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For fixed $\alpha >0$ and nonzero bounded $K$, the reconstruction vanishes when $\Vert K^{\ast }f{\Vert }_{\infty }\le \alpha$, which includes a neighborhood of zero. If it were linear it would vanish everywhere. Choose a coordinate $e_j$ with $K{e}_j\ne 0$ and data $f=tK{e}_j$ with $t\Vert K{e}_j{\Vert }^2>\alpha$; the optimality inequality at zero fails, so its minimizer is nonzero. Thus the map is nonlinear. The qualification $K\ne 0$ is essential: $K=0$ gives the zero linear map.

There are no weight-four cusp forms at level two. A nonzero form would have a weight-twelve coset norm of a modular form with modular cusp order at least two, contradicting the simple modular cusp order and interior nonvanishing of the modular discriminant. The two independent displayed Eisenstein series span the weight-four space, since its cusp-constant map embeds it in a two-dimensional space.

The ramification polynomial records the normalised differences of conjugates of a uniformizer, including the zero difference from the identity automorphism. Its quotient by $X$ is the reduced ramification polynomial. In the usual coefficient-index convention, the finite slopes of its lower Newton polygon are negatives of the valuations of its nonzero roots; the coefficient at the zero root is treated as having infinite valuation.

A polynomial may fail the Eisenstein polynomial test while a translate $f(T+a)$ passes it. Translation does not change the generated extension: if $f(\alpha )=0$, then $f(T+a)$ vanishes at $\alpha -a$. For $f(T)=T^3-m$, the translate by $m$ is Eisenstein at $3$ whenever $3\nmid m$ and $m\not\equiv \pm 1(\mathrm{mod}9)$.

Let a real symmetric matrix $M$ have ordered eigenvalues ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n$, with $n\ge 2$. If $A$ is obtained by deleting one matching row and column, its largest eigenvalue ${\mu }_1$ satisfiesThe Rayleigh quotient of $A$ is that of $M$ restricted to vectors whose deleted coordinate is zero. This gives the upper bound. Some nonzero linear combination of the top two orthonormal eigenvectors has that coordinate zero; normalizing it gives a quotient at least ${\lambda }_2$. The argument does not require distinct eigenvalues.

Geometric multiplicity is the dimension of an eigenspace; algebraic multiplicity is its multiplicity as a root of the characteristic polynomial. For a real symmetric matrix they coincide because it has an orthonormal eigenbasis.

The spectral abscissa is the largest real part of a matrix's eigenvalues. It describes asymptotic exponential rates of its matrix exponential, but a non-normal matrix can also have transient amplification. It is generally distinct from the Euclidean logarithmic norm, which gives an immediate Euclidean energy-growth bound.

For countably many unary predicates $P_i$, let $M$ contain infinitely many copies of each finite membership pattern and let $N$ add an element satisfying all predicates. Every finite vocabulary reduct has infinitely many elements in every Boolean cell in both structures, so $M\equiv N$. The Spoiler nevertheless wins the one-round full-language game by selecting the new element, whose complete atomic pattern has no counterpart in $M$.

A proper subvariety locally defined by one non-zero-divisor is an effective Cartier divisor. On an integral variety a nonzero local equation is automatically a non-zero-divisor. Its conormal sheaf is a line bundle, since $A/(f)\cong (f)/(f^2)$ via multiplication by $f$. Allowing the zero equation would include the whole variety and would not give this conclusion.

The terrestrial atmosphere has regions with opposite signs of the atmospheric lapse rate: the troposphere usually cools upward, the stratosphere warms upward through ultraviolet heating, the mesosphere cools, and the thermosphere warms. A monotone atmospheric pressure-temperature profile can approximate one such interval, but cannot describe all their transitions with one fixed branch.

For unwrapped grain and parent azimuths, define a signed lag $\Delta \theta ={\theta }_d-{\theta }_p$ and a positive trailing distance $s=-\Delta \theta$. A grain released without a velocity kick from a circular parent orbit has $\Delta \theta =2\beta (\sin {\theta }_d-{\theta }_d)+O({\beta }^2{\theta }_d)$. Keeping track of this sign avoids mistaking trailing motion for leading motion.

For a single-stream dust tail with injection rate $\dot{N}$, monotone angular distance $s(t)$, and survival probability $P(t)$, the number per unit angle is $n(s)=\dot{N}P(t(s))/\dot{s}$. A vanishing angular drift creates an idealized density enhancement; velocity dispersion rounds it.

For conjugate exponents $p,q\in (1,\infty )$, every bounded linear functional on an Lp space is uniquely $h\mapsto \int hgd\mu$ for $g\in L^q$, with functional norm $\Vert g{\Vert }_q$. This holds on arbitrary measure spaces. Apply the Radon-Nikodym theorem on finite-measure pieces, then use support localization of an Lp functional to obtain one global density without assuming sigma-finiteness of the entire measure.

For a dual pair in a monoidal category, the contraction $e:X\otimes Y\to I$. In a left closed monoidal category, evaluation also denotes the adjunction map $[M,N]\otimes M\to N$.

The unit-valued duality map $n:I\to Y\otimes X$ of a dual pair in a monoidal category, constrained together with evaluation by both snake identities.

The two triangular identities $(e\otimes 1_X)(1_X\otimes n)=1_X$ and $(1_Y\otimes e)(n\otimes 1_Y)=1_Y$, with canonical associators and unitors inserted, for a dual pair in a monoidal category.

For a linear map between finite-dimensional vector spaces, the image of its dual map is the annihilator of a vector subspace given by its kernel. Inclusion follows by composition. Conversely a linear functional vanishing on the kernel descends to the image and extends to the target by basis extension. The identity and the rank-nullity theorem show that a linear map and its dual have equal rank.

Every continuous group homomorphism ${\mathbb{R}}^n/\Lambda \to {\mathbb{C}}^{\times }$ is $x+\Lambda \mapsto e^{2\pi i⟨u,x⟩}$ for a unique $u$ in the dual lattice. Compactness forces unit modulus, and lifting the character to the simply connected cover makes its phase a real linear functional. These characters are the frequencies of torus Fourier series.

A double coset is generally not a subgroup, so Lagrange's theorem does not force its size to divide the ambient group order. In $S_3$, take $H=K=⟨(12)⟩$. Then $HeH$ has size two, while $H(13)H$ has size four, which does not divide six. The general size formula $|HgK|=|H||K|/|H\cap gK{g}^{-1}|$ explains why different double cosets can have different sizes.

For the order-eight dihedral group generated by $r$ of order four and $s$ of order two with $srs=r^{-1}$, let $H=K=⟨s⟩$. There are exactly three double cosets: $H$, $H{r}^{2}H$, and $HrH$, of sizes two, two, and four. Their number does not divide eight. This differs from the number of ordinary cosets, which divides group order by Lagrange's theorem.