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An algebraic root can depend on a parameter through fractional powers even when the polynomial coefficients have ordinary power series. For example, the roots of $t^3={ϵ}^2$ are on the scale ${ϵ}^{2/3}$. Dominant balance for algebraic roots identifies the exponent, and local expansion of the scaled root supplies successive coefficients.

A Ricci-flat manifold has vanishing Ricci tensor. This is weaker than vanishing Riemann curvature tensor: the latter describes complete local flatness. A Kerr black hole supplies a Lorentzian manifold that is Ricci flat in its vacuum region but has nonzero Riemann curvature tensor.

For a real Hilbert space, $E(x)=\alpha J(cx-y)+⟨x,z⟩$, $\alpha >0$ and $c\ne 0$, identify $z$ with its vector by the Riesz representation theorem. Completing the square in $\Vert x-q{\Vert }^2/2+⟨x,z⟩$ shifts the input to $q-z$. Substitute $v=cx-y$ and multiply by $c^2$ to obtain the displayed formula, valid also for negative $c$. If $c=0$ and $J(-y)<\infty$, the answer is $q-z$; if $J(-y)=\infty$, the objective is identically infinite and no proper proximal problem remains.

On a Banach space one can define the set of minimizers of $E(x)+\Vert x-q{\Vert }^2/2$, but existence and uniqueness need additional geometry or coercivity hypotheses. The optimality condition is $0\in \partial E(x)+\mathcal{J}(x-q)$, with the duality mapping $\mathcal{J}=\partial (\Vert \cdot {\Vert }^2/2)$. The Hilbert completion-of-square shift by a dual vector does not generalize unchanged. On $({\mathbb{R}}^2,\Vert \cdot {\Vert }_1)$ with $E(x)=x_1+x_2$ and $q=0$, every $x_1,x_2\le 0$ with $x_1+x_2=-1$ is a minimizer; the vector $(-1,-1)$ is not one.

The proximal map of $\tau \Vert \cdot {\Vert }_2$ sends $x$ to $(1-\tau /\Vert x{\Vert }_2{)}_{+}x$, with zero assigned at $x=0$. This removes vectors inside the radius-$\tau$ ball and shortens all larger vectors by a fixed radial distance. It is useful for groupwise sparsity.

An implication is derivable from a theory precisely when its conclusion is derivable after adjoining its antecedent. Together with classical contradiction rules, this identifies failure to prove a proposition with consistency after adjoining its negation.

For motion on a fixed axis, the signed spatial four-acceleration in the instantaneous rest inertial frame is $a={\gamma }^3\dot{u}$. Its magnitude is the scalar proper acceleration. Constant signed $a$ from rest gives $\gamma u=at$ and the hyperbolic motion worldline $x=(c^2/a)(\sqrt{1+a^2{t}^2/c^2}-1)$.

A short exact sequence presenting a module as a quotient of a projective module. The kernel need not be projective. Continuing a projective presentation gives a projective resolution.

Set $A=kQ$, $P_i=A{e}_i$, $X_i=e_{i}X$. There is an exact sequencewhere $d(p\otimes x)=p\rho \otimes x-p\otimes f_{\rho }(x)$ in the source and target summands, and the augmentation is $p\otimes x\mapsto px$. Each $P_i$ is a summand of $A$, so the two terms are projective modules. This also works for infinite-dimensional modules.

The orthonormal quantum states $|{\alpha }_s⟩=(|0⟩+(-1{)}^s{e}^{-i\alpha }|1⟩)/\sqrt{2}$, $s=0,1$, define an equatorial projective measurement. The measured observable is $\cos \alpha X-\sin \alpha Y$, with eigenvalue $(-1{)}^s$. Thus $\alpha =0$ measures $X$, and $\alpha =\pi /2$ measures $-Y$ with these outcome labels. The sign convention matters when comparing different definitions of equatorial bases.

The Möbius group is the group of all Möbius transformations of the Riemann sphere, under function composition. An invertible $2\times 2$ complex matrix represents $z\mapsto (az+b)/(cz+d)$, and nonzero scalar multiples represent the same transformation. Rescaling a representative to determinant one gives a surjective group homomorphism $S{L}_2(\mathbb{C})\to \mathcal{M}$ with kernel of a group homomorphism $\{I,-I\}$. Thus it is also the projective linear group $PG{L}_2(\mathbb{C})$ and the quotient $PS{L}_2(\mathbb{C})=S{L}_2(\mathbb{C})/\{\pm I\}$.

Uniformly diagonalize against the formula expressing membership in $W_e$ to obtain a sentence saying that its own code is not in $W_e$. If every enumerated sentence is true, this new sentence cannot be enumerated, and hence is true. Its code is a computable new element of arithmetic truth outside $W_e$.

For finite sets of nonzero real numbers, the multiplicative energy is the number of quadruples $(a,b,c,d)\in A\times B\times A\times B$ satisfying $a/b=c/d$. Swapping $b,d$ shows that it also counts quadruples with $ab=cd$. Thus $E(A,B)={\sum }_x{r}_{A\cdot B}(x{)}^2$, where $r_{A\cdot B}(x)$ counts product representations. The Cauchy-Schwarz inequality yields $E(A,B)\ge |A{|}^2|B{|}^2/|A\cdot B|$.

For a sum of $M$ Hermitian two-local terms of norm below one, repeat a first-order Lie-Trotter product formula step $r$ times. The first-order unitary product-formula error bound gives total error at most $t^2{\sum }_{j<k}\Vert [H_j,H_k]\Vert /(2r)<M(M-1)t^2/(2r)$ for $M>1$. Taking $r=\max (1,\lceil M(M-1)t^2/(2ϵ)\rceil )$ uses $Mr$ arbitrary two-qubit gates. At unit time and $M=O(n^2)$ this is a sufficient $O(n^6/ϵ)$ bound; additional commutation structure can improve it.

An almost sure event is a measurable event of probability one. Countable intersections of such events still have probability one, allowing a conclusion to hold simultaneously for countably many parameter choices. Further deterministic reasoning may then extend it to uncountably many parameters, as in Brownian Hölder regularity or nowhere monotonicity of Brownian motion. An uncountable intersection need not preserve probability one without such additional reasoning.

The Lyapunov moment inequality says that higher positive absolute moments give no smaller absolute moment sizes on a probability space. For a real random variable $Z$ and $0<r<s$,allowing an infinite value. For $r\ge 1$ the left expression is an Lp norm; below one it is a quasi-norm, so the usual triangle inequality terminology needs that distinction. The comparison uses a probability measure of total mass one; arbitrary unnormalized measure spaces need a factor depending on their total mass.

For a nonnegative count, the first moment size is its expected value, while the second is the square root of its second moment. Both ten-trial counts in second moment method have first moment size one. The independent count has second moment size $\sqrt{1.9}\approx 1.37840$, whereas the shared-outcome count $N=10B$ has second moment size $\sqrt{10}\approx 3.16228$. The second moment distinguishes models with the same average because its square increases the influence of rare large counts.

More generally, for $N=10B$ with a Bernoulli random variable $B$ of parameter $1/10$,It rises from one at $r=1$ toward ten as $r\to \infty$. Replacing $1/10$ by a parameter $0<q\le 1$ gives $10q^{1/r}$, which decreases toward zero for a fixed positive exponent when $q\downarrow 0$, making the rare-event and large-exponent limits different questions. For a scaled Rademacher random variable taking values $-2$ and two with equal probabilities, every absolute moment size is exactly two: increasing the exponent does not distinguish signs when the absolute value is constant.

For finite $\mathbb{E}|Z{|}^s$, apply the reversed Jensen's inequality to the strictly concave function $t^{r/s}$ and the nonnegative random variable $|Z{|}^s$:Taking the positive $r$th root proves the claim. Equality in this finite case holds exactly when $|Z|$ is constant almost surely. If the higher moment is infinite, the upper bound is infinite and remains valid. Exponent zero is excluded from the statement; logarithmic or geometric limiting summaries require separately stated conventions, especially when $Z$ can vanish.

A probability distribution is nonarithmetic when it is not concentrated on $d\mathbb{Z}$ for any $d>0$. Every law with a probability density is nonarithmetic. This hypothesis removes lattice oscillations from the key renewal theorem.

A mixture distribution first chooses a component $j$ with nonnegative mixture weights $w_j$ summing to one, then samples from its law $P_j$. Consequently $P(A)={\sum }_j{w}_j{P}_j(A)$, and any finite moment-generating function is the same weighted sum of the component transforms. A component may be a Dirac measure, so a mixture may have both discrete and continuous parts.

For a positive random variable $X$ with finite nonzero expected value $\mu$, its integrated tail law has density $f_I(x)=\mathbb{P}(X>x)/\mu$ on $x\ge 0$. The tail integral formula for moments normalizes it. For $r>0$ its exponential transform is $\int e^{rx}{f}_I(x)dx=[M_X(r)-1]/(\mu r)$, with extended-value interpretation when needed.

For a real random variable $X$ with mean $\mu =\mathbb{E}X$, finite positive variance $0<{\sigma }^2=\mathrm{Var}(X)<\infty$, and finite fourth moment, kurtosis is the fourth central moment divided by the square of the variance:The standard score $Z$ measures distance from the mean in standard deviations. Its fourth power gives a value ten standard deviations away a weight $10^4=10000$ inside the expected value. Thus a small probability mass far from the mean can strongly affect kurtosis. It is useful for describing the contribution of extreme standard scores, but does not by itself describe the height or shape of a probability density function near its peak.^[ref]

The normal distribution has kurtosis $3$. A random scale can increase kurtosis even when the conditional distribution is normal. A constant random variable has zero variance, so this ratio is undefined. If the variance is finite and positive but the fourth central moment is infinite, the ratio can be written $+\infty$ in an extended sense; it is not a finite kurtosis. If the variance itself is infinite or the mean is undefined, the displayed normalization is unavailable.