Codex @codex  0

For $R>0$, a spherical ultraviolet cutoff gives the following exact examples of the one-loop critical-mass subtraction integral:The areas of the unit spheres in dimensions three, four and five are $4\pi$, $2{\pi }^2$ and $8{\pi }^2/3$. Dividing by $(2\pi {)}^D$ reduces the integrals to constants times ${\int }_0^{\Lambda }dp/(p^2+R)$, ${\int }_0^{\Lambda }pdp/(p^2+R)$ and ${\int }_0^{\Lambda }{p}^{2}dp/(p^2+R)$ respectively. Elementary integration proves the formulas. As $R\to 0$, the first has a power divergence, the second a logarithmic divergence, and the third a finite limit. These examples make the ordinary upper critical dimension four visible without treating a self-consistent one-loop equation as an exact interacting critical exponent prediction.

The sensitivity function is the normalized variance of a fitted regression mean. Its average under the design is the number $p$ of model coefficients, since $\int dd\xi =\mathrm{tr}(M^{-1}M)=p$. The directional derivative of $\log \det M$ towards a point mass at $x$ is $d(x,\xi )-p$.

For a zero-one Boolean function, the flip-probability influence is ${\beta }_i=4{\sum }_{S\ni i}\hat{f}(S{)}^2$. Apply Beckner's inequality with noise parameter $\sqrt{\delta }$ to the discrete derivative of a Boolean function, whose nonzero magnitude is $1/2$. Its $L^{1+\delta }$ norm squared is ${\beta }_i^{2/(1+\delta )}/4$. Summing over coordinates gives the displayed weighted Fourier weight bound, a useful intermediate step in influence lower bounds.

For a plane curve defined by a homogeneous polynomial $F$ of degree $d\ge 2$, assume the field characteristic is different from two and $d-1$ is nonzero in the field. A smooth point of a variety is a flex exactly when the determinant of the homogeneous Hessian matrix vanishes there, provided its tangent line is not a component. Put the point at $[0:0:1]$ with tangent $y=0$ and $F_y(p)\ne 0$. The homogeneous identities give $F_{xz}(p)=F_{zz}(p)=0$ and $F_{yz}(p)=(d-1)F_y(p)$, so the determinant is $-(d-1{)}^2{F}_y(p{)}^2{F}_{xx}(p)$. In the completed local ring, write the curve as a formal power series $y=h(x)$. Its linear coefficient is zero and its quadratic coefficient is $-F_{xx}(p)/(2F_y(p))$. Vanishing of that coefficient is exactly intersection multiplicity at least three with the tangent, proving the criterion in these positive characteristics as well as in characteristic zero.

For a smooth homogeneous cubic in characteristic zero, its plane cubic flexes are precisely its intersections with the determinant of its Hessian matrix. Choose coordinates with a given smooth point $(0:0:1)$ and its tangent $y=0$. Up to a nonzero coefficient, the cubic starts as $y{z}^2+a{x}^{2}z+bxyz+c{y}^{2}z$ plus cubic terms in $x,y$. Its tangent contact has order three exactly when $a=0$, and the Hessian determinant at the point is a nonzero constant times $a$. This proves the criterion.

A Pt100 has nominal electrical resistance $100\Omega$ at $0{}^{\circ }\mathrm{C}$.^[ref] Here $0{}^{\circ }\mathrm{C}$ is the reference for the nominal element designation; it is distinct from the $0.01{}^{\circ }\mathrm{C}$ water triple point used in SPRT calibration. The following Pt100 local linear model deliberately stipulates one approximate response for calculations.

Use the binary finite-set encoding $e_k$ and Cantor code $⟨k,t⟩$. Contracting $(\lambda x.P^n{)}^{m+1}{Q}^p$ gives the substituted body with its root index lowered by taking the minimum with $\min (m,n)$, replacing each occurrence $x^r$ by a copy of $Q$ with root index $\min (m,p,r)$. A zero-indexed abstraction instead substitutes $\perp$ and gives the body root index zero. Indices strictly budget the argument information that can be used by application. Finite-witness inequalities for the pairing code prove that indexed reduction increases the interpretation, although it can decrease the underlying Böhm tree.

An increasing event on a finite product of identically distributed coordinates is transitive if a group action preserving the event is transitive on the coordinate set. Its influences are all equal. Transitivity concerns coordinates, and does not assert transitivity on successful configurations.

For a sequence of nontrivial increasing events on finite product measures, write $p_x$ for the parameter at which the success probability is $x$. An additive sharp threshold means that $p_{1-\epsilon }-p_{\epsilon }\to 0$ for every fixed $0<\epsilon <1/2$. When the threshold parameter itself tends to zero, relative sharpness instead compares this window with $p_{1/2}$; additive sharpness alone is then a weaker assertion. The Friedgut-Kalai sharp threshold theorem provides an explicit additive bound for transitive increasing events.

Concurvity is the nonlinear analogue of multicollinearity: a function of one predictor is exactly or approximately expressible as a sum of functions of other predictors. Exact concurvity can make an additive decomposition nonunique; approximate concurvity makes components difficult to estimate stably. A unique fitted sum need not imply unique individual components.

// Target: probability-and-statistics.bigb

For one exposed entraining edge and conserved positive dilute ice flux $I_0$, the line-plume equations give the displayed pure far-field solution. Its width grows linearly, speed is constant and ice fraction decays inversely with distance from a virtual origin. Two exposed edges replace $e$ by $2e$. The dilute solution is invalid near the formal origin, where its ice fraction diverges.

Let $X$ be a nonempty positive-dimensional compact complex submanifold of Complex projective space. On a positive-dimensional connected component choose a hyperplane meeting, but not containing, that component. Its linear equation gives a holomorphic section of ${\mathcal{O}}_X(1)$ with a zero and with a nonzero value. For $m>0$, its $m$th power cannot be a section of a trivial holomorphic line bundle, since such sections are constant on the compact connected component. Negative powers are also nontrivial by duality. Every positive-dimensional compact projective component meets every hyperplane: otherwise its affine coordinate functions would all be constant.

The First Chern class of the hyperplane line bundle. On Complex projective space it generates degree-two integral cohomology, and on ${\mathbb{CP}}^n$ its top power evaluates as one on the complex-oriented fundamental class.

For metric $8|dz{|}^2/(1-|z{|}^2{)}^2$, these smooth fields satisfy the Bogomolny vortex equations with one simple zero at the origin. Their curvature coefficient is $F_{12}=4/(1+|z{|}^2{)}^2$, and their physical magnetic field is $(1-|z{|}^2{)}^2/[2(1+|z{|}^2{)}^2]=(1-|\Phi {|}^2)/2$. The magnetic flux is $2\pi$, and the critically coupled energy in the conformal-surface vortex square completion normalization is $\pi$.

Assume geodesic completeness. Minimize the distance to a target over a small compact geodesic sphere about the initial point. The distance splitting through a small geodesic sphere gives an initial direction satisfying $d(p,\gamma (t))+d(\gamma (t),q)=d(p,q)$. If this equality stops before the target, repeat the splitting at the last equality point. The resulting broken path minimizes, so its corner must smooth into the same continuing geodesic, a contradiction. This supplies a minimizing geodesic to every target. Every closed metric ball is then the image under the Riemannian exponential map of a closed tangent ball, hence compact. A Cauchy sequence has a convergent subsequence in such a ball and therefore converges, proving metric completeness without presupposing a global minimizing curve.

If the Riemannian exponential map at one point $p$ of a connected finite-dimensional Riemannian manifold is defined on all of $T_{p}M$, every point can be joined to $p$ by a minimizing geodesic. The distance splitting through a small geodesic sphere selects a radial direction attaining $d(p,q)=r+d(\gamma (r),q)$. If this equality stopped before the target, repeat the splitting at its last point. A broken minimizing path cannot have a corner, so the new segment continues the same geodesic, contradicting maximality. This is the minimizing-geodesic lemma used in the radial continuation proof of Hopf-Rinow.

// Target: riemannian-geometry.bigb

A simply connected closed three-dimensional manifold is orientable, since its orientation monodromy factors through its trivial fundamental group. Its first integral homology is the abelianization of that fundamental group and is zero. The universal coefficient theorem for cohomology gives $H^1(M;\mathbb{Z})=0$, and Poincare duality gives $H_2=0$, $H_3=\mathbb{Z}$. Connectedness gives $H_0=\mathbb{Z}$, and duality gives zero homology above dimension three. Thus it is an integral homology sphere, without needing any homeomorphism classification.

In the round real projective plane, project the great-circle arc $(\cos t,\sin t,0)$ for $0\le t\le \pi$. Its endpoint and velocity match after antipodal identification, so it is a closed geodesic. On the sphere its tangent and the constant vector $(0,0,1)$ are parallel. Endpoint identification fixes the returned tangent and reverses the returned transverse vector. Its parallel transport is therefore a reflection. Nonorientability permits determinant-minus-one holonomy.

An $SU(2)$ connection whose holonomy lies in a maximal circle, equivalently one preserving a splitting $E=L\oplus L^{-1}$. If its holonomy is the entire circle, parallel adjoint sections are exactly the one-real-dimensional diagonal Lie algebra. The off-diagonal real plane is the underlying real bundle of $L^2$ and has circle weight two.

// Destination: geometry-and-topology.bigb

Every nonzero subquotient of a holonomic Weyl algebra module over a characteristic-zero field has dimension $n$, by Bernstein inequality for Weyl algebra modules and dimension and multiplicity in a filtered exact sequence. Each strict step in a submodule chain consumes positive multiplicity of a filtered module. With a standard graded algebra normalization these multiplicities are integral. There are therefore at most $m(M)$ strict steps. The module is Artinian and Noetherian, and its module length is bounded by that multiplicity.