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An affine minorant of $f$ is a function $x\mapsto ⟨a,x⟩+b$ that is everywhere at most $f(x)$. Every proper lower-semicontinuous convex function has such a minorant, by separating a point below its closed epigraph. Adding a positive quadratic then yields a coercive function, which establishes existence in proximal minimization.

Set $y_j=-\log (1-B_j)$ and let $e_C(a)=E(a,C)$. For positive integer capacities,The function $U$ is the carried load of an Erlang loss resource evaluated at its inverse blocking parametrization. It is continuous and strictly increasing from zero to $C$. Hence $F$ is a coercive function and a strictly convex function on the nonnegative orthant. Its first-order conditions are the fixed-routing Erlang fixed point equations, including zero prices at unused links.

For a finite support whose convex hull has full dimension, every interior target mean is realized by a unique natural parameter. Minimize the coercive function $\log {\sum }_n{e}^{\theta \cdot n}-\theta \cdot \mu$; its gradient is the mean discrepancy and its Hessian matrix is the positive-definite covariance matrix of the support.

Completing squares reveals the geometry of the sheared quartic double-well gradient flow:The critical points satisfy $2(x+y)=0$ and $4x^3-2x+2y=0$, so they are $(0,0)$, $(1,-1)$ and $(-1,1)$. The Hessian matrix isAt the origin its determinant is $-8$, so $(0,0)$ is a saddle point, with $V=0$. At either other critical point, the determinant is $16$ and the upper-left entry is $10$, so it is a positive-definite matrix. They are strict local minima; the completed squares show that both are global minima, with $V=-1$.

A contour at level $c$ obeysThere are no contours for $c<-1$, two isolated minimum points for $c=-1$, and two separate closed ovals for $-1<c<0$. At $c=0$ the two lobes meet at the saddle point; near the origin their tangents are $y=(-1\pm \sqrt{2})x$. For $c>0$ one closed contour surrounds both minima.

Along the gradient flow, the gradient-flow dissipation identity isAt the stated initial point, $V(1,-1/2)=-3/4$. Its trajectory stays in the invariant sublevel set $V\le -3/4$. This set is compact because $V$ is a coercive function. It also excludes every point with $x=0$, where $V=y^2\ge 0$; continuity therefore keeps the trajectory in its positive-$x$ component.

To justify its limit rather than merely infer it from the sketch, integrate the dissipation identity: ${\int }_0^{\infty }\Vert \nabla V{\Vert }^{2}dt$ is finite. On this compact set the gradient and its time derivative are bounded, so $\Vert \nabla V{\Vert }^2$ is uniformly continuous. A nonnegative uniformly continuous function with finite integral tends to zero: otherwise separated intervals of a fixed positive height and width would force an infinite integral. Every accumulation point is consequently a critical point. In the positive-$x$ component below this energy level the only one is $(1,-1)$. Compactness then implies convergence, andThis is a compact gradient-flow trapping criterion; the drawn trajectory illustrates, rather than replaces, the convergence proof.

The exact normalizing sum can be expensive on a large loss network. The Erlang fixed point approximation replaces joint resource acceptance by a product of marginal acceptances. Here consider unit requirements $A_{jr}\in \{0,1\}$, positive integer capacities and finitely many fixed routes. Let $B_j$ be resource blocking and $q_j=1-B_j$. A route-$r$ call contributes to the offered traffic at resource $j$ after surviving the other resources on its route. Thus the reduced-load approximation iswhere the Erlang B formula isThe estimated route acceptance is ${\prod }_{j\in r}{q}_j$. This is an independence approximation, not an alternative exact factorization of the stationary law in part (i).

Uniqueness follows from a convex potential for the Erlang fixed point. For a single resource, the carried load of an Erlang loss resource is $m_C(v)=v[1-E(C,v)]$. It increases strictly from zero to $C$ as $v$ increases: differentiating the expected value of its upper-truncated Poisson distribution with respect to $\log v$ gives the strictly positive occupancy variance. Blocking also increases strictly, as is evident on dividing the Erlang denominator by its final term.

Use $p_j=-\log q_j\ge 0$. Let $v_C(p)$ be the unique offered load with $1-E(C,v_C(p))=e^{-p}$, and set $g_C(p)=v_C(p)e^{-p}=m_C(v_C(p))$, with $g_C(0)=0$. This function is strictly increasing and tends to $C$. DefineEach exponential term is a convex function, and each integral is a strictly convex function because its derivative is strictly increasing. Therefore $F$ is a strictly convex function. It is also a coercive function on the nonnegative orthant: an unbounded coordinate makes its integral grow asymptotically linearly with positive slope $C_j$. A unique minimizer exists.

If resource $j$ carries some positive offered route, its inward derivative at $p_j=0$ is negative, so its minimizing coordinate is positive. At such a coordinate the first-order equation isDividing by $q_j$ gives precisely $v_j={\sum }_{r:j\in r}{\rho }_r{\prod }_{k\in r\setminus \{j\}}{q}_k$. A resource with no positive offered route uniquely has $p_j=0$, hence $B_j=0$. Thus the minimizer and the fixed point coincide, proving existence and uniqueness of the Erlang fixed point for fixed unit-resource routing. This does not by itself guarantee convergence of every simultaneous substitution algorithm; the uniqueness claim concerns the solution of the equations.

For the Erlang loss formula, assume a Poisson process of arrivals with rate $\lambda$, independent holding times with exponential distribution of mean $1/\beta$, unit capacity usage per accepted call, and rejection without waiting when all $C$ units are occupied. Here $C$ is a positive integer and the offered load is $a=\lambda /\beta$. The call population on $\{0,\dots ,C\}$ is a birth-death process; detailed balance givesBy Poisson arrivals see time averages, the fraction of arrivals rejected is the time probability of full occupancy. ThereforeThe paper's $\nu$ is the offered load, rather than necessarily an arrival rate. Exponential holding times suffice for this derivation; no unproved insensitivity of loss networks assumption is needed.

A loss network with fixed routing has capacities $C_j$, call types $r$, and a link-route incidence matrix $A$ with $A_{jr}=1$ exactly when a type-$r$ call requires link $j$. An accepted call holds one unit on every link of its route for its holding time; if any required link is full, the call is rejected everywhere. Take independent Poisson processes of calls, independent holding times, and route offered loads ${\nu }_r$. For the finite-variable formulation below, take $C_j\ge 1$; a zero-capacity link and all routes using it may be removed first.

The Erlang fixed point approximation treats blocking at different links as independent and thins the traffic offered to link $j$ by acceptance at the other links. Its reduced offered load isThe equality uses $A_{jr}\in \{0,1\}$: only routes with $A_{jr}=1$ contribute. Treating this reduced stream as Poisson leads to $B_j=E(a_j(B),C_j)$. Independence and Poisson thinning here are the approximation, not exact properties of a general loss network.

To construct the convex potential for the Erlang fixed point, write $e_C(a)=E(a,C)$ and $m_C(a)=a[1-e_C(a)]$, the carried load of an Erlang loss resource, equivalently the mean occupancy in the single-link stationary distribution. Differentiating its finite sums gives, for $a>0$,The variance is positive because the finite occupancy law is nondegenerate for $a>0$. Furthermore $e_C$ maps $[0,\infty )$ increasingly onto $[0,1)$, while $m_C$ increases from zero to $C$. SetIn particular $U(0,C)=0$, $U(\cdot ,C)$ is continuous and strictly increasing, and $U(z,C)\to C$ as $z\to \infty$.

Forthe first sum is convex, and each integrated strictly increasing function is strictly convex. Thus $F$ is a strictly convex function on $y\ge 0$. Since $U$ tends to a positive limit on every coordinate, $F$ is a coercive function there; it has a unique minimizer. Its coordinate derivative isThe Erlang fixed point equation is exactly ${\partial }_{j}F=0$. For an unused link, $a_j=0$ and the optimum is $y_j=0$; for a used link with positive offered load, the derivative at zero is negative and the optimum is positive. Hence every fixed point gives the unique minimizer, and the minimizer gives a fixed point. The Erlang fixed point is unique under these fixed-routing assumptions.

A convergent cyclic substitution for the Erlang fixed point updates one coordinate at a time, repeatedly cycling through all links:Always use the most recently available values of the other coordinates. Start, for example, with $B=0$. In $y$ coordinates this is exact coordinate descent for $F$: with other coordinates fixed, the displayed update is the unique coordinate minimizer.

Here is a convergence proof. All iterates stay in the initial compact sublevel set of the coercive function $F$, and their potential values decrease to a limit. Each single-coordinate update is continuous, since its reduced load is continuous and finite and $E(a,C)<1$. Thus a whole-sweep map $T$ is continuous. If a full-sweep subsequence converges to $z$, continuity gives $F(Tz)=F(z)$, because both consecutive potential values converge to the same limit. Strict coordinate convexity means equality can occur only when no coordinate changes during that sweep. Thus $z$ minimizes every coordinate, satisfies the nonnegative-orthant first-order conditions, and is the unique global minimizer of $F$. Every subsequential limit is therefore the same point, proving convergence of the full-sweep iterates and, by continuity of the updates, all intermediate iterates. This argument applies to sequential substitution; it does not assume simultaneous updates converge.

Use an ideal carrier-sense multiple access scheme on the interference graph. An active station finishes its transmission after an independent unit-rate exponential distribution waiting time. An inactive station has an independent attempt clock with an exponential distribution of rate $e^{{\theta }_r}$; an attempt succeeds only when all its neighbors are inactive. Attempts that are blocked do not change the state. Thus, on the independent sets represented by $S$, the continuous-time Markov chain has transition ratesThe empty independent set can be reached by successive deactivations, and any other independent set can be reached from it by successive activations. Hence this finite chain is an irreducible Markov chain. The weight $a_{\theta }(n)=e^{\theta \cdot n}$ satisfiesBy detailed balance for a continuous-time Markov chain, its unique stationary law isThis is a finite exponential family. Its natural parameter of an exponential family is $\theta$ and its mean parameter of an exponential family is the vector of active fractions.

The throughput region of an interference graph is exactly $\Lambda =\mathrm{conv}S$. To see the possibly less immediate inclusion, suppose a random feasible schedule has mean $v\ge \lambda$. Retain each of its active vertices $r$ independently with probability ${\lambda }_r/v_r$, interpreting zero coordinates as always deleted. A subset of an independent set remains an independent set; the thinned schedule has mean exactly $\lambda$. Thus $\lambda$ lies in the convex hull of $S$. The opposite inclusion follows from the definition using equality. Moreover $0,e_1,\dots ,e_R\in S$, so this convex hull has full dimension.

Now choose $\epsilon >0$ such that $\mu =\lambda +\epsilon 1$ remains in the interior of $\mathrm{conv}S$. Use interior moment matching in a finite exponential family. Consider the convex functionThere is a ball of radius $\delta >0$ around $\mu$ contained in $\mathrm{conv}S$. Maximizing a linear functional over the convex hull is the same as maximizing over its generating set, so, for every $\theta$,The last inequality uses the point $\mu +\delta \theta /\Vert \theta \Vert$ when $\theta \ne 0$. Hence $F$ is a coercive function and has a finite minimizer ${\theta }^{\ast }$. Differentiating the finite normalizing sum givesThis covariance matrix is a positive-definite matrix: a linear functional constant on all states of positive weight must be constant on $0,e_1,\dots ,e_R$, so its coefficients vanish. Thus $F$ is a strictly convex function; its minimizer is unique. At that minimizer the desired strict service margins areFor a backlogged station that transmits at unit speed whenever active, the long-run throughput is ${\mathbb{E}}_{\theta }[n_r]$, by the ergodic theorem for a finite continuous-time Markov chain. Therefore the ideal carrier-sense multiple access scheme can supply a strict service margin for every arrival vector in the interior of the throughput region of an interference graph. The local attempt rates can realize the whole interior capacity region in this sense. Boundary points may require parameters tending to infinity, and the argument gives no uniform delay or mixing-time bound near the boundary.

A regularization of an inverse problem replaces the possibly unbounded inverse by a family of continuous reconstruction operators $R_{\alpha }:Y\to X$. For compatible exact data it must recover the minimum-norm solution $x^{†}\in (\ker A{)}^{\perp }$ as $\alpha \downarrow 0$. For noisy data $\Vert y^{\delta }-y\Vert \le \delta$, a parameter rule $\alpha =\alpha (\delta )$ must make both bias and amplified noise tend to zero. The target is the minimum-norm solution because data cannot determine a component in the kernel of a linear map.

Tikhonov regularization minimizesThe quadratic norm penalty makes this functional strictly convex and a coercive function. Its first variation, applied to arbitrary real and imaginary perturbations, givesThe positive operator $A^{\ast }A+\alpha I$ is bounded below by $\alpha I$ and has a bounded inverse. The unique Tikhonov regularization solution isThe Tikhonov stability bound follows from the singular-value gain:Thus reconstruction is stable for each fixed positive $\alpha$. Stability is not uniform as $\alpha \downarrow 0$: that limit restores the unbounded inverse. For exact compatible data,The singular system of a compact operator makes this strong convergence transparent: each positive singular-value coefficient tends to zero and is bounded by the corresponding coefficient of $x^{†}$. Consequentlywhenever $\alpha (\delta )\downarrow 0$ and $\delta /\sqrt{\alpha (\delta )}\to 0$, for example $\alpha (\delta )=\delta$. This is a convergent regularization of an inverse problem, not merely a bounded formula.

In the Landau theory approximation, choose the global minimum of the Landau free energy as a function of the magnetization. Since ${\mathcal{A}}_6>0$, the polynomial is a coercive function and a global minimum exists. The candidates are stationary points satisfyingCheck their values of the free-energy density; a stationary point or a metastable local minimum alone need not be the equilibrium state. Positive curvature ${\mathcal{A}}_2+3{\mathcal{A}}_4{m}^2+5{\mathcal{A}}_6{m}^4$ ensures local stability; when it vanishes, higher terms must be checked. The selected global minimum is the equilibrium magnetization within this approximation.

At zero magnetic field, equal opposite minima describe distinct phases exhibiting spontaneous symmetry breaking. A spontaneous magnetization is obtained by taking the thermodynamic limit before $h\to 0^{+}$ or $h\to 0^{-}$. A symmetric finite-volume average can instead have zero expectation even when the minima are nonzero. Thus one must specify the phase when interpreting the minimizer as the equilibrium expectation.

First consider unit requirements, $A_{jr}\in \{0,1\}$. The Erlang loss formula for a resource with capacity $C$ and offered traffic $a$ isThe Erlang fixed point approximation assumes that resources block independently and that the traffic retained after screening at other resources can be treated as a Poisson process. If $B_j$ is the approximate blocking probability and $q_j=1-B_j$, the reduced-load approximation givesThe link whose load is being calculated is excluded from the screening product. Otherwise one would confuse its offered load with its carried load.

For existence, these equations define a continuous map from $[0,1{]}^J$ into itself. The Brouwer fixed-point theorem gives a fixed point. Assume $C_j\ge 1$; a zero-capacity resource forces rejection of every call needing it and can be removed together with those call types.

For uniqueness, let $K$ have probabilities proportional to $a^k/k!$, $0\le k\le C$, and write $m_C(a)=\mathbb{E}K$. Direct differentiation givesThese identities hold for $a>0$, and $m_C(0)=0$. Also $E(a,C)$ increases from zero to one and $m_C(a)$ increases from zero to $C$.

Put $p_j=-\log q_j$. Define $a_C(p)$ by $E(a_C(p),C)=1-e^{-p}$, and define $h_C(p)=m_C(a_C(p))$. This is a continuous, strictly increasing function on $[0,\infty )$, starting at zero and tending to $C$. Multiplying each Erlang fixed point approximation load equation by $q_j$ transforms it intoThese are exactly the zero-gradient conditions ofEach integral is a strictly convex function; the exponential terms are convex functions. Thus $\Phi$ is a strictly convex function. It is also a coercive function, since $h_{C_j}(p)\to C_j>0$, so it has a unique minimizer. At $p_j=0$, its partial derivative is negative if some positive-traffic route uses $j$, so that coordinate of the minimizer is positive. If no route uses $j$, the unique minimizing coordinate is zero. The minimizer therefore satisfies the equations in every coordinate. Conversely any fixed point has $B_j<1$, finite $p$, and these zero-gradient equations, so must equal that unique minimizer. Hence the Erlang fixed point approximation exists and is unique for fixed routing.

For integer requirements, the common generalized Erlang fixed point approximation usesMultiplication by $q_j$ gives the same equations for $\Phi$, so the existence and uniqueness argument also covers this generalized approximation. It remains an approximation, rather than the exact blocking law for a call requesting several units.

SetThe triangle inequality makes $f$ Lipschitz continuous and hence a continuous function. Since $\gamma$ is an isometric embedding,so $f(t)\to \infty$ as $|t|\to \infty$. The function is therefore coercive, and the extreme value theorem on a sufficiently large compact interval gives a minimizing parameter.

Suppose $t_1,t_2$ both minimize $f$, put $p=\gamma (t_1)$ and $q=\gamma (t_2)$, and letThe restriction of $\gamma$ between the two parameters is a geodesic from $p$ to $q$. Let $m$ be its midpoint. In the geodesic triangle with vertices $x,p,q$, the thin geodesic triangle condition gives a point $y$ on one of the other two sides with $d(m,y)\le \delta$. By symmetry suppose $y\in [x,p]$. ThensoBut $m$ lies on $\gamma (\mathbb{R})$ and $p$ is a closest point, so $R\le d(x,m)$. Therefore $L\le 4\delta$, which is stronger than the requiredThis is the closest point on a geodesic line in a hyperbolic metric space estimate.

Write each polynomial of degree at most $n$ as $T\mathbf{a}$ and defineThis is a continuous function, and the reverse triangle inequality together with part (a) givesConsequently $F$ is a coercive function. Choose $R$ so large that $F(\mathbf{a})>F(0)$ outside the closed Euclidean ball of radius $R$. That ball is compact, so the extreme value theorem supplies a minimizer ${\mathbf{a}}_{\ast }$ there. Then $P=T{\mathbf{a}}_{\ast }$ is a best uniform approximation to $f$ among polynomials of degree at most $n$.

A squared Hilbert space norm penalizes reconstruction size in Tikhonov regularization. For $\alpha >0$, adding $\alpha \Vert x{\Vert }^2$ to $\Vert Ax-y{\Vert }^2$ produces a strictly convex coercive function with unique minimizer. Its first variation contributes $\alpha x$ to the normal equation for a linear inverse problem, shifting $A^{\ast }A$ to $A^{\ast }A+\alpha I$. It selects a minimum-norm representative and controls amplification of noisy small-singular-value components.