FAT32 is a File Allocation Table filesystem variant using 32-bit table entries with fewer bits available for cluster numbers. A directory entry stores file size in a 32-bit unsigned field, limiting one ordinary file to $2^{32}-1$ bytes. This limit is distinct from a volume's capacity, which also depends on sector and cluster layout and implementation restrictions. FAT32 is widely interoperable but lacks native per-file security ACLs and metadata journaling of the kind provided by NTFS. Filesystem limits should be distinguished from a formatting tool's additional limits.^[ref]

// Target: computer-science.bigb

A triangular number counts rows of $1,2,\dots ,n$ objects, with $n$ a nonnegative integer:The first values are $0,1,3,6,10,15,21,28,36,45,\dots$. The empty arrangement contributes zero, by the empty sum convention. Adding one more row gives the triangular number recurrence $T_n-T_{n-1}=n$ for $n\ge 1$. The row lengths form an arithmetic progression, while their running totals are triangular numbers.

Pair the rows with a reversed copy: $1+n=2+(n-1)=\cdots =n+1$. There are $n$ such pairs, so two copies contain $n(n+1)$ dots. Dividing by two proves the formula. This rectangle picture explains the factor one-half without requiring a formula for a long summation. For example, one hundred rows contain $T_{100}=5050$ dots.

The same count appears whenever every pair of different objects must be considered once. Five participants make ten possible handshakes; five sensors have ten different pairwise comparisons; a complete graph on five vertices has ten edges. These are the triangular number pair-counting identity, with one fewer row than objects. The distinction prevents a common error: $T_5=15$ counts all pairs from six objects, whereas five objects give $T_4=10$.

The triangular number square-complement identity says that two neighboring triangular numbers fill a square. Combining that identity with the one-row difference makes the difference of squares $T_n^2-T_{n-1}^2$ equal to $n^3$. It leads to Nicomachus's theorem: the sum of the first $n$ cubes of numbers is the square of a number $T_n^2$. A change in the way one groups objects exposes a relation between summation, multiplication and geometry.

This geometric view is already visible in Nicomachus's Introduction to Arithmetic by Nicomachus (Hahn teacher edition): Book II, Chapter 8 describes growing triangular rows, Chapter 12 combines successive triangular counts into squares, and Chapter 20 groups successive odd integers into cubes of numbers.^[ref] The linked edition uses Martin Luther D'Ooge's translation, revised with modern teaching material by Nathanael Hahn. Our empty-row index $T_0=0$ is a modern convention; the historical passage starts from unity and calls three the first actual triangle.

K-essence is a cosmological scalar-field mechanism in which noncanonical kinetic dynamics supply negative pressure or accelerated expansion. Its action is a particular use of a P(X, phi) scalar field theory; that general action also includes theories without kinetic dark energy.

A field extension $L/K$ is algebraic if every element of $L$ is an algebraic element over $K$. Every finite field extension is algebraic, but algebraic extensions can have infinite degree.

A field extension obtained by adjoining finitely many square roots in characteristic different from two. If the classes of $a_1,\dots ,a_r$ are independent in $K^{\times }/(K^{\times }{)}^2$, the extension has degree $2^r$ and Galois group $(\mathbb{Z}/2\mathbb{Z}{)}^r$: each square root's sign can be changed independently. The products of distinct square roots form a basis over $K$.

The algebraic conjugates of an algebraic element over a field $K$ are the polynomial roots of its algebraic minimal polynomial over $K$ in an algebraic closure.

The algebraic conjugates of an algebraic number are the polynomial roots of its algebraic minimal polynomial, equivalently its images under field embeddings of its number field into an algebraic closure.

A field $K$ is Hilbertian if each finite family of polynomials irreducible over $K(T)$ and separable in its root variable remains irreducible of the same degree at infinitely many $T=a\in K$, after any finitely many prescribed bad parameter values are excluded. Equivalently, every finite separable extension of $K(T)$ has infinitely many unramified $K$-rational places whose residue extension retains its full degree.

For each positive integer $n$, the product of the preceding Fermat numbers is two less than the next:For example, $3\cdot 5\cdot 17=255=257-2$. The identity telescopes by repeated difference of squares: $(2^{2^j}-1)(2^{2^j}+1)=2^{2^{j+1}}-1$. Its usefulness is that a huge integer product exposes a tiny difference, two, from the next term.

The set of all feasible points of an optimization problem. It depends on the optimization constraints and the decision domain; the objective function ranks its members.

Think of three successive questions. Which decisions can be represented in the model? Which satisfy every optimization constraint? Which of those receive the best objective function value? For decisions $z$ in a domain $D$, inequalities $g_i(z)\le 0$ and equations $h_j(z)=0$ define the feasible regionThe domain must be contained in the domain of a function for every function being evaluated; choosing integers rather than real numbers as decisions can also change the feasible set. Constraint satisfaction determines membership of $F$. Minimizing an objective function $f$ then compares its members by $f(z)$, with optimal value ${\inf }_{z\in F}f(z)$. Changing the scores while keeping $D$ and the optimization constraints fixed leaves $F$ unchanged, provided the new objective function is defined throughout $D$. These distinctions separate an allowed decision from a best decision.^[ref]

Let $M_0\subset M_1$ be nested normal linear models with full-rank design matrices of ranks $p_0,p_1$, where $0\le p_0<p_1<n$, and $n$ observations. Under the reduced model,The difference of the orthogonal projection matrices onto the two model spaces is an orthogonal projection of rank $p_1-p_0$, orthogonal to the full-model residual projection. Independent normal distribution projections therefore give independent chi-squared numerators and denominators after division by ${\sigma }^2$. This proves the ratio law. The test evaluates the extra terms conditional on all reduced-model terms, and estimates the unknown common variance from the full model.

A set family on a finite set separates ordered pairs if each ordered pair of distinct points has a member containing the first point and omitting the second. If its members are $A_1,\dots ,A_m$, assign each point the characteristic vector of its incidences, or equivalently $T_i=\{a:i\in A_a\}$. Separation in both directions says precisely that the $T_i$ form an antichain in the Boolean lattice on $m$ coordinates. Thus the Sperner theorem gives $n\le \left(\genfrac{}{}{0ex}{}{m}{\lfloor m/2\rfloor }\right)$. Conversely, any antichain of $n$ distinct subsets of $[m]$ supplies such a set family by transposing its incidence table.

If all members of a set family on $[n]$ have odd size and all distinct pairwise intersections have even size, its characteristic vectors over the finite field ${\mathbb{F}}_2$ are linearly independent. Their pairwise dot products are zero, and each vector has dot product one with itself. Taking the dot product of a linear combination with each vector makes every coefficient vanish, so there are at most $n$ vectors. The set family of all singleton sets attains this bound.

If every member of a set family on $[n]$ and every distinct pairwise intersection has even size, its characteristic vectors span a self-orthogonal binary subspace. The standard dot product is nondegenerate, so $W\subseteq W^{\perp }$ implies $2\dim W\le n$. Thus the set family has at most $2^{\lfloor n/2\rfloor }$ members. All unions of $\lfloor n/2\rfloor$ disjoint pairs of points attain this bound.

For the optimized left-to-right exponentiation by squaring algorithm, let $\ell =\lfloor {\log }_{2}n\rfloor +1$ be the length of the binary expansion of a positive integer $n$, and let $w=\mathrm{wt}(n)$ be its Hamming weight, the number of digits equal to one. There are exactly $\ell -1$ digits after the leading one, and exactly $w-1$ remaining ones. Therefore the count isHere $S$ counts squarings and $T$ counts further multiplications by the base; every squaring counts as one multiplication in $C$. Assignment, digit reading, and any inversion are outside this count. The formula includes $n=1$, which requires zero multiplications; define $C(0)=0$ separately.

For a fixed length $\ell$, the bounds are $\ell -1\le C(n)\le 2\ell -2$. A power of two reaches the lower bound because its Hamming weight is one; $n=2^{\ell }-1$ reaches the upper bound because all digits are one. For $n=13$, $\ell =4$ and $w=3$, giving $C(13)=5$. Thus the arithmetic operation count is $O(\log (n+1))$ in Big O notation, but inputs of similar magnitude can have different counts.

This is a count for the specified algorithm, not a claim that it always minimizes multiplications among all methods. For instance, at $n=15$ the count is six, but the powers with exponents $1,2,3,6,12,15$ can be built in five multiplications: square, multiply by $a$, square, square, then multiply by the already stored $a^3$. Saving operations this way uses a different schedule and stored intermediate powers.

A stretched exponential has the form $e^{-c{x}^{\alpha }}$ with $c>0$ and $0<\alpha <1$. A boundary cost proportional to the square root of an enclosed area naturally produces exponent $\alpha =1/2$.

Methods of detecting exoplanets infer or resolve an exoplanet through its effects on the host star's light or through its own light. The observable determines what can be learned:

An exoplanet secondary eclipse measures the loss of the exoplanet's light when it passes behind its star, complementing exoplanet transit photometry. A detection method also selects which systems are easiest to find: deeper exoplanet transits favour larger radius ratios, and larger stellar radial-velocity semi-amplitudes favour more massive companions, shorter orbital periods, and larger $\sin i$. These preferences must be accounted for when inferring the underlying exoplanet population.

In an economy with a measure space of consumers, a consumption allocation is a measurable, integrable function assigning each consumer a feasible bundle. Exact aggregate feasibility means ${\int }_{A}fd\mu ={\int }_A\omega d\mu$.

A demand correspondence assigns a nonempty set of chosen consumption bundles to each price and wealth pair. Every chosen bundle satisfies the budget constraint. A utility function rationalizes the demand correspondence when the chosen bundles are exactly its maximizers over the budget set.

A price quasi-equilibrium with transfers is an exactly feasible consumption allocation $f$ and a nonzero price vector $p$ for which, almost everywhere, every strictly preferred bundle has cost at least $p\cdot f(a)$. The transfer-adjusted wealth is $p\cdot f(a)$; transfers $p\cdot (f(a)-\omega (a))$ sum to zero. Equal-cost strict improvements may remain, which distinguishes a quasi-equilibrium from a utility-maximizing equilibrium. Nonnegative prices need extra assumptions if goods may be undesirable.

Aggregate excess demand subtracts total endowment from the sum of consumers' demand correspondences. A market-clearing price satisfies $z(p)=0$. Under Walras law, one equation is redundant after price normalization.