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If $A(t)$ and $B(t)$ are perpendicular four-dimensional spatial vectors of equal fixed magnitude, every constant-time slice is a planar circle. Opposite rotations in two orthogonal coordinate planes can produce a Nambu–Goto action solution with constant radius and a rotating plane. The induced worldsheet metric can be constant even though the embedding moves. Rotating Cartesian axes clarify its shape without removing its kinetic target-space energy.

Choosing a positive system of a root system for a complex semisimple Lie algebra divides its root spaces into negative and positive ones. Their respective sums ${\mathfrak{n}}^{-}$ and ${\mathfrak{n}}^{+}$, together with the Cartan subalgebra $\mathfrak{h}$, give this direct sum of vector spaces. Both ${\mathfrak{n}}^{\pm }$ are nilpotent Lie algebras, and $\mathfrak{h}\oplus {\mathfrak{n}}^{+}$ is a Borel subalgebra. The Poincare-Birkhoff-Witt theorem turns this into an ordered vector-space decomposition of the universal enveloping algebra, useful for building Verma modules.

For bounded BV space data and $s<t$, nested ROF level-set formulation minimizing sets cannot share an interface of positive surface measure away from data jumps. On regular graph patches, upward and downward first variations use the BV traces on a hypersurface. At contacts outside $J_f$ the forcing BV trace on a hypersurface agrees, whereas the two levels differ by $t-s$. Graph regularity and contact differentiation give the contradiction. Bounded forcing is a hypothesis of this argument; arbitrary unbounded $L^2\cap BV$ data are not covered by this statement.

An Einstein manifold has Ricci curvature equal to a constant multiple of its metric. In the Riemannian setting, the round unit sphere has $\Lambda =n-1$. This condition is weaker than constant sectional curvature in higher dimensions. The definition can also be applied to pseudo-Riemannian metrics, but positivity is assumed in the completeness theorems discussed here.

Two representations of a functor have unique mutually inverse morphisms carrying their specified universal elements to one another. Composing those morphisms preserves each universal element, so injectivity of the representing bijection makes the composites identities. Arbitrary isomorphisms of the underlying objects need not preserve the specified elements.

The dual of $D_{\lambda }$ has dominant label $-w_0\lambda =(-{\lambda }_m,\dots ,-{\lambda }_1)$. Substitution $x_i\mapsto x_i^{-1}$ in the alternant character formula for the general linear group and reversal of determinant columns gives that character. The numerator and denominator reversal signs and common monomial factors cancel.

The finite-volume random-cluster weight monotonicity passes to consistently wired or free infinite-volume measures for finite increasing connection events. Decreasing these events to origin-to-infinity connectivity yields $\theta (p,q_1)\ge \theta (p,q_2)$ for $q_1\le q_2$. The corresponding supercritical parameter sets are nested, and hence $p_c(q_1)\le p_c(q_2)$.

Let $w\in BV\cap L^{\infty }$ and $g\in BV\cap L^2$ on a bounded domain, $F_g(v)=\frac{1}{2}\Vert v-g{\Vert }_2^2$, and $w_{\pm t}=(1-\theta )w+\theta w\circ {\Phi }_{\pm t}$ for a smooth compactly supported local flow. The displayed formula is exact to an $o(t)$ remainder, and $g$ need not be bounded. Writing $J_t=\det D{\Phi }_t$, its cross-term identity isThe remainder is $o(t)$ because $1-J_{-t}=O(t)$ and weighted $L^1$ convergence follows fromFirst let $t\to 0$, then $K\to \infty$. The mass change of $w^2$ under opposite flows is $O(t^2)\Vert w{\Vert }_2^2$. Combined with the BV jump-product limit with one bounded factor, this evaluates the jump contribution without a bounded fidelity derivative.

In the smooth zero-winding long-wave sector, integrate the density Gaussian integral by completing $g(\delta \rho +i{\partial }_{\tau }\varphi /g{)}^2/2$. Its field-independent normalization leaves the displayed action. The real-frequency pole is ${\omega }^2=g{\rho }_0{k}^2/m$, so the phonon speed is $\sqrt{g{\rho }_0/m}$. Keeping density gradients yields ${\omega }^2=g{\rho }_0{k}^2/m+k^4/(4m^2)$, matching the Bogoliubov spectrum.

If $Q\ll P$ and a continuous semimartingale is a semimartingale under both measures, its quadratic variations agree $Q$-indistinguishably. The same squared-increment sums converge uniformly on compacts in probability under both measures: absolute continuity transfers the original convergence, and uniqueness of the limit identifies the two continuous versions. Equivalence of measures is unnecessary.

Compare independent Poisson means with one shared mean under proper shape-rate gamma distribution priors. Adding the second mean as an inactive parameter under the shared-mean model gives a simple identity model switch. With equal model priors and identical active/inactive priors, its ratio from separate to shared means is $e^{{\gamma }_2-{\gamma }_1}({\gamma }_1/{\gamma }_2{)}^{x_2}$. Poisson-gamma conjugacy supplies exact within-model updates, and integrating the gamma kernels supplies an independent Bayes factor benchmark.

At fixed mass, the Press-Schechter halo mass function has abundance per logarithmic interval proportional to $\nu e^{-{\nu }^2/2}$. This increases as $\nu$ falls from infinity to one and then decreases as $\nu$ falls toward zero, reflecting incorporation of small dark-matter haloes into larger ones. A target below the maximum therefore has two mathematical crossings if the linear growth factor can increase without limit. The large-$\nu$ crossing describes first appearance; the small-$\nu$ crossing is late depletion. Late vacuum domination limits growth, so a formal future crossing need not be reached physically.

The Schur algebra is the commutant of place permutations on a tensor power. It is the linear span of the diagonal general linear group action, by Schur–Weyl duality. Modules over $S(m,n)$ correspond to homogeneous degree-$n$ polynomial representations. Over the complex numbers it is a semisimple algebra: decompose the tensor power as a module for the semisimple group algebra $\mathbb{C}{S}_n$, and take its endomorphism algebra.

For a partition of an integer $\lambda ⊢n$, applying a Young symmetrizer of that shape to the tensor power of each vector space gives a Schur functor. The construction respects linear maps because their tensor powers commute with place permutations. For single rows it gives symmetric powers, and for single columns exterior powers.

The symmetric algebra on $V\oplus {\Lambda }^{2}V$ contains every irreducible polynomial representation of $GL(V)$ exactly once. Its torus character is ${\prod }_i(1-x_i{)}^{-1}{\prod }_{i<j}(1-x_i{x}_j{)}^{-1}$, which equals ${\sum }_{\lambda }{s}_{\lambda }(x)$ by the Schur identity. Complete reducibility and linear independence of Schur characters identify the summands. This is a formal graded identity: each fixed scalar degree is finite, so no analytic convergence is needed. Multiplicity one refers to irreducible modules, not arbitrary reducible representations.

For positive group totals, the improper prior $du/u$ on an independent Poisson baseline yields the multinomial likelihood kernel after integration. This prior is flat in $\log u$. A proper finite-variance log-rate prior generally introduces an additional coefficient-dependent factor.

Reverse a handle decomposition of a compact $n$-dimensional manifold. Absolute $k$-handles become relative $(n-k)$-handles based on the boundary. The absolute boundary matrices count attaching sphere-belt sphere intersections; the corresponding relative matrices are their transposes over ${\mathbb{F}}_2$. The absolute chain complex is thus the complementary-degree dual of the relative chain complex. The universal coefficient theorem for cohomology proves Poincare-Lefschetz duality without an orientation hypothesis.

For independent bond percolation of density $0<p<1$ on the cubic lattice in dimension $d\ge 2$, let ${\beta }_n$ be the one-arm probability for reaching maximum-norm distance $n$. The weighted BK boundary-splitting estimate gives ${\beta }_{m+n}\le |\partial {\Lambda }_n|{\beta }_m{\beta }_n$. Since the logarithm of this boundary size is $o(n)$, the asymmetrically almost-subadditive sequence argument applies to $\log {\beta }_n$. The rate exists and lies in $[0,-\log p]$; positive rate gives exponential decay.

For $\sigma (M,0)=S{M}^{-q}$ with $q=(n+3)/6>0$, matching the Press-Schechter halo mass function to $A{M}^{q-2}\exp [-(M/M_{\ast }{)}^{2q}]$ requires $\sigma (M_{\ast },0)={\delta }_c(z)/\sqrt{2}$. The comoving normalization is $A=2q{\bar{\rho }}_{m,0}{M}_{\ast }^{-q}/\sqrt{\pi }$. Consequently both $A$ and $M_{\ast }$ evolve with the linear growth factor; $M_{\ast }\propto D^{1/q}$.

The two introductory requests can be settled before the lettered applications. In the functor category $[C,\mathbf{Set}]$, finite limits and finite unions of subobjects are pointwise. The component of the diagonal at $c$ is ordinary equality on $F(c)$. Its only possible complement isThis forms a subfunctor exactly when each $F(u)$ sends unequal elements to unequal elements, equivalently when every $F(u)$ is injective. In that case $D$ and the diagonal are disjoint and their union is $F^2$ at every component. Conversely, a diagonal complement must have these components and be stable under every transition map. Hence $F$ is decidable exactly when every transition map is injective.

Regard a monoid $M$ as a one-object category; a covariant set-valued functor is a left M-set. Give $M\times A$ the diagonal left action $t\cdot (w,a)=(tw,t\cdot a)$. LetRight multiplication on the first coordinate commutes with the diagonal left action, so $m\cdot f$ remains equivariant. The formula obeys $m\cdot (n\cdot f)=(mn)\cdot f$ and $1\cdot f=f$. Evaluation isIt is equivariant because $\mathrm{ev}(m\cdot f,m\cdot a)=f(m,m\cdot a)=m\cdot f(1,a)$.

For an equivariant $h:C\times A\to B$, defineThis is equivariant in $(w,a)$, and $\hat{h}(m\cdot c)=m\cdot \hat{h}(c)$. Evaluation recovers $h$. Conversely, currying the evaluation of a map $C\to E$ recovers that map by its equivariance. This proves the exponential universal property and the natural identificationwith exactly the stated action. In particular, decidability in a set-valued functor category says that a left M-set is decidable if and only if each of its action maps is injective.