hcju/proofwikips huggingface dataset:
Definition:RSA Algorithm
The '''RSA algorithm''' is a technique of encoding a message such that the method of encoding can be made public without compromising the security. Let Alice be sending a message to Bob. Alice and Bob both agree on two large primes $p$ and $q$, each having at least $100$ digits. Let $M = p q$. $M$ can be made public if they so wish. Let $K = \left({p - 1}\right) \left({q - 1}\right)$. $K$ is to be kept secret. Let Alice choose some number $a$ such that $a \perp K$. $a$ is made known to Bob, and may even be made public. Alice represents her message in a series of numbers in the range $0$ to $M$. Let $x$ be one such number. Alice calculates $y \equiv x^a \pmod M$. The sequence of numbers $y$ is sent to Bob as the coded message. Bob needs to know a number $b$ such that $a b \equiv 1 \pmod K$. This number needs to be kept secret. To decode $y$, Bob computes $y^b \pmod M$. This works because of Fermat's Little Theorem: :$y^b \equiv \left({x^a}\right)^b \equiv x^1 = x \pmod M$ The method works because: :$(1): \quad$ There are efficient tests to find large primes :$(2): \quad$ There are no known methods for finding the prime factors of large numbers efficiently. So making $p q$ public does not help $p$ and $q$ be found, and without those it is impossible to work out what $b$ is. {{NamedforDef|Ronald Linn Rivest|name2 = Adi Shamir|name3 = Leonard Max Adleman|cat = Rivest|cat2 = Shamir|cat3 = Adleman}}
Definition:Cryptography
'''Cryptography''' is the branch of mathematics which concerns the encoding of information for secrecy and security.
Definition:Real Function/Multivariable
Let $f: S_1 \times S_2 \times \cdots \times S_n \to \R$ be a mapping where $S_1, S_2, \ldots, S_n \subseteq \R$. Then $f$ is defined as a '''(real) function of $n$ (independent) variables'''.  The expression: :$y = \map f {x_1, x_2, \ldots, x_n}$ means: :(The dependent variable) $y$ is a function of (the independent variables) $x_1, x_2, \ldots, x_n$.
Definition:Derivative/Real Function/Derivative at Point/Definition 1
That is, suppose the limit $\displaystyle \lim_{x \mathop \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$ exists. Then this limit is called the '''derivative of $f$ at the point $\xi$'''.
Definition:Derivative/Real Function/Derivative at Point/Definition 2
That is, suppose the limit $\displaystyle \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$ exists. Then this limit is called the '''derivative of $f$ at the point $\xi$'''.
Definition:Secant Line
Let $f: \R \to \R$ be a real function. Let the graph of $f$ be depicted on a Cartesian plane. :400px A '''secant''' to $f$ is a straight line which intersects the graph of $f$ in (at least) two points. In the above diagram, the '''secant''' is the line $AB$ in $\color {blue} {\text {blue} }$.
Definition:Tangent/Analytic Geometry
Let $f: \R \to \R$ be a real function. Let the graph of $f$ be depicted on a Cartesian plane. :400px Let $A = \tuple {x, \map f x}$ be a point on $G$. The '''tangent to $f$ at $A$''' is defined as: :$\displaystyle \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$ Thus '''tangent to $f$ at $x$''' can be considered as the secant $AB$ to $G$ where: :$B = \tuple {x + h, \map f {x + h} }$ as $B$ gets closed and closer to $A$. By taking $h$ smaller and smaller, the secant approaches more and more closely the tangent to $G$ at $A$. Hence the '''tangent''' to $f$ is a straight line which intersects the graph of $f$ locally at a single point. :400px In the above diagram, the '''tangent''' is the straight line passing through $A$.
Definition:Celestial Mechanics
'''Celestial mechanics''' is the branch of astronomy which studies the motion of celestial bodies such as planets, stars and comets.
Definition:Orbit (Physics)
The '''orbit''' of body $A$ around body $B$ is the path taken by $A$ as it travels around $B$ under the influence of the force acting between $A$ and $B$. === Period of Orbit === {{:Definition:Orbit (Physics)/Period}}
Definition:Orbit (Physics)/Period
The '''period''' of an orbit of body $A$ around body $B$ is the length of time it takes for $A$ to travel once around $B$ and return to its original position.
Definition:Astrophysics
'''Astrophysics''' is the branch of physics which studies the internal nature of celestial bodies as opposed to their positions and movements.
Definition:Celestial Body
A '''celestial body''' is a more-or-less internally coherent body of matter that exists outside the Earth's atmosphere.
Definition:Planet
A '''planet''' is a celestial body which orbits a star (or the remains of a star) that: :$(1): \quad$ has enough mass so that its own gravitational field makes it approximately spherical :$(2): \quad$ does not have enough mass to trigger thermonuclear fusion :$(3): \quad$ has cleared its neighbouring region of planetesimals.
Definition:Star (Physics)
A '''star''' is a celestial body that consists of a luminous sphere of plasma held together by its own gravitational field.
Definition:Sun
The '''Sun''' is the star around which Earth (and the other planets of the solar system) is in orbit.
Definition:Earth
'''Earth''' is the planet on which we live. It is the third planet of the solar system from the sun. Its orbit lies between those of Venus and Mars.
Definition:Solar System
The '''solar system''' is the system of celestial bodies which are under the direct influence of the gravitational field of the sun.
Definition:Mercury (Planet)
'''Mercury''' is the innermost planet of the solar system. Its orbit lies within that of Venus.
Definition:Venus
'''Venus''' is the second planet of the solar system from the sun. Its orbit lies between those of Mercury and Earth.
Definition:Mars
'''Mars''' is the fourth planet of the solar system from the sun. Its orbit lies between those of Earth and Jupiter.
Definition:Jupiter
'''Jupiter''' is the fifth planet of the solar system from the sun. Its orbit lies between those of Mars and Saturn.
Definition:Saturn
'''Saturn''' is the sixth planet of the solar system from the sun. Its orbit lies between those of Jupiter and Uranus.
Definition:Uranus
'''Uranus''' is the seventh planet of the solar system from the sun. Its orbit lies between those of Saturn and Neptune.
Definition:Neptune
'''Neptune''' is the eighth planet of the solar system from the sun. Its orbit lies mainly between those of Uranus and Pluto, although the latter comes closer to the sun than '''Neptune''' on part of its orbit.
Definition:Pluto
'''Pluto''' is a dwarf planet of the solar system. Its orbit lies mostly outside that of Neptune.
Definition:Dwarf Planet
A '''dwarf planet''' is a celestial body which orbits a star (or the remains of a star) that: :$(1): \quad$ has enough mass so that its own gravitational field makes it approximately spherical :$(2): \quad$ does not have enough mass to trigger thermonuclear fusion :$(3): \quad$ has not cleared its neighbouring region of planetesimals.
Definition:Planetesimal
A '''planetesimal''' is a celestial body consisting of solid matter such that: :$(1): \quad$ it is held together by its own gravitational field :$(2): \quad$ it is not large enough for its gravitational field to significantly affect other celestial bodies nearby.
Definition:Time/Dimension
'''Time''' is one of the fundamental dimensions of physics. In dimensional analysis it is assigned the symbol $T$ or $\mathbf T$. It is a scalar quantity which can be mapped directly to the real number line.
Definition:Time/Unit Time
The usual units of '''time''' are seconds, and compound units consisting of multiples of seconds. However, when discussing a general physical process it is convenient to discuss the general situation, in which case the term '''unit time''' is used. Thus the specific '''units''' are not mentioned.
Definition:Time/Length
A quantity of '''time''' is usually referred to in natural language in the same terms as linear measure. Thus it is usual to refer to a '''length of time''' of an elapsed quantity of time.
Definition:Time/Unit
The units of measurement of time are universal in all systems of measurement. === Second === {{:Definition:Time/Unit/Second}} Derived units of time include: === Minute === {{:Definition:Time/Unit/Minute}} === Hour === {{:Definition:Time/Unit/Hour}} === Day === {{:Definition:Time/Unit/Day}} There are longer time periods, all of which are more or less approximate: === Week === {{:Definition:Time/Unit/Week}} === Fortnight === {{:Definition:Time/Unit/Fortnight}} === Month === {{:Definition:Time/Unit/Month}} === Lunar Month === {{:Definition:Time/Unit/Lunar Month}} === Season === {{:Definition:Time/Unit/Season}} === Quarter-Year === {{:Definition:Time/Unit/Quarter-Year}} === Year === {{:Definition:Time/Unit/Year}} === Decade === {{:Definition:Time/Unit/Decade}} === Century === {{:Definition:Time/Unit/Century}} === Millennium === {{:Definition:Time/Unit/Millennium}} === Great Year of Plato === {{:Definition:Time/Unit/Great Year of Plato}}
Definition:Derived Unit
'''Derived units''' are units of measurement created by taking specific multiples or fractions of fundamental units.
Definition:System of Measurement
A '''system of measurement''' is a set of fundamental units of measurement with which one can measure any measurable physical property.
Definition:Time/Unit/Minute
The '''minute''' is a derived unit of time. :$1$ '''minute''' $= 60$ seconds.
Definition:Time/Unit/Second/Symbol
The symbol for the '''second''' is $\mathrm s$.
Definition:Time/Unit/Minute/Symbol
The symbol for the '''minute''' is $\mathrm {min}$.
Definition:Time/Unit/Hour/Symbol
The symbol for the '''hour''' is $\mathrm {hr}$.
Definition:Time/Unit/Hour
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hour''' }} {{eqn | r = 60       | c = minutes }} {{eqn | r = 60 \times 60       | rr= = 3600       | c = seconds }} {{end-eqn}}
Definition:Time/Unit/Day
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''day''' }} {{eqn | r = 24       | c = hours }} {{eqn | r = 60 \times 24       | rr= = 1440       | c = minutes }} {{eqn | r = 60 \times 60 \times 24       | rr= = 86\, 400       | c = seconds }} {{end-eqn}}
Definition:Time/Unit/Day/Symbol
The symbol for the '''day''' is $\mathrm {day}$ or $\mathrm d$.
Definition:Sidereal Day
A '''sidereal day''' is (approximately) the length of time it takes for Earth to rotate once on its axis relative to the stars. The mean duration of a '''sidereal day''' is: : $23$ hours, $56$ minutes and $4.0916$ seconds : $23.9344699$ hours : $0.99726958$ (solar) days. The duration of any given '''sidereal day''' is not constant because of various astronomical effects.
Definition:Algorithm/Input
An '''algorithm''' has zero or more '''inputs'''. These are values supplied either: : before the algorithm starts : as the algorithm runs. These '''inputs''' are taken from specified sets of objects.
Definition:Algorithm/Output
An '''algorithm''' has one or more '''outputs'''. These are values which are specifically determined by the inputs.
Definition:Computational Problem
A '''computational problem''' is a mathematical object which represents a collection of questions that are to be solved by algorithmic techniques.
Definition:Algorithm/Finiteness
An '''algorithm''' must '''terminate''' after a finite number of '''steps'''.
Definition:Algorithm/Definiteness
Each '''step''' of an algorithm must be '''precisely defined'''.
Definition:Algorithm/Effectiveness
An '''algorithm''' is supposed to be '''effective'''. That is, its '''operations''' must be basic enough to be able to be done '''exactly''' and in a finite length of time by, for example, somebody using pencil and paper.
Definition:Algorithm/Analysis
There are two primary methods for analyzing '''algorithms''' formally: === Correctness === {{:Definition:Algorithm/Analysis/Correctness}} === Complexity === {{:Definition:Algorithm/Analysis/Complexity}}
Definition:Algorithm/Analysis/Complexity
An algorithm may be analysed in the context of its '''complexity'''. This is typically via measures of run-times using big-O, or big-omega, or big-theta notation. Complexity can be through 3 main types of analysis: # Average-Case Analysis, using a distribution of cases to find an average case of the algorithm run-time for analysis # Best-Case Analysis, using the best possible problem instance of the algorithm for analysis # Worst-Case Analysis,using the worst possible problem instance of the algorithm for analysis
Definition:Stationary
A body is '''stationary''' if its velocity (in a given frame of reference) is zero.
Definition:Infinitesimal
An '''infinitesimal''' is a mathematical object $\delta$ resembling a (real) number such that: :$(1): \quad \delta > 0$ :$(2): \quad \forall x \in \R_{>0}: \delta < x$ That is, an '''infinitesimal''' is a (strictly) positive real number which is smaller than every other (strictly) positive real number.
Definition:Solid of Revolution
Let $F$ be a plane figure. Let $L$ be a straight line in the same plane as $F$. Let $F$ be rotated one full turn about $L$ in $3$-dimensional space. The '''solid of revolution generated by $F$ around $L$''' is the solid figure whose boundaries are the paths taken by the sides of $F$ as they go round $L$.
Definition:Differential Equation/Order
The '''order''' of a '''differential equation''' is defined as being the order of the highest order derivative that is present in the equation.
Definition:Differential Equation/Ordinary
An '''ordinary differential equation''' (abbreviated '''O.D.E.''' or '''ODE''') is a '''differential equation''' which has exactly one independent variable. All the derivatives occurring in it are therefore ordinary. The general '''ODE''' of order $n$ is: :$\map f {x, y, \dfrac {\d x} {\d y}, \dfrac {\d^2 x} {\d y^2}, \ldots, \dfrac {\d^n x} {\d y^n} } = 0$ or, using the prime notation: :$\map f {x, y, y', y'', \ldots, y^{\paren n} } = 0$
Definition:Differential Equation/Partial
A '''partial differential equation''' (abbreviated '''P.D.E.''' or '''PDE''') is a '''differential equation''' which has: :one dependent variable :more than one independent variable. The derivatives occurring in it are therefore partial.
Definition:Differential Equation/Linear
A '''linear differential equation''' is a '''differential equation''' where all dependent variables and their derivatives appear to the first power. Neither are products of dependent variables allowed.
Definition:Differential Equation/Non-Linear
A '''non-linear differential equation''' is a '''differential equation''' which is not linear.
Definition:Differential Equation/System
A '''system of differential equations''' is a set of simultaneous '''differential equations'''. The solutions for each of the differential equations are in general expected to be consistent.
Definition:Differential Equation/System/Autonomous
A '''system of differential equations''' is '''autonomous''' if all of the '''differential equations''' which it comprises are themselves autonomous.
Definition:Differential Equation/Explicit
A '''differential equation''' is called '''explicit''' if it can be written in the form: :$y^{\left({n}\right)} = f \left({x, y, y', y'', \dots, y^{\left({n-1}\right)}}\right)$
Definition:Differential Equation/Autonomous
A '''differential equation''' is '''autonomous''' if none of the derivatives depend on the independent variable. The $n$th order '''autonomous differential equation''' takes the form: :$y^{\paren n} = \map f {y, y', y'', \dots, y^{\paren {n - 1} } }$
Definition:Legendre's Differential Equation
:$\displaystyle \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - 2 x \frac {\d y} {\d x} + p \paren {p + 1} y = 0$
Definition:Integral Curve
Let $\map f {x, y}$ be a continuous real function within a rectangular region $R$ of the Cartesian plane. Consider the first order ODE: :$(1): \quad \dfrac {\d y} {\d x} = \map f {x, y}$ :500px Let $P_0 = \tuple {x_0, y_0}$ be a point in $R$. The number: :$\paren {\dfrac {\d y} {\d x} }_{P_0} = \map f {x_0, y_0}$ determines the slope of a straight line passing through $P_0$. Let $P_1 = \tuple {x_1, y_1}$ be a point in $R$ close to $P_0$. Then: :$\paren {\dfrac {\d y} {\d x} }_{P_1} = \map f {x_1, y_1}$ determines the slope of a straight line passing through $P_1$. Let $P_2 = \tuple {x_2, y_2}$ be a point in $R$ close to $P_1$. Then: :$\paren {\dfrac {\d y} {\d x} }_{P_2} = \map f {x_2, y_2}$ determines the slope of a straight line passing through $P_2$. Continuing this process, we obtain a curve made of a sequence of straight line segments. As successive points $P_0, P_1, P_2, \ldots$ are taken closer and closer to each other, the sequence of straight line segments $P_0 P_1 P_2 \ldots$ approaches a smooth curve $C$ passing through an initial point $P_0$. By construction, for each point $\tuple {x, y}$ on $C$, the slope of $C$ at $\tuple {x, y}$ is given by $\map f {x, y}$. Hence this curve is a solution to $(1)$. Starting from a different point, a different curve is obtained. Thus the general solution to $(1)$ takes the form of a set of curves. This set of curves are referred to collectively as '''integral curves'''.
Definition:Parameter of Differential Equation
Let $f$ be a differential equation with general solution $F$. A '''parameter''' of $F$ is an arbitrary constant arising from the solving of a primitive during the course of obtaining the solution of $f$.
Definition:Substance
A '''substance''' is a particular kind of matter with properties specific to it.
Definition:First-Order Reaction
Suppose a body has a tendency to decompose spontaneously into smaller bodies at a rate independent of the presence of other bodies. Then the number of bodies that decompose in a single unit of time is proportional to the total number present. Such a reaction is called a '''first-order reaction'''.
Definition:Matter
'''Matter''' is the stuff of which the physical universe is composed. At a certain level, can be considered as a form of energy. '''Matter''' is (to a first degree of approximation) composed of atoms.
Definition:Chemical Element
A '''chemical element''' is a substance which is composed of atoms of one type. A '''chemical element''' is characterised uniquely by the number of protons it has in the nucleus of its atoms.
Definition:Atom (Physics)
An '''atom''' (in the context of physics and chemistry) is the smallest piece of matter that can exist of a particular type of substance. '''Atoms''' can be subdivided into smaller particles, but then it ceases to be that substance.
Definition:Radioactive Decay
'''Radioactive decay''' is a first-order reaction in which the atoms of a substance may spontaneously break apart, thereby turning into atoms of a different substance.
Definition:Radioactive Decay/Half-Life
The '''half-life''' of a radioactive isotope is the time it takes for exactly half of an arbitrary quantity of that isotope to undergo radioactive decay. Thus it is the time it takes for exactly half of an arbitrary quantity of that isotope to remain.
Definition:Radioactive Decay/Radioactive Element
A chemical element whose isotopes are ''all'' subject to '''radioactive decay''' is known as a '''radioactive element'''.
Definition:First-Order Reaction/Rate Constant
Let $S$ be a substance which decomposes spontaneously in a '''first-order reaction'''. By First-Order Reaction, this is governed by the equation: :$x = x_0 e^{-k t}$ where: :$x$ is the quantity of $S$ at time $t$ :$x_0$ is the quantity of $S$ at time $t = 0$ :$k$ is a positive number. The number $k$ is called the '''rate constant'''. $k$ is constant for a particular substance, and different substances, in general, have different '''rate constants'''.
Definition:Chemical Element/Isotope
'''Isotopes''' are instances of chemical elements which have the same number of protons in the nuclei of their atoms, but different numbers of neutrons. Thus it is usual to discuss '''an isotope''' of a particular element.
Definition:Atom (Physics)/Nucleus
The '''nucleus''' of an atom is the relatively dense core at its center. It consists (to a first degree of approximation) of a number of protons and a similar number of neutrons.
Definition:Elementary Particle
An '''elementary particle''' is one of the basic building blocks of matter. To a first degree of approximation, those building blocks can be considered to be protons, neutrons and electrons. Modern research in particle physics suggests the existence of smaller constituents of protons and neutrons, but at time of writing, electrons are believe to be truly, fundamentally indivisible.
Definition:Particle Physics
'''Particle physics''' is the branch of physics which studies the structure and interactions of the elementary particles of matter.
Definition:Radioactive Decay/Radioactive Isotope
An isotope of a chemical element which is subject to '''radioactive decay''' is known as a '''radioactive isotope'''.
Definition:Time/Unit/Year
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''year''' }} {{eqn | o = \approx       | r = 365 \cdotp 24219 \, 878       | c = days }} {{eqn | o = \approx       | r = 365 \cdotp 24219 \, 878 \times 24 \times 60 \times 60       | c = seconds }} {{end-eqn}} That is: {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''year''' }} {{eqn | o = \approx       | r = 365       | c = days, }} {{eqn | o =        | r = 5       | c = hours, }} {{eqn | o =        | r = 48       | c = minutes, and }} {{eqn | o =        | r = 45 \cdotp 9747       | c = seconds }} {{end-eqn}} It is defined to be equal to the orbital period of Earth round the sun.
Definition:Time/Unit/Year/Symbol
The symbol for the '''year''' is $\mathrm {yr}$ or $\mathrm y$. In some scientific contexts $\mathrm a$ (for '''annus''', Latin for '''year''') is used.
Definition:Time/Unit/Year/Calendar
A '''calendar year''' is the number of days between dates with the same designation in adjacent years. As a '''year''' is not an integral number of days, it is necessarily an approximation to the actual year. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''calendar year''' }} {{eqn | r = 12       | c = months usually, depending on the calendar used }} {{eqn | r = 365 \text{ or } 366       | c = days, usually, but it depends on the calendar used }} {{end-eqn}}
Definition:Radioactive Decay/Stable
An isotope of a chemical element which is ''not'' subject to '''radioactive decay''' is described as '''stable'''.
Definition:Density of Topological Space
Let $T$ be a topological space. The '''density''' of $T$ is the smallest cardinality of dense subset of $T$: :$d \left({T}\right) := \min \left\{{\left\vert{A}\right\vert: A \subseteq T \land A}\right.$ is dense$\left.{}\right\}$
Definition:Chemical Compound
A '''(chemical) compound''' is a (chemical) substance that consists of molecules composed of more than one type of chemical element.
Definition:Molecule
A '''molecule''' is an arrangement of atoms, possibly of different elements, which are held together in a configuration which requires the application of external energy to break apart. The '''molecule''' is the basic unit of a chemical compound.
Definition:Chemical Reaction
A '''chemical reaction''' is a physical process in which the atoms  composing the molecules of the chemical substances involved change their configurations into different molecules.
Definition:Second-Order Reaction
Let $\AA$ and $\BB$ be two chemical substances in solution which react together to form a compound $\CC$. Let the reaction occur by means of the molecules of $\AA$ and $\BB$ colliding and interacting as a result. Then the rate of formation of $\CC$ is proportional to the number of collisions in unit time. This in turn is jointly proportional to the quantities of $\AA$ and $\BB$ which have not yet transformed. Such a chemical reaction is called a '''second-order reaction'''.
Definition:Thermodynamics
'''Thermodynamics''' is the branch of physics which studies the behaviour of heat.
Definition:Heat
'''Heat''' is a form of energy which is held within a body in the form of the physical motion of the molecules or atoms of which that body is composed.
Definition:Energy
'''Energy''' is a property of physical bodies which can be defined as the ability to do work. Like time, it is difficult to define exactly what '''energy''' actually ''is''.