For an ideal $I\subseteq R$, the Rees ring is the graded subringIt packages all powers of $I$ into one algebra and turns stable filtrations into finitely generated graded modules.

Restrict a cubic equation to the affine line through two distinct singular points. At each point the equation and its first directional derivative vanish, so the restricted polynomial has two distinct double roots. A polynomial of degree at most three with this property is identically zero. Hence the whole line lies in the hypersurface. The multiplicity argument works in every characteristic.

An X-piece is a curvature-$-1$ four-holed sphere with geodesic boundary, formed by gluing two Y-pieces along equal-length cuffs. For a specified separating cuff and labelled outer boundaries, four outer lengths, the internal cuff length $s>0$, and a twist $\tau$ determine the metric. The gluing is orientation-reversing on boundary circles, with relative arclength displacement $\tau$ measured from chosen seam endpoints. Without markings, the gluing displacement is taken modulo $s$; with a marking, $\tau \in \mathbb{R}$ records Dehn twists as well. Changing the chosen separating curve can give different parameter descriptions of the same unmarked surface.

For a fixed positive integer $q$, $\cosh (1/q)={\sum }_{k\ge 0}{q}^{-2k}/(2k)!$ is irrational. If it were rational with denominator $b$, choose $N$ large enough that $b\mid (2N)!$. After multiplication by $q^{2N}(2N)!$, the finite partial sum is an integer and the remaining positive tail is bounded by a geometric series whose ratio is at most $1/[q^2(2N+1)(2N+2)]$. For sufficiently large $N$, this tail is strictly between zero and one, contradicting its integer value. Only a finite integer partial sum is used; convergence of an infinite series of rationals does not itself prove irrationality.

A catenary is the static shape of an ideal flexible, inextensible chain hanging under its own gravitational weight in a uniform gravitational field, with uniform linear mass density along its arc length. With the lowest point above $x=0$, one convenient vertical origin givesHere $a$ has units of length. The supports and the chain's arc length determine the scale and position; horizontal and vertical shifts give the general catenary. ^[ref]

For $a=1$, the three points at $x=-1,0,1$ have heights $\cosh 1\approx 1.543081$, $1$, and $\cosh 1\approx 1.543081$. If heights should instead be measured from the lowest point, subtract $a$:The same three heights are then approximately $0.543081$, $0$, and $0.543081$. The shift changes the vertical origin and preserves the catenary shape.

The scale also measures tension: if $H>0$ is its constant horizontal component, $\mu$ is the linear mass density, and $g$ is the magnitude of gravitational acceleration, then $a=H/(\mu g)$. A short force balance explains this. For the portion from the lowest point to $x\ge 0$, with arc length $s(x)$, vertical tension balances its gravitational weight and horizontal tension givesApplying differentiation yields $a{y}^{\prime \prime }=\sqrt{1+y^{\prime 2}}$. The proposed hyperbolic cosine solves this differential equation, because $y^{\prime }=\sinh (x/a)$ and $a{y}^{\prime \prime }=\cosh (x/a)=\sqrt{1+{\sinh }^2(x/a)}$. This is the mechanism behind the shape, under the stated ideal chain assumptions.

The same identity simplifies the arc length between symmetric supports at $x=\pm b$:For $a=b=1$, $L\approx 2.350402$, exceeding the horizontal span of two. A larger $a$ at fixed $b$ gives a shallower sag: the Taylor series gives $a(\cosh (x/a)-1)=x^2/(2a)+x^4/(24a^3)+\cdots$. Thus a parabola is a local approximation, while the exact ideal chain shape is a catenary. The useful connection is that the same hyperbolic Pythagorean identity controls its coordinates, force balance, and arc length.

For $g^{\prime }(0)\ne 0$, apply Jensen's formula with the smooth curvature term to the hyperbolic derivative density. Its distributional Laplacian is $H_g^2+2\pi \sum m_a{\delta }_a$, giving the displayed identity with ramification points defined by $g^{\prime }(a)=0$. If $g^{\prime }(z)=c{z}^m+\cdots$ at zero, use $m\log R$ as the origin counting term and replace the base constant by $\log {\lim }_{z\to 0}{H}_g(z)/|z{|}^m$. Without this regularization the basepoint-critical formula is undefined.

In a grey atmosphere with $T^4=T_0^4(1+3\tau /2)$ and $\kappa ={\kappa }_0{P}^{\alpha -1}{T}^{4-4\beta }$, hydrostatic equilibrium in optical depth gives $dP/d\tau =g/\kappa$. With $\alpha ,\beta >0$ and $P(0)=0$, integration gives the displayed pressure-temperature relation. At the matching surface $T=T_e$, the Eddington surface boundary condition gives $T_e^4=2T_0^4$, hence $P\kappa /g=4\alpha (1-2^{-\beta })/(3\beta )$.

Take an ideal gas grey atmosphere with $T^4=T_e^4(3\tau +2)/4$, constant surface gravity of a star $g$, vanishing external pressure and $\kappa ={\kappa }_0{P}^{1/2}{T}^8$. Integrating hydrostatic equilibrium in optical depth gives $P^{3/2}=8g[1/2-(3\tau +2{)}^{-1}]/({\kappa }_0{T}_e^8)$. Its logarithmic temperature-pressure gradient is $9\tau /16$, so the Schwarzschild criterion for a monatomic gas is crossed at ${\tau }_c=32/45$. The radiative continuation beyond that depth is unstable and must be replaced by convective transport.

The ionizing photon escape fraction is the fraction of photons capable of photoionization that leave their source environment. Only these escaping photons contribute to ionizing distant intergalactic medium. Its value can depend on frequency, source geometry, and time.

A spherical blackbody star emits $Q(E_0)=8{\pi }^2{R}_s^2(k{T}_s{)}^{3}F(E_0/k{T}_s)/(h^3{c}^2)$ photons per time above threshold $E_0$. Integrating specific intensity over the outward hemisphere gives flux $\pi B_{\nu }$; dividing the resulting spectral luminosity by photon energy and substituting $x=h\nu /k{T}_s$ proves the formula. The convergent series $F(y)={\sum }_{m\ge 1}{e}^{-my}(y^2/m+2y/m^2+2/m^3)$ follows by integrating the geometric-series expansion termwise.

For two same-height throats carrying the same light-layer volume flux and reduced gravity, the narrower throat has the larger critical minimum energy. A narrower downstream throat can back up and submerge an upstream throat. If the first throat is narrower, it can control discharge and feed a supercritical current through the second; a hydraulic jump followed by a second control is also possible when the available energy drop and intermediate geometry permit it. Equal throat widths are a marginal lossless case. Relative width alone does not fix jump position or mixing in the intervening reservoir.

For independent real sequences, put the faster sequence at the slower speed by slower-speed degeneration of a large deviation principle, then use the product large-deviation principle and contraction principle for large deviations for addition. Unequal speeds give a translate of the slower sequence's rate. Equal speeds give the infimal convolution of the two rates. Its unique zero is the sum of their zero points, and a translated positive finite point has positive finite cost, since compactness attains each finite fiber infimum. Thus the defining nondegeneracy is preserved.

// Target: queueing-theory.bigb

If $X$ is simply connected and $h_{\ast }:{\pi }_{\ast }(X)\otimes \mathbb{Q}\to {\tilde{H}}_{\ast }(X;\mathbb{Q})$ is onto, choose a homogeneous homology basis and representing sphere maps into $X_{\mathbb{Q}}$. Their wedge sum gives a rational homology equivalence, hence a rational homotopy equivalence by the rational Whitehead theorem. All positive cup products therefore vanish. In a minimal model, the induced map from cohomology to the indecomposable quotient is injective; replacing a set of generators by cocycle representatives produces a quasi-isomorphism to the square-zero cohomology algebra, proving formality of a topological space.

Let $f_0(t)$ be the initial instantaneous forward rate curve and choose $r_0=f_0(0)$. In the Hull-White model, take $\beta (t)=f_0^{\prime }(t)+a{f}_0(t)+{\eta }^2(1-e^{-2at})/(2a)$. To verify the fit, write $r_t=x_t+\varphi (t)$ with $d{x}_t=-a{x}_{t}dt+\eta d{W}_t^Q$, $x_0=0$, and $\varphi (t)=f_0(t)+{\eta }^2(1-e^{-at}{)}^2/(2a^2)$. The Gaussian exponential formula gives $-{\partial }_T\log P(0,T)=\varphi (T)-{\eta }^2(1-e^{-aT}{)}^2/(2a^2)=f_0(T)$.

At equally spaced distinct times, a mean profile is affine exactly when its consecutive second differences vanish. These form $p-2$ independent contrasts. At unequal times, compare adjacent slopes instead: $({\mu }_{i+1}-{\mu }_i)/(t_{i+1}-t_i)-({\mu }_i-{\mu }_{i-1})/(t_i-t_{i-1})$. A Hotelling test of linear hypotheses tests these restrictions jointly with unknown covariance.

In a local trivialization, a vector-bundle connection with matrix $A$ defines the displayed complement of the vertical tangent space. The frame-change law makes these complements independent of the trivialization. Their dependence on the fiber coordinate is linear; an arbitrary horizontal complement need not satisfy this condition and need not define a connection on a vector bundle. Horizontal lifts satisfy the ordinary differential equation defining parallel transport.

Let $P\to B$ be a principal bundle with principal connection. Fix a point $b\in B$ and a point $p\in P_b$. The horizontal lift of a piecewise smooth closed curve $\gamma$ based at $b$ starts at $p$ and ends at a unique $p{g}_{\gamma }$. This group element $g_{\gamma }$ is its holonomy relative to $p$.

Changing the starting point to $ph$ changes the group element to $h^{-1}{g}_{\gamma }h$, by equivariance of parallel transport. The conjugacy class is therefore independent of the chosen fiber frame. Transport along a path to another basepoint similarly conjugates the resulting group of holonomy elements.

For a flat principal connection, contractible closed curves have trivial holonomy, but noncontractible closed curves can return a different fiber frame. For example, the U(1) connection $A=\alpha d\theta$ on a trivial circle bundle over a circle has zero curvature form and holonomy $e^{2\pi i\alpha }$ around one circuit. This separates a local curvature condition from a global transport obstruction.

For $M$ binary units and desired patterns ${\xi }^{\mu }$, use these symmetric weights and set $w_{ii}=0$. For one pattern, the field at that pattern is $(M-1){\xi }_i/M$, so it is stable. Several patterns add cross-talk terms; storage by this rule alone does not guarantee stability of every arbitrary collection of patterns.

Writing $m_{\mu }=M^{-1}{\sum }_j{\xi }_j^{\mu }{x}_j$ givesfor $P$ stored patterns. The subtraction removes the forbidden self-couplings. It makes explicit how the desired association competes with all other stored patterns. An unintended stable state can therefore arise even though every update obeys asynchronous Hopfield energy descent.

For symmetric zero-diagonal weights and zero thresholds, update one binary unit to the sign of its current field, retaining it at a zero field. The displayed energy change is negative on every actual flip. There are finitely many states, so a fair asynchronous update schedule eventually reaches a fixed point that is a minimum against single-unit flips. Synchronous updates can instead cycle, and local minima need not be global minima.

To obtain the displayed identity, subtract the two energies. Terms not involving $x_k$ cancel, and symmetry combines the two copies of each pair. With thresholds included the result is $\Delta E=-(x_k^{\prime }-x_k)h_k$. The zero-threshold expression above is its specialization. At $h_k=0$, keeping the old state rules out motion along equal-energy states. Finiteness then excludes infinitely many strict decreases, while fairness ensures that an unstable unit cannot be ignored forever. A local minimum here means no single flip lowers energy; equality directions can remain.

The asynchronous condition matters even for two units: with $w_{12}=w_{21}=1$ and zero thresholds, simultaneous updates alternate $(1,-1)$ and $(-1,1)$ forever. Either single-unit update instead reaches an aligned stable state.

The leg consists of the cells strictly below $(i,j)$ in its column of the Young diagram; its length is $l_{\lambda }(i,j)={\lambda }_j^{\prime }-i$, where ${\lambda }^{\prime }$ is the conjugate partition. It excludes the anchor cell. At $(1,2)$ in $(3,2)$ the leg contains only $(2,2)$, so its length is one. Transposing the Young diagram exchanges the leg and the arm of a Young-diagram cell.