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An integral adds up contributions of a function across a domain of integration. For a continuous function $f$ on $[a,b]$, the Riemann integral is the common limit of its Riemann sums as the largest subinterval width tends to zero:Each product combines a value with the width over which it acts. When $f$ is nonnegative, the result is the area beneath its graph; values below the horizontal axis contribute with the opposite sign. The signed area under a graph can therefore vanish even when the graph encloses nonzero geometric area.

For a velocity $v(t)=1+2t$ on $0\le t\le 3$, use $n$ equal time steps. The left and right Riemann sums areBecause the function is increasing, they bound the integral. With six steps they give $10.5$ and $13.5$; with sixty steps they give $11.85$ and $12.15$. Both approach the same total, $12$. This is accumulation made exact by refining finite measurements.

An integral retains units: velocity times time gives displacement, and a probability density function integrated over an interval gives a probability. Weighting a density by a function gives its expectation. The same operation serves different subjects because it expresses a common pattern, value multiplied by extent, then accumulated.

The fundamental theorem of calculus often evaluates a definite integral by finding an antiderivative, but a finite sum or a numerical approximation is still useful when no convenient antiderivative is available. An indefinite integral denotes a family of antiderivatives; a definite integral denotes an accumulated value between specified endpoints. More general constructions, including Lebesgue integration, extend the class of functions and domains that can be handled.

The floor $\lfloor x\rfloor$ is the greatest integer not exceeding the real number $x$. Equivalently, $\lfloor x\rfloor =m$ exactly when $m\le x<m+1$, for an integer $m$. The floor function is constant on each such half-open interval and has a discontinuity of a function at each integer.

For nonnegative integers $N$ and positive integers $k$, $\lfloor N/k\rfloor$ is the number of complete groups of size $k$ that fit into $N$ items. With $N=102$ and $k=20$, five complete groups use 100 items. Euclidean division writes $102=20\cdot 5+2$, retaining the two-item remainder rather than treating the quotient $102/20=5.1$ as a number of complete groups.

If a finite group acts two-transitively on $X$, thenwhere $V$ is irreducible and nontrivial. The character norm is the number of orbits on $X\times X$, namely the diagonal and its complement.

If each component of a continuous map on a box has opposite weak signs on the corresponding pair of faces, the map has a zero.

If a polynomial equation can be rearranged as $z=F(z)$ with a continuous $F$ mapping a closed disk into itself, Brouwer's theorem guarantees a root in that disk.

Let $V$ be a finite-dimensional real inner-product space and let $F:V\to V$ be continuous. If $⟨F(x),x⟩<0$ on the sphere $\Vert x\Vert =R$, then $F$ has a zero in the open ball. Otherwise the radial projection of $F$ gives a map of the ball that contradicts the Brouwer fixed-point theorem.

When an inverse Laplace integrand contains $p^{-1/2}$ on the principal branch, closing the Bromwich contour to the left encloses isolated poles and wraps a branch cut along the negative real axis. Pole residues give persistent oscillatory terms, while the jump across the two sides of the cut gives a real decaying integral.

Letwhere $P$ has a simple zero at $a$, and let $A$ be the limiting value of $W$ at $a$ on one sheet. Analytic continuation once around $a$ changes the sign of the square root and hence of $W^{\prime }$. The continued primitive agrees at $a$ with the original one, so it is

A phonon is a bosonic quantum of a crystal's normal modes. Because phonon number is not conserved in thermal equilibrium, its chemical potential is zero.

Bose-Einstein condensation is the macroscopic occupation of the lowest one-particle state by bosons below a critical temperature.

The unit ball in ${\mathbb{R}}^n$ can be covered by at most $9^n$ balls of radius $1/4$. A maximal $1/4$-separated set supplies the centres, and disjoint radius-$1/8$ balls around them fit inside the ball of radius $9/8$.

Kahn and Kalai represented uniform subsets by quadratic sign vectors so that diameter pairs correspond to one forbidden intersection size. The Frankl-Wilson theorem bounds each smaller-diameter part, forcing exponentially many parts while the ambient dimension grows only quadratically.

For independent rate-one exponential variables $X_n$ and $M_n={\max }_{m\le n}{X}_m$,almost surely. The first follows by applying both Borel--Cantelli lemmas to thresholds $(1\pm \epsilon )\log n$; the second uses
$\mathbb{P}(M_n\le (1-\epsilon )\log n)\le e^{-n^{\epsilon }}$.

For a nonnegative random variable $X$, Tonelli theorem givesConsequently, for each $a>0$, $\mathbb{E}[X]=\infty$ exactly when ${\sum }_{n\ge 1}\mathbb{P}(X>an)=\infty$; compare the integral on successive intervals of length $a$.

A nonzero Boolean integral domain has only the elements zero and one, so it is ${\mathbb{F}}_2$. The quotient criterion then makes every prime quotient of a Boolean ring a field, and every prime ideal maximal.

A Boolean function is a function on a Boolean hypercube, often with codomain $\{-1,1\}$ or $\{0,1\}$.

The edge boundary of a vertex set $A$ consists of graph edges with exactly one endpoint in $A$. In a $d$-regular graph,where $e(A)$ counts edges with both endpoints in $A$.

If a nonzero function on the Boolean hypercube has degree at most $k$, then its support has measure at least $2^{-k}$. This sharp bound follows by induction on the dimension.

One form of the hypercontractive inequality is $\Vert T_{1/\sqrt{3}}f{\Vert }_4\le \Vert f{\Vert }_2$, where $T_{\rho }$ is the noise operator on the Boolean hypercube.

When $N$ is prime, simultaneous approximation of the frequencies shows that $B(\Gamma ,\rho )\subseteq \mathbb{Z}/N\mathbb{Z}$ contains a centered arithmetic progression of length at least