hcju/proofwikips huggingface dataset:
Definition:Labeled Tree for Propositional Logic/Hypothesis Set
The countable set $\mathbf H$ of WFFs of propositional logic is called the '''hypothesis set'''. The elements of $\mathbf H$ are known as '''hypothesis WFFs'''. The '''hypothesis set''' $\mathbf H$ is considered to be attached to the root node of $T$.
Definition:Finished Set of WFFs of Propositional Logic
Let $\Delta$ be a set of WFFs of propositional logic. Then $\Delta$ is '''finished''' iff: * $\Delta$ is not contradictory * For each WFF $\mathbf C \in \Delta$, either $\mathbf C$ is basic or one of the following is true: ** $\mathbf C$ has the form $\neg \neg \mathbf A$ where $\mathbf A \in \Delta$ ** $\mathbf C$ has the form $\left({\mathbf A \land \mathbf B}\right)$ where both $\mathbf A \in \Delta$ and $\mathbf B \in \Delta$ ** $\mathbf C$ has the form $\neg \left({\mathbf A \land \mathbf B}\right)$ where either $\neg \mathbf A \in \Delta$ or $\neg \mathbf B \in \Delta$ ** $\mathbf C$ has the form $\left({\mathbf A \lor \mathbf B}\right)$ where either $\mathbf A \in \Delta$ or $\mathbf B \in \Delta$ ** $\mathbf C$ has the form $\neg \left({\mathbf A \lor \mathbf B}\right)$ where both $\neg \mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$ ** $\mathbf C$ has the form $\left({\mathbf A \implies \mathbf B}\right)$ where either $\neg \mathbf A \in \Delta$ or $\mathbf B \in \Delta$ ** $\mathbf C$ has the form $\neg \left({\mathbf A \implies \mathbf B}\right)$ where both $\mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$ ** $\mathbf C$ has the form $\left({\mathbf A \iff \mathbf B}\right)$ where either: *** both $\mathbf A \in \Delta$ and $\mathbf B \in \Delta$, or: *** both $\neg \mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$; ** $\mathbf C$ has the form $\neg \left({\mathbf A \iff \mathbf B}\right)$ where either: *** both $\mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$ ** or: *** both $\neg \mathbf A \in \Delta$ and $\mathbf B \in \Delta$.
Definition:Finished Branch of Propositional Tableau
Let $T$ be a propositional tableau. Let $\Gamma$ be a branch of $T$. Then $\Gamma$ is '''finished''' iff: :$(1): \quad \Gamma$ is not contradictory :$(2): \quad$ Every non-basic WFF on $\Gamma$ is used at some node of $\Gamma$. That is, $\Gamma$ is '''finished''' iff the set $\Delta$ of WFFs of propositional logic which occur along $\Gamma$ is a finished set.
Definition:Finished Propositional Tableau
Let $T$ be a propositional tableau. Then $T$ is '''finished''' iff every branch of $T$ is either finished or contradictory.
Definition:Infix Notation
=== Binary Relations === {{:Definition:Infix Notation/Binary Relation}} === Binary Operations === {{:Definition:Infix Notation/Binary Operation}}
Definition:Steiner Triple System
A '''Steiner triple system of order $v$''' is a BIBD with block size $3$, and each pair of points occurring together in exactly $1$ block (called a '''triple''').
Definition:Quantifier
The universal quantifier $\forall$ and the existential quantifier $\exists$ are referred to collectively as '''quantifiers'''.
Definition:Atomic WFF of Predicate Logic
An '''atomic WFF''' of predicate logic is a WFF not containing any connectives or quantifiers. In the formal grammar of predicate logic, the '''atomic WFFs''' are precisely those  WFFs which can be formed by only applying the rule $\mathbf W ~ \mathcal P_n$.
Definition:Modulus of Complex-Valued Function
Let $f: S \to \C$ be a complex-valued function. Then the '''(complex) modulus of $f$''' is written $\left|{f}\right|: S \to \R$ and is the real-valued function defined as: :$\forall z \in S: \left|{f}\right| \left({z}\right) = \left|{f \left({z}\right)}\right|$.
Definition:Modulus (Geometric Function Theory)
In geometric function theory, the term '''modulus''' is used to denote certain conformal invariants of configurations or curve families. More precisely, the modulus of a curve family $\Gamma$ is the reciprocal of its extremal length: :$\mod \Gamma := \dfrac 1 {\map \lambda \Gamma}$ === Modulus of a Quadrilateral === Consider a quadrilateral; that is, a Jordan domain $Q$ in the complex plane (or some other Riemann surface), together with two disjoint closed boundary arcs $\alpha$ and $\alpha'$. Then the '''modulus''' of the quadrilateral $\map Q {\alpha, \alpha'}$ is the extremal length of the family of curves in $Q$ that connect $\alpha$ and $\alpha'$. Equivalently, there exists a rectangle $R = \set {x + i y: \cmod x < a, \cmod y < b}$ and a conformal isomorphism between $Q$ and $R$ under which $\alpha$ and $\alpha'$ correspond to the vertical sides of $R$. Then the modulus of $\map Q {\alpha, \alpha'}$ is equal to the ratio $a/b$. See Modulus of a Quadrilateral. === Modulus of an Annulus === Consider an annulus $A$; that is, a domain whose boundary consists of two Jordan curves. Then the '''modulus''' $\mod A$ is the extremal length of the family of curves in $A$ that connect the two boundary components of $A$. Equivalently, there is a round annulus $\tilde A = \set {z \in \C: r < \cmod z < R}$ that is conformally equivalent to $A$. Then: :$\mod A := \dfrac 1 {2 \pi} \map \ln {\dfrac R r}$ The modulus of $A$ can also be denoted $\map M R$.
Definition:Occurrence (Predicate Logic)
Let $\mathbf A$ be a WFF of predicate logic. Let $S$ be a string in the alphabet of predicate logic. Each place where $S$ appears in $\mathbf A$ is called an '''occurrence of $S$ in $\mathbf A$'''. Note that $S$ may consist of a single symbol, but may not be null. === Scope === {{:Definition:Scope (Logic)/Quantifier}} === Bound Occurrence === {{:Definition:Bound Occurrence}} === Free Occurrence === {{:Definition:Free Occurrence}} === Alphabetic Substitution === {{:Definition:Alphabetic Substitution}}
Definition:Well-Formed Part
Let $\FF$ be a formal language with alphabet $\AA$. Let $\mathbf A$ be a well-formed formula of $\FF$. Let $\mathbf B$ be a subcollation of $\mathbf A$. Then $\mathbf B$ is a '''well-formed part of $\mathbf A$'''  $\mathbf B$ is a well-formed formula of $\FF$. === Proper Well-Formed Part === {{:Definition:Well-Formed Part/Proper Well-Formed Part}}
Definition:Scope (Logic)/Quantifier
Let $\mathbf A$ be a WFF of the language of predicate logic. Let $Q$ be an occurrence of a quantifier in $\mathbf A$. Let $\mathbf B$ be a well-formed part of $\mathbf A$ such that $\mathbf B$ begins (omitting outer parentheses) with $Q x$. That is, such that $\mathbf B = \paren {Q x: \mathbf C}$ for some WFF $\mathbf C$. $\mathbf B$ is called the '''scope of the quantifier $Q$'''.
Definition:Bound Occurrence
Let $Q x$ be an occurrence of a quantifier in $\mathbf A$. Any occurrence of the variable $x$ in the scope of $Q$ is called a '''bound occurrence'''.
Definition:Freely Substitutable
Let $\mathbf C$ be a WFF of predicate logic. Let $x$ be a variable in $\mathbf C$. Let $\phi \left({y_1, \ldots, y_n}\right)$ be a term in which the variables $y_1, \ldots y_n$ occur. Then $\phi \left({y_1, \ldots, y_n}\right)$ is '''freely substitutable for $x$ in $\mathbf C$'''  no free occurrence of $x$ occurs in a well-formed part of $\mathbf C$ which is of the form: :$( Q y_i: \mathbf B )$ where $Q$ is a quantifier and $\mathbf B$ is a WFF.
Definition:Free Occurrence
An occurrence of a variable $x$ in $\mathbf A$ is said to be a '''free occurrence'''  it is not bound.
Definition:Classes of WFFs/Plain WFF
A '''plain WFF''' of predicate logic is a WFF with no parameters. Thus $WFF \left({\mathcal P, \mathcal F, \varnothing}\right)$ is the set of all '''plain WFFs''' with relation symbols from $\mathcal P$ and function symbols from $\mathcal F$. {{refactor|However immediate, this goes on a proof page}} It is immediate that a '''plain WFF''' is a WFF with parameters from $\mathcal K$ for ''all'' choices of $\mathcal K$.
Definition:Free Variable
Let $x$ be a variable in an expression $E$. $x$ is a '''free variable in $E$'''  it is not a bound variable. In the context of predicate logic, $x$ is a '''free variable in $E$'''  it has not been introduced by a quantifier, either: :the universal quantifier $\forall$ or :the existential quantifier $\exists$.
Definition:Wilson Prime
A '''Wilson prime''' is a prime number $p$ such that: :$p^2 \divides \paren {p - 1}! + 1$ where: :$\divides$ signifies divisibility :$!$ is the factorial operator.
Definition:Instance
Let $\mathbf C$ be a plain WFF in the language of predicate logic. Let $x_1, x_2, \ldots, x_n$ be the free variables of $\mathbf C$. Let $\mathcal M$ be a structure for predicate logic of type $\mathcal P$ whose universe set is $M$. Then an '''instance of $\mathbf C$ in $M$''' is the sentence with parameters from $M$ formed by choosing $a_1, a_2, \ldots, a_n \in M$ and replacing all free occurrences of $x_k$ in $\mathbf C$ by $a_k$ for $k = 1, \ldots, n$. The resulting sentence is denoted: :$\mathbf C \left({x_1, \ldots, x_n \,//\, a_1, \ldots, a_n}\right)$ Thus $\mathbf C \left({x_1, \ldots, x_n \,//\, a_1, \ldots, a_n}\right) \in SENT \left({\mathcal P, M}\right)$. If $\mathbf C$ is a plain sentence, then no parameters are needed, and $\mathbf C$ is already an instance of itself.
Definition:Heronian Triangle
=== Definition 1 === {{:Definition:Heronian Triangle/Definition 1}} === Definition 2 === {{:Definition:Heronian Triangle/Definition 2}}
Definition:Pseudometric
A '''pseudometric''' on a set $A$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \text M 1         | q = \forall x \in A         | m = \map d {x, x} = 0 }} {{axiom | n = \text M 2         | q = \forall x, y, z \in A         | m = \map d {x, y} + \map d {y, z} \ge \map d {x, z} }} {{axiom | n = \text M 3         | q = \forall x, y \in A         | m = \map d {x, y} = \map d {y, x} }} {{end-axiom}} The difference between a '''pseudometric''' and a metric is that a '''pseudometric''' does not insist that the distance function between distinct elements is ''strictly'' positive. === Pseudometric Space === {{:Definition:Pseudometric/Pseudometric Space}}
Definition:Derived Set
Let $T = \struct {S, \tau}$ be a topological space. Let $X \subseteq S$ be a subset of $S$. The '''derived set''' of $X$ is the set of all limit points of $X$. It is often denoted $X'$.
Definition:Perfect Set
=== Definition 1 === {{:Definition:Perfect Set/Definition 1}} === Definition 2 === {{:Definition:Perfect Set/Definition 2}} === Definition 3 === {{:Definition:Perfect Set/Definition 3}}
Definition:Independent Variable
=== Real Function === {{:Definition:Independent Variable/Real Function}} === Complex Function === {{:Definition:Independent Variable/Complex Function}}
Definition:Differential Equation
A '''differential equation''' is a mathematical equation for an unknown function of one or several variables relating: :$(1): \quad$ The values of the function itself :$(2): \quad$ Its derivatives of various orders.
Definition:First Order Ordinary Differential Equation
A '''first order ordinary differential equation''' is an ordinary differential equation in which any derivatives with respect to the independent variable have order no greater than $1$.
Definition:Dependent Variable
=== Real Function === {{:Definition:Dependent Variable/Real Function}} === Complex Function === {{:Definition:Dependent Variable/Complex Function}}
Definition:Initial Condition
Let $\Phi = \map F {x, y, y', y'', \ldots, y^{\paren n} }$ be an ordinary differential equation. An '''initial condition''' is an ordered pair $\tuple {x_0, y_0}$ which any solution of $\Phi$ must satisfy. That is, an '''initial condition''' is the additional imposition that a solution $y = \map y x$ of $\Phi$ satisfy: :$\map y {x_0} = y_0$
Definition:Initial Value Problem
Let $\map y x$ be a solution to the first order ordinary differential equation: :$\dfrac {\d y} {\d x} = \map f {x, y}$ which is subject to an initial condition: $\tuple {a, b}$. That is, it is required that $y = b$ when $x = a$, that is $\map y a = b$. The finding of the resulting particular solution is known as an '''initial value problem'''.
Definition:Homogeneous Function
Let $V$ and $W$ be two vector spaces over a field $F$. Let $f: V \to W$ be a function from $V$ to $W$. Then $f$ is '''homogeneous of degree $n$''' : :$f \left({\alpha \mathbf v}\right) = \alpha^n f \left({\mathbf v}\right)$ for all nonzero $\mathbf v \in V$ and $\alpha \in F$.
Definition:Homogeneous Differential Equation
A '''homogeneous differential equation''' is a first order ordinary differential equation of the form: :$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ where both $M$ and $N$ are homogeneous functions of the same degree.
Definition:Exact Differential Equation
Let a first order ordinary differential equation be expressible in this form: :$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ such that $M$ and $N$ are ''not'' homogeneous functions of the same degree. However, suppose there happens to exist a function $\map f {x, y}$ such that: :$\dfrac {\partial f} {\partial x} = M, \dfrac {\partial f} {\partial y} = N$ such that the second partial derivatives of $f$ exist and are continuous. Then the expression $M \rd x + N \rd y$ is called an '''exact differential''', and the differential equation is called an '''exact differential equation'''.
Definition:Integrating Factor
Consider the first order ordinary differential equation: :$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ such that $M$ and $N$ are real functions of two variables which are ''not'' homogeneous functions of the same degree. Suppose also that: :$\dfrac {\partial M} {\partial y} \ne \dfrac {\partial N} {\partial x}$ Then from Solution to Exact Differential Equation, $(1)$ is not exact, and that method can not be used to solve it. However, suppose we can find a real function of two variables $\map \mu {x, y}$ such that: :$\map \mu {x, y} \paren {\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} } = 0$ is exact. Then the solution of $(1)$ ''can'' be found by the technique defined in Solution to Exact Differential Equation. The function $\map \mu {x, y}$ is called an '''integrating factor'''.
Definition:Pervushin's Number
'''Pervushin's number''' is the Mersenne number $M_{61} = 2^{61} - 1$. {{NamedforDef|Ivan Mikheevich Pervushin|cat = Pervushin}}
Definition:Linear First Order Ordinary Differential Equation
A '''linear first order ordinary differential equation''' is a differential equation which is in (or can be manipulated into) the form: :$\dfrac {\d y} {\d x} + \map P x y = \map Q x$ where $\map P x$ and $\map Q x$ are functions of $x$.
Definition:Bernoulli's Equation
'''Bernoulli's equation''' is a first order ordinary differential equation which can be put into the form: :$\dfrac {\d y} {\d x} + \map P x y = \map Q x y^n$ where $n \ne 0$ and $n \ne 1$.
Definition:Boundary (Topology)
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. === Definition from Closure and Interior === {{:Definition:Boundary (Topology)/Definition 1}}
Definition:Probability Space
A '''probability space''' is a measure space $\struct {\Omega, \Sigma, \Pr}$ in which $\map \Pr \Omega = 1$. A '''probability space''' is used to define the parameters determining the outcome of an experiment $\EE$. In this context, the elements of a '''probability space''' are generally referred to as follows: :$\Omega$ is called the sample space of $\EE$ :$\Sigma$ is called the event space of $\EE$ :$\Pr$ is called the probability measure on $\EE$.  Thus it is a measurable space $\struct {\Omega, \Sigma}$ with a probability measure $\Pr$. === Discrete Probability Space === {{:Definition:Probability Space/Discrete}} === Continuous Probability Space === {{:Definition:Probability Space/Continuous}}
Definition:Measure Space
A '''measure space''' is a triple $\struct {X, \Sigma, \mu}$ where: :$X$ is a set :$\Sigma$ is a $\sigma$-algebra on $X$ :$\mu$ is a measure on $\Sigma$. Thus it is a measurable space $\struct {X, \Sigma}$ with a measure.
Definition:Outer Measure
Let $X$ be a set, and let $\mathcal P \left({X}\right)$ be its power set. An '''outer measure (on $X$)''' is a mapping: : $\mu^*: \mathcal P \left({X}\right) \to \overline \R_{\ge 0}$ that satisfies the following conditions: :$(1): \quad \mu^* \left({\varnothing}\right) = 0$ :$(2): \quad \mu^* \left({A}\right) \le \mu^* \left({B}\right)$ for all $A, B \in \mathcal P \left({X}\right)$ with $A \subseteq B$ (that is, $\mu^*$ is monotone) :$(3): \quad \displaystyle \mu^* \left({\bigcup_{i \mathop \in \N} A_i}\right) \le \sum_{i \mathop = 1}^\infty \mu^* \left({A_i}\right)$ for all sequences $\left({A_i}\right)_{i \mathop \in \N} \in \mathcal P \left({X}\right)$ (that is, $\mu^*$ is countably subadditive) where $\overline{\R}_{\ge 0}$ denotes the set of positive extended real numbers.
Definition:Limit Superior of Sequence of Sets
=== Definition 1 === {{:Definition:Limit Superior of Sequence of Sets/Definition 1}} === Definition 2 === {{:Definition:Limit Superior of Sequence of Sets/Definition 2}}
Definition:Limit Inferior of Sequence of Sets
=== Definition 1 === {{:Definition:Limit Inferior of Sequence of Sets/Definition 1}} === Definition 2 === {{:Definition:Limit Inferior of Sequence of Sets/Definition 2}}
Definition:Limit of Sets
Let $\Bbb S = \set {E_n : n \in \N}$ be a sequence of sets. Let the limit superior of $\Bbb S$ be equal to the limit inferior of $\Bbb S$. Then the '''limit of $\Bbb S$''', denoted $\ds \lim_{n \mathop \to \infty} E_n$, is defined as: :$\ds \lim_{n \mathop \to \infty} E_n := \limsup_{n \mathop \to \infty} E_n$ and so also: :$\ds \lim_{n \mathop \to \infty} E_n := \liminf_{n \mathop \to \infty} E_n$ and '''$\Bbb S$ converges to the limit'''.
Definition:Convergence Almost Everywhere
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.  Let $D \in \Sigma$. Let $f: D \to \R$ be a $\Sigma$-measurable function. Let $\left({f_n}\right)_{n \in \N}$ be a sequence of $\Sigma$-measurable functions $f_n: D \to \R$. Then $\left({f_n}\right)_{n \in \N}$ is said to '''converge almost everywhere''' (or '''converge a.e.''') on $D$ to $f$ : :$\mu \left({ \left\{{x \in D : f_n \left({x}\right) \text{ does not converge to } f \left({x}\right) }\right\} }\right) = 0$ and we write $f_n \stackrel{a.e.}{\to} f$. In other words, the sequence of functions converges pointwise outside of a $\mu$-null set.
Definition:Almost Uniform Convergence
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $D \in \Sigma$. Let $\sequence {f_n}_{n \mathop \in \N}, f_n: D \to \R$ be a sequence of $\Sigma$-measurable functions. Then $\sequence {f_n}_{n \mathop \in \N}$ is said to '''converge almost uniformly''' (or '''converge a.u.''') on $D$ : :For all $\epsilon > 0$, there is a measurable subset $E_\epsilon \subseteq D$ of $D$ such that: ::$(1): \quad \map \mu {E_\epsilon} < \epsilon$; ::$(2): \quad \sequence {f_n}_{n \mathop \in \N}$ converges uniformly to $f$ on $D \setminus E_\epsilon$.
Definition:Convergence in Measure
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\left({f_n}\right)_{n \in \N}, f_n: X \to \R$ be a sequence of $\Sigma$-measurable functions. Then $f_n$ is said to '''converge in measure''' to a measurable function $f: X \to \R$ iff: :$\displaystyle \forall \epsilon > 0: \lim_{n \to \infty} \mu \left({ \left\{ {x \in D : \left|{ f_n \left({ x }\right) - f \left({ x }\right) }\right| \ge \epsilon }\right\} }\right) = 0$ for all $D \in \Sigma$ with $\mu \left({D}\right) < + \infty$. To express that $f_n$ '''converges to $f$ in measure''' one writes $f_n \stackrel{\mu}{\longrightarrow} f$ or $\displaystyle \operatorname{\mu-\!\lim\,} \limits_{n \to \infty} f_n = f$.
Definition:Vector Length
The '''length''' of a vector $\mathbf v$ in a vector space $\struct {G, +_G, \circ}_K$ is defined as $\norm V$, the norm of $V$.
Definition:Dimension (Topology)/Locally Euclidean Space
Let $M$ be a locally Euclidean space.  Let $\left({U, \kappa}\right)$ be a coordinate chart such that:  :$\kappa: U \to \kappa \left({U}\right) \subseteq \R^n$ for some $n \in \N$. Then the natural number $n$ is called the '''dimension of $M$'''.
Definition:Congruence (Metric Spaces)
Two subsets $A, B \subset X$ of a space $X$ are said to be '''congruent''' if there exists an isometry $f: X \to X$ such that $f \left({A}\right) = B$. Such an isometry is called a '''congruence'''.  byxjucn6es49hx80reyrkzddur0unqq
Definition:Outer Product
Given two vectors $\mathbf u = \tuple {u_1, u_2, \ldots, u_m}$ and $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$, their '''outer product''' $\mathbf u \otimes \mathbf v$ is defined as: :$\mathbf u \otimes \mathbf v = A = \begin{bmatrix} u_1 v_1 & u_1 v_2 & \dots & u_1 v_n \\  u_2 v_1 & u_2 v_2 & \dots & u_2 v_n \\  \vdots & \vdots & \ddots & \vdots \\ u_m v_1 & u_m v_2 & \dots & u_m v_n \end{bmatrix}$
Definition:Harshad Number
A '''harshad number''' is a positive integer which is divisible by the sum of its digits base $10$. Hence a '''harshad number''' is a Niven number base $10$.
Definition:Number Base
=== Integers === {{:Definition:Number Base/Integers}} === Real Numbers === {{:Definition:Number Base/Real Numbers}} === Integer Part === {{:Definition:Number Base/Integer Part}} === Fractional Part === {{:Definition:Number Base/Fractional Part}} === Radix Point === {{:Definition:Number Base/Radix Point}}
Definition:Digit
Let $n$ be a number expressed in a particular number base, $b$ for example. Then $n$ can be expressed as: :$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0 . r_{-1} r_{-2} \ldots}_b$ where: :$m$ is such that $b^m \le n < b^{m+1}$; :all the $r_i$ are such that $0 \le r_i < b$. Each of the $r_i$ are known as the '''digits of $n$ (base $b$)'''.
Definition:Niven Number
A '''Niven number''' (in a given number base $b$) is a positive integer which is divisible by the sum of its digits in that given base $b$. That is, $N$ is a '''Niven number base $b$''' : :$\displaystyle \exists A \in \Z: N = \sum_{k \mathop = 0}^m r_k b^k = A \sum_{k \mathop = 0}^m r_k$ where $\displaystyle \sum_{k \mathop = 0}^m r_k b^k$ is the representation of $N$ in base $b$ as defined according to the Basis Representation Theorem.
Definition:All-Harshad Number
An '''all-Harshad number''' (or '''all-Niven number''') is a positive integer which is a Niven number in '''all''' number bases.
Definition:Total Derivative
Let $f \left({x_1, x_2, \ldots, x_n}\right)$ be a continuous real function of multiple variables. Let each of $x_1, x_2, \ldots, x_n$ be continuous real functions of a single independent variable $t$. Then the '''total derivative of $f$ with respect to $t$''' is defined as: :$\displaystyle \frac {\mathrm d f}{\mathrm d t} = \sum_{k \mathop = 1}^n \frac {\partial f} {\partial x_k} \frac {\mathrm d x_k}{\mathrm d t} = \frac {\partial f} {\partial x_1} \frac {\mathrm d x_1}{\mathrm d t} + \frac {\partial f} {\partial x_2} \frac {\mathrm d x_2}{\mathrm d t} + \cdots + \frac {\partial f} {\partial x_n} \frac {\mathrm d x_n}{\mathrm d t}$ where $\dfrac {\partial f} {\partial x_k}$ is the partial derivative of $f$ with respect to $x_k$. Note that in the above definition, nothing precludes $t$ from being one of the instances of $x_k$ itself. So we have that the '''total derivative of $f$ with respect to $x_k$''' is defined as: :$\dfrac {\mathrm d f} {\mathrm d x_k} = \dfrac {\partial f} {\partial x_1} \dfrac {\mathrm d x_1}{\mathrm d x_k} + \dfrac {\partial f} {\partial x_2} \dfrac {\mathrm d x_2} {\mathrm d x_k} + \cdots + \dfrac {\partial f} {\partial x_k} + \cdots + \dfrac {\partial f} {\partial x_n} \dfrac {\mathrm d x_n}{\mathrm d x_k}$
Definition:Second Chebyshev Function
The '''Second Chebyshev Function''' $\psi \left({x}\right)$ is defined as follows: :$\displaystyle \psi \left({x}\right) = \sum_{k \mathop \ge 1} \sum_{p^k \mathop \le x} \ln p$ where, for each $k$, the sum extends over all powers of prime numbers $p$ such that $p^k \le x$.
Definition:First Chebyshev Function
The '''First Chebyshev Function''' $\vartheta \left({x}\right)$, or $\theta \left({x}\right)$, is defined as:  :$\displaystyle \vartheta \left({x}\right) = \sum_{p \mathop \le x} \ln p$  where the sum extends over all prime numbers $p$ such that $p \le x$. {{NamedforDef|Pafnuty Lvovich Chebyshev|cat=Chebyshev}}