The bino is the spin-one-half gaugino of the hypercharge $U(1{)}_Y$ gauge group. Its adjoint representation is the singlet $(1,1,0)$, so it has no hypercharge of its own. After electroweak symmetry breaking, it can mix with the neutral wino and higgsinos into neutralinos.

With positive canonical kinetic terms and nonzero coupling $g$, expansion of the covariant scalar kinetic term gives the displayed gauge-boson mass matrix. It is a Gram matrix: for real $c_a$, $c^T{M}_A^{2}c=g^2|{\sum }_a{c}_a{T}_a{\varphi }_0{|}^2$. Its kernel is the Lie algebra of the stabilizer subgroup of the vacuum. Thus its rank is $\dim G-\dim H$, and exactly that many gauge-field combinations acquire nonzero squared masses. Removal of these gauge-orbit directions does not remove accidental zero eigenvalues of the scalar mass matrix on the transverse complement.

Gas outside the uniform-temperature bulb of a gas thermometer contributes to its conserved particle inventory. An idealized calculation divides the connecting tube into small regions with volumes $V_j$ and local temperatures $T_j$. Neglect hydrostatic pressure offsets and thermal transpiration, assume a common pressure $P$ and a local ideal gas law in each region, and let the bulb have volume $V_b$ and temperature $T_b$. Adding the particle inventories givesThus even an ideal gas in a perfectly rigid sealed apparatus need not obey $P\propto T_b$: some of its particles remain in regions with other temperatures. For a stipulated bulb of $100\mathrm{c}{\mathrm{m}}^3$, a connecting volume of $1\mathrm{c}{\mathrm{m}}^3$ maintained at $300\mathrm{K}$, and bulb reference and test temperatures of $300$ and $400\mathrm{K}$, conservation of $N$ givesThese are toy dimensions and a calculated bias, not measured data. Reducing $N$ reduces both pressures proportionally and leaves this ratio unchanged: extrapolating away gas nonideality does not eliminate the connecting-volume error. Actual apparatus also requires the tube's thermal expansion, a measured temperature profile, and the pressure corrections omitted in this model.

For a slowly varying nuclear factor and penetration factor $\exp [-2\sqrt{E_B/E}]$, the thermal integral is ${\int }_0^{\infty }{e}^{-\Phi (E)}dE$, where $\Phi (E)=E/(k_{B}T)+2\sqrt{E_B/E}$. The Gamow peak occurs at $E_0=[E_B(k_{B}T{)}^2{]}^{1/3}$, with $\Phi (E_0)=3E_0/(k_{B}T)$ and ${\Phi }^{\prime \prime }(E_0)=3/(2k_{B}T{E}_0)$. The Gaussian integral gives $\sqrt{4\pi k_{B}T{E}_0/3}{e}^{-3E_0/(k_{B}T)}$. Combined with the Maxwellian prefactor $(k_{B}T{)}^{-3/2}$ this produces the displayed thermonuclear reaction rate at fixed reactant density. The approximation assumes $E_B\gg k_{B}T$ and excludes strong resonances or rapidly varying nuclear factors.

For $Y_i\mid {\theta }_i\sim \mathrm{Poisson}({\theta }_i)$ and ${\theta }_i\mid \psi \sim \mathrm{Exp}(\psi )$, a scale-invariant prior $d\psi /\psi$ induces the Haldane prior on $\varphi =(1+\psi {)}^{-1}$. Integrating out each local mean gives $p(y_i\mid \varphi )=(1-\varphi ){\varphi }^{y_i}$. Thus $\varphi \mid y\sim \mathrm{Beta}(s,n)$ for $s={\sum }_i{y}_i>0$. The gamma full conditional distribution has mean $(y_i+1)\varphi$, so the tower property gives $(y_i+1)s/(s+n)$, equal to the displayed partial pooling formula. For $s=0$, the posterior is improper and this expectation is undefined.

A gamma distribution translated by a real constant $k$. With shape $\alpha >0$ and rate $\nu >0$, its support is $(k,\infty )$ and its first three cumulants are $k+\alpha /\nu$, $\alpha /{\nu }^2$, and $2\alpha /{\nu }^3$. Its skewness is $2/\sqrt{\alpha }$. A negative shift permits negative values; introducing a shift does not automatically respect the nonnegative support of an insurance loss.

For a nearest-neighbour random walk on $\{0,\dots ,N\}$ with right-step Probability $p$, left-step Probability $q=1-p$, and Absorption time $\tau$, first-step conditioning gives $u_k=s(p{u}_{k+1}+q{u}_{k-1})$, $u_0=1$, $u_N=0$. For $0<s<1$, the two positive distinct characteristic roots are $r_{\pm }=(1\pm \sqrt{1-4pq{s}^2})/(2ps)$. Solving the boundary equations gives $u_k=(r_{+}^N{r}_{-}^k-r_{-}^N{r}_{+}^k)/(r_{+}^N-r_{-}^N)$. The maximum principle $\Vert u-v{\Vert }_{\infty }\le s\Vert u-v{\Vert }_{\infty }$ proves uniqueness for equal boundary data. This is the joint generating function of the Absorption time and the lower-end absorption event.

For a simple symmetric random walk absorbed at $0,r$, the Probability of absorption at $0$ is $h_i=(r-i)/r$. Conditioning on that absorption gives the Doob h-transform with transition probabilities $(r-i+1)/[2(r-i)]$ downwards and $(r-i-1)/[2(r-i)]$ upwards. If $g_i={\mathbb{E}}_i[\tau 1_{\{{\tau }_0<{\tau }_r\}}]$, first-step analysis gives $g_i=h_i+(g_{i-1}+g_{i+1})/2$, $g_0=g_r=0$. Its solution is $g_i=i(r-i)(2r-i)/(3r)$, and dividing by $h_i$ gives the conditional duration.

For a function $p:A\to B$, order each power set by inclusion. The inverse image map $p^{\ast }(V)=p^{-1}(V)$ has a left adjoint ${\exists }_p(U)=p(U)$ and a right adjoint ${\forall }_p(U)=\{b\in B:p^{-1}(\{b\})\subseteq U\}$. The two adjunctions follow from $p(U)\subseteq V⟺U\subseteq p^{-1}(V)$ and $p^{-1}(V)\subseteq U⟺V\subseteq {\forall }_p(U)$. Empty fibers satisfy the second condition vacuously, so points outside the image of $p$ belong to ${\forall }_p(U)$ for every $U$.

A Fourier truncation retains a finite set of boundary-compatible Fourier modes and projects the PDE onto their span by the Galerkin method. Nonlinear products can generate unretained harmonics, so their coefficients are discarded by projection rather than declared zero pointwise. Agreement with a perturbation expansion through cubic order requires retention of every quadratically forced mode that feeds back on the critical mode at cubic order.

// Destination: dynamical-systems.bigb

The Petrov-Galerkin method chooses an approximate solution $u_h$ in a finite-dimensional trial vector space $V_h$ and tests a weak formulation against a possibly different test space $W_h$. For a bilinear form $a$ and a linear functional $\ell$, it imposes $a(u_h,v_h)=\ell (v_h)$ for every $v_h\in W_h$. Taking $W_h=V_h$ recovers the usual Galerkin method. Different test functions can alter stability and accuracy; the Galerkin orthogonality is then against the test space and need not be an orthogonal projection in an energy norm.

In a spherical Newtonian gravitational potential, use binding energy $E=\psi -v^2/2$ and specific angular momentum $L$. The displayed stationary galactic distribution function has locally Gaussian radial and transverse velocities: ${\sigma }_r^2={\sigma }^2$, ${\sigma }_{\theta }^2={\sigma }_{\varphi }^2={\sigma }^2/(1+r^2/r_a^2)$. Its mass density is $\rho ={\rho }_c{e}^{[\psi (r)-\psi (0)]/{\sigma }^2}/(1+r^2/r_a^2)$. The velocity-anisotropy parameter is ${\beta }_{\mathrm{an}}=r^2/(r^2+r_a^2)$, so the core is nearly isotropic and the halo nearly radial. These exact Gaussian integrals use the untruncated distribution over all velocities; imposing an escape-energy cutoff changes them.

For a galactic distribution function $f(\mathbf{x},\mathbf{v})$ with tracer number density $\nu =\int f{d}^{3}v$, a velocity moment is the weighted integral of a velocity monomial. The first moment gives the mean velocity ${\overline{v}}_i$; the second moment satisfies $\overline{v_i{v}_j}={\overline{v}}_i{\overline{v}}_j+{\sigma }_{ij}^2$, separating ordered motion from the random velocity dispersion tensor. Multiplying the Collisionless Boltzmann equation by $v_i$ and integrating yields the Jeans equation, with vanishing velocity-space boundary terms when the distribution decays sufficiently rapidly.

A lowered Maxwellian velocity distribution subtracts the value of a Maxwell-Boltzmann velocity distribution at the escape velocity and sets the result to zero beyond escape. This gives a nonnegative phase-space distribution function vanishing continuously at the boundary. If $A=n_0/(2\pi {\sigma }^2{)}^{3/2}$ and $x_e=v_e/(\sqrt{2}\sigma )$, its actual number density is $n=n_0[\mathrm{erf}{x}_e-2x_e{e}^{-x_e^2}/\sqrt{\pi }-4x_e^3{e}^{-x_e^2}/(3\sqrt{\pi })]$. Thus $n_0$ is a normalization parameter, not generally the integrated density. Integrating only speeds below $V<v_e$ gives the same bracket with the first two terms at $X=V/(\sqrt{2}\sigma )$ and the last term $-4X^3{e}^{-x_e^2}/(3\sqrt{\pi })$.

Put a simple closed curve in a surface collar and use its surface framing. Filling its deleted tubular neighborhood along $\mu \pm \lambda$ inserts one full positive or negative twist of the annulus between the collar levels. Thus surgery of coefficient $\pm 1$ changes the gluing by a Dehn twist or its inverse. The exact sign depends on the orientations of the seam and normal direction.

Define the indicator function of the rational numbers on the real line by $f(x)=1$ for rational $x$ and $f(x)=0$ otherwise. Every rational shift preserves rationality and irrationality, so every nonzero rational number is a period of a function. Conversely, an irrational shift sends $x=0$ from value one to value zero, so it is not a period. The period group of a function is exactly $\mathbb{Q}$, and the positive periods $1/n$ approach zero without a smallest one. The function is nonconstant and discontinuous everywhere; this is why continuity matters in continuous periodic functions have a least positive period.

For a real-valued continuous function, the period group of a function is a closed subgroup: if periods $T_n$ tend to $T$, continuity passes $f(x+T_n)=f(x)$ to the limit. If positive periods could tend to zero, integer multiples of those periods would approach any prescribed displacement $y-x$, forcing $f(y)=f(x)$. A nonconstant function therefore has a positive lower bound for its positive periods. Their infimum $T_0$ is itself a period by closedness. Subtracting an integer multiple of $T_0$ from any other period leaves a period in $[0,T_0)$, which must be zero. This proves the stated structure without assuming differentiability.

Writing $Y=GZ$ with an invertible matrix-valued function gives $Z^{\prime }=\Lambda Z$ exactly when $\Lambda =G^{-1}AG-G^{-1}{G}^{\prime }$. For a formal power series transformation at an irregular singular point, coefficient comparison recursively removes off-diagonal terms when leading diagonal eigenvalues are distinct. This is a differential-system change of frame, not specifically an electromagnetic gauge transformation.

Glue the boundary of a Möbius band to the projective line in the real projective plane by a mapping degree $n$. The band is the mapping cylinder of the degree-two boundary covering of its core. Collapsing the connecting tree in the resulting double mapping cylinder gives a CW complex with one vertex, edges $a,b$, and two attaching words $a^2$ and $a^{-n}{b}^2$. Its cellular chain complex has degree-two boundary matrix $\left(\begin{array}{cc}2& -n\\ 0& 2\end{array}\right)$. Its determinant is four, so $H_2=0$. The Smith normal form gives $H_1\cong \mathbb{Z}/4$ for odd $n$ and $(\mathbb{Z}/2{)}^2$ for even $n$; $H_0=\mathbb{Z}$ and higher homology vanishes. The space cannot be homotopy equivalent to a closed manifold: mod-two homology forces a potential manifold to be a surface, and its Euler characteristic is one, whereas the only such closed connected surface is the projective plane, with different integral $H_1$.

A presheaf on a category $\mathcal{C}$ is a functor $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$. A morphism $f:C\to D$ induces the restriction map $X(f):X(D)\to X(C)$, reversing direction. The indexing category need not be small. For the category of open subsets of a topological space, this specializes to a presheaf of sets on a topological space.