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Points ${\theta }_r$ have circular spacing at least $\delta$ when $\Vert {\theta }_r-{\theta }_s\Vert \ge \delta$ for distinct $r,s$, where $\Vert t\Vert ={\min }_{k\in \mathbb{Z}}|t-k|$. The distance is measured on the circle group $\mathbb{R}/\mathbb{Z}$, so points near zero and one can be close.

If ${\theta }_r$ have circular spacing at least $\delta$, thenMultiply the exponential sum by $e(-Mt)$, apply the Sobolev–Gallagher inequality on disjoint arcs of length $\delta$, and sum. The finite-interval Parseval identities and Cauchy-Schwarz inequality bound the derivative contribution by $2\pi N\sum |a_n{|}^2$.

For coefficients $b_n$ supported on an interval of $H$ consecutive integers, set $B=\sum b_n$ and $B_p(a)={\sum }_{n\equiv a(\mathrm{mod}p)}{b}_n$. ThenThe orthogonality of roots of unity gives $p{\sum }_a|B_p(a)-B/p{|}^2={\sum }_{r=1}^{p-1}|{\sum }_n{b}_{n}e(rn/p){|}^2$. The distinct fractions $r/p$ have circular spacing at least $Q^{-2}$; now apply the exponential-sum large sieve.

For coefficients supported on $N$ consecutive integers,The star restricts to primitive Dirichlet characters. Their finite Fourier transform of a primitive Dirichlet character has normalization of absolute value $\sqrt{q}$. Orthogonality of Dirichlet characters therefore bounds the weighted contribution for one modulus by ${\sum }_{\begin{array}{c}a\mathrm{mod}q\\ (a,q)=1\end{array}}|{\sum }_n{a}_{n}e(an/q){|}^2$. Sum over $q$ and use the exponential-sum large sieve on the distinct reduced fractions of denominators at most $Q$.

A weakly inaccessible cardinal is an uncountable regular cardinal that is a limit cardinal. It need not be a strong limit cardinal. Under the Generalized continuum hypothesis, it is a strongly inaccessible cardinal, since every smaller infinite $\lambda$ satisfies $2^{\lambda }={\lambda }^{+}<\kappa$.

An inaccessible cardinal $\kappa$ whose inaccessible ordinals below it form a stationary set. Equivalently, every predicate expansion of $V_{\kappa }$ has an inaccessible elementary rank level below $\kappa$.

A linear differential-equation solution may be sought as ${\int }_C{e}^{xt}f(t)dt$. Integration by parts transfers the factor $x$ to derivatives of $f$, giving an equation for the amplitude and boundary terms. Admissible contours must cancel those terms and make the integral converge.

An axisymmetric harmonic potential outside a sphere of radius $r_0$, with far field $v_{0}r\cos \theta$ and zero radial derivative on the sphere, isThe regular spherical expansion in Legendre polynomials fixes all coefficients. A spatial constant remains arbitrary if only the far-field gradient is specified, and is fixed by prescribing the potential difference at infinity.

Initialize Landweber iteration by Tikhonov regularization, then use step $1/\alpha$. Writing $B=A^{\ast }A$ and $R=I-B/\alpha$, the iterate is $x_n=R^n(B+\alpha I{)}^{-1}{A}^{\ast }y+{\alpha }^{-1}{\sum }_{j=0}^{n-1}{R}^j{A}^{\ast }y$. For nonzero $A$, the sufficient convergence condition is $\alpha >\Vert A{\Vert }^2/2$. This explicit method differs from Iterated Tikhonov regularization, whose update is $(B+\alpha I)x_{n+1}=\alpha x_n+A^{\ast }y$.

For nonzero compact operator $A$, step $0<\tau <2/\Vert A{\Vert }^2$, and data admitting a minimum-norm least-squares solution $x^{†}$, zero-initialized Landweber iteration satisfiesThe Picard criterion makes the majorant summable, so the dominated convergence theorem proves this limit. No mesh-independent or singular-value-independent strict contraction factor is required. At $\tau =2/\Vert A{\Vert }^2$, the top singular mode has multiplier $-1$ and may not converge. Without solvable projected data, the iterates need not have a limit in the solution Hilbert space, even when their residuals approach the least-squares infimum.

The Landau approximation evaluates a statistical field's partition function at a stable stationary configuration, omitting fluctuation-loop corrections to the free energy. Functional differentiation of that stationary value still gives a connected linear response: the inverse of the saddle's quadratic fluctuation operator. Freezing fields before differentiating would lose this response. A Gaussian field theory, in contrast, permits exact integration of its quadratic fluctuations.

For a real statistical source convention $Z[h]=\int \mathcal{D}\varphi e^{-H_0[\varphi ]+\int h\varphi }$, define $F=-\log Z$ and the displayed Legendre transform. On a differentiable stable branch, $\delta \Gamma /\delta m=h$. In the Landau approximation, the branch expression is ${\Gamma }_{\mathrm{L}}[m]=H_0[m]$. The exact Legendre effective free energy is convex; a bare nonconvex Landau free energy is a local approximation and requires branch or coexistence interpretation.

For a real unrestricted scalar order parameter and $A=A_2{m}^2/2+A_3{m}^3/3+A_4{m}^4/4$ with $A_3\ne 0$ and $A_4>0$, the disordered global minimum is replaced discontinuously by a nonzero minimum at the displayed phase coexistence condition. At coexistence $A=A_4{m}^2[m+2A_3/(3A_4){]}^2/4$. Ordered stationary points first appear at the spinodal point $A_2=A_3^2/(4A_4)$, while the disordered point loses local stability at $A_2=0$; neither local limit is the equilibrium transition condition.

A steady spatial forcing at twice the critical wavenumber permits coupling of the negative critical Fourier mode to the positive critical mode. A weakly nonlinear expansion then permits a term $\beta \overline{A}$ in the Landau amplitude equation. Its coefficient is found by projecting the resonant forcing-advection terms onto the adjoint eigenfunction. Reflection-symmetric forcing permits real coefficients. The symmetry permission does not prove a nonzero coefficient: it can vanish for a particular model or mode structure.

A lambda abstraction $\lambda x.t$ binds the variable $x$ in the lambda term $t$. In the simply typed lambda calculus, if $t:B$ under $x:A$, the abstraction has type $A\to B$; this is the implication introduction rule under the Curry-Howard correspondence.

A Lagrangian relaxation moves selected constraints into the objective with appropriately signed Lagrange multipliers. Maximizing the relaxed objective over a larger easy set gives an upper bound for a constrained maximization problem. Optimizing that bound need not solve the original integer problem.

The Lagrange dual function is the infimum of a Lagrangian over its primal variables. It is concave in the multipliers, even before convexity of the primal problem is assumed. Maximizing it over sign-compatible multipliers gives a lower bound on a minimization problem by weak duality; appropriate convex qualifications give strong duality.

The stationary squared process has the positive series $X_t^2={\alpha }_0{\sum }_{k\ge 0}{\alpha }_2^k{\prod }_{j=0}^k{\epsilon }_{t-2j}^2$. Even and odd observations therefore use disjoint families of independent random variables. This explains zero lag-one covariance of the squares despite dependence at lag two.

The doubly infinite ladder graph has graph vertices $\mathbb{Z}\times \{0,1\}$, horizontal edges $\{(j,s),(j+1,s)\}$, and vertical edges $\{(j,0),(j,1)\}$.

For $0<c\le w_k<\infty$, interpret the nonnegative sum as an extended-valued functional on the l2 sequence space. Finiteness implies $u\in {\ell }^1$, but unbounded weights can make it infinite even there. It is proper and convex. Every finite partial sum is norm-continuous, so their supremum is sequentially lower semicontinuous and also weakly lower semicontinuous. This is useful as a sparsity penalty in variational regularization.