Satici Lab – Emergence in Complex Systems

Rare Stability in the Chaos

Only five positions in the Sun–Earth system allow a stable equilibrium relative to both bodies — the Lagrange points.

This is a rare stable solution to the restricted three-body problem.

Is the generalized three-body problem solvable analytically? No — and that gap is the subject of this lecture.

The five Lagrange points of a two-body system (e.g., Sun–Earth).

Open with a genuinely surprising fact: out of the continuum of possible positions in space relative to the Sun and Earth, only five — L1 through L5 — permit a stable (or quasi-stable, in the case of L1–L3) equilibrium where a third, negligible-mass body can sit indefinitely without drifting away. This is the “restricted” three-body problem (one body’s mass is negligible), and it has this rare closed-form family of solutions.

The hook for the lecture: the general three-body problem — three bodies of comparable mass, mutually interacting — has no closed-form analytical solution at all. Poincaré proved this in the 1890s while working on a related stability question, and it was one of the first rigorous demonstrations that deterministic systems can be fundamentally unpredictable in closed form, even though every underlying law (Newtonian gravity) is perfectly well understood. That’s the thread this whole lecture pulls on: analytic predictability is the exception, not the rule, even in systems built from beautifully simple physical laws — and stability, when it does occur, is rare and often precarious.

Newton and the Roots of “Complex”

Sir Isaac Newton (1642–1727).

Newton’s laws describe planetary motion, fluid dynamics, and — through sensitivity to initial conditions — the seeds of chaos theory.

The three-body problem is a direct consequence of Newtonian mechanics applied to more than two interacting bodies.

Newton’s laws are exact. Their consequences, for three or more interacting bodies, are not analytically solvable.

Newton’s laws of motion and universal gravitation are the founding language of complex systems, even though Newton himself wasn’t thinking in those terms. Planetary motion is the classic case: multiple bodies interacting gravitationally is already a complex system by our working definition, and the three-body problem from the previous slide is the direct, unavoidable consequence of applying F = ma and universal gravitation to three or more mutually interacting bodies instead of two.

The same equations show up as the foundation of fluid dynamics (particles interacting via pressure and viscosity — think weather systems and ocean currents) and, more subtly, as the origin of chaos theory: chaotic sensitivity to initial conditions in systems like weather prediction is a consequence of nonlinear Newtonian dynamics, not a departure from it. The point to land: Newton’s laws are exactly correct for each pairwise interaction, yet the aggregate behavior of three or more bodies governed by those exact laws is not analytically solvable. Exact micro-laws do not guarantee analytically tractable macro-behavior — this is the first hint of a theme the entire course returns to.

Laplace’s Demon

“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which… knew all forces… and all positions of all items… would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom… for such an intellect nothing would be uncertain and the future, just like the past, would be present before its eyes.”

— Pierre-Simon Laplace, A Philosophical Essay on Probabilities

Pierre-Simon Laplace (1749–1827).

This is the classical, pre-complexity worldview stated in its most extreme and honest form — worth reading aloud slowly. Laplace’s claim, now known as “Laplace’s Demon,” is pure determinism: if a hypothetical intellect knew the exact position and momentum of every particle in the universe, along with all the forces acting between them, the entire future (and past) would be perfectly computable — no uncertainty anywhere, ever.

Laplace earned the right to make this claim: beyond the philosophical essay, he extended Newton’s celestial mechanics to analyze the long-term stability of the entire solar system (not just three bodies), pioneered probability theory and what became Bayesian updating, developed analytical mechanics as a general toolkit for systems of interacting particles, worked out the physics of tides from Earth-Moon-Sun gravitational interaction, and developed perturbation theory to handle the small deviations in planetary orbits caused by other planets’ gravity — a genuinely early and sophisticated engagement with what we’d now call a many-body complex system. His demon is the perfect target for what complex systems science spends the rest of this lecture dismantling: not because determinism is philosophically wrong, but because even perfectly deterministic systems can be, in practice and even in principle, unpredictable in closed form.

Analytic vs. Algorithmic

Definition

Physics is generally analytic; complex systems are generally algorithmic. Both can produce quantitative predictions that are experimentally testable.

Complex systems exhibit a rich phase structure and huge variety of macrostates that often cannot be inferred from the elements’ properties alone — this is emergence
The theory of complex systems is the quantitative, predictive, and experimentally testable science of generalized matter interacting through generalized interactions

Reference

Thurner, Hanel, & Klimek (2018), Introduction to the Theory of Complex Systems.

This is the methodological pivot of the whole lecture, and it’s worth being precise about the vocabulary. “Analytic” means: write down equations, solve them in closed form, get an exact prediction — this is the physics paradigm from Newton through Laplace, and it works beautifully for two-body gravity or simple circuits. “Algorithmic” means: you cannot write down a closed-form solution, but you can specify the local rules precisely enough to simulate the system forward and expect — reliably, reproducibly, testably — certain macro-level patterns to emerge, even without ever deriving them by hand.

Crucially, both paradigms can be genuinely scientific: algorithmic predictions from simulating cellular automata, agent-based models, or genetic algorithms are just as quantitative and experimentally falsifiable as analytic predictions from solving a differential equation — they’re just derived differently. The formal definition on this slide (from our course reference, Thurner, Hanel, and Klimek) is worth having students copy down verbatim: complex systems theory is the quantitative, predictive, testable science of generalized matter interacting through generalized interactions — deliberately abstract enough to cover particles, neurons, bees, or traders with the same mathematical machinery.

Algorithmic Thinking in Practice

Cellular automata: Conway’s Game of Life — simple grid rules produce unpredictable, complex patterns
Agent-based modeling: simple agent rules → emergent traffic, market, or ecosystem dynamics
Genetic algorithms: evolution-inspired search over vast, hard-to-navigate solution spaces

Network algorithms: e.g. PageRank — navigating and ranking complex networks
Self-organizing systems: local rules → global order, with no external guidance (animal coat patterns, galaxy structure)
Chaos & fractals: small changes in initial conditions → vastly different, self-similar outcomes

Walk through these six as a toolkit, not a laundry list — the point is that “algorithmic” isn’t one technique, it’s a family of related approaches united by the same philosophy: specify simple local rules, run them forward, observe what emerges. Cellular automata (Conway’s Game of Life is the canonical teaching example) show how astonishingly rich, unpredictable global patterns — gliders, oscillators, entire self-replicating structures — emerge from a handful of simple neighbor-counting rules on a grid.

Agent-based modeling generalizes this to heterogeneous agents with individual decision rules, used across economics, sociology, and ecology to simulate traffic patterns, market dynamics, and ecosystem interactions. Genetic algorithms borrow evolution’s variation-and-selection loop as a general-purpose optimization procedure for spaces too vast to search exhaustively. Network algorithms like PageRank show that even ranking and navigating a network (rather than simulating dynamics on it) is itself an algorithmic, not analytic, problem. Self-organizing systems — animal coat patterns, galactic structure, social organization — show global order emerging with no central planner. And chaos and fractals tie back to the very first slide’s Lagrange point discussion: extreme sensitivity to initial conditions and self-similar structure across scales are hallmarks of the same algorithmic, non-analytic regime.

Machine Learning and Emergence in Simulation

Predictive algorithms: deep learning models complex systems by learning patterns directly from data — climate models, stock markets, language
Emergence in simulation: flocking birds follow only local rules (spacing, alignment with neighbors) — yet the flock exhibits coordinated, complex group movement no individual bird “decides”

The macro-behavior is not contained in any individual agent’s rule set. It only exists at the level of the whole.

Two closing examples for the algorithmic-thinking arc. First, machine learning generalizes “algorithmic, not analytic” prediction to its modern extreme: deep learning models make no attempt at a closed-form physical theory of, say, stock markets or climate — they learn statistical patterns directly from data and generate predictions without ever producing an interpretable analytic law. It’s the algorithmic paradigm taken to its limit.

Second, and more conceptually important for this course, flocking birds is the cleanest possible demonstration of true emergence: each bird follows only a few simple local rules — keep some distance from your nearest neighbors, roughly align your heading with theirs, steer toward the local average position. No bird has any notion of “the flock” or “coordinated group movement” — that concept exists only at the system level, produced by, but not contained in, any individual bird’s behavior. This is the sharpest way to state emergence: the property exists at a level of description that the components themselves don’t have access to. Hold onto this example — it’s the cleanest one to reach for whenever a student asks “but isn’t emergence just a vague word for ‘complicated’?”

Biology: Living Matter Is Never at Equilibrium

Living matter constantly uses energy and performs work, driven by energy gradients, always out-of-equilibrium.

Evolution increases and destroys diversity — both a creative and a destructive process.

The Darwinian scenario alone fails to explain boom-and-bust phases, where diversity changes radically over short periods.

The fossil record: a story of both radiation and mass extinction.

Biology is the second of the three classical disciplines we revisit (after physics), and the framing here is sharper than in the last lecture: living matter is fundamentally, permanently out of thermodynamic equilibrium — it survives precisely by continuously importing energy and using it to do work, without letting itself relax to equilibrium (which, for a living organism, is death). This out-of-equilibrium requirement, plus the need for self-replication and localized “Carnot-cycle”-like energy processing, is a deep structural feature of biological complex systems, not an incidental detail.

Evolution itself gets a more nuanced treatment than “survival of the fittest”: it’s explicitly framed as both creative (generating novel diversity through variation) and destructive (extinguishing lineages), and simple Darwinian gradualism — smooth, continuous adaptation — doesn’t actually explain the fossil record’s real pattern, which includes sudden boom phases of rapid diversification and sudden crash phases of mass extinction, visible in the timeline pictured. This “boom and bust” irregularity is itself a complex-systems signature: it looks a lot more like the punctuated, threshold-driven dynamics we’ll see in self-organized criticality shortly than like smooth optimization.

Evolutionary Dynamics Break Physics’ Rules

Evolutionary dynamics differs from physics in two fundamental ways:

Boundary conditions can’t be fixed. A wolf pack’s fate depends on climate, disease, and human activity — none of which are stable “boundary conditions” you can hold constant.
The phase space itself isn’t fixed. New elements (traits, species, technologies) emerge and change the space of future possibilities for everyone else in the system.

This slide draws the sharpest possible contrast between the classical-physics worldview from the first half of this lecture and the biological/evolutionary worldview. In physics — even complicated physics like the three-body problem — you can typically write down a fixed phase space (the space of all possible states) and fixed boundary conditions, and then ask how the system evolves within that fixed space. Evolutionary systems break both assumptions.

First, boundary conditions can’t be fixed: take a wolf pack — even given complete knowledge of initial genetics, environment, and resources, you cannot predict the population’s future trajectory, because the “boundary conditions” (climate shifts, disease outbreaks, human encroachment) are themselves unpredictable and external, not fixed parameters you get to hold constant. Second, and more radically, the phase space is not well-defined and changes over time: consider a bird population’s phase space of traits (beak size, wing shape, coloration) — if the food source shifts, the fitness landscape over that same trait space reshapes itself; new traits or even new species entering the ecosystem can open up entirely new dimensions of what’s possible. This is fundamentally different from a fixed configuration space with dynamics evolving on top of it — the “board” itself is being redrawn as the game is played.

The Adjacent Possible

The set of all states the world could reach in the next step, given its present state.

Introduces path-dependence into the stochastic dynamics of phase space: what’s reachable next depends entirely on where you are now.

A game of Go mid-play: the legal next moves — the “adjacent possible” — depend entirely on the current board.

Example — Building a House

A single-level base’s adjacent possible: add a second level, add windows, add a roof. Once you have two levels and windows, the adjacent possible shifts — now it includes a chimney, a balcony, a garage.

Stuart Kauffman’s “adjacent possible” is the formal concept that makes the previous slide’s “changing phase space” idea precise and useful. At any given moment, the adjacent possible is the (typically much smaller) set of states reachable in the very next step from wherever the system currently is — not the set of all possible states ever, just the next-door neighbors of the current state. This automatically builds path-dependence into the system’s dynamics: two systems with identical eventual reachable state-spaces can have completely different near-term trajectories depending on their current position, because their adjacent possibles differ.

The Go board is a clean, literal illustration: at any point in the game, the legal next moves are a small, well-defined set determined entirely by the current board position — not by the space of all possible Go positions. The construction example makes it concrete for something less game-like: a single-level structure’s adjacent possible includes adding a second level, windows, or a roof; but once you’ve added a second level and windows, the adjacent possible has now shifted to include a chimney, a balcony, or a garage — options that weren’t reachable, and arguably weren’t even meaningful, from the earlier state. This concept becomes important later in the course when we talk about how architectural decisions constrain what future decisions are even available.

Robust and Adaptive — Not a Contradiction

Definitions

Robustness = maintaining function despite disturbance.
Adaptability = changing structure or behavior in response to disturbance.

At first glance these look mutually exclusive — but they operate at different levels and timescales.

Forest ecosystem: species diversity provides redundancy (robustness) — if a disturbance harms one species, others fill its ecological role. The same diversity lets new species establish when conditions change (adaptability).

This is the “apparent paradox” that structures the rest of the lecture. On the surface, robustness (resisting change, maintaining function under stress) and adaptability (embracing change, reorganizing in response to stress) sound like opposites — a system that resists change can’t simultaneously be one that readily changes. The resolution is that they’re not properties of the same level of the system at the same timescale; they coexist by operating at different levels and different timescales simultaneously.

A forest ecosystem is the cleanest illustration: at the ecosystem level and on a moderate timescale, species diversity provides redundancy — if one species is hit hard by a disturbance (a pest, a disease), other species with overlapping ecological roles compensate, and the forest as a whole keeps functioning (robustness). But that same diversity is what lets the system reorganize over a longer timescale: after a larger disturbance like a wildfire, different species — better suited to the new post-fire conditions — can establish and gradually become dominant (adaptability). The forest is robust in the short run precisely because it retains the diverse raw material needed to adapt in the long run. This “different levels and timescales” resolution is the key idea to carry into the CO₂ case study next.

Case Study: Vehicle CO₂ Emissions as a Coupled System

A single anthropogenic input — vehicle CO₂ — propagates through a coupled human–natural system at three levels:

Component (plant physiology): altered photosynthesis, stomatal regulation — short-term adaptive plasticity
Ecosystem (forest): diversity provides redundancy; species respond unevenly; succession shifts over time
Earth system (climate–carbon coupling): feedback loops (temperature ↔︎ CO₂ ↔︎ vegetation ↔︎ albedo), nonlinear interactions, possible tipping points

Local adaptation may coexist with global instability. Robustness at one scale does not guarantee stability at another.

This case study operationalizes “robust and adaptive at different levels and timescales” with a real, high-stakes example, and it’s worth taking slowly since it has three nested levels. Vehicle emissions raise atmospheric CO₂ — a persistent external forcing entering the Earth system through the transportation subsystem’s coupling with climate and the biosphere. At the component level, individual plants respond on short timescales: elevated CO₂ changes photosynthesis rates and stomatal regulation — genuine adaptive plasticity, but purely local. At the ecosystem level, forest species diversity provides redundancy (some functional robustness), but different species are affected unevenly, so the forest’s species composition and successional trajectory shift over time — a mix of structural adaptation and functional robustness, but not “no change.” At the Earth-system level, feedback loops between temperature, CO₂, vegetation, and albedo become dominant, interactions turn strongly nonlinear, and the system becomes vulnerable to threshold and tipping-point behavior.

The systems-engineering takeaway is the one to really land: this is not a simple “plants adapt, problem solved” story. It’s a coupled nonlinear system where local adaptation (plants coping fine at the component level) can coexist with global instability (the Earth system as a whole drifting toward a tipping point) — robustness observed at one scale tells you nothing reliable about stability at another scale. This directly previews why “systems thinking,” which we return to constantly in this course, insists on examining multiple levels simultaneously rather than declaring victory after checking just one.

Self-Organized Criticality

Dynamical, out-of-equilibrium systems that evolve toward a critical point as an attractor — with no external tuning required.

Characterized by (approximate) scale invariance: no characteristic scale, often showing up as power laws in the associated probability distributions.

Grains poured onto a growing sand-pile — the canonical SOC demonstration.

The system is simultaneously robust and adaptive — it returns to its critical angle of repose after every avalanche, by reorganizing.

Self-organized criticality (SOC) is the single richest idea in this lecture, so plan to spend real time here. The defining feature: the system drives itself to a critical state purely through its own internal dynamics — no external agent tunes any parameter to put it there. Once at that critical state, the system shows scale invariance: there’s no characteristic size to its events — small, medium, and enormous avalanches all occur, following a power-law distribution, rather than clustering around some typical “average” event size the way a bell curve would predict.

The sand-pile is the canonical teaching demonstration, pictured here: pour sand grains slowly and continuously onto a pile. Early on, grains just accumulate. Eventually the pile’s slope reaches a critical angle (the angle of repose) and avalanches begin — some tiny (one grain sliding), some pile-reshaping. Critically, the system doesn’t need anyone to tune it to this critical slope; continuous slow driving (adding grains) plus local instability rules is sufficient for the system to find and maintain criticality on its own. This is also where “robust and adaptive” from two slides ago becomes concrete and quantitative: the pile is robust because it reliably returns to the same characteristic angle of repose after every avalanche, and adaptive because it reorganizes its exact shape every single time — robustness in the aggregate statistic, adaptability in the microscopic detail, simultaneously, in the same system.

Slow Driving, Sudden Release

Sand-pile: grains added slowly and continuously; each grain adds stress to the system.

Amazon delivery network: customer orders arrive continuously (the slow driving signal); each order adds workload/stress to the network.

Both systems are continuously driven by external input they do not themselves control — and both can respond with avalanches of any size.

This slide is a deliberate analogy-building exercise: take the sand-pile mechanism just introduced and map it onto a system students interact with regularly, to show SOC isn’t just a physics-lab curiosity. In the sand-pile, grains arrive slowly and continuously, each one adding local stress somewhere in the pile, until a threshold is crossed somewhere and an avalanche redistributes stress — of unpredictable size.

In a delivery network like Amazon’s, customer orders play the exact same structural role as sand grains: they arrive continuously (the slow driving input), each one adds workload/stress to some part of the fulfillment and logistics network, and the network absorbs this gradually — until, occasionally, a capacity threshold is crossed somewhere (a hub, a shipping lane, a warehouse) and disruption cascades through connected parts of the network, an “avalanche” whose size is unpredictable in advance. The parallel is not just cute — it’s the same underlying mathematics (self-organized criticality) applied to a socio-technical system instead of a physical granular one, and it’s exactly why real supply chains sometimes suffer disruptions wildly disproportionate to whatever small triggering event kicked them off.

Criticality, Robustness, and Adaptability Together

Emergence of criticality: a characteristic slope forms naturally as grains accumulate — the system is progressing toward avalanches without being told to
Structural robustness: the pile maintains integrity despite local disturbances, reorganizing to preserve overall slope and shape
Adaptive reshaping: the pile continuously reshapes as new grains arrive — stable global behavior emerging from constant local change

This slide is a synthesis checkpoint before moving to social science — make sure students can state all three properties back to you before continuing. Criticality is the process (the pile organizes itself toward the critical angle without external tuning); robustness is the aggregate outcome (whatever local damage an avalanche does, the pile’s overall statistical behavior — its characteristic slope — is preserved); and adaptive reshaping is the mechanism underneath both (the pile’s literal, grain-by-grain shape is constantly changing, and it’s precisely that constant local reshaping that produces stable global behavior).

The general lesson to extract, applicable well beyond sand: don’t confuse “stable” with “static.” A self-organized critical system looks stable at the macro/statistical level (same characteristic slope, same power-law avalanche-size distribution) while being in constant flux at the micro/structural level. This is a recurring pattern across complex systems — global order and local churn are not in tension; local churn is often exactly how global order is maintained.

Social Science: Qualitative, Path-Dependent

The science of social interactions and their implications for society — usually neither quantitative nor predictive in the way physics is; largely qualitative and descriptive.

Social processes are hard to model mathematically: they’re evolutionary, path-dependent, out-of-equilibrium, context-dependent, high-dimensional, and multi-level.

Early social science built its methods around exactly these difficulties.

Return to social science with a more honest accounting of its methodological challenges than the first lecture gave. Unlike physics, social science generally does not produce sharp, quantitative, experimentally falsifiable predictions in the Newton/Laplace sense — and this isn’t a failure of the field, it’s a structural consequence of what it studies. Social processes are evolutionary (path-dependent history matters, there’s no reset to a clean initial state), out-of-equilibrium (societies are always in flux, never settling), context-dependent (the “same” intervention can produce different outcomes in different cultural or historical settings), high-dimensional (huge numbers of relevant variables), and operate on multiple interacting levels simultaneously (individual, group, institutional, societal).

All of these are exactly the features that make analytic, closed-form prediction (the Laplace’s-Demon ideal from earlier) essentially impossible for social systems — which is precisely why social science leans on qualitative and descriptive methods rather than closed-form equations. This sets up the payoff of the next slide: even though social systems resist classical analytic treatment, complexity science’s algorithmic, network-based toolkit gives us a genuinely new way to make progress on them quantitatively.

Social Systems as Evolving Networks

Social systems can be modeled as time-varying, multilayer (multiplex) networks:

Nodes: individuals or institutions
Links: interactions of different types
Interactions change over time

A multilayer network: the same nodes, connected differently across layers (relation types) and time.

This is the payoff promised at the end of the last slide: complexity science’s network toolkit, first introduced in the previous lecture as a way to represent structure, turns out to be exactly the right formalism for social systems too, despite social science’s general resistance to quantitative treatment. Model a social system as a time-varying multilayer (or “multiplex”) network: nodes are individuals or institutions, but instead of a single kind of link, there are multiple simultaneous layers of links, each layer representing a different type of relationship — a friendship layer, a professional layer, a family layer, a financial-transaction layer — all sitting on the same set of nodes.

Critically, none of this is static: the diagram’s “levels” and “aspects” (types of relations, points in time) both vary, meaning the very structure of who’s connected to whom, in what way, is itself dynamic — mirroring the evolving phase space we discussed for evolutionary biology earlier in the lecture. This is a genuinely useful research tool in practice: multilayer network models let researchers ask questions like “does influence in the professional layer predict influence in the friendship layer six months later” — a quantitative, testable question about a fundamentally social phenomenon, made tractable precisely because complexity science gave social science a new representational vocabulary (networks) instead of demanding physics-style closed-form equations.

Emergence, Revisited

Definition

Emergence: the whole exhibits behaviors and properties not predictable from the sum of its parts.

Traditional analytical methods (physics, biology, social science alone) are insufficient to predict emergent behavior
Emergent patterns can be expected through an algorithmic approach
Complex systems science is a developing field, working to make emergence quantifiable, predictable, and experimentally testable

Examples across domains: Conway’s Game of Life · flocking birds · ant colony optimization · ecosystems · financial market bubbles and crashes · human cognition · social-network misinformation spread · urban development

Close the loop on the lecture’s central concept by restating it with everything students now have under their belt. Emergence: system-level behaviors and properties that cannot be predicted from the parts alone, even with complete knowledge of every part — precisely because these properties exist only at the collective level of description, as the flocking-birds example made vivid earlier. No single traditional discipline, analyzed alone, is sufficient to predict this behavior; but algorithmic methods (simulate the local rules, observe what reliably emerges) can expect it, even without deriving it analytically.

The example list is worth reading through quickly as a “spot the pattern” exercise: Conway’s Game of Life and flocking birds are direct callbacks to earlier in this lecture; ant colony optimization (pheromone trails → collectively near-optimal foraging paths, without any ant knowing the whole picture) inspired real routing and scheduling algorithms; ecosystem biodiversity and resilience emerge from species interactions, not from any species in isolation; financial bubbles and crashes emerge from the aggregate of individually simple trading rules; human cognition and consciousness emerge from billions of neurons each following simple firing rules; the spread of ideas or misinformation through a social network emerges from the network’s structure, not from any one person’s intent; and urban development patterns emerge from countless individual and institutional decisions with no central planner. Complex systems science, as a field, is still actively working to make all of this quantifiable and testable — this isn’t settled science, it’s an open, active research frontier, which is itself worth telling students explicitly.

References

Reference

Thurner, S., Klimek, P., & Hanel, R. (2018). Introduction to the Theory of Complex Systems. Oxford University Press.

Close the lecture on the same reference that anchored the first one, reinforcing that both lectures draw from a single coherent theoretical framework rather than being a loose collection of anecdotes. Everything covered across these two lectures — the analytic/algorithmic distinction, self-organized criticality, the adjacent possible, evolving networks, emergence — traces back to this text’s unified treatment of complex systems as the quantitative, predictive, testable science of generalized matter interacting through generalized interactions. From here, the course moves into Part I proper: System Thinking, starting with a formal definition of architecture and the four-task discipline (form, function, relationships, emergent properties) that structures how we’ll analyze every system for the rest of the semester.