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A tail of a sequence consists of its terms from some index $N$ onward, retaining their order and repetitions. Discarding finitely many initial terms does not change its limit of a sequence or whether it is a Cauchy sequence. A tail can also be considered as the set of values $\{x_n:n\ge N\}$ when forming an infimum, supremum, or tail diameter of a sequence; that set alone forgets the original order and repetitions.

A sequence indexed from zero with an integer period $p\ge 1$ satisfying the displayed identity for all $n\ge 0$. Repeating a finite cycle is an example. The Cesàro mean article explains why its cumulative averages approach the one-cycle arithmetic mean.

A constant sequence has the same value $a$ at every index. It converges to $a$ by convergence of a sequence in a topological space.

An algebra homomorphism onto a semisimple Banach algebra is continuous. To see the spectral mechanism, suppose $a_n\to 0$ and $T(a_n)\to b$. For any $c\in B$, choose $u,v\in A$ with $T(u)=b$ and $T(v)=c$, and set $d=cb$, $d_n=T(v{a}_n)$. Apply the spectral-radius three-circle inequality to $T((1-z)v{a}_n+zvu)$. On the outer circle its norm tends uniformly to $\Vert d\Vert$; on the circle of radius $1/R$ its spectral radius is bounded by $(1+1/R)\Vert v{a}_n\Vert +\Vert vu\Vert /R$. Hence $r(d{)}^2\le \Vert d\Vert \Vert vu\Vert /R$ for every $R>1$, so $r(cb)=0$. The unit criterion for the Jacobson radical gives $b\in J(B)=0$, and the closed graph theorem finishes the proof.

For a uniform piecewise-linear Galerkin heat discretization, the consistent mass matrix is $M=(h/6)\mathrm{tridiag}(1,4,1)$ and the stiffness matrix is $K=h^{-1}\mathrm{tridiag}(-1,2,-1)$. The generalized eigenvalues are $6(1-\cos \theta )/[h^2(2+\cos \theta )]$. Applying forward Euler method requires $k\lambda \le 2$, giving the displayed uniform bound for $\mu =k/h^2$. For $J$ intervals with Dirichlet endpoints the exact finite-grid upper bound is $(2-\cos (\pi /J))/[3(1+\cos (\pi /J))]$, which approaches $1/6$. Replacing the consistent mass matrix by a lumped one changes the stability restriction.

For a nearly $D$-regular $s$-uniform hypergraph, sample each edge independently with probability $\alpha /D$, where $\alpha$ is small. Retain the isolated sampled edges as a matching in a hypergraph, and delete every vertex incident to any sampled edge before the next round. A vertex survives with probability $(1-\alpha /D{)}^{d(v)}\approx e^{-\alpha }$; a sampled edge meets another sampled edge with probability at most $s\alpha (1+o(1))$. Thus accepted edges cover order $\alpha$ of the vertices and collision waste is order ${\alpha }^2$. Small hypergraph codegrees control the overlap terms in residual-degree calculations. Repeating a bounded number of rounds while tracking near-regularity is the characteristic Rödl nibble mechanism.

For an isothermal gas slab, a horizontal Fourier mode has $\delta \rho =-ik\rho {\xi }_x-(\rho {\xi }_z{)}_z$, $W=c_s^2\delta \rho /\rho +\varphi$, ${\omega }^2{\xi }_x=ikW$, ${\omega }^2{\xi }_z=W_z$ and $({\partial }_z^2-k^2)\varphi =4\pi G\delta \rho$. Multiply momentum by $\rho {\mathbf{\xi }}^{\ast }$ and integrate by parts, then use the Poisson equation for Newtonian gravity. Vanishing pressure-work and gravitational boundary terms give the displayed real quadratic identity, with $I=\int \rho |\mathbf{\xi }{|}^{2}dz>0$. Thus every admissible squared eigenfrequency is real; a negative value gives gravitational growth rather than an oscillatory instability.

For a self-gravitating isothermal slab with $\rho ={\rho }_0{\mathrm{sech}}^2(z/H)$ and $H^2={\sigma }^2/(2\pi G{\rho }_0)$, hydrostatic equilibrium gives the displayed Newtonian gravitational potential. A test particle has constant in-plane velocity and conserved vertical energy $E_z={\dot{z}}^2/2+\Phi (z)-\Phi (0)$. Its vertical motion is periodic for every finite $E_z>0$, with turning height $H\mathrm{arcosh}\exp [E_z/(2{\sigma }^2)]$. Near the plane it is a harmonic oscillator of squared frequency $4\pi G{\rho }_0$; at large height the force tends to the constant $2\pi G\Sigma$, where $\Sigma =2{\rho }_{0}H$. The ideal infinite slab has no finite vertical escape energy.

If $B$ is a positive riskless numéraire of finite variation, $S$ the ex-dividend price vector and $D$ its cash-dividend rate, a self-financing portfolio with risky holdings $\pi$ satisfies $d(X/B)=\pi \cdot dY$. This follows by differentiating $X/B$, substituting $dX=\varphi dB+\pi \cdot (dS+Ddt)$ and using $X=\varphi B+\pi \cdot S$. If $Y$ is a martingale under an equivalent measure, discounted wealth is a local martingale. A deterministic lower bound on discounted wealth makes it a supermartingale, ruling out arbitrage. The cumulative dividends here are discounted at their own payment times.

For $p(x,t)=xg(x,t)+B_{t}h(x,t)$ and deterministic $d{B}_t=\rho B_{t}dt$, Itô formula shows that a self-financing portfolio must have $p_x=g$. This is equivalent to $x{g}_x+B_t{h}_x=0$. Differentiating it gives $p_{xx}=g_x$, so the remaining gain identity is $x{g}_t+B_t{h}_t+{\sigma }^2{x}^2{g}_x/2=0$. Conversely these two identities give $dp=gdS+hdB$. A value PDE alone does not verify arbitrary prescribed holdings.

A constant-proportion portfolio holds a fixed fraction $\gamma$ of its portfolio wealth in the stock. In the Black-Scholes model with bank rate $\rho$, initial wealth $V_0$, and no consumption, its value isThe stock holding is $\gamma V_t/S_t$ and the remaining value $(1-\gamma )V_t$ is invested in the bank account. The Itô formula verifies $d{V}_t/V_t=[\rho +\gamma (\mu -\rho )]dt+\gamma \sigma d{W}_t$ when the physical stock drift is $\mu$.

In the Black-Scholes model, take $B_t=e^{-\rho (T-t)}$ as a zero-coupon bond numéraire. Holding the fixed stock value $\theta B_t$ requires $g_t=\theta B_t/S_t$ shares. The self-financing conditions for smooth stock and bond holdings giveunits of the bond. Since $V_t=g_t{S}_t+h_t{B}_t$, this yields the displayed wealth. Under the risk-neutral measure, $V_t/B_t=w_0/B_0+\theta \sigma W_t^Q$: the discounted gain is a constant multiple of Brownian motion.

For the orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^4$, define ${\eta }_{ij}^a={ϵ}_{aij}$, ${\eta }_{i4}^a={\delta }_{ai}$, and extend antisymmetrically. The three forms $\frac{1}{2}{\eta }_{\mu \nu }^{a}d{x}^{\mu }\wedge d{x}^{\nu }$ are self-dual. Reversing the sign of the components involving index four gives the anti-self-dual symbols $\bar{\eta }$. They identify the two three-dimensional eigenspaces of the Hodge star on differential two-forms and give compact formulas for the BPST instanton.

For a uniform elastic filament clamped below and free above, its own weight gives filament tension $T(z)=-\lambda g(L-z)$. With filament bending modulus $B=EI$, the static slope $u$ satisfies $B{u}^{\prime \prime }+\lambda g(L-z)u=0$, $u(0)=0$, and $u^{\prime }(L)=0$. Setting $\eta =(2/3)\sqrt{\lambda g/B}(L-z{)}^{3/2}$ gives $u={\eta }^{1/3}{J}_{-1/3}(\eta )$ up to amplitude; the $J_{1/3}$ branch violates the zero-moment condition. The first positive zero $j_{-1/3,1}$ therefore yields $\lambda g{L}_c^3/B\simeq 7.83735$. This is a self-buckling of a vertical rod threshold, not a constant-end-load Euler threshold.

A densely defined semibounded symmetric operator on a Hilbert space has a canonical semibounded extension that is a self-adjoint operator obtained by closing its quadratic form. After shifting the lower bound to zero, complete the initial form domain in the norm $(\Vert u{\Vert }^2+q[u]{)}^{1/2}$; the associated closed form determines the Friedrichs extension. For the positive Laplace-Beltrami operator on a polygonal surface, this specifies the finite-energy realization at corners. Different choices of the initial form domain impose Dirichlet boundary conditions, Neumann boundary conditions, or mixed boundary conditions; the extension does not choose the boundary condition independently of that domain.

Let $\ddot{q}=Fq$ on a Hilbert space, with a time-independent self-adjoint operator $F$ that is a bounded linear operator, and conserved quadratic energy $E=\Vert \dot{q}{\Vert }^2/2-⟨q,Fq⟩/2$. If $⟨q_0,F{q}_0⟩>0$, set $\dot{q}(0)=\alpha q_0$ with ${\alpha }^2=⟨q_0,F{q}_0⟩/\Vert q_0{\Vert }^2$ and $\alpha >0$. This gives $E=0$. For $N=\Vert q{\Vert }^2$, $N^{\prime }=2\mathrm{Re}⟨q,\dot{q}⟩$ and $N^{\prime \prime }=4\Vert \dot{q}{\Vert }^2$, soby the Cauchy-Schwarz inequality. Since $(\ln N{)}^{\prime }(0)=2\alpha$, $N(t)\ge N(0)e^{2\alpha t}$ for $t\ge 0$. Positivity of $N$ follows from $N^{\prime }(0)>0$ and $N^{\prime \prime }\ge 0$. Thus a negative direction of the restoring energy $W=-⟨q,Fq⟩/2$ yields an exponentially growing solution without a normal-mode expansion or a variational spectral theorem. If $W$ is positive definite, conserved $E$ instead controls the energy norm; a bound in the original Hilbert space norm additionally follows from a coercive lower bound on $W$. For an unbounded self-adjoint operator, the same estimate applies to sufficiently regular solutions for which the energy identity is justified; existence for arbitrary initial data must be established separately.

Variation of the small-slope bending energy of an elastic filament gives endpoint work $A[h^{(2)}\delta h^{(1)}-h^{(3)}\delta h]$. Arbitrary endpoint displacement and slope variations force the bending moment and transverse force to vanish. These are $h^{(2)}=h^{(3)}=0$. They define a self-adjoint nonnegative fourth-order bending operator on a finite interval; its kernel contains rigid translation and tilt.

A response model and a dropout model share latent random effects. Conditional on those effects and observed predictors, the two processes may be independent, yet marginally dropout is missing not at random because the shared effects also predict unseen outcomes. Joint likelihood fitting integrates the product of both models over the shared effects; integrating the outcome model alone is insufficient. This is an explicit identifying model, and its latent-dependence assumptions should be assessed by sensitivity analysis.

If the displayed fiber bundle had a section $s$, its projection $p$ would satisfy $s^{\ast }{p}^{\ast }=1$. The cohomology ring of complex projective space is $\mathbb{Z}[h]/(h^4)$ with $|h|=2$. For a generator $u\in H^4(S^4;\mathbb{Z})$, write $p^{\ast }u=m{h}^2$. But $s^{\ast }h=0$ since $H^2(S^4;\mathbb{Z})=0$, giving $u=s^{\ast }{p}^{\ast }u=0$. This contradiction proves that no section exists, without needing the value of $m$.

For a nonnegative random variable $N$ with positive expected value and finite second moment, write $N=N{1}_{\{N>0\}}$. The Cauchy-Schwarz inequality giveswhich proves the bound in second moment method. It is the zero-threshold case of the Paley-Zygmund inequality. Equality occurs when $N$ is a fixed positive value on the event $N>0$ and zero elsewhere, almost surely. For $N=10B$ with $B$ a Bernoulli random variable of parameter $1/10$, $(\mathbb{E}N{)}^2/\mathbb{E}{N}^2=1/10$ equals the exact positivity probability.

The variance identity $\mathbb{E}{N}^2=(\mathbb{E}N{)}^2+\mathrm{Var}(N)$ turns the lower bound into a statement about variance relative to the squared mean. A small ratio forces positivity to be likely. A positive expected value alone does not do so: keeping $\mathbb{E}N=1$ while taking $N=1/q$ with probability $q$ and zero otherwise makes the positivity probability as small as any chosen $q>0$.