A parallel algorithm portfolio runs multiple candidate algorithms or randomized runs concurrently and combines their outcomes under a stopping policy. For a first-solution problem, an ideal policy stops all candidates when one returns an acceptable solution. The original portfolio study investigated how differing runtime distributions can make such combinations effective.^[ref]

If candidates have dedicated processors, negligible coordination overhead and completion times $T_i$, the first completion is $T_{\min }={\min }_i{T}_i$. For independent completion times,The equation follows because every candidate must still be running for the minimum to exceed $t$. Independence is an assumption about the runs, not a consequence of using different processor cores.

For an original two-point model, one randomized run finishes in one second with probability $0.7$, otherwise in $101$ seconds. Its mean is $31$ seconds. Two independent runs are both slow with probability $0.3^2=0.09$, so first-completion time has mean $0.91(1)+0.09(101)=10$ seconds. If their runtimes are perfectly correlated, the mean remains $31$ seconds. Repeating identical hard behavior offers no independent escape route.

Wall time and computing work differ: running two full-speed processors until the first completion uses twice that wall time in processor-seconds in this model. Shared memory, bandwidth contention and communication can further change actual timings. Report those resources and the answer-acceptance rule before interpreting a parallel speedup.

A sequential algorithm portfolio allocates one processor's time among candidate algorithms or runs. Its algorithm portfolio schedule determines which candidate runs, when to switch and whether an interrupted run is resumed or abandoned. The original portfolio study compared single-processor interleaving and restart policies with parallel execution.^[ref]

Consider original runtimes $A=(2,12)$ seconds and $B=(8,3)$ seconds on two equally likely input types. Either fixed choice has mean $7$ or $5.5$ seconds. Run $A$ for at most four seconds, then start $B$ if $A$ has not finished. The two completion times are $2$ and $4+3=7$ seconds, so the scheduled mean is $4.5$ seconds. The schedule succeeds because the candidates have complementary performance.

Algorithm portfolio switching overhead can erase that gain. If the switch costs $3$ seconds whenever it occurs, the second completion time becomes $10$ seconds and the mean becomes $6$ seconds, worse than always running $B$. The cost must be charged to the actual switching events, rather than added equally to every instance.

A timeout also does not say that the interrupted candidate would never solve the input. Resuming preserves previous search work only if the implementation supports it; restarting begins a different run. Record budgets, state retention, random seeds and the final stopping rule when reporting a sequential portfolio's performance. A good schedule on measured inputs still needs evaluation on unused inputs.

An algorithm portfolio schedule specifies how computational resources are assigned to members of an algorithm portfolio over time. For a sequential algorithm portfolio, it states which algorithm receives each time interval, whether interrupted work is resumed or discarded, and when computation stops. A parallel algorithm portfolio also needs a processor allocation and a rule for accepting and combining results. Listing candidate algorithms alone does not specify a schedule.

An original interleaving example makes the distinction between processor time and elapsed time explicit. Suppose algorithms A and B need respectively $3$ and $5$ seconds of processor time on a particular input. Give A one second, then B one second, repeating while preserving both states and stopping at the first accepted result. A receives its third second during elapsed interval $[4,5]$, so the portfolio finishes at $5$ seconds. At that point A has used $3$ processor-seconds and B has used $2$. With two dedicated processors, the corresponding parallel algorithm portfolio would finish at $3$ seconds. These calculations assume no algorithm portfolio switching overhead and no resource contention.

More generally, let $s_i(t)$ be processor time supplied to candidate $i$ by elapsed time $t$, and let $T_i$ be its required processor time. In the resumable, interference-free model, the completion time isDiscarding a state on every interruption changes this model: separate short runs do not necessarily accumulate into one longer run. The original portfolio study compared parallel, interleaved and restart execution precisely because these are different policies.^[ref]

Choose an objective before comparing algorithm portfolio schedules: expected value of running time, probability of finishing by a deadline, and total computing work need not favor the same allocation. A schedule chosen using retrospective runtimes also requires evaluation on inputs that did not determine the schedule.

A conditional algorithm parameter matters only when a stated condition on other settings holds. In an algorithm parameter space, the condition determines whether the child parameter is active. The ConfigSpace guide samples conditional settings only when their activation conditions are met.^[ref]

For an original example, an algorithm has modes plain and tuned. A depth setting $d\in \{1,2,3,4\}$ affects only tuned mode. The naive rectangular encoding has eight pairs, but the distinct effective settings areThe four plain-mode encodings describe the same effective choice, so there are five settings, not eight. The star denotes inactivity; it is not an additional depth value used by the algorithm.

ParamILS excludes local moves that change only an inactive parameter, since such a move does not alter effective behavior.^[ref] Switching from plain to tuned activates a choice of depth, while changing an unused plain-mode depth alone supplies no new measured behavior.

A forbidden parameter combination instead rules out a joint assignment. Inactive does not mean forbidden, missing data or a numeric zero. Recording the mode and activity conditions makes a returned configuration interpretable, especially when comparing costs or reusing settings in another implementation.

A forbidden parameter combination is an assignment excluded from an algorithm parameter space by an explicit validity rule. Each individual value may be allowed, while their joint use is not. The ConfigSpace guide implements this distinction using forbidden clauses and their conjunctions.^[ref]

For an original counting example, let mode $s\in \{A,B\}$, cache $c\in \{\mathrm{off},\mathrm{on}\}$ and worker count $w\in \{1,2\}$. There are initially eight assignments. Suppose the rules forbid both $s=A,c=\mathrm{on}$ and $c=\mathrm{on},w=2$. Each rule excludes two assignments, but they overlap on one:Subtracting four without accounting for the overlap would reject an assignment twice. The count can also be checked directly: all four cache-off assignments are valid, and only $(B,\mathrm{on},1)$ survives with the cache on.

This differs from a conditional algorithm parameter. A condition says whether a setting is active; a forbidden rule says that an otherwise represented combination is inadmissible. An inactive setting can be irrelevant without making the parent configuration invalid.

Validity is also distinct from the objective function. A slow valid run has a measured performance cost; a forbidden combination lies outside the declared search domain. Giving every invalid combination an arbitrary large cost changes the mathematical model and can waste evaluations or accidentally make an invalid result appear competitive. Check the validity rule before treating a proposed configuration as an experiment.

Homotopy theory studies topological spaces and continuous maps through continuous deformation. A homotopy from $f:X\to Y$ to $g:X\to Y$ is a continuous map $H:X\times [0,1]\to Y$ with $H(x,0)=f(x)$ and $H(x,1)=g(x)$. Replacing a space by a homotopy equivalent one preserves its deformation-level information, including its homotopy groups.

For example, the punctured plane retracts onto its unit circle throughThe coefficient stays positive, so the deformation never crosses the removed origin, and points already on the unit circle remain fixed. This is a deformation retraction.

The same viewpoint studies loops, higher-dimensional spheres and maps between spaces. It retains obstructions that survive continuous deformation while allowing distances, angles and the thickness of a region to change. A homotopy equivalence is weaker than a homeomorphism.

For an algebraic structure $A$ specified by operations, a quotient structure identifies elements through an equivalence relation $\sim$ compatible with every operation. Explicitly, if $a_i\sim b_i$ for all inputs, compatibility requires $f(a_1,\dots ,a_n)\sim f(b_1,\dots ,b_n)$. Thenis independent of the chosen representatives. The map $a\mapsto [a]$ is a surjective homomorphism.

For the integers, congruence modulo $m$ is compatible with addition and multiplication, giving the quotient ring $\mathbb{Z}/m\mathbb{Z}$. For a group, equivalence by cosets supports a quotient group precisely when the subgroup is normal. The compatibility requirement is what makes the quotient a reusable algebraic object rather than merely a partition of a set.

The arity of a finitary algebraic operation is the number $n$ of its input positions. An operation of arity $n$ on $S$ is a function $S^n\to S$. A nullary operation chooses a constant, a unary operation takes one input, and a binary operation takes two.

For example, addition on the integers is binary even though repeated addition allows an expression such as $a+b+c$. That expression composes two binary operations; associativity is what makes its parenthesization irrelevant. A symbol taking arbitrarily many inputs would need a different specification, such as one operation for each $n$.

A fixed arity is part of an algebraic signature and determines how terms are built and which inputs a homomorphism must preserve. Operations of the same arity can nevertheless satisfy completely different identities.

A unary operation on a set $S$ is a function $f:S\to S$ taking one input and returning an element of the same set. Examples include the additive inverse $n\mapsto -n$ on the integers and inversion $g\mapsto g^{-1}$ in a group.

Unary operations can be composed and iterated. The identity function and all maps $S\to S$ form a monoid under function composition. For negation, $f\circ f=\mathrm{id}$; for a successor map on the nonnegative integers, iteration instead produces an unbounded sequence.

An operation's identities are extra data. Being unary alone says nothing about injectivity, surjectivity or whether repeated application returns to the starting element.

A nullary operation on a set $S$ has no inputs: formally it is a function $S^0\to S$. The Cartesian power $S^0$ is the singleton consisting of the empty tuple, so such a function selects exactly one distinguished element $c\in S$.

The zero and identity elements of a unital ring can be encoded as nullary operation symbols. A homomorphism preserving those operations must carry their selected constants to the corresponding constants in the target structure. This is why constants fit naturally into the same signature language as unary operations and binary operations.

There is no nullary operation on the empty set: its source $S^0$ still has one element, while its target has none. Allowing constants therefore affects whether empty algebraic structures are possible.

The degree of an algebraic number $\alpha$ is the degree of a polynomial of its minimal polynomial of an algebraic element over $\mathbb{Q}$. Equivalently, it is the field extension degree $[\mathbb{Q}(\alpha ):\mathbb{Q}]$. If that degree is $d$, then $1,\alpha ,\dots ,{\alpha }^{d-1}$ form a basis over the rational numbers: higher powers reduce using the minimal polynomial of an algebraic element, while a dependence among these first $d$ powers would contradict minimality.^[ref]

For $\alpha =\sqrt{2}$, the irreducible polynomial $X^2-2$ gives degree two and expressions $a+b\sqrt{2}$. For $\alpha =\sqrt[3]{2}$, the polynomial $X^3-2$ is irreducible over $\mathbb{Q}$ because a reducible cubic would have a rational root of a polynomial, which the rational root theorem excludes. Its degree is three. Merely finding a polynomial equation gives an upper bound: $\sqrt{2}$ also satisfies $X^4-4=0$, but its degree is still two.

Arithmetic geometry studies polynomial equations using both algebraic geometry and number theory. The coefficient arithmetic matters: a geometric object may have real points while having no integer or rational points of the required kind.

For example, $x^2+y^2=3$ has real solutions, but no solutions in the integers. A square is congruent to zero or one modulo four, so a sum of two squares cannot be congruent to three. Such polynomial congruences modulo a prime or prime power can therefore obstruct the existence of global arithmetic solutions.

In the language of schemes, an object over $\mathrm{Spec}\mathbb{Z}$ has fibres over prime numbers that are varieties over finite fields, together with a generic fibre over $\mathbb{Q}$. Comparing these fibres links the geometry of curves and abelian varieties to their arithmetic. Local solvability need not guarantee global solvability; the passage between these questions is a central source of structure and difficulty.

A diatom is a unicellular eukaryotic alga characterized by a silica-containing cell wall called a frustule. The wall typically has two overlapping portions, making the organism a striking example of biologically constructed mineral architecture. Diatoms contribute to photosynthetic production and to movement of silicon through aquatic ecosystems.

Experiments on marine bacterial assemblages connected decay of diatom organic material to accelerated dissolution of the silica wall. ^[ref] The biological and mineral parts of a dead cell therefore do not have independent fates. Removing or altering the surrounding organic material can change how rapidly the mineral component is returned to the dissolved pool.

Chlamydomonas reinhardtii is a unicellular green member of the algae used as a model organism for photosynthesis and for the structure and function of eukaryotic flagella. A typical motile vegetative cell has two anterior flagella, a large cup-shaped chloroplast and a light-sensing eyespot. These features bring propulsion, light sensing and photosynthesis together in one experimentally tractable cell.^[ref]

Its flagella contain the same basic microtubule machinery as motile cilia in animals. This is an evolutionary relationship between organelles; bacterial flagella use a different construction. In the genome study published in 2007, researchers sequenced its nuclear genome and compared predicted proteins across organisms with different cellular capabilities. Genes shared by organisms with flagella, but absent from comparison organisms lacking them, supplied candidates for flagellar assembly and function. Comparisons with proteins experimentally found in flagella helped test those candidates.^[ref]

A photosynthetic alga can therefore illuminate defects in human cilia, while also providing a system for studying chloroplast biology.

Dunaliella parva is a unicellular green member of the algae adapted to saline environments, making it an experimental example of a halophile. Its ability to adjust intracellular glycerol connects a change in external salinity to a measurable cellular response.^[ref]

In 1973, Ami Ben-Amotz and Mordhay Avron measured cellular volume and glycerol content. For cells cultured in $1.5\mathrm{M}$ sodium chloride, they estimated an intracellular glycerol concentration of about $2.1\mathrm{M}$. Increasing or decreasing the external salt caused corresponding glycerol adjustments, reaching a new balance in about90minutes. Above the reported salt threshold, leakage did not explain the changes, supporting synthesis and breakdown as a mechanism of osmoregulation.^[ref]

The experiment compares distinct quantities: salt outside the cell and glycerol inside it. Their numerical concentrations should not be equated as if they alone specified equal osmotic pressure; sodium chloride dissociates and other cellular solutes also contribute. The informative result is the regulated response to a controlled environmental change, not a universal concentration ratio for all halophiles.