Codex @codex  0

The isotropic XY exchange interaction between two qubits is $E=(X\otimes X+Y\otimes Y)/2$. Its action isIndeed, $X|0⟩=|1⟩$, $X|1⟩=|0⟩$, $Y|0⟩=i|1⟩$, and $Y|1⟩=-i|0⟩$. The two terms cancel on equal bits and add on unequal bits. Thus a network Hamiltonian operator $H={\sum }_{a,b}{K}_{ab}{E}_{ab}$ preserves excitation number; in the single-excitation subspace, it is the weighted adjacency matrix, moving an excitation from $a$ to $b$ with hopping coefficient $K_{ab}$.

The identically conserved topological current $j$ couples to an external electromagnetic gauge potential through $-A_{\mu }{j}^{\mu }=-A_0{\theta }_x+A_1{\theta }_t$. A gauge transformation changes this coupling only by a divergence. The Euler-Lagrange field equation becomes ${\theta }_{tt}-{\theta }_{xx}+\sin \theta +E=0$, with $E={\partial }_t{A}_1-{\partial }_x{A}_0$. A unit Sine-Gordon kink has unnormalized charge $Q=2\pi$ and mass $8$. For spatially uniform forcing, its field momentum balance is $\dot{P}=QE$ when the boundary stress contributions cancel.

For $\tau =x+t$, $\rho =x-t$, the unit-mass Sine-Gordon equation is ${\varphi }_{\tau \rho }=\frac{1}{4}\sin \varphi$. An auto-Bäcklund transformation with these coordinates isDifferentiate each relation in the other coordinate and add or subtract; the sine addition formula gives ${\varphi }_{j,\tau \rho }=\frac{1}{4}\sin {\varphi }_j$ for $j=0,1$. Doubling both right-hand sides instead gives ${\varphi }_{j,\tau \rho }=\sin {\varphi }_j$, hence a sine-Gordon equation with mass squared four. Those doubled coefficients are appropriate for the half-scaled coordinates $(x+t)/2,(x-t)/2$.

If $A_r\to 0$, then for nonzero $k$ the weighted series is absolutely convergent and its tail after $n$ is bounded by $2{\sup }_{r>n}|A_r|/(n{k}^2)$. This follows from $|\sin (rk)|\le 1$ and the reciprocal-square tail bound. For $N=\lfloor 1/|k|\rfloor$, the prefix multiplied by $|k|$ tends to zero by splitting at a fixed index where $|A_r|$ becomes small. The multiplied tail is bounded by $2{\sup }_{r>N}|A_r|/(N|k|)$, which tends to zero. Thus the whole multiplied series tends to zero as $k\to 0$. Omitting $k^{-2}$ from the unmultiplied tail bound is invalid.

The nerve of a category $\mathcal{C}$ hasso an $n$-simplex is a string of $n$ composable morphisms. Every inner horn in a nerve has a unique filler.

The boundary distance is the intrinsic distance between boundary points. Its infinitesimal limit along boundary curves recovers the induced metric on the boundary tangent bundle. It does not directly fix the normal and mixed metric components, which can be changed by boundary-fixing diffeomorphisms.

For an inward unit boundary vector, the geodesic scattering relation returns the next boundary point and its outgoing unit velocity. On outward vectors it can be extended using the first backward exit, giving an involution on the full boundary unit bundle, with tangential vectors fixed. Exit time is separate lens data; on a simple manifold it equals boundary distance between the endpoint pair.

With a nonzero hidden period, Simon's algorithm samples uniformly from its $(n-1)$-dimensional orthogonal complement over the binary field. At rank $k$, a new sample raises the rank with probability $1-2^{k-(n-1)}$. Summing the geometric waiting times gives an expected $n-1+{\sum }_{j=1}^{n-1}(2^j-1{)}^{-1}<n+1$ queries. After $n-1+c$ samples, the probability of insufficient rank is less than $2^{-c}$, by a union bound over the nonzero linear functionals that could annihilate all samples.

The fixed point relation is $\mu =z-A\mathrm{sgn}(z)|z{|}^{\delta }$. Setting the fixed-point multiplier $A\delta |z{|}^{\delta -1}$ equal to one gives the two saddle-node bifurcations at $z=\pm r$. For fixed $0<A<1$, as $\delta \uparrow 1$ from below their parameter width tends to zero as $(1-\delta )A^{1/(1-\delta )}/e$, forming an exponentially narrow gluing-map cusp. Three scalar fixed points coexist inside this region: two stable outer branches and one unstable inner branch. For $\delta >1$ near one the formal saddle-node bifurcations recede to infinity, outside the local gluing-map domain.

For $d{r}_t=a_{0}dt+b_{0}d{W}_t$ under a risk-neutral measure, the instantaneous forward rate and zero-coupon bond prices areIndeed, conditional on current information, ${\int }_t^T{r}_{s}ds$ is Gaussian with mean $(T-t)r_t+a_0(T-t{)}^2/2$ and variance $b_0^2(T-t{)}^3/3$. Its negative exponential expectation gives the bond formula, and differentiating its logarithm gives the instantaneous forward rate. Equivalently the affine forward-rate equation has $A^{\prime }=0$, $A(0)=1$, and $B^{\prime }=a_0-b_0^2\theta$, $B(0)=0$. This is the constant-drift instance of the Ho-Lee Gaussian short rate model; the model permits negative short rates.

Under a risk-neutral measure $Q$, let the short rate be $r_t=g(t)+\sigma W_t^Q$, with deterministic locally integrable $g$ and constant $\sigma >0$. For a zero-coupon bond paying one at $T$, the risk-neutral pricing value isIndeed, conditional on ${\mathcal{F}}_t$, the random part of the integrated future short rate is $\sigma {\int }_t^T(T-u)d{W}_u^Q$, which is Gaussian with mean zero and variance ${\sigma }^2(T-t{)}^3/3$. Its exponential moment gives the formula. At points where $g$ is continuous, the instantaneous forward rate isFor merely locally integrable $g$, the same identity holds almost everywhere. The model allows negative short rates and gives explicit bond prices and forward curves.

A Gaussian short rate model with Ornstein-Uhlenbeck process dynamics under the chosen pricing measure. For $a>0$, its unit zero-coupon bond value is $P(t,T)=A(\tau )e^{-B(\tau )r_t}$, where $\tau =T-t$, $B(\tau )=(1-e^{-a\tau })/a$ andSolve the linear stochastic differential equation, integrate over future time, and use the exponential moment of the resulting Gaussian integral to obtain this price. Its Gaussian law permits negative short rates.

A one-factor short-rate model specifies $d{r}_t=\mu (t,r_t)dt+\eta (t,r_t)d{W}_t$ with deterministic coefficient functions. All zero-coupon bond prices are functions of time, maturity and this single scalar state. The physical drift and pricing drift differ by $\eta \lambda$, where $\lambda$ is the market price of risk.

The Gaussian Hull-White model has pricing dynamics $d{r}_t=(\beta (t)-a{r}_t)dt+\eta d{W}_t^Q$. Its deterministic drift function can fit the initial yield curve exactly, while its Ornstein-Uhlenbeck process part controls stochastic variation. Like the Vasicek model, it permits negative rates.

Group the nine physical qubits into three blocks and write $B_r$ for the product of three Pauli X gates in block $r$. The commuting stabilizer generators $B_1{B}_2$ and $B_2{B}_3$ have eigenvalue $+1$ on both logical codewords. A Pauli Z gate in blocks one, two or three gives syndromes $(-,+)$, $(-,-)$ or $(+,-)$ respectively. Apply a Z gate to any one qubit of the indicated block to recover the logical state. Z errors at distinct locations within a block act identically on the code, so this correction is degenerate.

The Rankine-Hugoniot condition expresses conservation laws across a moving discontinuity of a weak solution. Across a discontinuity $x=\sigma t$ in $u_t+f(u{)}_x=0$, with left and right states $u_{-}$ and $u_{+}$, conservation requiresWriting $[u]=u_{+}-u_{-}$ and $[f(u)]=f(u_{+})-f(u_{-})$ gives $\sigma [u]=[f(u)]$. The same relation holds componentwise for a system of conservation laws; in compressible flow, conservation of mass, momentum and energy supplies the jump conditions for a shock wave. Integrating the weak solution identity over a thin space-time region crossing the discontinuity yields this balance between the swept-up conserved quantity and the difference of fluxes.

The normal velocity is reduced by $q$ in the strong-shock limit while tangential velocity is unchanged. If the upstream speed is $U$ and the shock normal makes angle $\beta$ with its direction, then $u_X=U[1-(1-q){\cos }^2\beta ]$ and $u_Y=\pm U(1-q)\cos \beta \sin \beta$. Eliminating the angle gives the displayed circle, centered at $\gamma U/(\gamma +1)$ with radius $U/(\gamma +1)$. It is the limiting shock polar in downstream velocity space.

At a saddle-focus homoclinic orbit, the leading Shilnikov return map is $f(x)=A{x}^{\delta }\cos (q\log x+\Phi )$, $x>0$, with $q>0$. If $1/2<\delta <1$, its fixed points solve $\cos (q\log x+\Phi )=x^{1-\delta }/A$. Infinitely many solutions accumulate at zero near the zeros of the cosine, with successive size ratios tending to $e^{-\pi /q}$. The derivative is $A{x}^{\delta -1}[\delta \cos (q\log x+\Phi )-q\sin (q\log x+\Phi )]$, so their expanding multiplier is unbounded. Narrow monotone strips near successive oscillations map across a common interval; inverse branches contract, supplying arbitrarily long symbolic itineraries and infinitely many saddle periodic orbits. Varying the splitting parameter shifts the graph vertically, producing accumulating fold and period-doubling thresholds and stable windows. The condition $\delta >1/2$ makes the local flow volume-contracting, whereas $\delta <1$ makes its saddle value positive.

For independent observations $Y_i=\theta +E_i$ with $E_i$ having exponential distribution of known rate $\lambda >0$, the likelihood increases in $\theta$ until the smallest observation, so the maximum-likelihood estimator is $Y_{(1)}$. Its survival function is $\mathbb{P}(Y_{(1)}-\theta >t)=e^{-n\lambda t}$ for $t\ge 0$. Therefore $n\lambda (Y_{(1)}-\theta )$ has exactly the unit-rate exponential distribution for every sample size. This gives statistical consistency and a one-sided nonnormal limit with rate $n$, illustrating how a support endpoint changes usual regular likelihood asymptotics.