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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.

Neumann boundary data are prescribed values of the outward normal derivative of a function on the boundary of a boundary value problem. They enter a Neumann boundary condition as ${\partial }_{n}u=g_N$. Directional derivatives defined using a fixed coordinate direction must be converted to the outward-normal convention: for example, on the bottom of an upper domain, ${\partial }_{n}u=-u_y$.

In a directed network with no negative-cost directed cycle, add a source having zero-cost arcs to all vertices and compute shortest-path distances $d_i$. The inequalities $d_j\le d_i+c_{ij}$ make ${\pi }_i=-d_i$ dual feasible for uncapacitated minimum-cost flow. Shortest-path tree arcs are tight. An arbitrary spanning tree also fixes tight candidate potentials up to a constant, but it does not guarantee the non-tree inequalities; these must be checked.

The view factor $F_{i\to j}$ is the fraction of uniformly distributed, diffuse outgoing radiant flux from surface $i$ that directly reaches surface $j$. It depends on geometry, including obstruction, rather than on temperature or emissivity. All destinations in a closed enclosure have fractions summing to one. For a convex body fully enclosed by walls, $F_{\mathrm{body}\to \mathrm{walls}}=1$; the reverse view factor is generally smaller. Reciprocity gives $A_i{F}_{i\to j}=A_j{F}_{j\to i}$. Two unobstructed spheres with outer radius ten times the inner radius have $F_{\mathrm{inner}\to \mathrm{outer}}=1$ and $F_{\mathrm{outer}\to \mathrm{inner}}=1/100$. Do not multiply both directions by an assumed identical view factor.^[ref]

For the volume-preserving family $F=\mathrm{diag}(\lambda ,{\lambda }^{-1/2},{\lambda }^{-1/2})$, equal dead nominal tension $T$ gives potentialIts derivative factors as ${\Pi }^{\prime }=(1-{\lambda }^{-3/2})[\mu (\lambda +{\lambda }^{-1/2})-T]$. The nontrivial branch has a fold at $\lambda =2^{-2/3}$, $T/\mu =32^{-2/3}$, and meets the trivial branch at $T/\mu =2$. Stability in this one-parameter family is determined by ${\Pi }^{\prime \prime }$, not by branch count alone. At $T/\mu =2$ there is only one nontrivial positive solution, $\lambda =(3-\sqrt{5})/2$; the other branch value is $\lambda =1$.

The dielectric response differs along and perpendicular to the nematic director. In Gaussian units the orientational electric contribution at prescribed field is $-\Delta ϵ(\mathbf{E}\cdot \mathbf{n}{)}^2/(8\pi )$; positive anisotropy favors alignment with the electric field. The electric term does not receive the additional one-half convention used in writing the Frank elastic terms.

The Fréedericksz transition is a field-driven orientational instability of a confined nematic liquid crystal. Elasticity and surface anchoring resist the electric field or magnetic torque; a lowest allowed distortion mode becomes unstable at a finite threshold. Positive dielectric anisotropy of a nematic permits a normal electric field to tilt a planar anchored director.

The twisted nematic field effect switches a nematic liquid crystal between a twisted optically rotating state and a largely field-aligned state. With adiabatic optical following in a twisted nematic and crossed polarizers aligned to the two planar surface directors, a quarter-turn cell passes light at low voltage and suppresses it at high voltage. A rear reflector permits reflective operation with little power expended on light generation.

Let $k$ be a positive integer, $0<p<1$, and $\lambda >0$. For independent rate-$\lambda$ exponential distribution claims and a zero-based negative binomial distribution count of shape $k$ and parameter $p$, the random-sum transform identity gives $M_S(t)=[p(\lambda -t)/(p\lambda -t){]}^k$. This equals the transform of a compound binomial distribution with count $\mathrm{Bin}(k,1-p)$ and rate-$p\lambda$ claim sizes. Conditional on its positive count $n$, the total has Erlang distribution, whose survival function is the inner exponential-polynomial sum. Conditioning yields the displayed finite mixture distribution formula for $x>0$. It replaces an infinite mixture and includes the zero-total atom $p^k$ through the omitted $n=0$ term.

For $k=1$, the formula reduces to $(1-p)e^{-p\lambda x}$ for $x>0$: there is an atom $p$ at zero and a rate-$p\lambda$ exponential distribution positive component. This checks both the omitted zero-count term and the parameter scaling.

A uniform algebra is natural exactly when functions with no common zero generate the unit ideal. If a generated ideal is proper, it is contained in a maximal ideal. That maximal ideal is closed: the closure of a proper ideal stays proper, since an element within distance less than one of the identity is invertible by the Neumann series. The Gelfand-Mazur theorem identifies the maximal-ideal quotient with the complex numbers, yielding a character of an algebra. Naturality would make that character a common-zero evaluation, a contradiction. Conversely, the asserted unit-ideal property makes the zero sets of every finite collection in a character's kernel intersect. Compactness gives a point common to all these zero sets; applying this to $f-\phi (f)1$ proves $\phi (f)=f(x)$. For $C(K)$, one can directly take $g_j=\overline{f_j}/{\sum }_k|f_k{|}^2$.