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A matrix pencil is an affine one-parameter family of matrices. For square matrices of size $m$, its determinant is a polynomial of degree at most $m$. If $B$ is invertible, its leading coefficient is $\det B\ne 0$, so at most $m$ distinct parameters give a singular matrix. This translates linear dependence in a family of bases into polynomial root counting.

A matrix norm is a norm on a vector space of matrices; common choices include induced operator norms and the Frobenius norm. A submultiplicative matrix norm additionally satisfies $\Vert AB\Vert \le \Vert A\Vert \Vert B\Vert$. The matrix 2-norm is the norm induced by Euclidean norms and equals the largest singular value.

A Cauchy matrix has entries $1/(x_i+y_j)$ with nonzero denominators. If $x_i=y_i>0$ are distinct, it is positive definite: $C_{ij}={\int }_0^{\infty }{e}^{-x_{i}s}{e}^{-x_{j}s}ds$ is the Gram matrix of linearly independent exponential functions. This form occurs in finite-rank inverse scattering transform reconstructions.

An integer matrix is a matrix whose entries are integers. Sums and products of compatible integer matrices remain integer matrices, but a nonsingular square integer matrix can have a nonintegral inverse matrix. The unimodular matrix condition characterizes exactly when that inverse matrix also has integer entries.

In a singular first-order equation with coefficient $x+ϵ$, an outer correction proportional to $ϵ\log x$ cannot be evaluated at the endpoint. The inner scale $x=ϵX$ changes it to $ϵ\log ϵ+ϵ\log (1+X)$. Matching retains the logarithmic constant, which is larger than a plain order-epsilon correction. For $(x+ϵ)y^{\prime }=ϵy$, $y(1)=1$, the exact solution $[(x+ϵ)/(1+ϵ){]}^{ϵ}$ verifies the displayed endpoint asymptotic and its remainder.

A decaying algebraic tail in an inner asymptotic expansion can become a fractional-order outer correction after rescaling. For an inner tail proportional to $r^{-1/2}$ and outer variable $x=ϵr$, the outer correction is proportional to ${ϵ}^{1/2}{x}^{-1/2}$. If the next outer correction contains $ϵ\log x$, expressing it in the inner variable produces both $ϵ\log r$ and $ϵ\log ϵ$. Omitting the latter loses a term larger than a plain $O(ϵ)$ correction. Matching must include all these scales and constants, rather than only the leading algebraic powers.

Multiplication by $x^2$ makes this example exactly integrable once: $ϵx^{2}y\prime +(x^2-2{ϵ}^3)y=C$. Its homogeneous solution is $\exp (-x/ϵ-2{ϵ}^2/x)$. A solution with $y(1)={ϵ}^3$ that remains uniformly bounded needs $C\sim {ϵ}^3$, hence $y(0)\sim -1/2$. The scales $x=O(ϵ)$ and $x=O({ϵ}^2)$ connect the small outer solution to the order-one endpoint value. Bounded homogeneous contributions remain invisible to algebraic matching at the right boundary.

For fixed $k>0$ and endpoint values $y(0)=0$, $y(1)=1$, the outer solution is $y\sim {\epsilon }^2{k}^2$. The left boundary layer has width $\epsilon$. At the right, a layer of width $\sqrt{\epsilon }$ carries the order-one rise, and a translated layer of width $\epsilon$ connects its inner edge to the small plateau. The dominant balance depends on both amplitude and derivative scale; using the same scale at both ends loses a necessary match.

An asymptotic matching condition requires the inner expansion and outer expansion, re-expressed in a common overlap region, to agree to the retained order. Both function values and slopes may constrain the free constants. In the square-root reaction problem with nested boundary layers, the shared small plateau sets the first integral of a square-root reaction layer, while its large-field fourth-power tail fixes the neighboring broad layer.

For $q>0$, matching $Y\to -q$, $Y^{\prime }\to 0$ fixes the first integral to $(Y^{\prime }{)}^2=(q-2Y)(Y+q{)}^2/3$. Hence endpoint values above $q/2$ cannot match this outer branch. The upper family is $Y=-q+(3q/2){\mathrm{sech}}^2[\sqrt{q}(\xi -{\xi }_0)/2]$. An endpoint value strictly between $-q$ and $q/2$ admits two translations, one initially approaching $-q$ and one initially moving towards the turning value $q/2$. Values below $-q$ give the unique lower family $Y=-q-(3q/2){\mathrm{csch}}^2[\sqrt{q}(\xi +{\xi }_0)/2]$ with ${\xi }_0>0$. Matching to zero instead gives $(Y^{\prime }{)}^2=-q{Y}^2-2Y^3/3$, which forbids a nonconstant real approach to zero. These are leading boundary layer classifications, not exact multiplicity theorems at degenerate endpoint data.

A growing linear mode can make a formally small derivative nonlinearity order one at large distance. After the crossover scaling, the equation $X{H}^{\prime \prime }-H^{\prime }+(H^{\prime }{)}^2=0$ reduces to a logistic differential equation in $\log X$ for $H^{\prime }$. Matching $H\sim X^2/2$ at small $X$ gives $H^{\prime }=X/(1+X)$ and the displayed primitive with $H(0)=0$. It changes from quadratic growth to linear growth without a finite turning singularity.

For $ϵy^{\prime \prime }-(1+x)y^{\prime }+y=0$, $y(0)=1$, $y(1)=1+ϵ$, the exact outer solution is $1+x$ and the mismatch is at the right endpoint. In $X=(1-x)/ϵ$, matching gives $Y_0=2-e^{-2X}$ and $Y_1=-X+(1-X-X^2/2)e^{-2X}$. Differentiation at $X=0$ gives the displayed endpoint slope. The order-one term requires the first inner correction; differentiating only the leading boundary layer loses it.

For a free massless p-form gauge field with $0\le p\le D-2$, its field equation makes $\ast dA$ closed. The Poincaré lemma writes it locally as $dB$, giving a dual potential of degree $D-p-2$. The original Bianchi identity for an Abelian p-form becomes the dual field equation, and the original equation becomes the dual Bianchi identity. The two descriptions have equal little group counts $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$. Global fluxes and top forms require separate treatment.

// Target: quantum-field-theory.bigb

For a free massive p-form field, the displayed dual has degree $D-p-1$. The massive field equation reconstructs $A$ locally from $m^{-1}\ast dB$ with the signature-dependent sign. It also gives $\delta B=0$ and $\delta dB+m^{2}B=0$. The polarization counts agree because $\left(\genfrac{}{}{0ex}{}{D-1}{p}\right)=\left(\genfrac{}{}{0ex}{}{D-1}{D-p-1}\right)$. The construction depends on nonzero mass and therefore differs from massless p-form duality by one potential rank.

// Target: physics.bigb

For $p\ge 1$, introduce a $(p-1)$-form $C$ and replace the mass term of a massive p-form field by one built from $A-m^{-1}dC$. The transformations $A\mapsto A+d\Lambda$, $C\mapsto C+m\Lambda$ leave this combination and $dA$ invariant. Gauge fixing $C=0$ recovers the original massive theory. Its massless-component counts add as $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)+\left(\genfrac{}{}{0ex}{}{D-2}{p-1}\right)=\left(\genfrac{}{}{0ex}{}{D-1}{p}\right)$, explaining the longitudinal degrees of freedom and the corresponding assembly of nonzero Kaluza-Klein modes.

The massive N=1 vector multiplet contains spins $1,1/2,1/2,0$. Its massless helicity content splits into a CPT-complete vector multiplet with helicities $(1,1/2)$ and $(-1/2,-1)$, and a CPT-complete chiral multiplet with helicities $(1/2,0)$ and $(0,-1/2)$. The longitudinal vector polarization becomes one of the two scalar polarizations; the other comes from the original real scalar. This is the representation-theoretic inverse of the supersymmetric Higgs mechanism.

Let independent increments ${\xi }_k$ equal $4^k$ or $-4^k$ with probabilities $2^{-k-1}$ each, and zero otherwise. Every partial sum $M_n={\sum }_{k\le n}{\xi }_k$ is integrable and is a martingale. The Borel-Cantelli lemma makes only finitely many increments nonzero almost surely, so $M_n$ has a finite almost-sure limit $Y$. Yet $\mathbb{E}|Y|=\infty$: the disjoint events that only the $k$th increment is nonzero have probabilities at least $c{2}^{-k}$ for $c={\prod }_j(1-2^{-j})>0$, and contribute at least $c{2}^k$ each. This does not contradict the martingale convergence theorem, whose integrability conclusion requires a uniform first-moment bound.

// Target: markov-process.bigb

A trace plot displays an observable along successive iterations of a Markov chain Monte Carlo algorithm. Its horizontal axis is iteration and its vertical axis is the observed parameter or function value. Trends suggest an initial transient or slow exploration, separated regimes suggest transitions between posterior regions, and exact plateaus show repeated states. The plot should be read with acceptance statistics, autocorrelation and multiple starting states; its visual appearance alone cannot prove stationarity or identify the unique cause of poor mixing.

For a chain map $f:C\to D$, use $M_i=C_{i-1}\oplus D_i$ and $d(x,y)=(d_{C}x,(-1{)}^{i}f(x)+d_{D}y)$. The chain map identity makes $d^2=0$. The short exact sequence of chain complexes $0\to D\to M\to C_{\ast -1}\to 0$ has connecting homomorphism $(-1{)}^i{f}_{\ast }$ in degree $i$. Its long exact sequence in homology shows that the homology of $M$ vanishes precisely when every $f_{\ast }$ is an isomorphism.

A spectrum homotopy between compatible maps $D\to E$ is a map from the cylinder spectrum $D\wedge [0,1{]}_{+}$ restricting to them at the endpoints. Thus the homotopies at all levels must themselves respect the structure maps.