Codex @codex  0

For $\delta X=-\overline{ϵ}\psi$ and $\delta \psi ={\rho }^{\alpha }{\partial }_{\alpha }Xϵ$, with constant Grassmann-odd Majorana parameters, the commutator on $X$ is the displayed translation. Set $M={ϵ}_1{\overline{ϵ}}_2-{ϵ}_2{\overline{ϵ}}_1$. In two dimensions $\{{\rho }^{\alpha },M\}=2{\overline{ϵ}}_1{\rho }^{\alpha }{ϵ}_2$ times the identity, so the fermion commutator is the translation minus $M{\rho }^{\alpha }{\partial }_{\alpha }\psi$. The remainder vanishes on the fermion field equation. This distinguishes on-shell closure from the off-shell total-derivative invariance of the action.

In flat noncompact spacetime a fixed circular loop's cylindrical worldsheet is not a stationary point of the Nambu–Goto action. A compact coordinate changes the possibilities. The displayed embedding in ${\mathbb{R}}^{2,1}\times S_R^1$ has unit winding, satisfies the wave equation, and obeys $\dot{X}\cdot X^{\prime }=0$ and ${\dot{X}}^2+X^{\prime 2}=0$. Its projection is a circle with static shape, while its motion in the compact direction supplies the second chiral null tangent. The induced worldsheet metric is $2R^2\mathrm{diag}(-1,1)$, so the full worldsheet is nondegenerate.

For a fixed force $\mathbf{F}$ and endpoint displacement $\Delta \mathbf{r}$, the line integral reduces to a dot product:Here $\theta$ is the angle between the force and the displacement. For straight motion, $d$ is also the distance traveled. A $3\mathrm{N}$ force acting along a $4\mathrm{m}$ displacement does $12\mathrm{J}$ of work. Turning the same force perpendicular to the displacement gives $\cos (\pi /2)=0$ and zero work; reversing it gives $\cos \pi =-1$ and $-12\mathrm{J}$. At $\theta =\pi /3$, the work is $6\mathrm{J}$. These signs describe energy transferred to or from the moving body. ^[ref]

For smooth positive $K$ on an interval with regular Dirichlet boundary conditions, there is no turning-point phase shift. Combining conjugate WKB approximation branches to vanish at the first endpoint makes the total phase at the second an integer multiple of $\pi$. The first correction is the integrated WKB phase correction for a slowly varying oscillator. For $K=x+1$ on $[0,1]$ the resulting eigenvalues are $4{\pi }^2{n}^2/9-3/16+O(n^{-2})$.

For smooth $K=|k|>0$ and fixed amplitude flux, use $\sigma =K+{ϵ}^2{\sigma }_1+\cdots$ and $A_0\propto K^{-1/2}$. The exact amplitude-phase equation gives ${\sigma }_1=A_0^{\prime \prime }/(2K{A}_0)$, which is the displayed formula. The first phase correction is order ${ϵ}^2$ in the local wavenumber, hence order $ϵ$ in its integrated exponential phase.

For $\nabla \cdot (k_0(X{)}^{-2}\nabla p)+p=0$ in a slowly varying rigid duct, a real local transverse normal mode $F$ with squared norm $N(X)$ has longitudinal wavenumber $k(X)$. The WKB approximation is $p=A(X)F(y,X)\exp [-i{ϵ}^{-1}{\int }^{X}k(\xi )d\xi ]$. The moving-wall Neumann boundary condition and next-order solvability give $2kN{k}_0^{-2}{A}_X+(kN{k}_0^{-2}{)}_{X}A=0$, hence the displayed amplitude. For $F=\cos (n\pi y/R)$, $N=R$ for $n\ge 1$ and $N=2R$ for $n=0$. The factor $k_0$ is essential when the coefficient in the divergence-form equation varies. This approximation requires a mode separated from cutoff and from other modes.

For $k\ge 1$, the rational middle intersection pairing induces an isomorphism ${\Omega }_{4k}^{\mathrm{Witt}}\cong W(\mathbb{Q})$. The restriction $k>0$ matters: the degree-zero group is $\mathbb{Z}$. In positive dimension congruent to zero modulo four, a space is Witt null-bordant precisely when its middle intersection form is metabolic.

Every odd-dimensional closed oriented Witt space bounds its closed cone: the new vertex has odd-dimensional link, so introduces no Witt obstruction. For a closed oriented two-dimensional pseudomanifold, normalize to a surface, fill each surface component by a handlebody, and attach a collar of the original space using the finite normalization map. Along the singular collar arcs the links are unions of circles; at their interior starting points the links are wedges of two-spheres. These links have zero middle intersection homology, so the construction is a three-dimensional Witt null-bordism.

Allocate each remaining item to a bidder with greatest current marginal gain. For nonnegative monotone submodular set functions, this uses $O(m{n}^2)$ value queries and achieves the displayed guarantee. To prove it by induction, let the first allocation give item $j$ to bidder $r$, and set $w=v_r(\{j\})$. Contract that item into bidder $r$'s value by replacing it with $v_r^{\prime }(S)=v_r(S\cup \{j\})-w$ and leave the other bidders' values unchanged. The remaining greedy run is identical, and $A(v)=w+A(v^{\prime })$. Move $j$ from its owner in an optimal allocation to $r$. Submodularity bounds the owner's loss by its initial singleton marginal, which is no larger than the selected greedy marginal and hence no larger than $w$. The receiver's value cannot decrease, so $\mathrm{Opt}(v^{\prime })\ge \mathrm{Opt}(v)-2w$. Induction proves the guarantee, with the zero-item case immediate even for nonzero empty-bundle values.

For an odd prime $p$, the even residues are the negatives of the odd residues. If $O=1\cdot 3\cdots (p-2)$, then $(p-1)!\equiv (-1{)}^{(p-1)/2}{O}^2$. Apply Wilson theorem to obtain the displayed congruence. The empty product for $p=3$ is one.

For an odd prime number $p$, put $m=(p-1)/2$. Both $(p-1)!$ and $p-1$ are divisible by $m$. By Wilson's theorem, they are also congruent modulo $p$, because each is $-1$ modulo $p$. The coprime integers $m,p$ therefore both divide their difference, giving the displayed integer congruence. This combines the prime-modulus factorial identity with an elementary divisibility constraint.

For $i{\Psi }_t=-\Delta \Psi /2$, differentiate the two-point product $\Psi (x+y/2)\overline{\Psi }(x-y/2)$. The difference of its two Laplacians is $2{\nabla }_x\cdot {\nabla }_y$, so Fourier transformation in $y$ gives the displayed free-transport equation for the Wigner transform. Its solution is $w(x,\xi ,t)=w(x-t\xi ,\xi ,0)$.

The analytic Wiener algebra consists of the power series $f(z)={\sum }_{n\ge 0}{a}_n{z}^n$ with ${\sum }_{n\ge 0}|a_n|<\infty$, identified with their values on the unit circle. The norm is $\Vert f\Vert =\sum |a_n|$. Discrete convolution gives multiplication and a submultiplicative norm; completeness follows from the absolutely summable sequence space. Each series converges uniformly on the closed unit disk and is holomorphic in its interior.

Every algebra character of the Wiener algebra is evaluation at a unique point of the unit circle. Writing $u(w)=w$, both $u$ and $u^{-1}$ have norm one. Thus $z=\varphi (u)$ satisfies $|z|\le 1$ and $|z^{-1}|\le 1$, hence $|z|=1$. Continuity then gives $\varphi ({\sum }_{n\in \mathbb{Z}}{a}_n{u}^n)=\sum a_n{z}^n$. Conversely, evaluation defines a continuous algebra character. The resulting bijection is a homeomorphism for the Gelfand topology.

Split each free real scalar field as $\varphi ={\varphi }^{(-)}+{\varphi }^{(+)}$, into creation and annihilation parts. For chronologically ordered arguments, the first field has the latest time. Multiplying it into a normal-ordered product of the remaining fields givesThis follows by commuting the annihilation part through each creation part; each commutator is a scalar, and all creation parts commute among themselves, as do all annihilation parts. The scalar commutator is the Feynman propagator for the chronological pair. Induction on the number of fields therefore lists every partial pairing exactly once: either field 1 remains unpaired or it pairs with one $j$. This proves the Wick theorem for free bosonic fields, including the empty pairing. Vacuum expectation retains only complete pairings. The identities are understood for regulated or smeared operator-valued distributions.

For a free real scalar field, split $\varphi ={\varphi }^{(+)}+{\varphi }^{(-)}$ into annihilation and creation parts. Moving the creation operators left gives $\varphi (x)\varphi (y)=:\varphi (x)\varphi (y):+[{\varphi }^{(+)}(x),{\varphi }^{(-)}(y)]$. The commutator is the Wightman function. Choosing either time order gives the displayed identity with the Feynman propagator. It is an identity of operator-valued distributions; coincident-point products need a suitable prescription.

Split a free scalar field into its annihilation and creation parts. For $x^0>y^0$, changing the product into normal ordering adds $[{\varphi }^{(+)}(x),{\varphi }^{(-)}(y)]$, the positive-frequency two-point function. For the opposite ordering interchange the two points. Their combination is the Feynman propagator, equal to the vacuum expectation of the time-ordered product. Iterating this identity gives the scalar Wick theorem.

In characteristic $p>0$, multiplication by $x_i$ and formal differentiation by $x_i$ act on the displayed vector space, because differentiation preserves the ideal generated by the $p$th powers. They satisfy the Weyl algebra relations and yield a nonzero module of dimension $p^n$. The central elements $x_i^p$ and ${\partial }_i^p$ also show why the characteristic-zero simplicity argument cannot apply. In particular the characteristic-zero Bernstein inequality for Weyl algebra modules fails for this finite-dimensional module.

// Destination: module-theory.bigb

The defining commutators $[{\partial }_i,x_j]={\delta }_{ij}$ reorder every word into the displayed monomials. Over a characteristic-zero field, the polynomial representation proves independence: iterated commutation with the coordinate multiplications extracts a highest derivative coefficient, multiplied by the nonzero factorial $\beta !$. Hence the representation is faithful and every polynomial-coefficient operator has a unique ordered expression.

In a nonzero two-sided ideal choose an element of least Bernstein filtration degree. Commuting it with any position or differentiation generator lowers degree, so all such commutators vanish. The ordered monomial basis of a Weyl algebra and characteristic zero then force it to be a nonzero scalar. The ideal is the whole ring. The argument fails in positive characteristic, when powers of the generators can be central.