For $p>n$, the Hölder continuous function representative $u^{\ast }$ of $u\in W^{1,p}({\mathbb{R}}^n)$ has a classical Fréchet derivative at almost every point, equal to its weak derivative. At a point $x$ where the $p$-mean oscillation of $Du$ tends to zero, apply the Morrey inequality on a cube to $u^{\ast }(y)-u^{\ast }(x)-Du(x)\cdot (y-x)$. On a cube of side comparable to $|y-x|$, its oscillation is at mostThe required points have full measure by the Lebesgue differentiation theorem.

For a smooth map between manifolds $f:M\to N$, the differential $d{f}_x:T_{x}M\to T_{f(x)}N$ is the linear map obtained by differentiating in charts. Intrinsically it sends the tangent vector of a curve $\gamma$ through $x$ to the tangent vector of $f\circ \gamma$. The chain rule makes it independent of charts and gives $d(g\circ f{)}_x=d{g}_{f(x)}\circ d{f}_x$.

A tubular neighborhood of a smooth embedded submanifold is a neighborhood identified with a neighborhood of the zero section in its normal bundle. Radial deformation of punctured fibers gives the sphere bundle, and excision identifies the associated pair with a normal disk-bundle pair.

On overlapping manifold charts, a transition map changes one coordinate representation to the other. In a smooth atlas it and its inverse are smooth on the corresponding open coordinate domains.

A smooth projective surface is a two-dimensional smooth variety admitting a closed embedding in projective space. Its integral curves are Cartier divisors, and their intersections, arithmetic adjunction and point blowups control its birational geometry.

For an integral curve $C$ on a smooth projective surface, $2p_a(C)-2=C\cdot (C+K_X)$. Apply Riemann–Roch theorem for algebraic surfaces to the divisor restriction exact sequence $0\to {\mathcal{O}}_X(-C)\to {\mathcal{O}}_X\to {\mathcal{O}}_C\to 0$ to derive it. This arithmetic version applies also to singular curves and in arbitrary characteristic.

In units of elastohydrodynamic penetration length and inverse forcing frequency, $y_t=-y_{xxxx}$. Prescribing $y(0,t)=y_0\cos t$, zero bending moment $y_{xx}(0,t)=0$, and decay at infinity givesThe two components have phase velocities $-1/S$ and $1/C$. Their unequal spatial attenuation produces a nonreciprocal shape cycle even though the endpoint actuation is reciprocal.

The anisotropic-drag contribution to axial force from a small-slope transverse motion is $F=(c_{\perp }-c_{\parallel })\int y_x{y}_{t}dx$. Using the small-slope elastohydrodynamic filament equation and integration by parts givesThus this propulsive contribution is determined by endpoint slope, bending moment, and shear. A separately imposed axial translation adds ordinary longitudinal drag. Exact second-order longitudinal motion required by inextensibility also contributes to instantaneous total drag, but its time-derivative contribution averages to zero over a deformation cycle.

For a rational matrix $\gamma$ of positive determinant, use the positive real power of $\det \gamma$ in the displayed slash operator for modular forms. It satisfies $(f{|}_k\gamma ){|}_k\eta =f{|}_k(\gamma \eta )$ by the automorphy factor identity and multiplicativity of the determinant. It agrees with the usual operator on the modular group. The determinant factor is essential for the stated normalization of the Hecke operators.

The unit Skyrmion has a spherically symmetric ansatz with Pauli matrices $\mathbf{\sigma }$, $f(0)=\pi$ and $f(\infty )=0$. The baryon number isThe radial profile is obtained from the Skyrme model variational equation. Spatial rotations and isospin rotations act on the same hedgehog orientation, leaving only three independent rotational collective coordinates.

A complex matrix $Z=-Z^T$ can be put by unitary congruence into two-by-two skew blocks and zero blocks; the magnitudes $|z_r|$ are its singular values, each repeated twice. One proof uses the antilinear map $A(v)=Z\bar{v}$. Because $Z^{†}=-\bar{Z}$, $A^2=-Z{Z}^{†}$. On a nonzero eigenspace of $Z{Z}^{†}$ with eigenvalue $s^2$, any unit vector $v$ has orthogonal partner $A(v)/s$, and $A(A(v)/s)=-sv$. These orthonormal pairs give the blocks; the orthogonal complement is invariant, so the procedure repeats. Phases of the paired vectors set the block phases.

A skeleton of $\mathcal{C}$ is a full subcategory with exactly one object from each isomorphism class. The axiom of choice supplies skeletons of small categories. Conversely, if every small category has a skeletal equivalent, apply this to the groupoid on ${\coprod }_i{A}_i$ with exactly one morphism between objects in the same nonempty fibre. A quasi-inverse from the skeletal category selects one object per fibre, giving a choice function. This converse uses an equivalence of categories with specified quasi-inverse functors, not just the existence of a full essentially surjective functor in one direction.

A left singular vector $u_j$ is a unit vector in the output space satisfying $A{A}^{\ast }{u}_j={\sigma }_j^2{u}_j$. For positive singular values, the pairing also satisfies $A^{\ast }{u}_j={\sigma }_j{v}_j$. Thus $u_1=(1,0{)}^T$ and $u_2=(0,1{)}^T$ describe the output axes of the preceding ellipse, while the right singular vectors describe its input directions. At zero singular values, division by ${\sigma }_j$ is impossible: complete the orthonormal bases independently in the null spaces of $A$ and $A^{\ast }$. Repeated singular values allow different matched orthonormal bases within the corresponding eigenspaces. The same positive pairing extends to a compact operator through its singular value system.

Even distinct singular values do not fix arrow signs: replacing both $u_j$ and $v_j$ by their negatives preserves $A{v}_j={\sigma }_j{u}_j$ and the outer product $u_j{v}_j^{\ast }$. Reversing only one arrow would change that term's sign. The paired directions, rather than independently chosen arrow orientations, encode the matrix.

A right singular vector $v_j$ is a unit vector in the input space satisfying $A^{\ast }A{v}_j={\sigma }_j^2{v}_j$. For positive singular values, $A{v}_j={\sigma }_j{u}_j$ determines the matching left singular vector. In the ellipse example, $A{v}_1=(5,0{)}^T$ is not parallel to $v_1$: a right singular vector need not be an eigenvector of $A$. Indeed, the characteristic polynomial of $A$ is ${\lambda }^2-(18/5)\lambda +5$, with eigenvalues $(9\pm i\sqrt{44})/5$ and no real eigenvectors. Nevertheless, its real right singular vectors give a complete orthonormal basis. Eigenvectors describe preserved directions; right singular vectors describe orthogonal input channels of stretching.^[ref]

An arbitrary input mixes these channels. For $x=c_1{v}_1+c_2{v}_2$, orthogonality gives $\Vert Ax{\Vert }_2^2=25|c_1{|}^2+|c_2{|}^2$, while $\Vert x{\Vert }_2^2=|c_1{|}^2+|c_2{|}^2$. Equal contributions, $c_1=c_2=1/\sqrt{2}$, produce output Euclidean norm $\sqrt{13}$ from a unit vector. Concentrating the entire input along $v_1$ produces the maximum $5$. This is why the right singular vectors give a direct reading of how any input will behave. For a compact operator, the same definition describes the positive terms of its singular value system.

A Betti number is the dimension of a real singular homology group; on a smooth manifold it is also the dimension of the corresponding de Rham cohomology group. For a finite-dimensional cohomology group on a closed Riemannian manifold, it equals the dimension of its space of harmonic representatives. Betti numbers can increase under a finite cover, so equality is a separate hypothesis, not a general covering-space theorem.

For $T_A=T_m+\Delta T\cos (t/\tau )$ and $F_O={\lambda }_O(T_O-T_m)$, put $r=F_O/({\lambda }_A\Delta T)$. Open-water freezing requires $r<1$ and starts at $t_1=\tau \arccos (-r)$. Neglecting ocean heat during growth is a bulk approximation when $r(1+{\lambda }_A{h}_m/k)\ll 1$. It cannot hold pointwise at the onset or maximum, where the atmospheric removal and ocean input are comparable. The actual maximum satisfies $-\cos (t_2/\tau )=r[1+{\lambda }_{A}h(t_2)/k]$; neglecting $F_O$ gives $t_2=3\pi \tau /2$.

In a cycle starting at the atmospheric temperature maximum, positive thickness at the next maximum is a peak-temperature survival test. It is not sufficient for perennial ice: with sinusoidal forcing, surface melting continues for another quarter-cycle. If ocean heat was neglected during winter growth, the leading thickness at the next temperature maximum isLiteral year-round survival must be tested through the end of the subsequent melt season, rather than at the temperature maximum alone. For the specified forcing with annual mean atmospheric temperature $T_m$, the zero-ocean-heat upper bound on winter growth obeys $h_m^2/2+h_m/a=2H^2$, where $a={\lambda }_A/k$ and $H^2=\kappa \tau /\mathcal{S}$, with $\mathcal{S}$ the inverse Stefan number. Thus $h_m<2a{H}^2$, already less than the atmospheric melt over the full subsequent warm half-cycle. Positive ocean heat only strengthens the failure to produce literal perennial ice from an ice-free start.

The probability simplex consists of vectors of nonnegative weights summing to one. It is the convex hull of the coordinate vectors. A linear function $r^{T}x$ has maximum ${\max }_i{r}_i$ on this simplex; equality holds exactly when the weights are supported on maximizing indices.

For $u_t+(c_0+bu)u_x=-\alpha u$, $b,\alpha >0$, the characteristic curves are $u=e^{-\alpha t}{u}_0(\xi )$ and $x=c_{0}t+\xi +(b/\alpha )(1-e^{-\alpha t})u_0(\xi )$. If $m=\max [-u_0^{\prime }]>0$, finite-time gradient blow-up occurs precisely for $\alpha <bm$, at $t_s=-{\alpha }^{-1}\log (1-\alpha /(bm))$. Equality $\alpha =bm$ still gives a positive Jacobian at each finite time; only the limiting map degenerates.

If a real matrix $B$ has a positive diagonal matrix $W$ such that $WB$ is a symmetric matrix, then $H=W^{1/2}B{W}^{-1/2}$ is a symmetric matrix related to $B$ by a similarity transformation: $B^{T}W=WB$ gives $H^T=H$. Thus $B$ is a diagonalizable matrix with real eigenvalues. If ${\mathbf{z}}^{T}WB\mathbf{z}\le 0$ for all real vectors $\mathbf{z}$, its eigenvalues are nonpositive. This converts symmetry in a weighted inner product into Euclidean symmetry.