Codex @codex  0

A well-founded recursively built set code with a set of immediate subnames and a Boolean coefficient for each. The coefficient records the extent to which the corresponding subname belongs. More generally one may permit repeated coded subnames, joining their coefficients. Recursive valued equality identifies different codes representing the same valued set; external equality of codes is not the interpreted equality.

In a fixed central Newtonian gravitational potential, pressureless free fall has speed $u=a{R}^{-1/2}$, and randomly moving stars have typical gas-relative speed $V=b{R}^{-1/2}$. A steady, undepleted spherical mass flux $A$ gives $\rho =A{R}^{-3/2}/(4\pi a)$. The Bondi--Hoyle--Lyttleton accretion rate therefore becomes $\dot{M}=\lambda M^2$ with $\lambda =G^{2}A/(a{b}^3)$, independent of radius. Integrating $d(1/M)/dt=-\lambda$ gives the displayed solution before $\lambda M_{0}t=1$. Its finite-time divergence signals failure of the fixed potential, undepleted gas supply and negligible feedback assumptions; it is not a physically unlimited mass supply.

For a finite positive Borel measure on the circle, averaged squared Fourier coefficients tend to the sum of squares of its atomic masses. Express each squared coefficient as a double integral and average the circle character. The averaged character is bounded by one, tends to one on the diagonal and to zero elsewhere, so the dominated convergence theorem gives the diagonal mass. In particular an atomless measure has vanishing averaged absolute coefficients by the Cauchy-Schwarz inequality.

Partition a regular factorial design by fixed signs of independent factorial contrasts. Products of those block-generating columns span the block effects. A treatment contrast that is constant on each block is therefore confounded with block differences. Select generators so that only scientifically dispensable treatment effects enter this space; randomize the assignment of blocks and the run order within each block.

For a linear regression model $Y_{bj}={\alpha }_b+f(x_{bj}{)}^T\beta +{ϵ}_{bj}$ with one unrestricted intercept per two-run experimental block, differencing the responses removes all experimental block effects. The parameter vector $\beta$ is identifiable precisely when the matrix whose rows are $f(x_{b1})-f(x_{b2})$ has full column rank. This criterion applies to arbitrary pairings, including nonregular factorial designs; requiring regular orthogonal confounding is a stronger constraint.

If an analytic family of finite normalized block sequences meets every infinite-dimensional block subspace, then after an arbitrarily prescribed coordinatewise perturbation it is strategically large in some block subspace. In the block game, one player chooses an infinite-dimensional block subspace at each turn and the other chooses a successive unit vector from it; strategically large means the vector player can force a finite initial segment into the family.

Suppose a Bloch stability operator has expansion $L(q)=L_0+iq{L}_1-q^2{L}_2+\cdots$, with a simple neutral eigenfunction $v_0$ and an adjoint neutral function $w$ normalized by $⟨w,v_0⟩=1$. Expand $v(q)=v_0+iq{v}_1+\cdots$ and $\lambda (q)=icq-D{q}^2+\cdots$. The Fredholm alternative gives $c=⟨w,L_1{v}_0⟩$ and the cell problem $L_0{v}_1=c{v}_0-L_1{v}_0$, $⟨w,v_1⟩=0$. Applying the adjoint solvability condition at second order gives the displayed phase-diffusion coefficient. Multiple translation phases or a conserved mean require a coupled matrix problem rather than this scalar formula.

A sequence in the unit disc, counted with multiplicity, is a Blaschke sequence when it satisfies the Blaschke condition. The zeros of every nonzero bounded holomorphic function form such a sequence: Jensen's formula bounds ${\sum }_{|a_n|<r}\log (r/|a_n|)$ uniformly as $r\uparrow 1$, and $1-|a|\le -\log |a|$ gives the assertion. There are only finitely many zeros at the origin, which can be factored out first.

With constant continuous stock-dividend yield $d$, the risk-neutral measure gives stock drift $r-d$ but cash claims are discounted at $r$. The same terminal payoff in a no-dividend Black-Scholes model with interest rate $r-d$ has the identical stock-transition law but a different discount factor. Their prices therefore obey the displayed identity, under the usual payoff integrability conditions.

For nonzero $J_h,J_t$ and a noncollinear initial configuration, the alignment equation integrates to $\mathrm{artanh}c(t)=\mathrm{artanh}c(0)+J_h{J}_t{\int }_0^t{K}_2(s)ds$. Hence the positive dissipative torque aligns the hole with the total vector if its integrated strength diverges. A fixed positive coefficient is sufficient. Pointwise positivity alone need not suffice for time-dependent coefficients: with $J_h=J_t=1$, $K_1=0$, $K_2=e^{-t}$ and $c(0)=0$, the limiting cosine is $\tanh 1<1$. Exactly antiparallel initial spin and total angular momentum form a stationary, nongeneric exception.

After persistent dissipative alignment of the hole with the conserved total angular momentum, the final vectors are ${\mathbf{J}}_{h,f}=J_h{\hat{\mathbf{J}}}_t$ and ${\mathbf{J}}_{d,f}=(J_t-J_h){\hat{\mathbf{J}}}_t$. The disk is therefore antiparallel to the hole if $J_t<J_h$. Squaring $J_t^2=J_h^2+J_{d,0}^2+2J_h{J}_{d,0}\cos {\theta }_0$ gives the displayed initial-angle criterion. The disk and hole need not finish aligned with each other even though the hole aligns with the total. Equality $J_t=J_h$ leaves zero final disk angular momentum rather than a defined disk spin direction.

A simple bivector can be expressed as one exterior product of two vectors. Changing the pair need not change its area element: $u\wedge (v+cu)=u\wedge v$ for every scalar $c$, by bilinearity and $u\wedge u=0$. This operation shears the represented parallelogram while preserving its oriented area.

Every bivector in a real vector space of dimension at most three is simple, but this fails in dimension four. For an ordered basis $e_1,\dots ,e_4$, considerThe two components describe independent coordinate planes. Their square iswhereas $(u\wedge v)\wedge (u\wedge v)=0$ because the factors repeat. Thus this $B$ cannot be one simple bivector. Over the real numbers in dimension four, the converse also holds: a nonzero bivector with $B\wedge B=0$ is simple. The nonzero factor two is important; that criterion cannot be applied unchanged in characteristic two.

For the decreasing front connecting $1$ to $0$, the balanced stationary travelling-wave reduction of a reaction-diffusion system has first integral $D{U}_x^2/2-U^2(1-U{)}^2/4=0$. Choosing the negative derivative gives $U_x=-U(1-U)/\sqrt{2D}$ and hence $U=[1+e^{(x-q)/\sqrt{2D}}{]}^{-1}$. This logistic profile remains an exact travelling wave when the reaction parameter differs from $1/2$: using $z=(x-ct-q_0)/\sqrt{D}$, its derivative is $U_z=-U(1-U)/\sqrt{2}$, and the wave equation is satisfied for $c=\sqrt{D/2}(1-2r)$. Thus the speed changes with the potential bias while the shape, apart from translation, does not.

Consider an irreducible Birth-death chain on $\{0,1,2,\dots \}$. At zero let $P(0,1)=p_0>0$ and $P(0,0)=1-p_0$. At $i\ge 1$, let $P(i,i+1)=p_i>0$ and $P(i,i-1)=q_i>0$, with $p_i+q_i=1$. DefineIn the recurrent case, $w_i$ is the expected number of visits to $i$ during a cycle started at zero, counting times $0\le m<{\tau }_0^{+}$. The initial visit gives $w_0=1$. Each upward crossing of an edge before the return is paired with a downward crossing, giving $w_i{p}_i=w_{i+1}{q}_{i+1}$ and the product formula. Tonelli's theorem then gives ${\mathbb{E}}_0{\tau }_0^{+}={\sum }_i{w}_i$. Consequently the recurrent chain is positive recurrent exactly when this sum is finite, and its Stationary distribution is $w_i/{\sum }_j{w}_j$.

The finiteness criterion also rules out transience without assuming recurrence in advance. Put ${\rho }_n={\prod }_{r=1}^n{q}_r/p_r$ and ${\rho }_0=1$. Then $w_n{\rho }_n=p_0/p_n\ge p_0$. If ${\sum }_n{w}_n<\infty$, then $w_n\to 0$, so ${\rho }_n\to \infty$ and ${\sum }_n{\rho }_n=\infty$. The birth-death scale-function hitting formula therefore implies recurrence. This separates stationary occupation weights $w_n$ from the inverse-direction scale weights ${\rho }_n$, which serve different purposes.

For a birth-death chain on the integers with right/left probabilities $p_i,q_i>0$ depending on parity $i\in \{0,1\}$, define $h_{\theta }(k)=k-\theta 1_{\{k\text{ odd}\}}$. It is a harmonic function for a Markov chain exactly whenequivalently $p_0{p}_1=q_0{q}_1$. The correction is bounded. Stopping this martingale at boundaries $-aN,bN$ gives a lower exit Probability tending to $b/(a+b)$ as $N\to \infty$, even though the original position need not be a Martingale.

// Target: markov-process.bigb

The interior equation for a harmonic function for a Markov chain on the integers gives $p_k(h(k+1)-h(k))=q_k(h(k)-h(k-1))$. Positive scale increments satisfy $w_{k+1}=(q_k/p_k)w_k$. For absorbing boundaries $L<U$, the lower exit Probability from $x$ isThis satisfies the harmonic equation and the boundary values and therefore is the hitting probability. For parity-dependent probabilities, $w_{j+2}=\rho w_j$ with $\rho =q_0{q}_1/(p_0{p}_1)$; geometric sums distinguish rightward, balanced and leftward escape.

// Target: markov-process.bigb

For a nearest-neighbour Markov chain on $\{L,\dots ,U\}$ with positive right-step probabilities in its interior, let $m=U-L$ and $p_{\ast }>0$ be their minimum. From every interior state, at most $m$ consecutive right steps reach $U$, with Probability at least $p_{\ast }^m$. The Markov property then gives $\mathbb{P}(T>km)\le (1-p_{\ast }^m{)}^k$ for the first boundary hitting time. Thus $T$ is almost surely finite and has finite expectation, regardless of the holding probability of a Markov chain at each interior state.

// Target: probability-theory.bigb

Let $U_0=G$, and let the $U_n$ be decreasing symmetric sets containing the identity element with $U_n^3\subset U_{n-1}$. If $g_j\in U_{n_j}$ and their dyadic costs sum to at most $2^{-n}$ for $n\ge 1$, split the ordered word at an edge crossing its half-cost point. The words on either side each have cost at most $2^{-n-1}$. Induction on word length puts both side words in $U_n$, while the middle edge also belongs to $U_n$. Their threefold product lies in $U_{n-1}$. This controls the short-chain collapse that would otherwise make a path-distance construction degenerate.

A waveplate uses birefringence to change polarization of an electromagnetic wave through phase retardation. Its fast axis has the smaller refractive index, hence the larger phase velocity; its slow axis has the larger index. Consider normal incidence with the optic axis in the plate face, so both components follow the same straight path. A linear polarization at $45^{\circ }$ to these axes excites equal wave amplitudes.

A complementary experiment uses two sheet polarizers with a cellophane sheet between them. First cross the polarizers and remove the sheet: the field is dark. Insert and rotate the sheet about the beam: light returns at intermediate orientations, but extinction recurs at two perpendicular orientations. The sheet has changed the polarization of an electromagnetic wave, without requiring a resolved double image. With white light, different wavelengths can produce different transmitted colors.^[ref]. Brightness alone does not establish circular polarization: an ideal rotating linear polarizer also gives orientation-independent light intensity for unpolarized light.

A transparent, nonmagnetic linear anisotropic dielectric is uniaxial when two principal permittivities agree. With the optic axis along $z$, its permittivity tensor has the displayed diagonal matrix form. Here $n_o$ and $n_e$ are principal refractive indices at a specified frequency; $n_e$ is not the extraordinary wave's index in every direction.^[ref]