A pile top in patience sorting is the currently exposed value on one pile. With the rule “place the next value $x$ on the leftmost pile whose top is at least $x$,” pile tops remain strictly increasing from left to right. Replacing the chosen top preserves that invariant: all earlier tops are smaller than $x$, and later tops exceed the old chosen top and hence $x$.

More strongly, top $i$ is the smallest possible last value of an increasing subsequence of length $i+1$ among the entries seen so far. A smaller last value leaves more options for extending that subsequence. For 3,1,2, the top arrays are [3], [1], [1,2]. binary search finds each replacement position in logarithmic time. The final top array gives the length but need not itself be a subsequence: 2,3,1 ends with tops [1,3], although1 appeared after3. Predecessor pointers are needed to reconstruct a witness.

A mechanical energy storage function assigns a nonnegative stored-energy quantity $E$ to the state of a mechanical model so that its increase never exceeds entering work. For a one-port passive mechanical network with entering power $Fv$, the mechanical dissipation inequality isThis state-based accounting includes energy present before the measurement begins. A temporary outward power is compatible with release of stored energy.^[ref]

For an ideal linear spring, $E_k=\frac{1}{2}k{x}^2$ is elastic potential energy. For an ideal inerter, $E_b=\frac{1}{2}b{v}^2$ is kinetic energy associated with terminal relative motion. An ideal dashpot need not store mechanical energy: its entering power $c{v}^2$ is dissipated. If these elements share terminals, their state variables obey $v=\dot{x}$ andThis equality proves the inequality for $c\ge 0$. In a larger network, summing element balances cancels internal transfers, leaving power through the external terminals and total dissipation.

For an original numerical check, let $k=800N/m$, $b=5\mathrm{kg}$ and $c=10\mathrm{N}s/m$. Start at $x=0.050\mathrm{m}$ and $v=0.20m/s$, so $E=1.10\mathrm{J}$. Maintain that relative velocity for $0.10\mathrm{s}$: displacement reaches $0.070\mathrm{m}$ and stored energy becomes $2.06\mathrm{J}$. Dissipation is $c{v}^2\Delta t=0.040\mathrm{J}$, so input work is $0.96+0.04=1.00\mathrm{J}$. Counting all entering work as newly stored energy would overestimate storage.

The mechanical available storage is the maximum net work that could be recovered under allowed future terminal motions; it is a related operational quantity, rather than automatically every form of internal physical energy. A model's storage function is a certificate that must respect its accessible states and port variables. With zero external power, a positive-definite storage function can supply a Lyapunov function, but nonincreasing energy alone does not establish that every motion decays to rest: a lossless spring–inerter oscillation keeps a constant nonzero stored energy.

Tilman Plehn develops quantitative particle-physics treatments of dark matter. His notes with Martin Bauer connect candidate models with relic abundance, direct detection, indirect detection and collider searches.^[ref]

These are different tests of a proposed interaction. A parameter choice that supplies the cosmological abundance need not automatically evade a scattering experiment or produce a collider signal. Placing the calculations in one framework makes those competing constraints explicit rather than treating “dark matter” as one uniquely specified particle.

Iain W. Stewart is a theoretical nuclear and particle physicist at the Massachusetts Institute of Technology, where his research develops and applies effective field theory to interactions of elementary particles, especially quantum chromodynamics. MIT's biographical sketch records his joining its physics faculty in January2003. His research description emphasizes choosing the relevant degrees of freedom and symmetries, then organizing calculations through power counting.^[ref]

A2001 paper with Christian W. Bauer, Sean Fleming and Dan Pirjol constructs an effective theory for energetic quarks interacting with collinear and soft gluons. This is part of the development of soft-collinear effective theory. The primary paper identifies an energetic scale $Q$, a transverse-momentum scale $p_{\perp }$ and a lower scale $p_{\perp }^2/Q$. It also matches currents and uses renormalization group equations to organize the calculation.^[ref]

An original scale-accounting example makes the hierarchy concrete. In units where the speed of light is1, choose illustrative values $Q=100\mathrm{GeV}$ and $p_{\perp }=10\mathrm{GeV}$. ThenThese numbers are not experimental inputs from the paper. They show why an interaction can involve several parametrically different momentum scales even when it belongs to one collision process. Power counting tracks powers of a small ratio such as $\lambda$, making the size of retained and omitted terms explicit within the chosen framework. It does not replace comparison with data or justify an approximation outside its scale assumptions.

Stewart also authored Advanced Classical Mechanics by Iain W. Stewart, whose treatment of moving holonomic constraints distinguishes an actual displacement during elapsed time from a virtual displacement at fixed time.^[ref] This connects his teaching to the careful definition of mathematical variations used in mechanics.

Christian W. Bauer is a theoretical particle physicist associated with Lawrence Berkeley National Laboratory. His laboratory theory-group entry and personal research page identify his work in high-energy physics.^[ref]

With Sean Fleming, Dan Pirjol and Iain W. Stewart, Bauer coauthored a2001 paper constructing an effective field theory for energetic quarks interacting with soft and collinear gluons. The paper matches currents between theories and evolves their Wilson coefficients using the renormalization group.^[ref] This is a foundational contribution to soft-collinear effective theory, which organizes strongly interacting processes involving separated momentum scales.

An original toy expansion illustrates the general logic of retaining controlled low-energy information. Consider an exchange contribution proportional toThe geometric series separates a leading coefficient from corrections with additional powers of momentum. If $q^2/M^2=0.01$, replacing the exact expression by $1/M^2$ gives a relative error of1percent; retaining the next term reduces that relative error to0.01percent. This simple exchange model is not the Lagrangian or a numerical result of the2001 paper. It shows why specifying an expansion parameter and truncation order gives an approximation more content than merely calling it a low-energy model.

In the actual collinear problem, an energetic particle can still have a small invariant mass. The relevant low-energy description therefore retains selected energetic modes instead of removing every field carrying a large energy. The explicit hierarchy and degrees of freedom in soft-collinear effective theory encode this distinction.

Sean Fleming is a theoretical physicist whose work develops and applies effective field theory to quantum chromodynamics. The University of Arizona's physics profile identifies him with research on separating the disparate energy scales that occur in strong-interaction measurements.^[ref]

With Christian W. Bauer, Dan Pirjol and Iain W. Stewart, he coauthored a2001 construction of a theory for energetic quarks coupled to collinear and soft gluons.^[ref] His2006 overview discusses soft-collinear effective theory and applications to particular meson decays and a radiative-decay spectrum.^[ref] These works connect a scale-organized field theory with quantities inferred from particle observations.

An original calculation illustrates why power counting is useful. Suppose a dimensionless observable is modeled byFor chosen values $A_0=1$, $c_1=2$, $c_2=-1$ and $\lambda =0.1$, the first three displayed terms give1.19; retaining only the leading term gives1. The small parameter organizes corrections, but the coefficients still matter. At $c_1=20$ instead, the first correction is2, so a small $\lambda$ alone would not make that correction negligible. These invented coefficients are not a fit or a prediction from Fleming's papers.

A controlled expansion therefore needs both a scale hierarchy and a specification of which contributions have been retained. Fleming's institutional research statement describes effective theories as an organizing tool for disentangling scales rather than treating a complicated measurement as a one-scale calculation.^[ref]

Dan Pirjol is a researcher whose work spans theoretical particle physics and mathematical finance. Stevens Institute of Technology's profile describes research in derivatives pricing, hedging and risk management using asymptotic analysis and applied probability. Its linked publication record directly associates him with the2001 physics paper by Christian W. Bauer, Sean Fleming, Pirjol and Iain W. Stewart.^[ref]

That paper develops an effective field theory for energetic quarks and soft/collinear gluons and uses scale-separated calculations of heavy-to-light processes. It forms part of the development of soft-collinear effective theory.^[ref] The institutional link is important here: the physics author and the financial-mathematics researcher are not being identified by name similarity alone.

An original payoff calculation illustrates one mathematical feature of the option models named in his research profile. An arithmetic-average Asian option with observation values $S_1,\dots ,S_n$ and strike $K$ can have payoffFor stipulated observations90,90,90and130 with $K=100$, the arithmetic mean is100 and this payoff is zero. A European option using only the terminal value130 instead has call payoff30. Four observations130,130,130and130 instead give Asian payoff30 with the same terminal price. Thus identical terminal prices need not determine an average-dependent payoff. This is an explanatory calculation, not a pricing result attributed to Pirjol or a recommendation about a financial product.

His profile's focus on asymptotic analysis concerns mathematical approximations to such stochastic-model quantities, while the earlier physics work organizes approximations through scale hierarchies. The applications differ, but both require an explicit model and a justified regime for the approximation.

A subsolution satisfies a one-sided version of a partial differential equation, with the inequality convention tied to the operator's sign. For the Dirichlet problem $-\Delta u=f$ with boundary data $g$, a classical subsolution $v$ satisfiesIf a suitable comparison principle for harmonic functions holds and $u$ is a solution, then $v\le u$. Supersolutions reverse both inequalities.

In the unit ball of ${\mathbb{R}}^n$ with $f=g=0$, take $v(x)=|x{|}^2-1$. It vanishes on the boundary and $-\Delta v=-2n\le 0$, so it lies below the zero harmonic solution. The maximum principle explains the general comparison: $v-u$ is subharmonic and nonpositive on the boundary.

Weak subsolutions interpret the inequality against nonnegative test functions; viscosity subsolutions instead use touching smooth test functions. These are specified solution notions, not an automatic pointwise interpretation for a nonsmooth function.

The complex Monge-Ampère equation prescribes the determinant of the complex Hessian of a real-valued function. On a domain in ${\mathbb{C}}^n$, one local normalization isThe positivity condition means that the complex Hessian is positive semidefinite; where it is positive definite the equation is elliptic. The determinant is nonlinear when $n>1$, unlike the Laplace operator, which takes a trace.

For $u(z)={\sum }_j|z_j{|}^2$, the complex Hessian is the identity and $f=1$. In one complex dimension, $u_{z\overline{z}}=\frac{1}{4}\Delta u$, so the equation reduces to a Poisson equation. On a Kähler manifold, its geometric version prescribes the volume form $(\omega +i\partial \overline{\partial }\phi {)}^n$ of a positive metric. Different conventions for $i\partial \overline{\partial }$ change constants, so a determinant normalization should always accompany a coordinate formula.^[ref]

A stochastic partial differential equation describes an unknown random field through spatial differential operators and random forcing. A model form is $du=(\kappa \Delta u+F(u))dt+G(u)d{W}_t$. The meaning of the noise, boundary data and solution is part of the specification; writing a formal derivative of Brownian motion does not define an ordinary function-valued forcing term.^[ref]

A simple exact example avoids the complications of noise at every spatial point. On $0<x<1$ with zero boundary values, considerThe solution is $u(t,x)=Y_t\sin (\pi x)$, whereHere $\kappa >0$. Spatial diffusion damps a single mode while noise continually excites it. More spatially irregular forcing can yield solutions that are distributions rather than pointwise functions, so regularity must be proved for the chosen dimension and noise.

Adaptive capping reduces the cost of algorithm configuration by ending an evaluation when a valid bound already rules out winning the comparison. For nonnegative runtimes, completed runs and elapsed time in the current run can provide a lower bound on the final cost. ParamILS distinguishes trajectory-preserving capping from a more aggressive heuristic policy.^[ref]

Consider an original comparison of mean runtime over $N=4$ fixed runs. The incumbent has mean $c_{\ast }=5$ seconds, or total $20$ seconds. A candidate has completed runs taking $6$ and $7$ seconds, and its third run has already taken $8$ seconds. Even if all remaining work took zero time, the final mean would satisfyContinuing that candidate cannot improve this comparison. The partial total itself supplies the certificate; no prediction of the remaining runtime is needed.

By contrast, after only the first two runs, their observed mean is $13/2=6.5>5$, but rejecting solely on that partial mean would be invalid: the four-run lower bound is only $13/4=3.25$. Two sufficiently fast remaining runs could still win. The denominator must correspond to the comparison objective.

A censored algorithm runtime is therefore not an observed completed runtime. Record the cutoff and what bound it establishes. Timeout penalties, changing instance sets or non-runtime objectives require a bound compatible with their own scoring rules. A heuristic early-stopping rule may be useful without preserving the same search decisions as full evaluation.

BasicILS is the fixed-evaluation variant of ParamILS. In $\mathrm{BasicILS}(N)$, competing parameter settings are compared using the same $N$ benchmark runs, with matched instances and random seeds.^[ref] The comparison rule supplies the cost estimates used by the surrounding iterated local search.

For mean runtime, writewhere $r_i$ is the recorded cost on the specified run. Consider invented costs $A=(1,1,19)$ seconds and $B=(5,5,5)$ seconds. With only the first run, $A$ appears better, $1<5$. Across all three runs, its mean is $7$ seconds while $B$ remains at $5$ seconds. Choosing $N$ therefore affects both evaluation expense and how much instance variation the estimate captures.

Using the same run list makes that example a comparison on common evidence. Comparing $A$ only on easy instances with $B$ on harder ones would answer a different question. Even a matched finite sample can be unrepresentative of future workloads, so the final configuration still needs evaluation on unused instances.

FocusedILS changes the allocation of runs rather than requiring the same fixed $N$ for every explored candidate. Adaptive capping can avoid completing an already-lost runtime comparison under a valid lower-bound rule. Neither modification makes a finite observed mean a proof of the best possible setting in the entire algorithm parameter space.