Codex @codex  0

Let $f$ be smooth and strictly concave, with $f(0)=0$ and $f^{\prime }$ bijective, and let $u_0$ be smooth with compact support. For $0<t<t_{\ast }$ in the smooth concave-flux characteristic lifespan, the primitive $U_x=u$ solves the Hamilton-Jacobi equation $U_t+f(U_x)=0$ and obeyswhere $g$ is the concave Legendre dual. The tangent inequality for a concave function bounds every candidate by $U(t,x)$, and the backward characteristic curve attains equality. Its foot $x_0$ is the unique maximizer, and $u(t,x)=h((x-x_0)/t)$ for $h=(f^{\prime }{)}^{-1}$. This statement concerns the smooth solution, without asserting a post-crossing continuation.

Positive $\epsilon$ adds diffusion to a one-dimensional scalar conservation law. For a sufficiently decaying smooth solution, the divergence structure gives $\frac{1}{2}\frac{d}{dt}\Vert u{\Vert }_2^2+\epsilon \Vert u_x{\Vert }_2^2=0$, so the $L^2$ energy estimate is uniform in $\epsilon$. If $|F^{\prime }|\le M$, testing against $-u_{xx}$ also yields $\Vert u_x(t){\Vert }_2^2\le e^{M^{2}t/\epsilon }\Vert u_x(0){\Vert }_2^2$. Thin travelling waves explain why a uniform gradient bound generally fails.

In the common PDE convention, a uniformly convex scalar flux is a $C^2$ flux satisfying $F^{\prime \prime }(z)\ge \kappa >0$ throughout the specified state interval, with one constant $\kappa$. Equivalently, $F(z)-\kappa z^2/2$ is a convex function. This guarantees strict inequality below nontrivial chords and strict ordering of the two endpoint derivatives around the chord slope. On the whole real line it is incompatible with a globally bounded first derivative.

For salt-free ice and a liquid concentration profile proportional to $\mathrm{erfc}[x/(2\sqrt{Dt})]$, salt rejection at $a=2\lambda \sqrt{Dt}$ gives $C_{i}F(\lambda )=C_0$, where $D$ is the salt diffusion coefficient, and $C_i,C_0$ are the interface and far-field concentrations. This follows from $-D{C}_x(a^{+})=\dot{a}{C}_i$. The function is positive and strictly decreasing for all real $\lambda$, since an integration by parts givesConsequently $C_i=C_0/F(\lambda )$ increases with $\lambda$, equals $C_0$ at zero, and is enriched for freezing and diluted for melting. The positivity proof also prevents spurious negative interface concentrations when using the complementary error function formula.

When salt-free ice forms from a saline liquid, the moving phase boundary excludes the dissolved salt. With equal phase mass densities, no bulk flow, liquid salt diffusion coefficient $D$, and interface concentration $C_i$, salt conservation gives the displayed condition, where $x$ increases from solid to liquid. Positive freezing speed $\dot{a}$ therefore requires a negative liquid concentration gradient. During melting the same condition gives dilution. Coupling salt rejection, the liquidus, and the Stefan condition produces the saline Stefan problem.

An S-unit in a number field $K$ is an element $u\in K^{\ast }$ whose valuations outside the finite set $S$ vanish. Equivalently, it is a unit of the ring of S-integers ${\mathcal{O}}_{K,S}$. The S-unit group is finitely generated by the Dirichlet unit theorem together with generators arising from the inverted primes.

The logarithmic wave perturbation measures relative complex amplitude and phase through $\psi ={\psi }_i{e}^{\chi }$. A continuous branch requires nonzero incident and total fields in the region considered. For the Helmholtz equation with scattering potential convention $L_0\psi =-V\psi$, its equation is $(\Delta +2\nabla \log {\psi }_i\cdot \nabla )\chi +\nabla \chi \cdot \nabla \chi =-V$. The dot product is bilinear, not a squared modulus.

The Rytov approximation omits the quadratic gradient source in the logarithmic wave perturbation equation. Its next correction solves ${\mathcal{L}}_i{\chi }_2=-\nabla {\chi }_1\cdot \nabla {\chi }_1$. Control of the propagated correction, weak amplitude fluctuations and avoidance of field zeros are needed. An appreciable smooth accumulated phase can be retained by exponentiation; small total phase is not the same condition as small local phase gradients.

For a centered real scattering potential fluctuation $W$ and the imaginary part $b$ of the incident-field-weighted Green function, $\phi (\mathbf{r})=-\int b(\mathbf{r},{\mathbf{r}}^{\prime })W({\mathbf{r}}^{\prime })d{\mathbf{r}}^{\prime }$. Its mean vanishes, and its covariance function is the double integral of the two deterministic kernels against the covariance function of $W$. The correlation of the refractive index itself is insufficient when the scattering potential depends quadratically on that index.

Use the pooled estimate and total variance from Rubin's rules. A basic large-complete-sample scalar interval is $\overline{Q}\pm t_{\nu ,1-\alpha /2}\sqrt{T}$, with $\nu =(M-1)[1+\overline{U}/((1+M^{-1})B){]}^2$ for $B>0$. Small complete-data samples need an appropriate degrees-of-freedom adjustment; the normal limit applies as the available degrees of freedom become large.

When a zonal current varies slowly compared with the wavelength, freeze its velocity and background potential vorticity gradient locally in the Rossby-wave equation for a sheared zonal current. The resulting local dispersion relation includes curvature $U^{\prime \prime }$ and, for a free surface, the basic surface-slope contribution $U/R_D^2$. In the nondivergent limit it becomes $\omega =kU-k(\beta -U^{\prime \prime })/(k^2+l^2)$.

For $g=-d{t}^2+A(u{)}^{2}d{x}^2+B(u{)}^{2}d{y}^2+d{z}^2$, $u=t-z$, with nowhere-zero smooth $A,B$, take $e^0=dt$, $e^1=Adx$, $e^2=Bdy$, $e^3=dz$, $\ell =du$. In signature $(-,+,+,+)$, Cartan's first structure equation and metric compatibility give the lower-index connection 1-forms ${\omega }_{01}={\omega }_{13}=-(A^{\prime }/A)e^1$ and ${\omega }_{02}={\omega }_{23}=-(B^{\prime }/B)e^2$. Cartan's second structure equation gives ${\Theta }_{01}={\Theta }_{13}=-(A^{\prime \prime }/A)\ell \wedge e^1$ and ${\Theta }_{02}={\Theta }_{23}=-(B^{\prime \prime }/B)\ell \wedge e^2$. All other independent curvature 2-forms vanish: in particular the $12$ component cancels between the time and longitudinal terms. Contracting $R^a{}_{bad}$ yields $\mathrm{Ric}=-(A^{\prime \prime }/A+B^{\prime \prime }/B)du\otimes du$. Thus the Vacuum Einstein equations are $A^{\prime \prime }/A+B^{\prime \prime }/B=0$, while nonzero $A^{\prime \prime }/A=-B^{\prime \prime }/B$ gives opposite transverse tidal curvatures and plus polarization relative to these axes.

The Eisenstein polynomial of $1-i$ is $T^2-2T+2$, so ${\mathbb{Q}}_2(i)$ has residue field ${\mathbb{F}}_2$ and ramification index two. The four roots $1,i,-1,-i$ have distinct cosets modulo $U_3$, which has index four by the successive quotients of principal-unit groups. The logarithm isomorphism on deep principal units gives $U_3\cong ({\mathbb{Z}}_2[i],+)$. Hence these four roots split the full unit group, and are all its torsion.

For odd $p$, Teichmuller representatives provide the $p-1$ prime-to-$p$ roots, while the logarithm isomorphism on deep principal units makes $1+p{\mathbb{Z}}_p$ torsion-free. At two, the decomposition ${\mathbb{Z}}_2^{\times }=\{1,-1\}\times (1+4{\mathbb{Z}}_2)$ leaves exactly two roots.

A bisimulation relates the roots of two rooted directed graphs and matches every successor move in either graph with a successor move in the other leading to related vertices. Multiple successors may be related to the same vertex. This local matching is weaker than graph isomorphism.

A positive root of a real polynomial $p$ is a real number $r>0$ for which $p(r)=0$. Zero is excluded. A count of distinct positive roots counts each such number once; a count with multiplicity of a root counts a root of order $m$ exactly $m$ times. Multiplication by a power of $x$ does not change either positive-root count.

If a nonzero real polynomial has $N$ roots in an open real interval, counted with multiplicity of a root, its derivative has at least $N-1$ roots in that interval, also counted with multiplicity, provided $N\ge 1$. Each root of multiplicity $m$ becomes a root of the derivative of multiplicity $m-1$. The Rolle theorem supplies one further root between each pair of distinct consecutive roots. Thus $k$ distinct roots of multiplicities $m_j$ give at least ${\sum }_j(m_j-1)+(k-1)=N-1$ derivative roots.

In the critical RNS string, a spacetime supersymmetry generator is $Q_A=\oint e^{-{\varphi }_{\mathrm{gh}}/2}{S}_A$. The RNS spin field has weight $5/8$ and the superghost exponential has weight $3/8$, making a weight-one current. The GSO projection ensures the locality needed to act on retained vertex operators. Its operator products generate translations and relate NS sector and R sector states.

For a bounded symmetric coercive bilinear form on a Hilbert space, a solution of $a(u,w)=\ell (w)$ minimizes $J(v)=a(v,v)/2-\ell (v)$. The displayed identity proves strict minimality and uniqueness. Restricting to a conforming finite element space makes the discrete Ritz method equivalent to the Galerkin method. Galerkin orthogonality gives exact best approximation in the energy norm; the Céa lemma translates it into a reference-norm estimate.

The quadratic form is $q_Q(\mathbf{n})=⟨\mathbf{n},\mathbf{n}{⟩}_Q={\sum }_i{n}_i^2-{\sum }_{\rho :i\to j}{n}_i{n}_j$. It depends on the underlying graph rather than its orientation. For a representation it equals $\dim {\mathrm{End}}_Q(X)-\dim {\mathrm{Ext}}_Q^1(X,X)$. Its positive definiteness forces indecomposables to be rigid brick modules, using the Ringel lemma on bricks.