
Numatican is designed to restore this balance.
Absolutely! It was Andrew Wiles.
That's one of the most beautiful things in math, that the seemingly easiest theorems don't necessarily have the easiest demonstrations.😁
Fermat's Last Theorem is a deceptively simple-looking equation that states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. Do you know who proved it?
#MathType #math #mathematics
Amazing how such an apparently easy function has so many applications in engineering! The Heaviside function can also be defined as the integral of the Dirac delta function (be careful at x=0). For H[0] different values can be considered in the discrete domain.
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De Moivre's Formula is an expression that connects the world of #ComplexNumbers and #Trigonometry. Although the one-liner proof via Euler's Identity feels very intuitive and direct now, this formula was proven before Euler's Identity was known.
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Math fact: both the Jacobian matrix and its determinant are called “The Jacobian”. Confusing right?
The Jacobian matrix of a vector-valued function, named after Carl Gustav Jacobi, is the matrix of all its first-order partial derivatives.
#math #mathematics #mathfacts
Euler’s identity, beauty in a formula.
Sometimes called "the most beautiful equation in mathematics", Euler's Identity connects together 5 seemingly unrelated mathematical constants: e, π, i, 0, and 1. Do you know the proof of this identity?
#math #mathematics #mathfacts
Thales' theorem states that a triangle inscribed in a circle, with one side being the circle's diameter, is always a right triangle. This simple yet powerful theorem is a fundamental concept in geometry and has wide-ranging applications.
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#Fibonacci Numbers have fascinated mathematicians for centuries. The Fibonacci rabbit hole goes so deep that even to this day new insights are being gained from them and more properties, generalizations and applications in #NumberTheory are being discovered.
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