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For a smooth gauge field on ${\mathbb{R}}^3$ with finite positive magnetic energy, dilate its connection one-form by $A_i^{(s)}(x)=s{A}_i(sx)$. Its gauge curvature scales as $F_{ij}^{(s)}(x)=s^2{F}_{ij}(sx)$, so the energy scales as $s^4{s}^{-3}E=sE$. A static stationary configuration must have zero derivative at $s=1$, forcing $E=0$ and hence $F=0$. This Derrick scaling argument assumes the dilation preserves the boundary conditions and that boundary terms vanish. It does not exclude four-dimensional Yang-Mills instantons, whose Euclidean action is scale invariant.

For a smooth finite-energy static planar Skyrme model field, Derrick scaling gives zero quartic energy, hence $[R_1,R_2]=0$ pointwise. The quartic first variation then vanishes, leaving the sigma-model equation ${\partial }_i{R}_i=0$. The right-current Maurer-Cartan equation gives ${\partial }_1{R}_2-{\partial }_2{R}_1=[R_1,R_2]=0$. Each current component is therefore harmonic on the plane and square-integrable. The mean-value estimate on expanding disks makes each component zero, so $U$ is constant. This excludes smooth finite-energy planar static solutions of this potential-free SU(2) model, without misapplying the same conclusion to a pure curved-target sigma model.

Every derivative has the intermediate-value property, even when the derivative is discontinuous.

The right and left one-sided derivatives are the limits of the difference quotient restricted respectively to positive and negative increments:At an interior point, differentiability is equivalent to existence and equality of these two finite limits. For the absolute value function, $f_{+}^{\prime }(0)=1$ and $f_{-}^{\prime }(0)=-1$, so it is continuous at zero but not differentiable there.

If a continuous real function on an interval has the displayed limit zero at every interior point, it is affine. On any compact subinterval subtract its endpoint chord. If the difference were positive somewhere, adding $\epsilon (t-a)(t-b)$ for sufficiently small positive $\epsilon$ would retain a positive interior maximum. At a maximum every symmetric second difference is nonpositive, but its normalized limit would be $2\epsilon >0$. Thus the function lies below its chord; applying the same argument to its negative gives equality. Real and imaginary parts give the complex-valued version. Existence of ordinary second derivatives is unnecessary.

For a continuous function with affine formulas $At+B$ to the left of $c$ and $A^{\prime }t+B^{\prime }$ to its right, the quotient $[F(c+h)-2F(c)+F(c-h)]/h$ equals $A^{\prime }-A$ for positive $h$. If it tends to zero, the slopes agree, and continuity at $c$ makes the intercepts agree as well. This detects slope jumps that ordinary pointwise derivatives away from the corner cannot detect.

For a continuous function with affine formulas $At+B$ to the left of $c$ and $A^{\prime }t+B^{\prime }$ to its right, the quotient $[F(c+h)-2F(c)+F(c-h)]/h$ equals $A^{\prime }-A$ for positive $h$. If it tends to zero, the slopes agree, and continuity at $c$ makes the intercepts agree as well. This detects slope jumps that ordinary pointwise derivatives away from the corner cannot detect.

The symmetric second derivative is the displayed limit when it exists. For a twice continuously differentiable function it equals the ordinary second derivative by Taylor expansion; the definition itself does not require ordinary differentiability. A continuous function with vanishing symmetric second derivative throughout an interval must be affine, by vanishing symmetric second derivative forces an affine function. For a twice-integrated Fourier series, sinc-squared weights express this quotient, connecting the operation to Riemann summation of a convergent trigonometric series.

For a differentiable function, a local linear approximation replaces nearby values by its value plus its derivative times the input change. The error divided by that change tends to zero. It uses the first-order Taylor polynomial; the statement concerns a limit near the specified point, not a guaranteed approximation far from it.

A layer with a pronounced vertical change in salinity is a halocline. When relatively fresh surface water lies over saltier water, its contribution to density stratification inhibits vertical exchange. A cold Arctic halocline can separate sea ice from warmer Atlantic water; increased heat at depth does not imply an equally increased ocean heat flux at the ice unless the intervening exchange also changes.

The density gradient is the gradient of a spatial mass density field. With an upward vertical coordinate in the Boussinesq approximation, the squared buoyancy frequency is $N^2=-(g/{\rho }_{\ast }){\partial }_z{\rho }_0$. A negative vertical density gradient therefore gives stable density stratification.

A Haar density estimator estimates the Haar scaling function coefficients by sample averages. It is exactly the histogram with cells of width $2^{-j}$. Its expectation is the Haar projection of the density. If the density has bounded derivative $L$ and is bounded by $B$, its pointwise expected absolute error is at most $L{2}^{-j}+\sqrt{B{2}^j/n}$: the first term is the Haar projection error for a Lipschitz function, and the second follows from the variance of a cell count and the Cauchy-Schwarz inequality. Choosing $2^j\asymp n^{1/3}$ gives error $O(n^{-1/3})$ while preserving nonnegativity and integral one.

Add constants $c_0<c_1<\cdots$ to the theory of dense linear orders without endpoints. The three countable models have respectively a least upper bound for the named sequence, no upper bound, or upper bounds with no least one. The unbounded model is an atomic model; the bounded model with no least upper bound is an omega-saturated model. The remaining model omits the type $c_i<x<d$ over its least-upper-bound parameter $d$.

A sunflower is a delta-system: its pairwise intersections all equal one core, and its petals outside the core are pairwise disjoint. For an $s$-uniform set family, prove the displayed bound by induction on $s$. If a maximal disjoint subfamily has $r$ members, their core is empty. Otherwise its union has at most $s(r-1)$ points and meets every member. Some point belongs to more than $(s-1)!(r-1{)}^{s-1}$ members. Remove it and apply the induction hypothesis; putting it back enlarges the common core. The case $s=0$ is immediate.

If a response has mean $\mu$ and variance $V(\mu )$, the delta method gives $\mathrm{Var}(g(Y))\approx g^{\prime }(\mu {)}^{2}V(\mu )$. Choosing $g^{\prime }(\mu )$ proportional to $V(\mu {)}^{-1/2}$ makes this first-order variance independent of $\mu$. For $V(\mu )=k{\mu }^2$ on positive means, the resulting transformation is the natural logarithm. This is a local approximation and requires the transformation to be defined on the observations; it does not make arbitrary normal responses positive or preserve an exact normal distribution.

Every closed connected orientable three-manifold is obtained by integer Dehn surgery on a finite framed link in the three-sphere. Compare a Heegaard splitting with a standard splitting of the sphere, factor their gluing difference using the Lickorish-Dehn theorem, and realize each twist by unit surgery on a distinct level of a collar of the splitting surface.

Two projective plane curves of degrees $m$ and $n$ with no common irreducible component have total intersection multiplicity $mn$. In particular, two nonempty projective plane curves always intersect.

A smooth compact Riemann surface embedded in Complex projective space ${\mathbb{CP}}^2$ determines an integral homology class $d[{\mathbb{CP}}^1]$. Since a projective line has self-intersection one, $d$ is the intersection number with a transverse projective line. Every transverse intersection of complex curves has positive sign, because a complex-linear change of tangent basis has positive real determinant. For the Fermat curve $x^d+y^d=z^d$, the line $y=0$ meets it transversely in the $d$ points $[\zeta :0:1]$, ${\zeta }^d=1$, directly verifying its homological degree.

Suppose a map $S^m\to M$ of closed connected oriented manifolds has nonzero mapping degree. A product decomposition with $0<k<m$ would have oriented closed factors: an orientation-reversing loop in either factor, with the other coordinate fixed, would reverse the product orientation. Pull back their top orientation classes to obtain $\alpha \in H^k(M;\mathbb{Z})$ and $\beta \in H^{m-k}(M;\mathbb{Z})$ with $\alpha ⌣\beta$ a generator of $H^m(M;\mathbb{Z})$. Both pullbacks to the sphere vanish in their intermediate degrees, contradicting the nonzero-degree pullback of their product.

For a map $f:M\to N$ between closed connected $n$-manifolds, the mod-two degree is defined by $f_{\ast }[M{]}_2={\mathrm{deg}}_2(f)[N{]}_2$. Every manifold has a mod-two orientation class, so no orientability assumption is needed. It is invariant under homotopy. A nonsurjective map has mod-two degree zero: mapping the fundamental classes into the local relative homology at a missed target point makes the source contribution zero but leaves the target local generator nonzero.