# Wikidata:Database reports/Constraint violations/P2534

Constraint violations report for defining formula (Discussion, uses, items, changes, related properties): mathematical formula representing a theorem or law. Maximum length: 400 characters
Data time stamp: (UTC) — Items processed: 3,484
The report is generated based on the settings on Property talk:P2534.
When incremental dumps and the bot work as planned, items fixed before 07:00 UTC disappear in the next update. The report is not updated if only the item count changes.
The report can include false positives. No need to "fix" them.

## "Single value" violations

Violations count: 91

• force (Q11402): \vec{F} = m\vec{a}, \vec{F}=\frac{d\vec{p}}{dt}
• sphere (Q12507): x^2+y^2+z^2=r^2, ||\vec{x}||^2 = r^2
• radian (Q33680): \text{angle in degrees} = \text{angle in radians} \cdot \frac {180^\circ} {\pi}, \text{angle in radians} = \frac{\text{length of arc}}{r}
• bel (Q50098): \text{B} = \frac{1}{2} \cdot \ln{10} \cdot \text{Np}, B = \ln \sqrt{10} Np
• candela (Q83216): I_\mathrm{v}(\lambda)= 683\ \mathrm{lm/W} \cdot \overline{y}(\lambda) \cdot I_\mathrm{e}(\lambda), cd = \frac{lm}{sr}
• multiplication theorem (Q98831): \frac{d \xi}{d t} + \nabla \cdot f(\xi) = 0, \Gamma (z)\;\Gamma \left(z+{\frac {1}{2}}\right)=2^{{1-2z}}\;{\sqrt {\pi }}\;\Gamma (2z).\,\!, \Gamma (z)\;\Gamma \left(z+{\frac {1}{k}}\right)\;\Gamma \left(z+{\frac {2}{k}}\right)\cdots \Gamma \left(z+{\frac {k-1}{k}}\right)=(2\pi )^{{{\frac {k-1}{2}}}}\;k^{{1/2-kz}}\;\Gamma (kz)\,\!, k^{{m}}\psi ^{{(m-1)}}(kz)=\sum _{{n=0}}^{{k-1}}\psi ^{{(m-1)}}\left(z+{\frac {n}{k}}\right), k\left[\psi (kz)-\log(k)\right]=\sum _{{n=0}}^{{k-1}}\psi \left(z+{\frac {n}{k}}\right)., k^{s}\zeta (s)=\sum _{{n=1}}^{k}\zeta \left(s,{\frac {n}{k}}\right),, k^{s}\,\zeta (s,kz)=\sum _{{n=0}}^{{k-1}}\zeta \left(s,z+{\frac {n}{k}}\right), \zeta (s,kz)=\sum _{{n=0}}^{{\infty }}{s+n-1 \choose n}(1-k)^{n}z^{n}\zeta (s+n,z)., F(s;q)=\sum _{{m=1}}^{\infty }{\frac {e^{{2\pi imq}}}{m^{s}}}=\operatorname {Li}_{s}\left(e^{{2\pi iq}}\right), 2^{{-s}}F(s;q)=F\left(s,{\frac {q}{2}}\right)+F\left(s,{\frac {q+1}{2}}\right)., k^{{-s}}F(s;kq)=\sum _{{n=0}}^{{k-1}}F\left(s,q+{\frac {n}{k}}\right).
• normal distribution (Q133871): f(x | \mu, \sigma^2) = \frac{1}{\sigma\sqrt{2\pi} } \; e^{ -\frac{(x-\mu)^2}{2\sigma^2} }, \mathcal{N}(\mu,\,\sigma^2)
• permutation (Q161519): \sigma: \{1, 2, \ldots, n\} \rightarrow \{1, 2, \ldots, n\}, \sigma : \begin{pmatrix} 1 & 2 & \dots & n \\ 1 & 2 & \dots & n\end{pmatrix}
• moment of inertia (Q165618): I=\int\limits _{M}r^{2}\mathrm {d} m, I=\sum_{n}r_n^{2} m_n
• Hooke's law (Q170282): F= k X, \sigma = E \varepsilon
• Avogadro's law (Q170431): V \propto n, \frac{V}{n}=k, \frac{V_1}{n_1} = \frac{V_2}{n_2}
• De Morgan's laws (Q173300): \neg(P \land Q) \vdash (\neg P \lor \neg Q)., \neg(P \lor Q) \vdash (\neg P \land \neg Q).
• Cartesian product (Q173740): A\times B = \{\, (a,b) \mid a \in A \and \ b\in B \,\}, \Pi_{i=1}^n A_i = \{\, (a_1,...,a_n) \mid \forall i \in \{1,...,n\}, a_i \in A_i \,\}
• Boyle's law (Q175974): P \propto \frac{1}{V}, PV = k, P_1 V_1 = P_2 V_2
• metric space (Q180953): d : X \times X \to [0,\infty), d\left(x,y\right) = 0 \Leftrightarrow x = y, d\left(x,y\right) = d(y,x), d\left(x,y\right) \leq d(x,z) + d(z,y)
• Fermat's little theorem (Q188295): a^p \equiv a \pmod{p}, a^{p-1} \equiv 1 \pmod{p}
• interval (Q189962): n = 1200 \cdot \log_2 \left( \frac{f_2}{f_1} \right), n = 12 \cdot \log_2 \left( \frac{f_2}{f_1} \right)
• ordered pair (Q191290): (a,b) = \{ \{a\}, \{a,b\} \}, (a_1, b_1) = (a_2, b_2)\quad \Leftrightarrow \quad a_1 = a_2\text{ and }b_1 = b_2, ( a, b ) := \{\{ \{a\},\, \emptyset \},\, \{\{b\}\}\}, (a,b) := \{\{a,1\},\{b,2\}\}
• Charles's law (Q193523): V \propto T, \frac{V}{T}=p, \frac {V_2}{V_1} = \frac{T_2}{T_1}, V_1 T_2 = V_2 T_1 y9 n0
• wave equation (Q193846): \frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u, \Box_c u = 0
• Gay-Lussac's law (Q202151): {P}\propto{T}, \frac{P}{T}=k, \frac{P_1}{T_1}=\frac{P_2}{T_2}, {P_1}{T_2}={P_2}{T_1}
• Laplace operator (Q203484): \nabla \cdot \nabla, \nabla^2
• natural logarithm (Q204037): e^{\ln(x)} = x, \qquad x > 0, \ln x=\log_e x
• Zipf's law (Q205472): f(k;s,N)=\frac{1/k^s}{\sum_{n=1}^N (1/n^s)}, f(k;s,N)=\frac{1}{k^s H_{N,s}}
• Dolbear's Law (Q206043): T_F = 50 + \left ( \frac{N_{60}-40}{4} \right ), T_C = 10 + \left ( \frac{N-40}{7} \right )
• cissoid of Diocles (Q206079): y^2=\frac{x^3}{2a-x}, \rho=\frac{2a\sin^2\varphi}{\cos\varphi}
• cardioid (Q207726): r=a(1-\cos\varphi), (x^2 + y^2 + 2 a x)^2 - 4 a^2 (x^2 + y^2) \, = \, 0
• electrical conductance (Q309017): G = \frac{1}{R}, G=\frac{I}{V}
• Rayl (Q359151): {\rm 1~Rayl_{CGS} = 1~\frac{dyn \cdot s}{cm^3} = 1~\frac{ba \cdot s}{cm} = 1~\frac{g}{s \cdot cm^2}}, {\rm 1~Rayl_{MKS} = 1~\frac{N \cdot s}{m^3} = 1~\frac{Pa \cdot s}{m} = 1~\frac{kg}{s \cdot m^2}}
• Taylor circle (Q365086): \sin \alpha(\cos \alpha - \cos 2\alpha \cos(\beta - \gamma)) : \sin \beta(\cos \beta - \cos 2\beta \cos(\gamma - \alpha)) : \sin \gamma (\cos \gamma - \cos 2\gamma \cos(\alpha - \beta) ), R_T = R\sqrt{\sin^2\alpha\sin^2\beta\sin^2\gamma+\cos^2\alpha\cos^2\beta\cos^2\gamma}
• complex conjugate (Q381040): \overline{z} = \text{Re}(z) - i \text{Im}(z) \text{sign}(\text{Im}(z)), \overline{z} \text{ complex conjugate of } z \Leftrightarrow z\overline{z} = \|z\|^2
• Bézout's identity (Q513028): \forall a, b \exists x, y: GCD(a, b)=x \cdot a+y \cdot b, ax+by=d
• Maurer–Cartan form (Q552787): \omega = g^{-1} \mathrm dg, \mathrm d\omega + \frac12 [\omega\wedge\omega] = 0
• Cauchy–Riemann equations (Q622741): \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
• uniform space (Q652446): U=(X, \Phi), U=(X, (\{(a,b)\in X\times X|P_n(a,b)\})_n)
• modus ponens (Q655742): P \to Q,\; P\;\; \vdash\;\; Q, \frac{P \to Q,\; P}{\therefore Q}
• magnification (Q675287): \mathrm{MA}=\frac{\tan \varepsilon}{\tan \varepsilon_0}, M = {f \over f-d_o}, M= {f_o \over f_e}
• lemniscate of Bernoulli (Q736896): \rho^2=2c^2\cos 2\varphi, (x^2+y^2)^2=2c^2(x^2-y^2)
• Modus tollens (Q844118): \frac{P \to Q, \neg Q}{\therefore \neg P}, P\to Q, \neg Q \vdash \neg P
• Euler characteristic (Q852973): \chi=V-E+F, \chi=b_0 - b_1 + b_2 - b_3 +\, ..., \chi=k_0-k_1+k_2-...,
• Helmholtz free energy (Q865821): A=U-TS, A \equiv U-TS
• Doppler broadening (Q902598): \Delta f_{\text{FWHM}} = \sqrt{\frac{8kT\ln 2}{mc^2}}f_{0}, \sigma_{f} = \sqrt{\frac{kT}{mc^2}}f_0
• Lemoine point (Q940537): a^2 : b^2 : c^2, a:b:c = \sin(A):\sin(B):\sin(C)
• acceleration of the universe (Q1049613): \frac{\ddot{a}}{a}=-\frac{4{\pi}G}{3}( \rho + \frac{3P}{c^2}), H^2={\left ( \frac{\dot{a}}{a} \right )}^2=\frac{8{\pi}G}{3}\rho-\frac{{\Kappa}c^2}{R^2a^2}
• Strophoid (Q1050391): y^2 = x^2 \left| \frac{a + x}{a - x}\right|, \rho = - \frac{a \cos2 \phi}{ \cos \phi}
• basic hypergeometric series (Q1062958): \;_{j}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_{j} \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum_{n=0}^\infty\frac {(a_1, a_2, \ldots, a_{j};q)_n} {(b_1, b_2, \ldots, b_k,q;q)_n} \left((-1)^nq^{n\choose 2}\right)^{1+k-j}z^n, (a_1,a_2,\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \ldots (a_m;q)_n
• 1 − 2 + 4 − 8 + … (Q1064181): \sum_{n=0}^\infty(-2)^n, \sum_{k=0}^{n} (-2)^k
• Combined gas law (Q1077153): \frac {PV}{T}= k, \frac {P_1V_1}{T_1}= \frac {P_2V_2}{T_2}
• Y-Δ transform (Q1110301): R_{ac} = \frac{R_aR_b + R_bR_c + R_cR_a}{R_b}, R_{ab} = \frac{R_aR_b + R_bR_c + R_cR_a}{R_c}, R_{bc} = \frac{R_aR_b + R_bR_c + R_cR_a}{R_a}, R_a = \frac{R_{ac}R_{ab}}{R_{ac} + R_{ab} + R_{bc}}, R_c = \frac{R_{ac}R_{bc}}{R_{ac} + R_{ab} + R_{bc}}, R_b = \frac{R_{ab}R_{bc}}{R_{ac} + R_{ab} + R_{bc}}
• condition number (Q1147936): K(x) = \sup_{x \in D(g)} \left \{ \frac{\lVert g(\overline{x})-g(x) \rVert }{\lVert \overline{x}-x \rVert} \frac{\lVert x \rVert}{\lVert g(x) \rVert}\right \}, \text{ where } D(g) \text{ is the domain of }g., K(x) \approx \lVert g'(x) \rVert \frac{\lVert x \rVert}{\lVert g(x) \rVert} \text{ if } g \text{ is differentiable.}
• isogonal conjugate (Q1156867): x : y : z \longrightarrow x^{-1} : y^{-1} : z^{-1}, x : y : z \longrightarrow \frac{a^2}x : \frac{b^2}x : \frac{c^2}x
• hippopede (Q1205428): (x^2 + y ^2)^2 - (2m^2 + c)x^2 + (2m^2 - c)y^2 = 0, (x^2+y^2)^2=cx^2+dy^2
• Jefimenko's equations (Q1321084): \mathbf{E}(\mathbf{r}, t) = \frac{1}{4 \pi \epsilon_0} \int \left[ \left(\frac{\rho(\mathbf{r}', t_r)}{|\mathbf{r}-\mathbf{r}'|^3} + \frac{1}{|\mathbf{r}-\mathbf{r}'|^2 c}\frac{\partial \rho(\mathbf{r}', t_r)}{\partial t}\right)(\mathbf{r}-\mathbf{r}') - \frac{1}{|\mathbf{r}-\mathbf{r}'| c^2}\frac{\partial \mathbf{J}(\mathbf{r}', t_r)}{\partial t} \right] \mathrm{d}^3 \mathbf{r}', \mathbf{B}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \left[\frac{\mathbf{J}(\mathbf{r}', t_r)}{|\mathbf{r}-\mathbf{r}'|^3} + \frac{1}{|\mathbf{r}-\mathbf{r}'|^2 c}\frac{\partial \mathbf{J}(\mathbf{r}', t_r)}{\partial t} \right] \times (\mathbf{r}-\mathbf{r}') \,\mathrm{d}^3 \mathbf{r}'
• Eulerian number (Q1373849): A(n,m)=\sum_{k=0}^{m+1}(-1)^k \binom{n+1}{k} (m+1-k)^n, A_{n}(t) = \sum_{m=0}^{n} A(n,m)\ t^{m}
• conchoid of de Sluze (Q1509478): r=\sec\theta+a\cos\theta, (x-1)(x^2+y^2)=ax^2
• isotomic conjugate (Q1511235): x : y : z \longrightarrow x^{-1} : y^{-1} : z^{-1}, x : y : z \longrightarrow \frac1{a^2x} : \frac 1{b^2y} : \frac 1{c^2z}
• potential flow (Q1543991): \nabla\times\vec{v}=\vec{0}, \vec{v}=\nabla\phi
• Balance wheel (Q1928854): T = 2 \pi \sqrt{ \frac {I}{\kappa} } \,, T = 2 \pi \sqrt{ \frac {I}{\kappa} }
• sinusoidal spiral (Q2064156): r^n = a^n \sin(n \varphi), r^n = a^n \cos(n \theta)
• Zeller's congruence (Q2140717): h = \left(q + \left\lfloor\frac{13(m+1)}{5}\right\rfloor + K + \left\lfloor\frac{K}{4}\right\rfloor + \left\lfloor\frac{J}{4}\right\rfloor - 2J\right) \bmod 7, h = \left(q + \left\lfloor\frac{13(m+1)}{5}\right\rfloor + K + \left\lfloor\frac{K}{4}\right\rfloor + 5 - J\right) \bmod 7
• asymmetric relation (Q2298831): \forall a, b \in X(a R b \Rightarrow \lnot(b R a)), \forall a, b \in X,\ a R b \Rightarrow \neg (b R a)
• Quadrifolium (Q2358628): (x^2+y^2)^3 = (x^2-y^2)^2, r=sin(2\phi)
• D'Alembert's formula (Q2388753): u_{tt}-c^2u_{xx}=0,\, u(x,0)=g(x),\, u_t(x,0)=h(x),, u_{tt}-c^2u_{xx}=0,\, u(x,0)=g(x),\, u_t(x,0)=h(x)
• gear ratio (Q2403634): v = r_A \omega_A = r_B \omega_B, \mathrm{MA} = \frac{N_B}{N_A}
• Homogeneous differential equation (Q2434565): M(x,y)\,dx + N(x,y)\,dy = 0, F (y, y', y, \ldots) = 0
• sequence of Lucas numbers (Q2503280): L(x) = \frac{2 - x}{1 - x - x^2}, L_{n}=L_{n-1}+L_{n-2}, L_1=1, L_0=2
• Boussinesq number (Q2650387): Bo_I = \frac{g \; \beta \; \Delta T \; L_{c}^3}{\alpha^2} = Ra \; Pr = Gr \; Pr^2, Bo_{II} = \frac{v}{\sqrt{2\, g\, L_c}}
• Linear classifier (Q2679259): y = f(\vec{w}\cdot\vec{x}) = f\left(\sum_j w_j x_j\right),, y = f(\vec{w}\cdot\vec{x}) = f\left(\sum_j w_j x_j\right)
• Padovan sequence (Q2706626): \begin{align} P_n = P_{n-2} + P_{n-3},\!\ \\ P_0=P_1=P_2=1\end{align}, f(x)= \frac{1+x}{1-x^2-x^3}
• cartesian oval (Q2983648): p_1r_1+p_2r_2=const, (x^2 + y^2 - 2ax)^2 = b^2(x^2 + y^2) + c
• trisectrix of Maclaurin (Q3294140): r= {a \over 2} (4 \cos \theta - \sec \theta), 2x(x^2+y^2)=a(3x^2-y^2), r = \frac{a}{2 \cos{\theta \over 3}}, {r \over \sin 3\theta} = {a \over \sin 2\theta}\!
• Tschirnhausen cubic (Q4243599): r=\frac{a}{\cos^3(\theta/3)}, 27ay^2 = (a-x)(8a+x)^2
• сapacity of a set (Q5034494): C(\Sigma, S) = - \frac1{(n - 2) \sigma_{n}} \int_{S'} \frac{\partial u}{\partial \nu}\,\mathrm{d}\sigma', C(K) = \int_{\mathbb{R}^n\setminus K} |\nabla u|^2\mathrm{d}x
• generalized hypergeometric function (Q5532481): \beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_{n \geqslant 0} \beta_n z^n, \,{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_p)_n}{(b_1)_n\cdots(b_q)_n} \, \frac {z^n} {n!}
• monomial basis (Q6901742): \sum_{i=0}^d a_ix^i, 1,x,x^2,x^3, \ldots
• Parry point (Q7139901): \frac{a^2}{2a^2-b^2-c^2} :\frac{b^2}{2b^2-a^2-c^2}:\frac{c^2}{2c^2-a^2-b^2}, \frac{a}{2a^2-b^2-c^2} :\frac{b}{2b^2-a^2-c^2}:\frac{c}{2c^2-a^2-b^2}
• Rogers–Ramanujan identities (Q7359380): G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty}=1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots, H(q) =\sum_{n=0}^\infty \frac{q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty}=1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots
• Spieker center (Q7577104): b+c:a+c:a+b, bc(b+c):ac(a+c):ab(a+b)
• Picard–Fuchs equation (Q7877856): \frac{d^2f}{dj^2} + \frac{1-1968j + 2654208j^2}{4j^2 (1-1728j)^2} f=0.\,, j=\frac{g_2^3}{g_2^3-27g_3^2}
• orthocentre (Q10621500): S_{BC}:S_{AC}:S_{AB}, \cos(B)\cos(C) : \cos(A)\cos(C) : \cos(A)\cos(B)
• Information gain in decision trees (Q17083041): IG(T,a) = H(T) - H(T|a), IG(T,a) = H(T)-\sum_{v\in vals(a)}\frac{|\{\textbf{x}\in T|x_a=v\}|}{|T|} \cdot H(\{\textbf{x}\in T|x_a=v\})
• Fibonacci series (Q23835349): \begin{align} F_n = F_{n-1} + F_{n-2},\!\ \\ F_1 = 1, F_2 = 1 \\ \text{or } F_0 = 0, F_1 = 1 \end{align}, F_n = \frac{1}{\sqrt{5}} \left( \left( \frac{1 + \sqrt{5} }{2}\right)^n -\left(\frac{1 - \sqrt{5}}{2}\right)^n \right), f(x)=\frac{x}{1-x-x^2}
• Circumgon (Q25304548): G_B = (3/2)G_A., G_B = (3/2)G_A
• absolute condition number (Q26922716): K(x) = \sup_{x \in D(g)} \left \{ \frac{\lVert g(\overline{x})-g(x) \rVert }{\lVert \overline{x}-x \rVert} \right \}, \text{ where } D(g) \text{ is the domain of }g., K(x) \approx \lVert g'(x) \rVert \text{ if } g \text{ is differentiable.}
• convergent numerical method (Q26926645): \begin{aligned} \forall \varepsilon > 0, \exist n_0 \in \mathbb{N}, \exist \delta_{n_0, \varepsilon}>0 \text{ such that } \\ \forall n>n_0, \forall l_n : \lVert l_n \rVert < \delta_{n_0, \varepsilon} \\ \Rightarrow \lVert g(x) - g_n(x+l_n) \rVert < \varepsilon \end{aligned}, \lim g_n= g \Rightarrow \left \{ F_n(x_n,g_n(x_n)) = 0 \right \}_{n \in \mathbb{N}} \text{ is convergent}
• GCJ02 (Q29043602): \phi_g = \phi_w + \frac{\Delta\phi}{\rm lat\_arclen_{sk42}}; \theta_g = \theta_w+ \frac{\Delta\theta}{\rm lon\_arclen_{sk42}}, \Delta\phi = -100 + 2 x + 3 y + 0.2 y^2 + 0.1 xy + 0.2 \sqrt{|x|} + \frac{20(2\sin(6 \pi x) + 2\sin(2 \pi x) + 2\sin(\pi y) + 4\sin(\frac{\pi y}{3}) + 16\sin(\frac{\pi y}{12}) + 32\sin(\frac{\pi y}{30}))}{3}, \Delta\theta = 300 + x + 2 y + 0.1 x^2 + 0.1 xy+ 0.1 \sqrt{|x|} + \frac{20(2 \sin(6\pi x) + 2 \sin(2\pi x) +2 \sin(\pi x) + 4 \sin(\frac{\pi x}{3}) +15 \sin(\frac{ \pi x}{12}) + 30 \sin(\frac{ \pi x}{30}))}{3}, x = {\rm degrees}(\theta_w) - 105, y = {\rm degrees}(\phi_w) - 35, {\rm psin}(x) = \sin(\pi x)
• BD09 (Q29043632): \phi_b = r\times\cos (t) + 0.0065; \theta_b = r\times\sin(t) + 0.0060, t = {\rm atan2} (\phi_g, \theta_g) + 0.000003 \times \cos (3000\theta_g), r = \sqrt{{\rm degrees}(\phi_g)^2+{\rm degrees}(\theta_g)^2} + 0.00002 \times \cos (3000\phi_g)
• characteristic equation (Q33104580): \det(A-\lambda I) = 0, \det(\lambda I-A) = 0
• set of prime numbers (Q47370614): \mathbb{P}, \mathbf{P}
• Binet's formula (Q54539168): F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right], F_n = \frac{\phi ^n -(-\phi)^{-n}}{\sqrt{5}}

## "Conflicts with instance of (P31)" violations

Violations count: 0

## "Unique value" violations

Violations count: 69