For arrays of values in $[0,1]$, let $U_N$ average a product of $d$ entries over all ordered index tuples from $N$ coordinates, and let $V_N$ average it over the tuples with distinct indices. Decompose the first average as $U_N=\alpha V_N+(1-\alpha )W_N$, where $\alpha =(N{)}_d/N^d$ and the repeated-index average $W_N$ lies in $[0,1]$. This gives $|U_N-V_N|\le 1-\alpha$. For uniformly sampled indices, a union bound over the $\left(\genfrac{}{}{0ex}{}{d}{2}\right)$ pairs bounds the collision probability by $\left(\genfrac{}{}{0ex}{}{d}{2}\right)/N$. This deterministic estimate is the bridge between empirical-product limits and the De Finetti theorem.

On the Dahlquist test equation, the symmetric two-step two-derivative formula has amplification polynomial $F_z(w)=(z^2-7z+15)w^2-16zw-(z^2+7z+15)$. For every real $z<0$, its leading coefficient is positive and $F_z(-1)=2z<0$, whereas $F_z(w)\to +\infty$ as $w\to -\infty$. The intermediate value theorem therefore gives a real root below minus one. Thus the method is not A-stable, without any appeal to a multistep order barrier.

On the Dahlquist test equation, write the amplification polynomial as $D{\xi }^2-16z\xi -A$, where $D=z^2-7z+15$ and $A=z^2+7z+15$. The identity $|A{|}^2-|D{|}^2=28\mathrm{Re}z(|z{|}^2+15)$ shows that both roots of a polynomial can lie inside the open unit disk only if $\mathrm{Re}z<0$. But the complex quadratic Schur criterion would then require $32|\mathrm{Re}z|(|z{|}^2+15)<28|\mathrm{Re}z|(|z{|}^2+15)$, which is impossible. Thus the strict linear stability domain is empty. This concerns robust damping of all modes, rather than convergence on fixed time intervals or the bounded-root stability set of the symmetric two-derivative formula.

For the symmetric two-step two-derivative formula, the closed-disk root condition for a multistep method fails off the imaginary axis. Positive real part makes the product of root moduli exceed one; negative real part fails the complex quadratic Schur criterion, as in the empty strict decay domain of the symmetric two-derivative formula proof. On $z=iy$, write $D=15-y^2-7iy=|D|e^{i\delta }$ and set $\xi =i{e}^{-i\delta }\eta$. The amplification polynomial becomes ${\eta }^2-16y\eta /|D|+1=0$. Its two roots are distinct and on the unit circle exactly when $|16y/|D||<2$, or $y^4-45y^2+225>0$. Thus the bounded-root set consists of the imaginary intervals $|y|<\sqrt{(45-15\sqrt{5})/2}$ and $|y|>\sqrt{(45+15\sqrt{5})/2}$, including $y=0$ and excluding the double-root endpoints. In particular, the set is unbounded. Unit-modulus oscillations are bounded but do not tend to zero.

A symmetric tensor invariant under rotations about a fixed nonzero vector $y$ has the form $\alpha I+\beta y{y}^T$. Choose the third coordinate axis parallel to $y$: rotation by a half-turn makes the mixed axial-transverse entries vanish, and all transverse rotations make the transverse block a scalar multiple of the identity. Its two transverse eigenvalues are $\alpha$ and its axial eigenvalue is $\alpha +\beta |y{|}^2$. The trace and the quadratic contraction $y^{T}Ty$ determine these two coefficients. Full isotropy holds exactly when $\beta =0$.

For $0<a(x)\le \beta$, the explicit conservative diffusion update has weights $k{a}_{m-1/2}/h^2$, $1-k(a_{m-1/2}+a_{m+1/2})/h^2$, and $k{a}_{m+1/2}/h^2$. Under the displayed restriction they are nonnegative and sum to one. With zero Dirichlet boundary conditions, each new value is a convex combination of old interior and boundary values, proving contraction in the discrete maximum norm. The spatial matrix is also symmetric with $v^T{L}_{h}v=-h^{-2}\sum a_{m+1/2}(v_{m+1}-v_m{)}^2$. This puts its eigenvalues in $[-4\beta /h^2,0]$, so the same restriction gives L2 norm contraction. A stable recurrence need not be consistent with a different proposed differential equation.

For forms of degrees $m,n$, the $r$th transvectant contracts $r$ pairs of indices using the invariant alternating form. Explicitly it is a scalar normalization of ${\sum }_{j=0}^r(-1{)}^j(\genfrac{}{}{0ex}{}{r}{j})({\partial }_x^{r-j}{\partial }_y^{j}f)({\partial }_x^j{\partial }_y^{r-j}g)$. These contractions realize the irreducible summands in the symmetric-power tensor-product decomposition.

Let $V$ be an elementary abelian $r$-group and let $r\nmid |H|$, with $H$ soluble. In each Sylow basis of $V⋊H$, $V$ is the unique Sylow $r$-subgroup and the product of the remaining Sylow subgroups is a complement to $V$. All complements are $V$-conjugate, and the stabilizer of $H$ under this conjugation is its fixed-vector space $C_V(H)$. Each complement has $b(H)$ Sylow bases, proving the formula. For the permutation module $V={\mathbb{F}}_5^4$ of $S_4$, the fixed-vector space is the one-dimensional constant-coordinate subspace, so the number is $5^3\cdot 12=1500$.

The sphere spectrum is $\mathbb{S}={\Sigma }^{\infty }{S}^0$. It is the unit of the smash product of spectra. Its integer suspensions of spectra represent stable homotopy groups.

For an Erlang distribution claim probability density function $f(x)={\beta }^{2}x{e}^{-\beta x}$ and $r=\lambda /c$, write $H=\varphi \ast f$. The survival integro-differential equation gives ${\varphi }^{\prime }=r(\varphi -H)$. The convolution satisfies $(D+\beta {)}^{2}H={\beta }^2\varphi$, because $H={\beta }^2{e}^{-\beta u}{\int }_0^u(u-t)\varphi (t)e^{\beta t}dt$. Applying $(D+\beta {)}^2$ to the first equation eliminates $H$ and gives the displayed third-order equation. The original integral equation also supplies ${\varphi }^{\prime }(0)=r\varphi (0)$ and ${\varphi }^{\prime \prime }(0)=r^2\varphi (0)$; these conditions must not be lost when solving the higher-order equation.

For a geometric morphism, the characteristic map of $f^{\ast }\top$ induces a comparison $f^{\ast }{\Omega }_{\mathcal{E}}\to {\Omega }_{\mathcal{F}}$ whose adjoint transpose is $\lambda$. Composing with a characteristic map classifies the inverse-image subobject. Thus monicity of $\lambda$ is equivalent to reflection of equality of subobjects under $f^{\ast }$, and hence to faithfulness of the inverse image: apply reflection to equalizers of parallel arrows. Conversely faithfulness reflects when a pulled-back monomorphism is an isomorphism, by comparing its characteristic map with truth; intersections then reflect equality of arbitrary subobjects.

The time translation is normalized at infinity. For the roots $r_{\pm }^2=M\pm \sqrt{M^2-Q^2}$, evaluate $V^{\prime }(r)/2$ to get the signed horizon values. The outer Hawking temperature in natural units is $\sqrt{M^2-Q^2}/(\pi r_{+}^3)$. The extremal limit has zero surface gravity; the neutral case has only the outer positive-radius horizon and temperature $1/(2\pi \sqrt{2M})$.

An object of a regular category is well-supported when its support of an object in a regular category is the terminal object, equivalently when $A\to 1$ is a regular epimorphism. Finite products of well-supported objects are well-supported, by pullback and composition stability of regular epimorphisms. A regular epimorphism has the same support as its codomain, and an object with a point $1\to A$ is well-supported.

A regular category is almost totally supported if every object is a well-supported object or a strict initial object. The case in which every object is well-supported is allowed. There can be at most one strict initial object up to isomorphism. In the Category of sets, these two cases distinguish nonempty sets from the empty set. A meet-semilattice regarded as a category satisfies the condition precisely when it has at most two objects up to isomorphism.

The first term uses the induced worldvolume metric from the supervielbein; the second is the Wess-Zumino brane coupling to the super-three-form. Its fermionic dependence is retained before bosonic truncation. The relative charge magnitude equals the brane tension, while $q=\pm 1$ sets the orientation and matching kappa symmetry projector sign. In an on-shell eleven-dimensional supergravity superspace, this action has kappa symmetry.

If $\kappa$ is a supercompact cardinal, each true ${\Sigma }_2$ statement, allowing parameters in $V_{\kappa }$, holds in $V_{\kappa }$. Write it as an existential witness followed by a ${\Pi }_1$ assertion. Choose a supercompact elementary embedding whose transitive target contains the witness in $V_{j(\kappa )}$. Downward absoluteness of the universal bounded-matrix assertion puts the witness in that rank segment, and elementarity reflects this to $V_{\kappa }$.

Cesàro summation assigns a value to a series when the arithmetic means of its partial sums approach a finite limit of a sequence. Here we use the first-order method, often written $(C,1)$. For a real or complex series beginning at index zero, defineIt is Cesàro summable to $L$ when ${\sigma }_N\to L$. The averaging is applied to the partial sums $s_n$, rather than to the original terms $a_n$. This distinction is essential: the terms can average to zero while their accumulated sum keeps growing. ^[ref]

For Grandi's series $1-1+1-1+\cdots$, the partial sums alternate between one and zero. The series is not a convergent series, but the two values occupy asymptotically equal proportions of the averaging window. In the declared index convention,so its first-order Cesàro sum is $1/2$. This names a particular summation method; it does not turn the oscillating partial sums into an ordinarily convergent sequence.

Why is this useful? An oscillating approximation can retain stable information when its cumulative average settles. Because the Cesàro theorem for convergent sequences preserves an existing limit of a sequence, this adds some new cases without changing the sum of any ordinarily convergent series. For Fourier series, averaging the Fourier partial sums gives a Fejér sum; the existing Fejér kernel and approximation articles explain its effect on reconstruction. Averaging does not guarantee success for every oscillating sequence, as the bounded partial sums need not be Cesàro summable example shows.

One historical motivation was the multiplication of series. In 1918, A. F. Andersen recalled Ernesto Cesàro's product theorem and cited his 1890 publication at the start of a study of products involving Cesàro summable series. This connects the theory with preserving useful operations on sums, as well as with assigning a value to an isolated example. His paper treats several orders of summability; the examples here use only $(C,1)$.^[ref]

A summation method that includes every ordinarily convergent series and assigns it its ordinary sum. The Cesàro theorem for convergent sequences makes first-order Cesàro summation regular. Regularity alone does not justify every rearrangement or multiplication of a divergent series.

If $p_0<p<p_1$, the displayed split of an $L^p$ function gives $\Vert g{\Vert }_{p_0}\le a^{1-p/p_0}\Vert f{\Vert }_p^{p/p_0}$ and $\Vert h{\Vert }_{p_1}\le a^{1-p/p_1}\Vert f{\Vert }_p^{p/p_1}$. Weighted multiplication with $1/p=(1-\theta )/p_0+\theta /p_1$ cancels the threshold, yielding $\Vert g{\Vert }_{p_0}^{1-\theta }\Vert h{\Vert }_{p_1}^{\theta }\le \Vert f{\Vert }_p$. Choosing $a=\Vert f{\Vert }_p$ also gives the continuous inclusion $\Vert f{\Vert }_{L^{p_0}+L^{p_1}}\le 2\Vert f{\Vert }_p$. The analogous split works if the larger endpoint is infinity.

The proper-divisor sum, or aliquot sum, adds the positive proper divisors:In particular $s(1)=0$, an empty sum. Removing $n$ from the sum-of-divisors function is essential: $\sigma (6)=12$, but its proper-divisor sum is $6$.