The Hall effect is a transverse electric potential difference produced when an electric current flows through a material in a transverse magnetic field. The contact orientation fixes its measured sign. The Drude conductivity and Hall experiment gives an explicit convention and the limitations of inferring a number density from it.

For bad events $A_i$, a correlation graph is sufficient for the Lopsided Lovász local lemma when, for every set $S$ of nonneighbors and every positive-probability conditioning event, $\mathbb{P}(A_i\mid {\bigcap }_{j\in S}{A}_j^c)\le \mathbb{P}(A_i)$. A dependency graph of events satisfies this with equality. The condition concerns simultaneous avoidance, not just signs of pairwise correlations.

A simple closed curve on the boundary of a three-manifold which is nullhomotopic in the manifold bounds a properly embedded disk. The boundary curve is retained; the Loop theorem can instead find a different simple essential curve in a nontrivial kernel.

Start with a singular disk whose boundary represents a prescribed nontrivial class in a boundary subsurface. Repeatedly lift it through double covering spaces of regular neighbourhoods of its image. The number of distinct image simplexes strictly increases and is bounded by the number of domain simplexes, so the tower ends. At the top, vanishing first mod-two homology forces every boundary component to be a sphere. An embedded disk can then be constructed and projected down the tower. Resolving double curves preserves a boundary class outside a chosen normal subgroup, giving an embedded compression disk at the bottom.

On an infinite regular triangular grid, height data constant along one edge direction remain constant along that direction under Loop subdivision. Across rows the induced rule is $g_{2j}^{\prime }=(g_{j-1}+6g_j+g_{j+1})/8$, $g_{2j+1}^{\prime }=(g_j+g_{j+1})/2$, the centered Cardinal cubic B-spline rule.

// Target: analysis.bigb

For the three unit-grid edge vectors $v_1,v_2,v_3$ and a multivariate polynomial $f$ of total degree at most three, sampled regular-grid Loop subdivision generates the graph of $f+\frac{1}{12}{\sum }_{r=1}^3(v_r\cdot \nabla {)}^{2}f$. Thus affine polynomials are reproduced exactly and all polynomials through degree three retain their degree. Fourth-degree data constant along one edge direction reduce to a Cardinal cubic B-spline, whose limit need not be a polynomial.

Such a continuous projective unitary representation admits strongly continuous rotations whose self-adjoint generator is bounded below and whose adjoint action rotates the loops. In the basic fermionic construction, integer one-particle modes are filled below zero. The rotation energy counts nonnegative particle energies and positive hole energies. It has finite-dimensional integer eigenspaces and implements circle rotations. Tensor products give higher nonnegative integer levels.

Assume a nonzero invariant Loitsyansky integral, self-preserving large-scale correlations, and high-Reynolds number freely decaying isotropic turbulence. Then $I\propto u^2{\ell }^5$ and $-d(u^2)/dt\propto u^3/\ell$ imply the displayed decay and growth laws. The assumptions on correlation shape and the dissipation coefficient are needed in addition to conservation of the integral.

For a confined incompressible flow with no velocity flux across the boundary, an integrated divergence identity gives $⟨|\int \mathbf{x}\times \mathbf{u}dV{|}^2⟩=-\iint r^2⟨\mathbf{u}\cdot {\mathbf{u}}^{\prime }⟩dVd{V}^{\prime }$. In a large approximately homogeneous region with short correlations, the latter is approximately $VI$. Conservation of angular momentum in an isolated torque-free cloud then motivates conservation of the Loitsyansky integral. A fixed subvolume of infinite turbulence is not an isolated cloud: advective transport, pressure torques, wall stresses and long-range correlations can invalidate the argument.

For nonzero design entries the Bernoulli logistic-regression model has strictly concave log likelihood, since $-{\ell }^{\prime \prime }={\sum }_i{x}_i^2{\mu }_i(1-{\mu }_i)>0$. A unique finite estimate exists exactly when its score is positive at minus infinity and negative at plus infinity. Perfect agreement of binary outcomes with the signs of the design can instead give a supremum approached at an infinite coefficient. If every design entry is zero, the coefficient is unidentifiable. Newton method uses the score divided by this observed information, with damping when needed.

Complete separation means that a predictor direction $v$ strictly separates the binary outcomes as displayed. In logistic regression, replacing $\beta$ by $\beta +tv$ increases each observed-outcome probability as $t$ increases, so the log-likelihood has no finite maximizer. Weak inequalities with some equalities can similarly produce quasi-complete separation. Finite coefficient estimates and ordinary information-matrix inference therefore require an existence check rather than relying only on the formal score equations.

For ordered response categories, ordered thresholds ${\theta }_l$ and a common slope vector define this cumulative logistic regression. Increasing $x^T\beta$ shifts probability toward higher categories. A covariate contrast multiplies all corresponding cumulative odds by the same factor. This model is not a canonical baseline-category multinomial logistic regression, so the latter's design-weighted count margins are not generally sufficient for it.

The forward Euler method applied to the logistic differential equation $y^{\prime }=ry(1-ay)$ becomes the logistic map after $u=a(\lambda -1)y/\lambda$, $\lambda =1+r\Delta t$. The positive equilibrium point of the differential equation has perturbations decaying as $e^{-rt}$. Its discrete fixed-point multiplier is $1-r\Delta t=2-\lambda$: for $r\Delta t>2$, the discretization has an artificial oscillatory instability, leading initially to alternation on successive steps. This contrasts with the positive perturbation multiplier $e^{-r\Delta t}$ of the exact time-step map.

Suppose $|a_n|\le (\log (2n){)}^k$ and $A(x)={\sum }_{n\le x}{a}_n\log (x/n)=O(x(\log x{)}^{\beta })$, with $0\le \beta \le k$. For $0<h\le 1$, direct subtraction givesThe last term has absolute value $O((x{h}^2+h)(\log x{)}^k)$. Choose $h=(\log x{)}^{(\beta -k)/2}$ to obtain the displayed bound, with an additional $O((\log x{)}^k)$ term harmless as $x\to \infty$ for fixed $k$. This elementary finite-difference argument converts a logarithmically smoothed Perron formula bound to cancellation in its original partial sum.

For a stock with observed dynamics $d{S}_t/S_t=\sigma d{\hat{W}}_t+\sigma m_{t}dt$ and constant riskless interest rate $r$, the finite-horizon pricing density for Gaussian drift learning with $k=r/\sigma$ gives ${\zeta }_t=e^{-rt}{D}_t$. The terminal logarithmic utility condition $1/w_T^{\ast }=\lambda {\zeta }_T$ and the state-price budget constraint give $\lambda =1/w_0$. Replication then gives $w_t^{\ast }=w_0/{\zeta }_t$ at every time. The Itô formula for $1/\zeta$ makes the optimal dollar stock fraction $(m_t-r/\sigma )/\sigma$. Its finite expected log return follows from finite posterior second moments on finite horizons.

Let a portfolio hold $x\in [0,T]$ in a risk-free asset with net return $r$ and $T-x$ in an asset returning $g$ or $b$, with probabilities $p$ and $1-p$. Assume $T>0$, $-1<b<r<g$, $0<p<1$, and $pg+(1-p)b>r$. Its state wealths are $w_g=T(1+g)-(g-r)x$ and $w_b=T(1+b)+(r-b)x$. The expected utility $p\log w_g+(1-p)\log w_b$ has strictly negative second derivative. Setting its first derivative to zero gives the displayed unconstrained root; the derivative at $x=T$ is negative, so only the lower constraint can bind. Thus the displayed formula is the unique optimum. All equity is optimal exactly when $p\ge (r-b)(1+g)/[(g-b)(1+r)]$. The positivity of both state wealths is part of the logarithmic utility domain.

A fixed positive excess over the Turan theorem threshold forces logarithmic balanced complete multipartite graphs. There is an elementary induction using common neighbourhood from bipartite density: first obtain a logarithmic $r$-partite subgraph from the induction hypothesis, and select disjoint transversal $r$-cliques in it. Each such clique has at least $r\epsilon n$ common neighbours, by the union bound on non-neighbour sets. Apply the bipartite common-neighbour count to these cliques as left-side objects. Their selected union and a large common neighbourhood form the extra part.

In an instantaneous-transition approximation, match the superhorizon adiabatic CDM mode in radiation domination and its first derivative to the radiation-era logarithmic solution at ${\tau }_h=1/k$. Its value is $A_R/k^2$ and its derivative is $2A_R/k$. The two matching equations give the displayed expression. Its coefficient two belongs to this approximation; a smooth radiation forcing through entry changes the constants while preserving the logarithmic CDM growth after radiation-era horizon entry scaling.

A logarithmic scale places a positive quantity $x$ according to a logarithm, so equal ratios occupy equal distances. For example, $1,2,4,8$ are equally spaced under the natural logarithm because every successive difference is $\ln 2$. An ordinary linear scale instead gives equal distances to equal differences: $1,2,3,4$.

For a base $b>1$ and a positive reference $x_0$, define the axis positionThe reference has the same units as $x$, making $x/x_0$ a dimensionless quantity. The distance between two positions isMultiplying both values by the same positive factor preserves that distance. Replacing $x_0$ by $c{x}_0$ translates every position by $-{\log }_{b}c$; changing the base rescales positions through ${\log }_{b}z=\ln z/\ln b$. Neither change alters which ratios receive equal gaps.

In base ten, the values $1,10,100,1000$ occupy positions $0,1,2,3$ when $x_0=1$. The differences between neighboring values grow from nine to ninety to nine hundred, but every step spans one order of magnitude. The values $2,20,200$ occupy positions $0.30103,1.30103,2.30103$: the same spacing shifted by ${\log }_{10}2$. This lets one plot small and large positive quantities together while keeping equal proportional changes equally visible.

A logarithmic scale is a rule for position, not an instruction to relabel the tick marks with their logarithms. A mark labelled $100$ still means the original value one hundred. Ordinary real logarithms exclude zero and negative inputs. Omitting such values or clipping them to a plotting boundary does not give them genuine logarithmic positions; the plotting choice needs to be clear.^[ref]

For $0<u<1$, the power series for the logarithm gives $-\log (1-u)-u={\sum }_{n=2}^{\infty }{u}^n/n<\frac{1}{2}{\sum }_{n=2}^{\infty }{u}^n=u^2/[2(1-u)]$. The inequality is strict because $1/n<1/2$ for every $n\ge 3$.