For the square-grid nine-point finite-difference stencil for the Laplacian,If $\Delta u=0$, both displayed correction terms vanish: the mixed sixth derivatives sum to $\Delta u_{xxyy}=0$, and the pure sixth derivatives sum to the negative of that sum. Expanding two orders further yields $D_{9}u=h^6{u}_{xxxxxxxx}/3024+O(h^8)$ for a smooth harmonic function. Thus the compact harmonic scheme has sixth-order normalized consistency of a numerical method and, with the discrete maximum principle inverse bound, sixth-order nodal convergence on a square with exact smooth Dirichlet boundary data. For example $\mathrm{Re}(x+iy{)}^8$ shows that the sixth-order term need not vanish.

For the Dirichlet discrete Laplacian on a square, let $T=h^{-2}\mathrm{tridiag}(1,-2,1)$. The standard axial five-point operator is $T\otimes I+I\otimes T$, and the axial-weight $2/3$, corner-weight $1/6$ nine-point operator additionally contains $h^{2}T\otimes T/6$. Its eigenvalues are therefore strictly larger than the five-point values for the same sine modes. Both lie above the corresponding negative continuum Laplacian eigenvalues, so the five-point operator has smaller spectral error. For fixed mode numbers the leading errors are respectively ${\pi }^4{h}^2(k^4+l^4)/12$ and ${\pi }^4{h}^2(k^2+l^2{)}^2/12$.

There are $q^{n^2-n}$ nilpotent matrices of size n over $F_q$. Every matrix has a unique invariant direct sum on which it is nilpotent and invertible. Counting those ordered decompositions gives $q^{n^2}/|G{L}_n(q)|={\sum }_{k=0}^n{a}_k/|G{L}_k(q)|$. Subtraction at n and n−1 yields $a_n=q^{n^2-n}$. Translation by the identity counts unipotent matrices.

For a real or complex two-by-two matrix $A$, the Cayley-Hamilton theorem reads $A^2-(\mathrm{tr}A)A+(\det A)I=0$. Hence $A^2=0$ exactly when $\mathrm{tr}A=\det A=0$. Conversely, if $A^m=0$ for any positive integer $m$, every eigenvalue is zero and the same identity gives $A^2=0$. Thus a nonzero nilpotent matrix of this size has nilpotency index two.

For a nilpotent ideal $I$, its nilpotence index of an ideal is the least positive integer $n$ for which $I^n=0$. In $k[X]/(g^e)$, with $g$ an irreducible polynomial, the unique maximal ideal $(g)$ has index $e$. In a discrete valuation ring with $p=u{\pi }^e$, the maximal ideal of its reduction modulo $p$ has the same index. Comparing these indices under a quotient ring isomorphism recovers the ramification exponent in the Kummer-Dedekind theorem.

Group commutators are taken coordinatewise in a direct product of groups, giving ${\Gamma }_j(A\times B)={\Gamma }_j(A)\times {\Gamma }_j(B)$. A finite direct product of nilpotent groups is nilpotent, with class the maximum of the factor classes.

// Target: algebra.bigb

When $|\Phi |\to 1$ and its gauge covariant derivative decays fast enough, on a large circle write $\Phi =\rho e^{i\chi }$ locally. The tangential derivative gives $A=d\chi -\mathrm{Im}(\overline{\Phi }D\Phi )/{\rho }^2$. The integrated error tends to zero, so Stokes theorem turns total magnetic flux into the phase change $2\pi N$. This integer is the winding number of the normalized Higgs field on the circle. For positive Bogomolny vortex equations it is the total zero multiplicity.

For a unit Abelian Higgs vortex, the piecewise profiles $h=r/R$ and $a=(r/R{)}^2$ inside $r=R$, with both equal to one outside, are continuous finite-energy trial fields. In the normalization $E=\int [B^2/2+|D\varphi {|}^2/2+(1-|\varphi {|}^2{)}^2/8]$, their radial energy is the displayed function. Its minimum has $R^4=48$ and $E=\pi (2/3+1/\sqrt{3})$, an upper-bound estimate above the exact Bogomolny bound $\pi$. Derivative jumps at the matching radius are allowed in the variational energy space; the trial profile is not an exact solution.

For independent exponential distribution observations with rate $\lambda$, the likelihood ratio for a larger simple rate is decreasing in their sum. The Neyman-Pearson lemma therefore selects a lower-tail sum threshold $c$. The sum has a gamma distribution, giving the displayed power; its derivative is $c{e}^{-\lambda c}(\lambda c{)}^{n-1}/(n-1)!>0$. Calibrating at ${\lambda }_0$ gives size $\alpha$ for the composite null $\lambda \le {\lambda }_0$ and a uniformly most powerful test against larger rates.

For two simple hypotheses with probability densities $f_0,f_1$, a critical region $C$ has sum of Type I error and Type II error probabilities ${\int }_C{f}_0+{\int }_{C^c}{f}_1$. Pointwise minimization chooses $C=\{f_1>f_0\}$, with either decision allowed on the equality set. Thus the optimal likelihood ratio threshold is $1$, and the minimum is $\int \min (f_0,f_1)=1-\frac{1}{2}\int |f_1-f_0|$. This criterion weights the two error probabilities equally; unequal costs or prior probabilities change the threshold.

For two unit-scale Laplace distributions with locations ${\theta }_0<{\theta }_1$, the likelihood ratio is nondecreasing in $X$, with constant tails and an increasing middle interval. Therefore an upper-tail statistical test of exact size of a statistical test $\alpha$ is a most powerful test. Its cutoff is $c_{\alpha }={\theta }_0-\log (2\alpha )$ for $\alpha \le 1/2$, and $c_{\alpha }={\theta }_0+\log (2(1-\alpha ))$ otherwise. Under location ${\theta }_1$, its statistical power is $\frac{1}{2}{e}^{{\theta }_1-c_{\alpha }}$ if $c_{\alpha }\ge {\theta }_1$, and $1-\frac{1}{2}{e}^{c_{\alpha }-{\theta }_1}$ otherwise. Constant likelihood ratio tails can make the most powerful test nonunique.

For a Newtonian central configuration, set ${\mathbf{x}}_i(t)=a(t){\mathbf{a}}_i$ with $a>0$. The gravitational force scales as $a^{-2}$, reducing all equations to the displayed scalar ordinary differential equation. Its first integral is ${\dot{a}}^2/2-\Lambda /a=k/2$. The total energy is $kI/2$, so the scalar energy constant exactly tracks the many-body energy for the fixed shape.

For equal initial temperatures and the same Newton's law of cooling coefficient $a>0$, compare a unit Dirac delta function cooling impulse at time one with unit-rate cooling on the interval from one to two. The impulse produces a temperature jump of minus one. The finite-duration forcing gives a continuous temperature with change $-(1-e^{-a(t-1)})/a$ while it is active. Relative to the common unforced cooling curve, the impulse contributes $-e^{-a(t-1)}$. Consequently the temperatures cross at $t=1+\log (1+a)/a$, strictly between one and two. After time two the impulse-cooled body is warmer: the later cooling receives less subsequent exponential attenuation.

For NBUE claims in the classical risk model, the equilibrium claim size $Y$ is stochastically bounded by $X$. The positive exponential tail integral formula for moments gives $M_Y(R)\le M_X(R)$. Since $M_Y(R)=1+\theta$ and $M_X(R)=1+(1+\theta )\mu R$, rearrangement proves the displayed bound.

A relativistic species decoupling before ordinary neutrino decoupling cools as $T_X\propto a^{-1}$. Cosmological entropy conservation in the remaining bath gives $T_X/T_{\nu }=[10.75/g_{\ast s,D}{]}^{1/3}$, where $g_{\ast s,D}$ counts that bath just after the early species has left it. Subsequent electron-positron annihilation in cosmology gives $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$. For the ideal high-temperature relativistic entropy content of the Standard Model, the resulting ratio is about $0.332$.

In natural units with unreduced Planck mass $M_P=G^{-1/2}$, the radiation-era Hubble parameter scales as $\sqrt{g_{\ast }}{T}^2/M_P$, while a low-energy weak interaction rate scales as $G_F^2{T}^5$. Their equality gives a decoupling scale of order one MeV. Distinct weak processes have distinct rate coefficients: neutrino decoupling and cosmological weak freeze-out of neutron-proton conversion are nearby events, not an identical precise temperature inferred from the power laws alone.

If all neutrinos are exactly massless, arbitrary unitary rotations among them preserve both their kinetic terms and their zero mass matrix. After diagonalizing the charged-lepton Yukawa matrix, choose the neutrino basis to coincide with that rotation. The resulting weak charged current has no physical lepton mixing matrix. With nondegenerate neutrino masses this extra basis freedom is lost, permitting observable neutrino oscillations and lepton mixing.

A recurrent network with symmetric interactions and an energy function. In the binary version, units take values $\pm 1$, self-couplings are zero, and asynchronous threshold updates can retrieve stored patterns from incomplete or corrupted cues. Fixed points are local energy minima, which need not be the intended memories or global minima.

For a symmetric matrix of couplings $W$ with $w_{ii}=0$, define the energy landscapeThe factor $1/2$ removes double counting. The local input is $h_k={\sum }_{j\ne k}{w}_{kj}{x}_j-{\theta }_k$. Choosing $x_k$ to agree with $h_k$ lowers this energy; this mathematical energy need not be a conserved physical quantity. Its pairwise form connects the model to the Ising model; competing couplings also motivate the comparison with a spin glass. Here retrieval seeks a useful nearby local minimum, so reaching the global minimum is not the goal.

John Hopfield's 1982 computer study stored random binary strings in neural networks of 30 and 100 units. Couplings were prepared from the stored patterns; asynchronous updates supplied the computational apparatus. Starting at each nominal memory, the readout counted Hamming distances after settling as storage load varied. Corrupted starting patterns separately tested recovery. Varying the load and cue damage exposed interference between memories. These simulations supported collective retrieval, not a measurement of biological memory capacity.^[ref]

An activation function converts a unit's weighted input to its output. A sigmoid function, hyperbolic tangent or rectifier can give nonlinear features. Replacing all hidden activation functions by the identity collapses consecutive affine layers into a single affine map, removing the representational benefit of those hidden layers.

A binary threshold unit outputs one when its weighted input reaches its threshold and zero otherwise. Its hard threshold is not differentiable, so ordinary backpropagation cannot directly train the exact threshold model. Smooth activation functions can approximate its computation.