If an equivalence relation is closed in a space's square and the quotient map is open, the quotient is Hausdorff. An inequivalent pair has an open rectangle avoiding the relation. The two factor images are open quotient neighborhoods and are disjoint, since an intersection would give equivalent representatives in that rectangle. Conversely, a Hausdorff quotient always has a closed equivalence relation by pulling back its closed diagonal.

For a normal subgroup $N$ of $G$, projection to the quotient group gives a bijection between subgroups of $G$ containing $N$ and subgroups of $G/N$. The maps are $H\mapsto H/N$ and inverse image under projection. Their two composites are identities because each relevant subgroup is a union of cosets of $N$. They preserve inclusions and normality; consequently a composition series in a quotient lifts to a subnormal chain above $N$ with the same composition factors.

A quasi-steady-state approximation for enzyme-substrate complexes can remain useful when total enzyme and initial substrate concentrations are comparable. The relevant parameter can involve the sum $s_0+K_m$, rather than $s_0$ alone. In a mechanism-based inhibition model with $s_0/K_m=O(\epsilon )$, bounded intermediate decay rates and an inactivation rate of order $\epsilon$, the fast complexes adjust in a layer of scaled width $O(\epsilon )$. On bounded slow-time intervals the main complex follows the substrate, the later complex and accumulated inactivated enzyme remain $O(\epsilon )$, and the substrate has leading exponential depletion. An outer algebraic complex concentration does not satisfy the unbound initial condition inside that initial layer.

For a convergent quasi-Newton method sequence $x_k\to x_{\ast }$ with $\nabla f(x_{\ast })=0$, nonsingular $H_{\ast }={\nabla }^{2}f(x_{\ast })$, and continuous Hessian nearby, the displayed condition implies q-superlinear convergence. Here $B_k{s}_k=-\nabla f(x_k)$ and $s_k=x_{k+1}-x_k$. The condition requires Hessian accuracy only along the actual steps, rather than convergence of the full matrix approximations. It is associated with John E. Dennis and Jorge Moré.

Indeed, the first-order expansion of the gradient gives $\nabla f(x_{k+1})=-(B_k-H_{\ast })s_k+o(\Vert s_k\Vert )=o(\Vert s_k\Vert )$. Nonsingularity of $H_{\ast }$ gives $\Vert x-x_{\ast }\Vert \le C\Vert \nabla f(x)\Vert$ locally. Consequently $\Vert e_{k+1}\Vert \le {\epsilon }_k(\Vert e_k\Vert +\Vert e_{k+1}\Vert )$ with ${\epsilon }_k\to 0$. Rearrangement proves $\Vert e_{k+1}\Vert /\Vert e_k\Vert \to 0$. Finite termination is understood separately from ratios with zero denominators.

Consider the closed SIR model in count variables, with $\dot{S}=-rSI$, $\dot{I}=rSI-aI$, transmission coefficient $r>0$ and recovery rate $a>0$. An ideal intervention stops all further transmission at the first upward crossing of $I=I_{\ast }$. This is a mathematical contact-removal rule; its instantaneous perfect effect is an assumption of the model.

Before intervention,so $I+S-(a/r)\log S$ is a first integral of an ordinary differential equation. At the trigger, the remaining susceptible individual count $S_{\ast }$ is a root of an equation satisfyingThis is a transcendental equation, with $S_{\ast }>0$, rather than a polynomial equation.

For an initially growing outbreak with $I_0>0$ and $S_0>a/r$, the unmodified SIR model reaches its infectious-count peak at $S=a/r$, withA trigger with $I_0<I_{\ast }\le I_{\mathrm{peak}}$ is therefore reachable on this growth branch. The function $S-(a/r)\log S$ is strictly increasing for $S>a/r$, so the trigger equation has exactly one solution in $a/r\le S_{\ast }<S_0$. A threshold above $I_{\mathrm{peak}}$ is never reached; choosing another mathematical root would not describe the first upward crossing.

Once the assumed intervention stops transmission, recovery can continue but $S$ no longer changes. The number of new infections among those initially susceptible individuals is thus $S_0-S_{\ast }$. The initial $I_0$ is excluded from that count. An intervention with residual transmission, or a model with unobserved exposed individuals, requires a different final-size calculation.

A ket $|\psi ⟩$ denotes a vector representing a quantum state, and its bra $⟨\psi |$ is the linear functional taking $|\varphi ⟩$ to $⟨\psi |\varphi ⟩$. The map from a ket to its bra is conjugate-linear. In the physics convention, $⟨\psi |\varphi ⟩$ is conjugate-linear in $\psi$ and linear in $\varphi$. The adjoint operator is characterized by $⟨\psi |A\varphi ⟩=⟨A^{\ast }\psi |\varphi ⟩$. Products such as $|\psi ⟩⟨\varphi |$ are rank-one operators, while $⟨\varphi |\psi ⟩$ is a scalar inner product.

Let orthonormal energy eigenvectors be ${\Psi }_1,{\Psi }_2$, and let a measured observable have eigenvectors $v_{+}=\cos \alpha {\Psi }_1+\sin \alpha {\Psi }_2$ and $v_{-}=-\sin \alpha {\Psi }_1+\cos \alpha {\Psi }_2$. If the initial quantum state is $v_{-}$, unitary time evolution multiplies each energy component by $e^{-it{E}_j/\hslash }$. Its amplitude in $v_{+}$ is $\sin \alpha \cos \alpha (e^{-it{E}_2/\hslash }-e^{-it{E}_1/\hslash })$. The Born rule gives the displayed probability. The energy gap fixes the oscillation frequency, while the angle between the measurement and energy bases fixes its amplitude. Energy degeneracy or coinciding bases makes the transition probability vanish.

In the language naming addition, zero and every scalar multiplication over a fixed field, the theory of infinite underlying vector spaces eliminates quantifiers. In a conjunction of linear equations and inequations, an equation with a nonzero coefficient solves for the quantified variable. Otherwise the variable is excluded from only finitely many values, so infinitude supplies a witness. Disjunction and induction finish elimination. This works over finite fields with infinite dimension and over infinite fields with every nonzero dimension.

Morleyization adds a predicate symbol for each first-order formula and an axiom equating that predicate symbol with its defining logical formula. Every original first-order model has a unique expansion. Translating an expanded logical formula back to the original first-order language and then using its named predicate symbol gives quantifier elimination. Completeness of the original first-order theory implies completeness of the expanded first-order theory.

A projective quadric is the zero set of a nonzero homogeneous quadratic polynomial in projective space. Over a field of characteristic different from two, represent it by a symmetric matrix $B$, up to a nonzero scalar. The gradient at a representative $v$ is $2Bv$, so the singular locus is the projectivization of $\ker B$. Thus the quadric is nonsingular exactly when $B$ is invertible. In the complex projective plane, ranks three, two, and one give respectively a smooth projective conic, two distinct projective lines, and a double projective line.

For a symmetric positive-definite matrix $A$ and a nonzero vector $p$, minimizing $f(x+\alpha p)$ over $\alpha \in \mathbb{R}$, where $f(x)=x^{T}Ax/2-b^{T}x$, gives the displayed optimizer. Put $r=b-Ax$. Expansion gives $f(x+\alpha p)=f(x)-\alpha p^{T}r+{\alpha }^2{p}^{T}Ap/2$, a strictly convex quadratic function of $\alpha$. At its minimum,Strict decrease is therefore equivalent to $p^{T}r\ne 0$. This is minimization over a full line, so the optimizer can be negative. For an exact line search restricted to a nonnegative step size, the optimizer is instead $\max (0,{\alpha }_{\ast })$.

The quadratic form gradient gives $D{q}_x(h)=2x^{T}Ah$. Since $A$ is invertible, this linear map is nonzero exactly when $x\ne 0$. The regular level set theorem therefore makes every nonzero point of $C_{\alpha }$ a smooth point of dimension $n-1$. If $\alpha \ne 0$, the entire level set is smooth.

If $\alpha =0$ and the level set contains a nonzero point, the form is an indefinite quadratic form. Its only singular point is the origin. To prove this rather than merely observe a vanishing derivative, use Sylvester's law of inertia to write $q=|u{|}^2-|v{|}^2$ with both sign spaces nonzero. The null vectors $e_i\pm f_j$ span the whole ambient space. Their straight-line curves through the origin lie in the cone. If a relatively open neighborhood of the origin in the cone were an embedded submanifold, its tangent space would contain all these directions and hence have dimension $n$. It would then be open in the ambient space, impossible for the zero set of this nonzero quadratic form. Thus the smooth locus is $C_0\setminus \{0\}$.

Apply the product rule to the scalar quadratic form and transpose its last scalar term: $x^{T}Adx=(dx{)}^T{A}^{T}x$. Only the symmetric part of $A$ contributes to its value or its derivative with respect to $x$. A varying matrix also contributes the independent term $x^T(dA)x$. No assumption that $A$ is symmetric is needed. Repeating this rule and the chain rule computes mixed derivatives when vector and matrix arguments depend on separate parameters.

A QR shift is a scalar ${\mu }_k$ subtracted before the QR factorization in a QR algorithm step and restored after reversing the factors:The final identity uses the orthogonality of $Q_k$, proving that the eigenvalues are preserved. Choosing a shift near an eigenvalue can accelerate block deflation in the QR algorithm. A common choice is the current bottom-right entry, but it can stall, as in Exact-shift diagonalization of a two-state swap matrix.

For $A=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, one shifted QR algorithm step with real QR shift $\mu$ gives, with a consistent choice of factor signs,To verify this, factor $A-\mu I=QR$ with $Q=(1+{\mu }^2{)}^{-1/2}\left(\begin{array}{cc}-\mu & -1\\ 1& -\mu \end{array}\right)$. Its columns are orthonormal, and $R=Q^T(A-\mu I)$ is upper triangular. Then $A_1=Q^{T}AQ$. Thus $A_1$ is diagonal exactly when $\mu =\pm 1$, the two eigenvalues. The bottom-right-entry shift is zero and leaves the matrix unchanged under the positive-diagonal factor convention. This illustrates why a poorly chosen QR shift can stall even a two-by-two symmetric eigenvalue computation.

The control state $q$, unread input word $w$, and stack contents $\alpha$ of a pushdown automaton. A fixed convention must say which end of $\alpha$ is the top. With the top on the left, replacing top $X$ by $\gamma$ changes $X\beta$ into $\gamma \beta$.

A pushdown automaton accepts a word if some finite run consumes all its input and reaches a designated final control state, regardless of the remaining stack contents. Reaching a final control state while leaving unread input is insufficient.

The unique positive square root of a density operator is a Hilbert-Schmidt operator $W$ with $\Vert W{\Vert }_{\mathrm{HS}}^2=\mathrm{tr}\rho =1$. It reproduces expectations by $\mathrm{tr}(A\rho )=\mathrm{tr}(W^{\ast }AW)$. In chosen orthonormal bases, vectorizing its entries gives a normalized bipartite vector $|W⟩⟩={\sum }_{r,s}{W}_{rs}|r⟩|s⟩$ whose reduced density operator is $W{W}^{\ast }=\rho$. Thus the positive operator amplitude and the vector purification are related but distinct objects.

More generally a coefficient matrix $C$ purifies $\rho$ when $C{C}^{\ast }=\rho$. For square nonsingular $C$, the matrix $U=C^{-1}\sqrt{\rho }$ obeys $U{U}^{\ast }=I$ and is unitary, so $\sqrt{\rho }=CU$. In an eigenbasis of $\rho$, $C=\mathrm{diag}(\sqrt{p_r})U^{\ast }$; its conjugated unitary columns give the second-factor Schmidt decomposition vectors. This is the matrix form of unitary freedom of purification.

Dilatation of the pupil; it is a sign requiring interpretation in context, rather than an identification of a unique poison.

This formal language is not a context-free language. For pumping length $p$, use $a^p{b}^{p^2}{c}^p$. A pumpable window of length at most $p$ cannot meet both outer blocks. If pumping adds $\alpha ,\beta ,\gamma$ copies of $a,b,c$, then $\alpha \gamma =0$ and $1\le \alpha +\beta +\gamma \le p$. Preserving the product constraint would require $\beta =p(\alpha +\gamma )$, impossible both when $\alpha +\gamma =0$ and when it is positive. This contradicts the pumping lemma for context-free languages.