At integer times, $B_n/\sqrt{n}$ has the same standard normal distribution for every $n$. The event that its limit superior exceeds a fixed finite level is a tail event of the independent unit-time Brownian increments, since changing any finite initial segment contributes only a term tending to zero. The Kolmogorov zero-one law applies. Its probability is positive because each fixed-time exceedance has the same positive probability, giving a positive probability of infinitely many exceedances by the decreasing union-of-tail-events argument. Thus its probability is one. Intersecting over integer levels proves the stated divergence, without the lower iterated-logarithm bound.

An increment of a Brownian motion between deterministic times $s<t$ is $W_t-W_s$. It has normal distribution $N(0,t-s)$ and is independent of the filtration up to time $s$. Increments on disjoint time intervals are independent. Their covariance is the length of overlap of the corresponding intervals; this also follows from the Brownian covariance kernel.

For the completed natural Brownian filtration, every event in the germ sigma-field of Brownian motion has probability zero or one. Indeed, an event measurable at every positive time is independent of all increments after each such time. Letting that time decrease to zero and using path continuity makes the event independent of the entire Brownian path sigma-field. Since it is itself measurable in that sigma-field, its probability equals its square.

A Wiener sausage is the region swept out by a ball of fixed radius along a Brownian motion path. A deterministic continuous displacement $f(s)$ gives the variant ${\bigcup }_{s\le t}\mathcal{B}(B_s+f(s),r)$. On each bounded time interval it has finite expected volume: the region lies inside a ball whose radius is the path's maximum displacement plus $r$, and the Brownian maximum has Gaussian tail bounds and finite moments of every order.

A Brownian motion starting outside a fixed affine subspace of codimension at least two almost surely never hits it. Choose a two-dimensional orthogonal projection annihilating the subspace's direction and with nonzero projected initial displacement. The projected process is planar Brownian motion starting away from zero, and a hit of the subspace would hit zero in that projection. The polar point for planar Brownian motion result excludes this. In three dimensions this shows avoidance of every fixed line when started outside it.

Porous-matrix drag depends on fluid velocity relative to the matrix, so it is not invariant under adding a uniform fluid velocity while leaving the matrix fixed. In a swimmer frame with stationary laboratory matrix, the drag is proportional to $\mathbf{u}-U{\mathbf{e}}_x$. A constant shift can alternatively be absorbed into a linear pressure term, whose physical interpretation must then be retained.

For $h_t=\kappa (h{h}_x{)}_x$ with a drained boundary $h(0,t)=0$ and a finite current vanishing at its nose, integration by parts gives $\dot{D}=-\kappa [h^2{]}_0^{\infty }/2=0$. The outlet can have finite volume flux because its contribution to the first-moment flux is multiplied by $x=0$. A positive prescribed boundary height instead gives $\dot{D}=\kappa h(0,t{)}^2/2$, so drainage is essential to this invariant.

For the unit-modulus hyperbolic scattering block with elementary mass $m$, a pair bound at rapidity difference $iu$ has mass $m_2=2m\cos (u/2)$. Bootstrap fusion with another elementary particle gives a further direct pole at $3iu/2$. It is inside the physical rapidity strip with positive imaginary residue only for $0<u<2\pi /3$. In that range the new bound state has rest-frame constituent rapidities $iu,0,-iu$, so its energy is $m(1+2\cos u)=m\sin (3u/2)/\sin (u/2)$. Charges add at the bound-state vertex. At $u=2\pi /3$ the apparent pole cancels; beyond that point it lies outside the physical strip. The formula is conditional on this particular scattering block and its direct-pole interpretation, not a universal three-body binding law.

Canonical diagonalization of $A_k{b}_k^{†}{b}_k$ plus an anomalous pair term gives the vacuum shift $(E_k-A_k)/2$ per momentum. Thus $H=C+\frac{1}{2}{\sum }_k(E_k-A_k)+{\sum }_k{E}_k{\alpha }_k^{†}{\alpha }_k$. The subtraction of $A_k/2$ accompanies the quasiparticle zero-point term; omitting it changes the ground-state energy.

For a finite physical supermultiplet at fixed positive energy, fermion parity anticommutes with each odd supercharge. Trace cyclicity makes $\mathrm{Tr}[(-1{)}^F\{Q,Q^{†}\}]=0$. Summing the Super-Poincaré algebra diagonal entries replaces the anticommutator by $4E$, giving equal bosonic and fermionic state counts. A zero-energy vacuum is an exception to the division by $E$, so it need not have a partner. The argument requires an invariant fixed-momentum representation, which is the step lost under explicit supersymmetry breaking.

A dilute condensed Bose system whose interaction is sufficiently weak for a small-depletion expansion. Its low-energy excitations are described by a Bogoliubov transformation rather than free particles. The homogeneous quadratic theory is controlled only while noncondensate occupation remains small compared with the condensate and its excitation energies are stable.

A Bose-Einstein condensate is a state with macroscopic occupation of a one-particle mode. A phase-selected mean-field description can use $⟨\psi ⟩\ne 0$, while a fixed-number finite-volume state need not have a nonzero field expectation. Bose-Einstein condensation and superfluidity are related but distinct notions; long-wavelength phase fluctuations can invalidate a uniform condensate saddle in low dimensions.

For an input $x$, count the largest family of pairwise disjoint nonempty sets of coordinates such that flipping each set individually changes the Boolean function value. Maximizing over inputs gives block sensitivity. Sensitive sets need not be contiguous and need not consist of single bits. The standard bounded-error quantum query complexity lower bound is $\Omega (\sqrt{\mathrm{bs}(f)})$. A certificate for $f(x)$ meets every sensitive set, so its size bounds their disjoint count.

Here $\tau (x,y)=P_{p_c}(x\leftrightarrow y)$ is critical two-point connectivity. The condition controls the intersection effects that spoil a branching approximation. It is a sufficient criterion for important mean-field percolation exponents, including the order-parameter, percolation susceptibility and cluster-tail exponents. Establishing detailed spatial asymptotics is a further task, commonly addressed by lace expansion.

If independent edges of a locally finite graph receive fair orientations, exploring outward from a root tests each fresh reached-to-unreached boundary edge with success probability $1/2$. Deferred decisions give exactly the exploration law of the root cluster in bond percolation of density $1/2$. Thus the two reachable graph vertex sets have the same distribution. An infinite out-cluster contains an infinite simple directed ray by the König infinity lemma. Geometrically biased orientations do not generally give the same uniform edge-success law.

A path in the underlying graph all of whose edges are open. Connectivity of open paths determines the percolation clusters. A ray in a graph with all edges open is an infinite ray in a graph.

For finite nonnegative measures with equal mass and first moment and finite second moments, the constant and linear terms of their Fourier transforms cancel. Taylor's remainder bounds their difference by a constant times $|\xi {|}^2$, making this distance finite. Fourier uniqueness gives definiteness. The origin is excluded from the supremum; the quotient need not have a direction-independent limit there.

The divisor contracted by a blowup of an algebraic variety to its centre. For the blowup of a smooth point on an $n$-dimensional smooth variety it is ${\mathbb{P}}^{n-1}$. Being a closed projective subvariety of positive dimension, it can obstruct affineness of a fibre containing it.

For a blowup of an algebraic variety $\pi :\tilde{X}\to X$ with centre $Z$, the strict transform of an algebraic subvariety of a subvariety $Y$ not contained in $Z$ is the closure of ${\pi }^{-1}(Y\setminus Z)$. It excludes components supported entirely in the algebraic exceptional divisor.

Constant areal velocity gives constant specific angular momentum $l=r^2\dot{\theta }$, hence zero transverse acceleration and a central force. For a focus-based ellipse $u=1/r=(1+\epsilon \cos \theta )/A$, one has $u_{\theta \theta }+u=1/A$. The Binet equation then gives the displayed radial acceleration, proving an attractive inverse-square law. An orbit's shape alone does not specify its acceleration without a time law.