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# L - labelled tree to LSLR (Least Squares Regression Line) - Mathematics Dictionary

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labelled tree: A tree in graph theory in which all vertices are assigned labels.

Lagrange multipliers: A method of finding the extrema of a multivariate function subjects to constraints (such as along a curve) by introducing extra variables and tranforming the graph representing a function with constraints to a higher dimensional graph with no constraints, so that a simpler method of simply finding the critical points can be used, since the critical points all satify the constraints.

Lagrange's theorem: Also known as Lagrange's Four Squares Theorem or Bachet's Conjecture, it is a theorem that states that any natural numbers can be exprressed as the sum of 4 squares of integers (or 4 squares of natural numbers including zero).

Lagrangian function: Also known simply as the Lagrangian, a function that is defined as the system's kinetic energy minus the potential energy, which summarizes a dynamical system - it can be used to deduce equations of motions under certain conditions.

Lagrangian mechanics: A classical system of mechanics that is an alternative to Newtonian mechanics by consideration of conservation of energy and momentum of a clsed system.

lamina: A ideal mathematical object with no thickness and zero curvature - a bounded plane with area and (possibly) mass but not volume.

lanczos method: Also known as Lanczos Algorithm, it is an algorithm for finding the eigenvectors and eigenvalues of a (necessarily) square matrix through iterative methods.

Laplace's equation: A second-order partial differential equation which states that the Laplacian (or generlised forms of the Laplacian differential operator) of a scalar function is zero.

Laplace transformation: More commonly known as a Laplace Transform, it is a linear operator which transform a function with non-negative real arguments into an integral of complex arguments.

Laplacian: Also known as the Laplace operator, it is a second-order differential operator finding the divergence of the gradient of a scalar function.

latent root: Also known as the characteristic root, or more commonly, the eigenvalue of a linear transformation or its representation (such as a matrix).

lateral: Also known as a lateral face or lateral side, it is the surfaces of a geometric figure (often polyhedra) not parallel to its base (or perpendicular to its base, depending on the author).

lateral area: The total area of all the lateral sides of a given geometric figure.

latus rectum: The chord of a conic section containing the ficus and parallel to a directrix - a line segment with endpoints on the conic section, through the focus, parallel to the directrix.

law of cosines: Also known as the cosine rule, an equation relating the lengths of all three sides and any one of the angles of any triangle.

law of sines: Also known as the sine rule, an equation relating the three ratios of sides to the sine value of the corresponding (opposite) angles.

laws of indices: A number of rules for the manipulation of indices in an exponentiation of algebraic expressions.

laws of large numbers: A theorem which states that the long term average of an experiment should be close to the expected value. (The avergae tends to the expected value - the greater the number of experiments, the higher the probability that it stays within a given neighbourhood of the expected value.) The probability of this average straying outside any given neighbourhood around the expected value tends towards zero (statistically) even if it is not impossible (physically).

laws of motion: Usually referring to Newton's 3 laws of motion.

LCD: A shorthand for the Lowest/least common denominator of a set of denominators.

LCM: A shorthand for the Lowest/least common multiple.

leading coefficient: The coefficient of of the leading term, that is the term of the highest-order within a polynomial - it is the leading coefficient in the conventional order of arranging terms of one variable (from highest to lowest degree).

leading diagonal: Also known as the main diagonal, it is the elements of a matrix whose indices are all the same; the set of elements between (and including) the top left and bottom right elements.

leading term: The term of the highest-order within a polynomial - it is the leading term in the conventional order of arranging terms of one variable (from highest to lowet degree).

least common denominator: The lowest/least common multiple of a set of denominators of fractions.

least common multiple: The smallest of the set of numbers which are multiples of all the numbers within a given set of positive numbers.

least squares: The method of finding the parameters corresponding to the minimum sum of squares of errors. It represents the most likely actual value, assuming that the errors are distributed normally.

least upper bound: Also known as a supremum. Since an upper bound is simply a value which is greater than any values in a set of numbers, there can be more than one upper bound for a given set. In such cases, the upper bound that is the lowest in value is known as the least upper bound.

lemma: A proven mathematical sentence, it differs from a theorem in the sense that a lemma is itself not the goal for the motivation of its proof. (The motivation is usually another result proven from the lemma.)

lemniscate: A curve with the shape of a "figure of eight", the etymology of the term originates from describing the shape as ribbons. length: 1. A property of any pair of points in a space with a metric.

2. The property of a geometric figure through 2 specific points within the figure.

3. A property related to 1, the limit of the piecewise-linear approximations of a curve.

lever: A object with a pivot used to manipulate the force applied to an object (relative to the force input into the system), or the distance for which a force is applied (relative to the distance over which the force input into the system).

L'Hôpital's rule: A method for finding the limit of a ratio with an indeterminate value through the use of derivatives.

life tables: A table listing the probabilities of an individual of a certain age (possibly with additional assumptions such as geography) surviving until the next "age" for actuarial purposes.

lift: The component of a force on an object from a fluid, perpendicular to the flow of the fluid (relative to the object), in a direction relative to the object that is usually (or designed to be) the opposite direction to gravity.

light year: An unit of distance travelled by light in a vacuum over a year.

limaçon of Pascal: Also known simply as a limaçon, an epitrochoid where the radii of the two circles are the same (in the case where neither centre is inside the other circle), or the moving circle is twice the radius of the fixed circle (in the case of the centre of the smaller/fixed circle being inside of the larger/moving circle - which also means that the centre of the larger/moving circle is on the circumference of the smaller/fixed circle).

limit: A mathematical object perceived to represent the "last" object in a sequence of the same type of mathematical object in the sense that there are truncated sequences for which all objects are within any specified neighbourhood of the limit.

limit comparison test: A method of determining the convergence or divergence of a sequence by comparing the ratio of the terms of a sequence under consideration, with the corresponding terms of a sequence of known status (of convergence/divergence).

limit inferior: Another class="d-title" name for the infimum.

limits of integration: The endpoints (a and b) of an interval over which a defnite integral (the following example) is performed. line: A straight line of infinite length, i.e. indefinitely extended at both ends. Note that a straight line of finite length is called a line segment, and a infinite straight line where only one end is extended indefinitely is usually called a half-line.

linear: 1. An equation related to the form that describes a line in its representation in a graph.

2. A property of an operation where addition and multiplication can commute with the operator in its evaluation, such that:

f (ax + by) = a f(x) + b f(y)

linear algebra: The branch of mathematics that studies vector spaces and linear functions and their representations (such as matrices) between vector spaces.

linear combination: Any mathematical object that is the sum of scalar multiples of the same type of objects. (e.g. transformation)

linear equation: An equation formed by the sums of linear expressions.

linear form: A linear map between 2 vector spaces.

linear function: A function represented by a linear expression, a first-order (degree 1) polynomial..

linear interpolation: A method of interpolation where the new data point is assumed to lie on the (straight) line between the points on either side.

linearly dependent, independent: 1. A vector is said to be linearly dependent of a set of vectors if it can be represented by a linear combination of the set of vectors, otherwise, the vector is said to be linearly independent of the set of vectors.

2. A set of vectors where one is linear dependent of the other vectors is said to be a linearly dependent set of vectors. If all vectors within the set is linearly independent of the other vectors then the set is sais to be linearly independent..

linear mapping: A mapping which is linear.

linear model: A number of different meanings applied to statistical models in statistics - referring to a linear regression model, a model where the expectation of the random variable is a linear expression of the parameters etc.

linear momentum: A quantity that is the product of the mass and (linear) velocity. As opposed to angular momentum.

linear regression: The modelling of the relationship between 2 variables by a linear function/expression.

linear scale: A scale where distances (parallel to a particular axis) always represent the same difference (between the values), regardless of the choice of the 2 points.

linear space: An algebraic structure generalising the idea of an affine space where there is a set of subsets (called lines) of elements (called points) where the intersection of 2 lines is exactly one point.

linear transformation: A linear mapping between 2 vector spaces.

line of best fit: The model of the highest likelihood representing the relationship between 2 variables by a linear function,

line segment: A straight line of finite length; a straight line with 2 endpoints. (i.e. neither end is extended indefinitely.)

litre: A metric (but not SI) unit of capacity that is defined to be one-thousandth (1/1000) of a cubic metre. (m3). The unit is considered a special unit compatible with other SI units and prefixes.

lituus: A type of spiral where the angle from the centre is (partially) inversely proportional to the square of the radius.

ln: The symbol for the natural logarithmic function, i.e. logarithmic function of base e.

load: An external force (usually weight) on a system from an object which is in equilibrium given the equilibruim of the system.

local behaviour: 1. A property which is the limit of a mathematical object in neighbourhood of a point. (e.g. locally linear)

2. A property of a mathematical function for all neighbourhoods within a certain neighbourhood of a point. (e.g. local extrema)

local maximum: The maximum point within a certain neighbourhood (and all smaller neighbourhoods).

local minimum: The minimum point within a certain neighbourhood (and all smaller neighbourhoods).

located vector: A vector with a specific origin (thus also destination).

location: The property of a mathematical object relative to other objects through the concept of adjacency.

locus: A set of points which all satisfy a set of conditions. (e.g. the locus of points equidistant from a point and a line)

logarithm: 1. The exponent to which a base number can be raised so that the result is a specified number.

2. A logarithmic function.

logarithmic coordinate system: A coordinate system using a scale where distances (parallel to a particular axis) always represent the same ratio (between the values), regardless of the choice of the 2 points.

logarithmic differentiation: A method of differentiation by first applying a logarithmic function to the operand. The aim is generally to turn an expression with a complicated exponent expression into one which has only products of complicated expressions, or similarly turning one with complicated products of expressions into one containing only sums of such expressions.

logarithmic function: A function of one variable. associated with a specified base, such that an argument of the result of exponentiation yields a value of the logarithm (the exponent to which the base number should be raised).

logarithmic series: The taylor series expansion of a logarithmic function.

logarithmic spiral: Another class="d-title" name for an equiangular spiral or logistic spiral.

logical equivalence: The relation between 2 sentences whose logical values are always the same.

logistic curve: A bounded function where the bounds are the 2 asymptotes (whose values cannot be attained) initially used for modelling population growth with limited resources.

logistic map: A non-linear difference equation, specifically, a second order (degree 2) recurrence relation which is used to demonstrate how simple non-linear systems can demonstrate complex behaviours.

logistic spiral: Another class="d-title" name for an equiangular spiral or logarithmic spiral.

lognormal distribution: The distribution of a random variable whose logarithm is normally distirbuted. Equivalently, the distribution of the exponentiation of a normally distrubuted random variable.

log-series distribution: logarithmic series distribution, also known as a logarithmic distribution. A discrete distribution derived from the logarithmic series.

longitude: One of the coordinates in the coordinate system used for Earth's surface. It's a directed number equivalent to the azimuth for a (assumed to be) fixed radius in the spherical coordinate system where Greenwich, London is used as a reference point of having 0o longitude. Its values ranges from 180 W (west) to 180 E (east), where the endpoints describe the same coordinate is generally given without the direction.

loop: A bounded curve with no endpoints such that the removal of almost all points result in a curve. (Almost all in the sense that only the removal of a finite number of points may result in more than one curve.)

Lorentz transformation: A linear transformation used to explain (or derive) results from special relativity.

Lorenz attractor: An example of a subset of a phase space of a (chaotic) dynamical system towards which the dynamical system tends.

lower bound: A value which is less than all elements of a given set.

lower limit: One of the limits of integration in the evaluation of a definite integral that is used as the argument in the indefinite integral that is used as the subtrahend (i.e. the number to be subtracted, not the number to subtract from).

lower triangular matrix: A square matrix where all values on or above the leading diagonal is zero. (i.e. a matrix M where mi,j = 0 for i ≥ j )

lowess: LOcally WEighted Scatterplot Smoothing - a method for modelling through regression.

LSRL: Least Squares Regression Line - a linear regression done by least squares methods.

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