Mathematics - Concept

Concavity

In mathematics, this describes the way the derivative of the function is changing.

Kurtosis

It is a statistical measure used to describe the degree to which scores cluster in the tails or the peak of a frequency distribution.

Stirling's Approximation

A method for approximating the value of large factorials that uses the mathematical constants e and π

Skew Hermitian

A square matrix A that satisfies the equation A^H = −A where A^H is the conjugate transpose of A

Chain Rule

the derivative of f o g is the product of the derivative of the outer function f (evaluated at g(x)) and the derivative of the inner function (evaluated at x). It is assumed that g is differentiable at x and that f is differentiable at g(x). This is also known as

Wronskian

is a determinant of a square matrix used in the study of differential equations, where it can sometimes show linear independence of a set of solutions

Divergence

It is a scalar-valued function that measures the rate at which "density" exits a point in the field. Mathematically, it is the dot product of the del operator ∇ and a vector field F.

Z-Transform

converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.

L’Hospital’s Rule/Bernoulli’s Rule

is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives

Orthogonal Matrix

if the product of a matrix and its transpose gives an identity value then this matrix is also known as

Negative Binomial Distribution

is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes occurs.

Mason’s Gain Formula

It is a technique used for finding the transfer function of a control system. Basically, a formula that determines the transfer function of a linear system by making use of the signal flow graph

Mean-Value Theorem

This states that there is some point on the graph between A and B where the tangent line is parallel to the secant line through A and B; that is, there is some number c in (a,b) such that:

f'(c) = (f(b)-f(a))/(b-a)

Cavalieri's Principle

This states that any two solids included between parallel horizontal planes; if every right section has the same area in both solids, then the volume of the solids are equal.

Cauchy-Euler Equation

A linear differential equation of the form

Where the coefficients an, a(n-1), . . . , a0 are constants is known as __________.

Hypocycloid

What do you call the curve traced by a point on the circumference of a circle which is rolling on the interior of another circle?

Type I Error

This error occurs when the null hypothesis is rejected when it is actually true. In practical terms, it means concluding that there is an effect or difference when none exists. The probability of committing this eror is denoted by alpha (α).

Second Theorem of Pappus

It states that the volume of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area of the lamina and the distance traveled by the lamina's geometric centroid. What is this statement?

Tridiagonal Matrix

What do you call a matrix whose entries on the main diagonal, super diagonal and sub-diagonal are non-zero?

Becquerel/Bq

Radioactivity is the emission of ionizing radiation or particles caused by the spontaneous disintegration of atomic nuclei. What is the SI unit of radioactivity?

Fibonacci Sequence

The first two terms of the _______ are 1 by definition. Every term after that is the sum of the two preceding terms.

216

What is the absolute value where the surface area and volume of a cube will be equal?

Kite

A convex quadrilateral having a symmetry axis. It thus consists of two adjacent sides of the same length and two adjacent sides are of the same length. Special case is rhombus.

Thales Theorem

States that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle.

Apothem

Perpendicular distance from the center of a regular polygons.

Arthur Cayley

Who is the British Mathematician who discovered matrices in the year 1858?

Cauchy's Mean Theorem

The geometric mean of several positive numbers is smaller than the arithmetic mean of these numbers.

Cevian

A line segment joining the vertex of a triangle to any given point on the opposite side is called?

Lerch's Theorem

It states that if L{F1(t)} = L{F2(t)}, then F1(t) - F2(t) = N(t), where N(t) is a null function. That is, an inverse Laplace transform is unique for the addition of an arbitrary null function.

Ordinary Differential Equation

It is an equation that contains one or several derivatives of an unknown function called y(x) and which we want to determine from the equation.

Rose Curves

The graphs of the equations of the forms r=asin(nθ) and r=acos(nθ) where n is a positive integer, greater than 1, are called _____.

Limacon

The graph of an equation of the form r=b+asinθ or r=b+acosθ is called a ________.

Ellipse

A/n ______ is the set of all points P in a plane such that the sum of the distances of P from two fixed points F and G of the plane is constant.

Laguerre Polynomials

Any differential equation of the form y=px+f(p) where f(p) contains neither x nor y explicitly is called a/n _______.

Canonical Variables

These variables are dimensionless combinations of the physical variable and parameters of the original

Fundamental Theorem of Algebra

This states that every integral rational equation has at least one root

Cologarithm

The logarithm of the reciprocal of a number is called _____.

Relatively Prime/Coprime

Two integers are said to be _______ if their greatest common divisor is 1.

Diophantine Equation

What do you call the equation which is a polynomial equation with integer coefficients and integer solutions?

J/K

What is the SI unit of Boltzmann Constant?

Axiom

A proposition that is not actually proven or demonstrated, but is considered to be self-evident and universally accepted as a starting point for deducing and inferring other truths and theorems, without any need of proof

Transcendental

A/an ______ number is a real or complex number that is not a root of any non-zero polynomial equations of rational coefficients.

Circumcenter

In Trigonometry, what refers to the point of concurrency of the perpendicular bisector of the sides of the triangle?

Prolate Spheroid

What solid is generated when an ellipse is revolved about its major axis?

Fourier Analysis

A technique for resolving complex repetitive waveforms into sine or cosine waves and a dc component is known as:

Euler's identity

The equation (e^(jπ))+1=0 is known as:

dilation

The sum of the strains in the three orthogonal directions (εx, εy, εz) in accordance with Poisson's ratio is

Plastic Number

A mathematical constant which is the unique real solution of the cubic equation x^3=x+1 is called:

Malthu's Law

A theory proposed in 1798 that population would grow at a geometric rate while the food supply grows at an arithmetic rate.

Planimeter

is a mechanical instrument used for measuring the area of a region by tracing its boundary curve

Nihonium/Nh

What is the newest element added in the periodic table with atomic no. 113

Green's Theorem

A theorem that gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C.

Myriagon

What is a shape with ten thousand sides called?

Corollary

A proposition that can be proven as a consequence of a theorem that has just been proved

Point of Inflection/Inflection Point

A point on the graph of a function at which concavity changes

Centroid

The point where the three medians of a triangle intersect. It acts as the triangle’s center of gravity and divides each median in a 2:1 ratio, with the longer segment closer to the vertex.

Orthocenter

The point where the three altitudes of a triangle meet.

Incenter

The point where the angle bisectors of a triangle meet. It is the center of the incircle (the circle inscribed within the triangle) and is equidistant from each side of the triangle.

Theorem

A statement that has been rigorously proven to be true based on previously established statements, such as axioms or other theorems.

Proposition

A statement that is true and proven, but generally less significant or foundational than a theorem.

Lemma

A preliminary or helper statement used to prove a larger theorem, simplifying complex proofs by breaking them down into smaller, manageable parts.

Proof

A logical and structured argument that establishes the truth of a statement by using definitions, axioms, and previously proven statements.

Conjecture

A statement believed to be true based on observations, but not yet proven, often the starting point for new theories or theorems.

Square Matrix

A matrix with the same number of rows and columns

Diagonal Matrix

A square matrix in which all non-diagonal elements are zero. Only the diagonal entries (from top left to bottom right) may be non-zero.

Identity Matrix

A diagonal matrix with all diagonal elements equal to 1. It acts as the multiplicative identity in matrix multiplication (denoted as I), so AI=IA=A for any matrix A.

Symmetric Matrix

A square matrix that is equal to its transpose, meaning A=A^T

Unitary Matrix

A complex square matrix U that satisfies U∗U=I

Upper Triangular Matrix

square matrix where all elements below the main diagonal are zero. Only elements on or above the main diagonal may be non-zero.

Lower Triangular Matrix

A square matrix where all elements above the main diagonal are zero. Only elements on or below the main diagonal may be non-zero.

Toeplitz Matrix

A matrix in which each descending diagonal from left to right is constant. This structure is often used in signal processing.

Diagonalizable Matrix

A matrix that can be expressed as A=PDP^−1, where D is a diagonal matrix, and P is an invertible matrix.

Nilpotent Matrix

A square matrix A for which some power k (where k≥1) results in the zero matrix: A^k=0.

Idempotent Matrix

A matrix A that satisfies A²=A. This property implies that applying the matrix operation twice has the same effect as applying it once.

Type II Error

This error occurs when the null hypothesis is not rejected when it is false. It means failing to detect an effect or difference that is present. The probability of committing this error is denoted by beta (β).

Sampling Error

This statistical error occurs due to the natural variability that arises when a sample is drawn from a population. It reflects the difference between the sample statistic and the actual population parameter.

Non-Sampling Error

A statistical error type encompasses all other errors that can occur in data collection and analysis, such as measurement errors, data processing errors, and response bias, which can also occur even with a complete census and can significantly affect results.

Measurement Error

This refers to the statistical error of inaccuracies in data collection that can result from faulty instruments, misinterpretation of questions, or respondent errors. This error can lead to biased or unreliable results.

Systematic Error

Also known as bias, this type of statistical error occurs consistently in the same direction, causing the results to deviate from the true values. It can arise from flawed measurement techniques, sample selection, or survey design.

Random Error

This statistical error arises from unpredictable fluctuations in the data due to chance. These errors can occur in any measurement process and are typically reduced by increasing the sample size or using more precise instruments.

Gradient

is a vector field that points in the direction of the greatest rate of increase of the scalar function. It is calculated as the vector of partial derivatives of the function.

Curl

It measures the rotation or swirling of the field around a point. It results in another vector field, calculated as the cross product of the del operator and the vector field.

Laplacian

It is a second-order differential operator that measures the divergence of the gradient of a scalar field. It indicates how a function spreads out from a point.

Oblate Spheroid

is a shape that resembles a flattened sphere, with a smaller polar radius than equatorial radius. It's the result of rotating an ellipse around its minor axis.