Molality is the amount of substance of a solute divided by the mass of the solvent: $b=n_{\mathrm{solute}}/m_{\mathrm{solvent}}$. Its SI unit is $\mathrm{mol}\mathrm{k}{\mathrm{g}}^{-1}$. Dissolving $0.10\mathrm{mol}$ of solute in $0.50\mathrm{kg}$ of solvent produces $b=0.20\mathrm{mol}\mathrm{k}{\mathrm{g}}^{-1}$.

The denominator contains solvent mass, rather than the volume of the resulting chemical solution. Thus molality avoids changes caused solely by thermal expansion of the chemical solution, making it useful in colligative properties and chemical thermodynamics. It must be distinguished from molar concentration, whose denominator is solution volume.

Probably approximately correct learning formalizes learning as a two-parameter guarantee. With probability at least $1-\delta$ over the training random sample and any learner randomness, the returned hypothesis must have true prediction error at most $ϵ$. In the realizable distribution-free setting, this must hold for every input probability distribution and every target in the specified concept class, using an appropriate polynomial number of examples; efficient PAC learning also requires polynomial running time.

“Probably” refers to the chance of obtaining a good hypothesis; “approximately” refers to that hypothesis's error on a fresh observation. For example, $ϵ=0.01$ and $\delta =0.05$ require a 95% chance of returning a predictor whose error is at most 1%, rather than a claim that 95% of individual predictions are right. For finite realizable classes, the sample complexity bound follows by excluding every high-error rule that could have survived the observations.^[ref]

Sample complexity is the number of training observations needed for a specified learning guarantee. It depends on the allowed prediction error $ϵ$, failure probability $\delta$, the hypothesis class and assumptions about data generation. It counts observations, whereas computational complexity theory measures the work of processing them.

For a finite class $H$ containing the true binary labeling rule, suppose examples are independent and identically distributed. A fixed hypothesis with error greater than $ϵ$ fits $m$ examples with probability at most $(1-ϵ{)}^m\le e^{-mϵ}$. The union bound makes the probability that any such bad hypothesis fits at most $|H|e^{-mϵ}$. Thus any consistent learner succeeds with probability at least $1-\delta$ whenThe derivation assumes realizability and establishes an information guarantee; it does not give an efficient search for a consistent hypothesis.

A computable predicate is a yes-or-no relation for which an algorithm halts on every input and returns the correct truth value. Equivalently, its indicator function takes values in $\{0,1\}$ and is a total computable function. For predicates of several natural numbers, a fixed effective tuple encoding reduces the inputs to one number if needed.

For example, $R(a,b)$ defined by $a\mid b$ is computable: with a stated convention for $a=0$, integer division or finite arithmetic decides it. The predicate that program $e$ halts on input $x$ within at most $t$ steps is also computable, by bounded simulation.

In contrast, the existence of some halting time, $\exists tR(e,x,t)$, is not a computable predicate in general: it is the halting problem. It is computably enumerable because a successful simulation eventually witnesses it. Thus an unbounded existential quantifier can turn a decidable relation into one whose false instances cannot be recognized by a terminating procedure.

A program index is a natural number encoding a program in a fixed effective enumeration of programs, often written $({\phi }_e{)}_{e\in \mathbb{N}}$ for the corresponding partial computable functions. The meaning of an index depends on the chosen programming model and enumeration; ${\phi }_e(x)$ may be undefined because program $e$ does not halt on input $x$.

The universal partial computable function simulates the indexed program: $U(e,x)={\phi }_e(x)$ whenever that computation halts. Two different indices can compute the same partial function, because different program texts can implement identical behavior. Equality of indices is therefore a syntactic test, whereas equality of computed functions is a semantic property.

The halting problem becomes the question whether ${\phi }_e(x)$ is defined. Encoding programs as numbers also permits diagonal arguments, such as constructing a computation that uses its own index as input. These arguments concern a specified effective enumeration, rather than assuming every sequence of functions can be uniformly simulated.

The composite matrix in a composite material is its continuous surrounding phase, distinguished from embedded fibres or particles. It holds the reinforcement together and transfers load through interfacial stresses; it can also protect the reinforcement from damage or the environment.

For a fibre-reinforced polymer, the polymer is the composite matrix even if the fibres carry much of the axial load. Matrix stiffness, interfacial adhesion and failure therefore affect the whole specimen. This materials meaning differs from a rectangular array of numbers in linear algebra.

The isostress rule of mixtures estimates a composite material’s Young's modulus by assuming that its constituent phases carry the same axial stress. For two phases with positive Young's moduli $E_1,E_2$ and volume fractions $\varphi ,1-\varphi$, the effective modulus is their weighted harmonic mean:This is the one-dimensional equal-stress specialization of the Reuss elastic approximation, rather than the equal-strain assumption of the longitudinal rule of mixtures.^[ref]

To derive it, put two ideal elastic segments in series, with the same sectional area $A$, lengths $L_1,L_2$, and a common axial force $F$. Force balance gives $\sigma =F/A$ in each segment. Under one-dimensional linear elasticity, ${\epsilon }_i=\sigma /E_i$. Since the extensions add,Equal sectional areas make the length fraction $L_1/(L_1+L_2)$ equal to the volume fraction $\varphi$. Dividing by the common stress proves the formula.

For an illustrative pair $E_1=100\mathrm{GPa}$, $E_2=2\mathrm{GPa}$ and $\varphi =0.8$,At $\sigma =10\mathrm{MPa}$, the segment strains are $0.0001$ and $0.005$, so the mean strain is $0.00108$. The softer phase supplies most of the extension despite occupying only one fifth of the volume. Placing the same proportions in ideal parallel load paths with equal strain instead gives $0.8E_1+0.2E_2=80.4\mathrm{GPa}$. Geometry has changed the result substantially.

The series derivation is exact within its one-dimensional model, with bonded segments and no bending or additional transverse constraints. It is not an exact prediction for every real fibre-reinforced composite material loaded across its fibres: stress and strain vary around the actual reinforcement geometry. The primary teaching resource explicitly distinguishes the equal-stress slab estimate from the more complicated transverse response.^[ref]

A complex analytic space is a locally ringed space locally isomorphic to the common zero set of finitely many holomorphic functions on an open subset of ${\mathbb{C}}^n$, equipped with the quotient sheaf of holomorphic functions by a coherent ideal sheaf. This sheaf is part of the structure: the same underlying set can carry different analytic structures.

The cusp $y^2=x^3$ gives a reduced singular example. Its local holomorphic functions are represented by ambient holomorphic functions modulo the equation $y^2-x^3$. A complex manifold is the special smooth local model, with no singular equations needed after choosing coordinates.

General complex analytic spaces can also be nonreduced. The quotient $\mathbb{C}\{z\}/(z^2)$ has underlying set consisting of one point, but retains a nonzero nilpotent class of $z$. This distinguishes an analytic space from only a zero set. The notion extends complex geometry to singularities and infinitesimal structure that ordinary manifold charts cannot describe.

Kähler geometry studies complex manifolds with a Hermitian metric $g$ whose associated real two-form $\omega (X,Y)=g(JX,Y)$ is closed. Here $J$ is the complex structure. Equivalently, the Levi-Civita connection preserves $J$. Thus the same metric simultaneously supports complex geometry, Riemannian geometry and symplectic geometry.^[ref]

On ${\mathbb{C}}^n$, the Euclidean metric gives $\omega ={\sum }_{j}d{x}_j\wedge d{y}_j$, which is closed and nondegenerate. Locally one can express a Kähler form as $\omega =i\partial \bar{\partial }\phi$ for a real Kähler potential $\phi$, using a stated normalization. On Complex projective space, $\phi =\log (1+{\sum }_j|z_j{|}^2)$ gives the Fubini-Study metric in an affine chart. The potentials need not glue as one global function; their forms glue. This is how locally elementary differentiation can define nontrivial global geometry.

Several complex variables studies holomorphic functions on open subsets of ${\mathbb{C}}^n$ and their geometry, especially when $n\ge 2$. A scalar-valued function is holomorphic when it is complex differentiable in the multivariable sense; locally it has a convergent power series in $z_1,\dots ,z_n$. For a continuously differentiable function, the equations $\partial f/\partial {\bar{z}}_j=0$ for every $j$ give the corresponding Cauchy-Riemann equations.

For example, $f(z_1,z_2)=z_1{z}_2$ is holomorphic, while $|z_1{|}^2$ is not holomorphic on any open set where $z_1$ varies. A polydisc admits an iterated Cauchy integral formula, reconstructing interior values by integrating over one boundary circle for each coordinate.

The higher-dimensional theory is not just the one-variable theory repeated independently. Hartogs's extension theorem shows that isolated holes in complex dimension at least two can be removable for holomorphic functions under conditions that have no one-dimensional analogue: $1/z$ obstructs such extension in one variable. Complex analytic spaces and complex analytic hypersurfaces provide the natural geometric objects.

A holomorphic matrix function on a domain $D\subseteq \mathbb{C}$ is a matrix-valued function $A(z)$ whose entries are holomorphic functions. Differentiation is entrywise, and products satisfy the ordered product rule $(AB{)}^{\prime }=A^{\prime }B+A{B}^{\prime }$. The order matters because matrix multiplication need not be commutative.

If $A(z)$ is square and $\det A(z)\ne 0$ throughout $D$, the adjugate formula shows its inverse is also holomorphic. Differentiating $A{A}^{-1}=I$ yieldsFor $A(z)=\left(\begin{array}{cc}1& z\\ 0& 1\end{array}\right)$, its inverse is $\left(\begin{array}{cc}1& -z\\ 0& 1\end{array}\right)$ and the identity is immediate. For a fixed matrix $B$, the resolvent $(zI-B{)}^{-1}$ is holomorphic off its spectrum. This links complex analysis with eigenvalue problems; it is distinct from applying a scalar function to a matrix by functional calculus.

An anaphylatoxin is a small signaling peptide released from a complement system component during activation. C3a and C5a are important examples: fragments derived from larger circulating proteins act as ligands for cell-surface receptors, connecting complement system activation with cellular responses involved in inflammation.^[ref]

The historical name is best understood through measurements of activity. Viktor A. Bokisch and colleagues isolated C3a and measured its ability to contract smooth muscle. Its activity could be inhibited or lost without simply equating every Complement C3 fragment with the same effect. This supplies a concrete example of a protein cleavage product acquiring a signaling role distinguishable from the remainder of its precursor.^[ref]

An anaphylatoxin is therefore a host-derived signal, rather than necessarily an externally introduced poison. A response to a fragment must be specified by the receptor, responding cell and experimental readout.

The membrane attack complex is an assembly of complement system proteins that can form an opening through a target lipid bilayer. Assembly begins with C5b and recruits complement C6, complement C7, complement C8 and multiple copies of complement C9. The resulting structure connects a soluble protein cascade to a physical change in membrane permeability.^[ref]

The scientific puzzle is how proteins carried in solution become a membrane-spanning assembly. A 2012 study determined the crystal structure of complement C6 and compared it with related structures. The authors proposed that coordinated conformational changes expose membrane-inserting regions as components join the growing complex. The directly measured C6 structure and the proposed assembly pathway are distinct levels of evidence.^[ref]

An illuminating comparison is between detecting complement proteins near a membrane and demonstrating a completed membrane-spanning structure. Recruitment is one step; assembly and insertion are further steps. Counting a precursor or one bound component alone therefore cannot establish that a fully formed membrane attack complex has changed the membrane's transport properties.

Data-flow analysis propagates facts about values or variable uses through a control-flow graph. Each program operation transforms incoming facts, and a join combines information from possible predecessor paths. A compiler can use the result to justify compiler optimization or issue warnings. Clang's introduction develops this method using sets of possible values and iteration to a fixed point.^[ref]

For example, live-variable analysis asks which current values might be read later before being overwritten. For a basic block $b$, with uses before any local definition in ${\mathrm{USE}}_b$ and definitions in ${\mathrm{DEF}}_b$, the backward equations areA use requires the incoming value; an intervening definition removes that requirement. Iterating from empty sets terminates for a finite graph and finite variable set because these equations only add facts. The result guides register allocation.

Register allocation maps temporary values or virtual registers in a compiler's representation to a limited supply of physical processor registers. A value's live range covers where it may still be needed. Overlapping live ranges constrain which values can share a processor register; register spilling moves some storage into computer memory when necessary.^[ref]

For example, in a=x+y; b=a*z, the processor register holding a can hold b after a's final use, if the target operation permits. If both values are needed later, overwriting a would lose required information. An interference graph represents simultaneous storage requirements, connecting register allocation to graph coloring. Real targets add constraints from register classes, reserved registers, calling conventions and particular machine instructions; minimizing the number of names alone does not solve the whole problem.

Linking combines separately produced program components and resolves references between them. A linker (computing) reads object files and library archives, assigns or adjusts locations through relocation (computing), and connects references to their definitions. GNU ld describes these as its central operations.^[ref]

For example, one object file contains a call to a procedure defined in a second object file. The first compiler invocation can leave a symbolic reference; linking supplies the connection once both components are available. A missing definition or conflicting definitions can cause an error even when both source files compiled separately. Depending on the toolchain, some connections are completed before execution and others when a shared software library is loaded.

An object file stores program components produced by a compiler or assembler, together with information needed to combine or load them. A relocatable object file commonly contains machine instructions and data divided into sections, named symbols, and relocation records for references whose final locations are not yet known.^[ref]

For example, compiling a file that calls square can produce an object file containing the call-site instruction and an unresolved reference to that symbol. The linker (computing) later supplies its definition from another input. The file thus contains more than the bytes of a self-contained runnable program: its metadata describes how incomplete components fit together.

A linker (computing) is a program that combines compiled or assembled components and connects references to their definitions. It reads object files and libraries, resolves named symbols, and applies relocation (computing) when locations become known. A typical output is an executable program or a shared software library.^[ref]

Suppose one source file calls square and another defines it. Their object files can be produced separately; the linker (computing) matches the call's unresolved symbol to the definition and arranges the final reference. A missing definition can therefore cause a link-time failure even when each source file compiled successfully. Some references can instead be resolved when a shared software library is loaded.

An unlabelled combinatorial structure is a combinatorial structure considered up to an appropriate isomorphism, so the names assigned to underlying elements do not distinguish objects. For graphs, this means a bijection of vertices preserving adjacency. A different drawing or vertex numbering can therefore represent the same unlabelled graph.

On three labelled vertices there are $2^{\left(\genfrac{}{}{0ex}{}{3}{2}\right)}=8$ simple graphs, but only four unlabelled types: the empty graph, one edge with an isolated vertex, the two-edge path, and the triangle. Simply dividing eight by $3!$ cannot work because these types have different numbers of automorphisms.

More generally, relabelling gives a group action of the symmetric group on labelled structures. Unlabelled structures are its orbits, and the orbit-stabilizer theorem explains how automorphisms affect orbit size. Burnside's lemma counts the orbits by averaging fixed-point counts.

A $t$-$(v,k,\lambda )$ combinatorial design is an incidence structure on $v$ points with blocks containing $k$ points, such that every $t$-element subset lies in exactly $\lambda$ blocks. Repeated blocks may be allowed; a simple design has no repetitions. With $b$ blocks, counting pairs consisting of a block and one of its $t$-subsets givesFor a two-design, each point lies in $r=\lambda (v-1)/(k-1)$ blocks.^[ref]

The seven triples $123,145,167,246,257,347,356$ on $\{1,\dots ,7\}$ form a $2$-$(7,3,1)$ design: every pair occurs once, and every point occurs three times. This is the Fano plane. The equation checks $7\cdot 3=1\cdot 21$. Such exact incidence balance makes combinatorial designs useful for arranging comparisons and constructing highly symmetric finite structures.