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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.

If $E$ is a ring spectrum, then $F(X_{+},E)$ is a ring spectrum. Its multiplication is adjoint to evaluating the two functions on the two copies supplied by the diagonal of $X$, then multiplying in $E$. Its unit is the constant unit function. Coassociativity of the diagonal and associativity in $E$ prove the associative identity; the counit and unit identities prove both unit laws.

A fundamental matrix for $F^{\prime }=RF/z$ on a chosen logarithm branch is $\exp (R\log z)$. A counterclockwise circuit changes the logarithm by $2\pi i$, giving the displayed monodromy matrix and the cyclic group it generates. Changing the solution basis conjugates the matrix. The residue eigenvalues exponentiate to the monodromy eigenvalues; the latter do not determine the former uniquely.

Suppose the matrix $B$ is holomorphic at zero and $0$ is an eigenvalue of $R$. Let $m$ be the largest nonnegative integer for which $-m$ is an eigenvalue of $R$. Seek $F=z^m{\sum }_{n\ge 0}{h}_n{z}^n$, with $h_0\in \ker (mI+R)$ nonzero. The Frobenius method gives $((m+n)I+R)h_n=-{\sum }_{j=0}^{n-1}{B}_j{h}_{n-1-j}$ for $n\ge 1$, where scalar terms multiply the identity. Every left-hand matrix is invertible by the choice of $m$. Its inverse is $O(1/n)$, so a scalar convergent majorant proves convergence of the series. This yields a nonzero analytic solution; it is necessarily nonconstant when $m>0$, but can be constant when $m=0$. For the displayed sign convention, indicial exponents are eigenvalues of $-R$.

A local series solution at a regular singular point, with exponent $r$ satisfying the indicial equation. Resonant exponents can require a logarithmic companion solution.

With $s=\sqrt{z}$ and $Y(s)=y(s^2)$, the displayed linear differential equation becomes $Y_{ss}+Y=0$ away from zero. Its Frobenius solutions have exponents $0$ and $1/2$: $\cos \sqrt{z}={\sum }_{n\ge 0}(-1{)}^n{z}^n/(2n)!$ and $\sin \sqrt{z}=z^{1/2}{\sum }_{n\ge 0}(-1{)}^n{z}^n/(2n+1)!$. The first is an entire function of $z$; the second needs a local square-root branch. This illustrates how a regular singular point can arise from a degenerate change of independent variable.