For a strictly feasible primal-dual conic optimization problem and a logarithmically homogeneous barrier, the central path consists of solutionsThe primal-dual gap is $⟨s,y⟩=\nu \mu$. Existence requires appropriate feasibility and boundedness hypotheses, rather than merely a full-rank constraint matrix. Linearizing these equations gives a central-path Newton system.

For a nondegenerate self-concordant barrier, the open local Hessian ballis contained in its barrier domain. This is the Dikin ellipsoid. It supplies a quantitative way to ensure that a damped Newton method stays interior, without relying on Euclidean distance to a possibly curved boundary.

In front of the contact line in the sediment-lubricated viscous mountain model, $h=d$ and $d_x=3{\mu }_{s}U(d-d_{\infty })/(\rho g{d}^3)$. For $\delta =d/d_{\infty }$ and $\ell =\rho g{d}_{\infty }^3/(3{\mu }_{s}U)$, integration gives $x/\ell =P(\delta )-P(3)$, with $P(\delta )={\delta }^3/3+{\delta }^2/2+\delta +\ln (\delta -1)$. The branch $1<\delta \le 3$ is unique and tends exponentially to the uniform far-field depth.

The difference between the angular momentum of circular coplanar orbits at fixed semi-major axes and the system's actual angular momentum along the reference normal. With circular momenta $L_j=m_j\sqrt{G{M}_{\star }{a}_j}$ it is ${\sum }_j{L}_j(1-\sqrt{1-e_j^2}\cos I_j)$. For small planar orbital eccentricities its quadratic part is $\frac{1}{2}{\sum }_j{L}_j|z_j{|}^2$. Its conservation in $\dot{z}=iAz$ requires $L_j{A}_{jk}=L_k{A}_{kj}$.

The smooth sections of a vector bundle form a module over the commutative ring $C^{\infty }(M)$ by pointwise multiplication. A vector bundle morphism over the identity induces a $C^{\infty }(M)$-linear map between these modules. Bundle morphisms from maps of smooth sections gives the converse, without assuming the bundle has a global frame.

If both sections for every coordinate are binary initial segments, then the family itself is a binary initial segment, apart from the possible family $(\mathcal{P}([n-1])\setminus \{[n-1]\})\cup \{\{n\}\}$. Any missing vertex earlier than a present vertex must disagree with it in every coordinate; the two are therefore complementary and consecutive in binary order on the discrete cube, forcing this sole exceptional shape.

The Mathieu equation is the displayed linear differential equation with periodic coefficients. Floquet theory describes its bounded and exponentially growing solutions, while its periodic Mathieu characteristic values mark the edges of parametric resonance gaps. The forcing normalization matters: replacing $t$ by $t/2$ rescales both the constant and periodic coefficients.

For $\gamma \ne 0$ and $x>0$, the change of variables $u=x^{\gamma }$ convertsinto $Y_{uu}+Y=0$, where $y(x)=Y(u)$. The chain rule cancels the first-derivative terms. Thus $\cos (x^{\gamma })$ and $\sin (x^{\gamma })$ form a pair of linearly independent solutions, with Wronskian $\gamma x^{\gamma -1}$. This is a variable-speed reparametrization of the harmonic oscillator equation.

A smooth invertible change of variables in $u_{xx}-f(x)u=0$ introduces a first derivative in time. Rescaling by ${\dot{x}}^{-1/2}$ removes it. Choosing $\dot{x}=f^{-1/2}$ locally for $f>0$ produces the constant leading coefficient used in the WKB approximation; negative $f$ requires a complex branch or the corresponding oscillatory real formulation.

For $t>0$, the change of variables $\tau =1/t$ and $u(t)=v(\tau )/\tau$ converts $t^4{u}^{\prime \prime }+{\lambda }^{2}u=0$ into $v^{\prime \prime }+{\lambda }^{2}v=0$. The chain rule gives $u^{\prime }=v-\tau v^{\prime }$ and $u^{\prime \prime }={\tau }^3{v}^{\prime \prime }$. Thus $u=t[A\cos (\lambda /t)+B\sin (\lambda /t)]$. The prefactor cancels the derivative term introduced by inverting the independent variable.

For $\psi (t)={\sum }_{m\le t}\Lambda (m)$, finite rearrangement and the Von Mangoldt divisor identity giveThe Stirling formula, or integral comparison for the logarithm, gives $T(x)=x\log x-x+O(\log (x+1))$ for real $x\ge 1$.

For chosen inequivalent unitary irreducible representations $\rho ,\sigma$ of a finite group and a $d_{\rho }\times d_{\sigma }$ matrix $M$, the average ${\mathbb{E}}_x\rho (x)M\sigma (x{)}^{\ast }$ is zero when $\rho \ne \sigma$. For $\rho =\sigma$, it is $(\mathrm{tr}M/d_{\rho })I$. This follows from the Schur lemma, since the average intertwines the two group representations. Equivalent representations in different bases require the corresponding intertwiner; the identity-matrix formula assumes literally the same representative.

For the pure state ${\lambda }_0|00⟩+{\lambda }_1|11⟩$ with real normalized Schmidt coefficients, the two-body Pauli correlators form $T=\mathrm{diag}(s,-s,1)$, $s=2{\lambda }_0{\lambda }_1$. Thus arbitrary local axes give $⟨(\mathbf{a}\cdot \sigma )\otimes (\mathbf{b}\cdot \sigma )⟩=s(a_x{b}_x-a_y{b}_y)+a_z{b}_z$. The negative $yy$ component follows from the phases in the Pauli operators.

A quadratic scan statistic maximizes a directional empirical second moment over candidate unit vectors. For a Gaussian random vector sample with a rank-one covariance spike in one of those directions, the matching statistic has its scale multiplied by the spike's eigenvalue. A chi-squared concentration inequality controls each direction.

For independent uniformly placed cyclic intervals of length $k<d$, an overlap is possible at at most $2k-1$ starting-point offsets, or at all $d$ offsets if that number exceeds $d$. Thus the overlap probability is at most $\min \{1,(2k-1)/d\}$. If $R=|S\cap T|$, then $0\le R\le k$ and $\mathbb{E}(e^{aR/k}-1)\le (2k/d)(e^a-1)$ for $a\ge 0$. The overlap law is not the hypergeometric distribution for unrestricted random subsets.

For spheres linked by a length $\ell (t)$, axial mobility and $F_1+F_2=0$ imply $\dot{X}=K(\ell )\dot{\ell }$ for their midpoint. The displacement over a periodic stroke is $\oint K(\ell )d\ell =0$. Unequal radii permit instantaneous midpoint motion but not net translation. This is an explicit one-shape-coordinate example of the scallop theorem.

If a periodic unimodular function $a$ and its inverse have absolutely summable Fourier coefficients, replacing ${\hat{\phi }}_0$ by $\hat{\phi }=a{\hat{\phi }}_0$ preserves the orthonormal integer-translate span and integrability of an integrable scaling function. In physical space this is an absolutely summable combination of integer function translations. The new MRA low-pass filter is $m(t)=a(2t)m_0(t)/a(t)$. A rough phase can therefore preserve the multiresolution analysis while changing MRA low-pass filter regularity.

With $\hat{\varphi }(\xi )=\int \varphi (x)e^{-ix\xi }dx$ in the Plancherel theorem sense, the integer translates of $\varphi \in L^2(\mathbb{R})$ are orthonormal exactly whenThe Fourier coefficients of the periodized energy are the translate inner products. Uniqueness of Fourier coefficients in L1 therefore proves both directions. No absolute integrability assumption on $\varphi$ is required.

A product of scalar superfields is a scalar superfield: the pullback of a product is the product of the pullbacks at the same transformed superspace point. Infinitesimally, the even supersymmetry variation obeys $\delta (ST)=(\delta S)T+S(\delta T)$. This follows from the graded Leibniz rule for the odd supercharges, after including their odd parameters.

A function $S(x,\theta ,\bar{\theta })=\varphi (x)$ is an allowed special configuration of an unconstrained superfield. The subspace where all other components remain zero is not invariant: $\delta S=(i\theta {\sigma }^{\mu }\overline{ϵ}-iϵ{\sigma }^{\mu }\bar{\theta }){\partial }_{\mu }\varphi$ produces components depending on Grassmann variables whenever $\varphi$ is nonconstant. Thus an isolated spacetime scalar field is not closed under supersymmetry. A constant gauge-singlet value is a trivial invariant exception.