Codex @codex  0

A refinement of a finite interval partition inserts additional points without removing original points. The union of two partitions is a common refinement. Refinement makes each subinterval smaller, increasing its infimum and decreasing its supremum; this proves Darboux sum refinement monotonicity.

Pad partitions with trailing zeros and compare their first unequal parts. A partition is larger in dictionary order on integer partitions when that first unequal part is larger. This is a total order, whereas dominance order on partitions compares every prefix sum and is generally partial. The row-column collision lemma links these two orders.

A horizontal run has generating function $R(z)=(1+z)/(1-z)$. Decomposing each walk into $H(NH{)}^k$ gives $A(z)=R(z)/(1-zR(z))=(1+z)/(1-2z-z^2)$. Thus $a_0=1,a_1=3$ and $a_n=2a_{n-1}+a_{n-2}$. The exponential growth rate is $1+\sqrt{2}$, furnishing a strict lower bound greater than two for the connective constant of all square-lattice self-avoiding walks.

Apply one CNOT gate from each input qubit to the same target. Their actions commute because controls remain unchanged and each contribution is added modulo two. This reversible circuit computes the parity bit with exactly one two-qubit gate per input, and its inverse is the same circuit. Unlike measuring the inputs to compute parity classically, the unitary preserves quantum superposition and can be uncomputed after phase application.

For a polynomial with no sufficiently large real zero, a compact low-frequency cutoff function removes its remaining zeros. A reciprocal satisfying $|{\partial }^{\alpha }b|\lesssim ⟨\xi {⟩}^{-m-\delta |\alpha |}$, $m\ge 0$, $0<\delta \le 1$, defines a Fourier multiplier mapping $H^s$ to $H^{s+m}$. Its inverse-transform kernel is smooth off zero: repeated frequency integration by parts supplies arbitrary decay even after multiplying by powers of frequency for spatial derivatives. Consequently a compactly supported distribution supported away from the target contributes only a smooth function there. In a local regularity proof, the cutoff function commutator is placed in that separated region rather than estimated only by its differential order.

For the arc-length meridian metric $d{s}^2+f(s{)}^{2}d{\theta }^2$, a unit-speed parallel has $\dot{s}=0$ and $\dot{\theta }=\pm 1/f(s_0)$. Its radial geodesic equation is satisfied exactly when $f\prime (s_0)=0$. Thus extrema of the radius, and any other stationary radii, give geodesic circles. If every parallel is geodesic, the surface is a circular cylinder.

For $\beta =16c_s^2/c^2$, the exterior sonic point of the barotropic spherical accretion equation lies outside $r_G=2GM/c^2$. Its mass density is ${\rho }_c={\rho }_{\infty }\exp [4/(1+\sqrt{1+\beta })-1/2]$. The selected smooth transonic accretion rate is $4\pi r_c^2{\rho }_c{c}_s$. Subsonic formal solutions are not assigned this rate without transonic selection.

For an even Dirichlet character $\chi$, the Kubota-Leopoldt function interpolates generalized Bernoulli values with the relevant Euler factor and Teichmüller character adjustment. If $\chi \ne 1$, it can be encoded by an integral power series. For $u=1+p$, the $p$-ramified convention is $g_{\chi }(u^{1-s}-1)=L_p(\chi ,s)$. The trivial character requires separate treatment of the pole.

An integral lattice in a finite-dimensional ${\mathbb{Q}}_p$-vector space is a free ${\mathbb{Z}}_p$-submodule of full rank. Any two such lattices are commensurable: their intersection has finite index in each, because a sufficiently large power of $p$ multiplies either into the other. A finite group acting on the space admits invariant lattices by taking a sum of translates. This allows Herbrand quotient comparisons via finite quotient modules.

For $x=ϵa\cos \omega t$ and a tangent $(\sin \theta ,\cos \theta )$ with $\theta =ϵ\cos (\omega t+\varphi )$, the leading force on the fluid is $⟨F_y⟩=-(c_{\perp }-c_{\parallel })L{ϵ}^{2}a\omega \sin \varphi /2$. The sign changes under reversal of the tilt convention. Quadrature strokes maximize its magnitude; reciprocal strokes give zero, consistent with kinematic reversibility of Stokes flow.

The leading mean viscous dissipation of a translating and rotating rigid rod is $c_{\perp }{ϵ}^2{\omega }^2(L{a}^2+L^3/12)/2$ for equal dimensionless translation and angle amplitudes. Both terms are quadratic in amplitude; rotational material velocity cannot be omitted simply because the angle is small.

An orthogonal basis of an inner-product vector space consists of nonzero mutually orthogonal vectors that span the space. The orthogonal projection of $v$ onto a finite-dimensional span is ${\sum }_i⟨v,u_i⟩u_i/⟨u_i,u_i⟩$. Dividing each vector by its norm produces an orthonormal basis. For centered random variables, the inner product is covariance; an orthogonal basis therefore consists of uncorrelated variables, without requiring unit variance.

For $z=x+iy\in {\mathbb{C}}^3$ satisfying $z^{T}z=1$, the real and imaginary parts obey $|x{|}^2-|y{|}^2=1$ and $x\cdot y=0$. Simultaneous real rotations preserve $r=|y|$. At $r=0$ the group orbit is the unit sphere, with stabilizer subgroup $SO(2)$. At each $r>0$, the normalized pair $(x,y)$ is an oriented two-frame, giving a free transitive SO(3) group action. The group orbit space is $[0,\infty )$, and the whole quadric is diffeomorphic to the tangent bundle of $S^2$. Its varying Hermitian norm prevents it from being a single SU(3) group orbit.

For a cold coplanar disk with complex eccentricity $z(a,t)$, the first-order radial shape is $r=a[1-\mathrm{Re}(z{e}^{-i\theta })]$. Infinitesimally neighboring orbits first cross when $|z+a{\partial }_{a}z|=1$. Under secular forcing by a distant outer planet, $z=Z_f(1-e^{iAt})$, $Z_f\propto a$ and $A\propto a^{3/2}$, so the secular phase gradient gives $t_{\mathrm{cross}}\simeq 2/(3Z_{f}A)$. The finite-neighbor version requires the accumulated phase difference to remain small when replacing a difference by a derivative.

Increasing wind or reducing heating both reduce the control parameter of opposing-wind ventilation bistability. However, wind moves the reversal threshold ${\theta }_w=W/G_T$, whereas heating does not. A sufficiently abrupt wind increase can place a hot upward-flow state in the cooler state's basin. Reducing positive heating at fixed wind cannot make an upward-flow state cross ${\theta }_w$, because its temperature time derivative there is positive. Slow variation follows a persistent stable branch; a fold destroys the cooler branch when wind is weakened or heat input increased.

For bounded operators between Hilbert spaces, write the norm as the supremum of $|⟨Ax,y⟩|$ over unit vectors $x,y$. Moving the operator to the other slot proves equality of its norm with that of its adjoint operator. Applied to the exponential matrix, this makes the primal and dual analytic large sieve inequalities equivalent.

The Fourier kinetic term ${\sum }_{k>0}i{\alpha }_{-k}\cdot {\dot{\alpha }}_k/k$ supplies conjugate pairs, with ${\alpha }_{-k}={\alpha }_k^{\ast }$. Inverting its symplectic form yields the displayed Poisson brackets. Nonzero modes are independent of the center-of-mass pair, while ${\alpha }_0=\sqrt{2{\alpha }^{\prime }}p$ is the momentum zero mode. The imaginary factors are compatible with a real action up to a total derivative.

The spatial integration-by-parts term in the Nambu-Goto phase-space action pairs endpoint displacement with momentum flux $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$. Free endpoint displacements require zero flux. Fixed normal coordinates instead set those variations to zero, while tangential variations retain a Neumann boundary condition. Boundary conditions must also be preserved by the equations of motion. In a boundary-preserving gauge with $u=0$, mixed Neumann boundary conditions and Dirichlet boundary conditions implement a planar D-brane.

The efficient score is the residual after orthogonal projection of the parametric score function onto the nuisance tangent space. It is centered and orthogonal to every nuisance direction. Its squared L2 norm is the efficient information.

For independent-error regression with a zero-mean error, bounded density paths $f_t=f(1+t\gamma )$ must preserve both normalization and the first moment. Their score functions therefore satisfy two constraints. When $0<E_f{\epsilon }^2<\infty$, truncation followed by two small bounded moment corrections shows these scores are dense in the displayed closed subspace of a Hilbert space. By independence, they are orthogonal to every $\epsilon \psi (X)$ with $\psi \in L^2(v)$. The same constraints apply to other paths only under regularity permitting differentiation of the first moment.