A $\pi$ rotation about the x-axis on one qubit conjugates $Z\otimes Z$ to its negative. With ideal instantaneous pulses, two equal free-evolution intervals separated by this pulse and its inverse cancel an Ising coupling exactly. Finite-duration pulses require accounting for the coupling during the pulses; the error is small only when $|J|$ times the pulse duration is small, or when the control is explicitly designed to compensate it.

A real group representation is absolutely irreducible if its complexification is irreducible. For a finite group this is equivalent to having only scalar real endomorphisms commuting with the representation. Independent coordinate sign changes force a commuting matrix to be diagonal; a transitive cyclic permutation of the coordinates then forces its entries equal. Alternatively, the coordinate projections and the cyclic permutation directly prove irreducibility over either field.

A group representation over a field is absolutely irreducible if it remains irreducible after every field extension. For a finite-dimensional representation it is enough to check an algebraic closure. Irreducibility over the original field alone is weaker: the two-dimensional real rotation representation of the cyclic group of order three splits into two lines over the complex numbers.

A steady incompressible jet at atmospheric pressure conserves Bernoulli's equation and volume flux. Its radius contracts as $R=a[u/u_0{]}^{-1/2}$. After impact, a thin nearly horizontal axisymmetric film has $2\pi rhv=\pi a^2{u}_0$. Evaluating Bernoulli's equation on its free surface gives $v^2=u_0^2+2g(H-h)$, hence $a^4/(4r^2{h}^2)=1+2g(H-h)/u_0^2$. This inviscid thin-film description applies away from the impact region and does not describe a dissipative hydraulic jump.

The expected value of a random financial return $R$. For asset return vector $R$ with mean $\mu$, a portfolio of weights $w$ has expected return $w^T\mu$ by linearity of expectation. This measures average payoff growth rather than a guaranteed realized gain.

Selling a borrowed asset and later buying it back to return it. A negative portfolio weight represents a short position. The position benefits from falling prices and loses from rising prices; its sale proceeds may be invested in the bank account. Unrestricted short positions are a modelling assumption, not a guarantee of unlimited practical borrowing.

Spreading a portfolio across assets can reduce risk that is not shared by them. For centered asset risks with covariance matrix $\Gamma$, the variance of the weighted risk is $w^T\Gamma w$. Equal weights across $n$ independent risks with common variance $s^2$ give variance $s^2/n$, whereas equal weights across risks that are all the same random variable leave variance $s^2$. Thus small individual weights do not by themselves remove common risk; the covariance criterion for diversification of factor residuals identifies sufficient dependence and weight conditions.

Define $h(x)=|x|\log (e\pi /|x|)$ on $[-\pi ,\pi ]$, set $h(0)=0$, and extend periodically. Concavity and monotonicity on $[0,\pi ]$ give $\omega (h,\delta )=\delta \log (e\pi /\delta )$ for $0<\delta \le \pi$. Its cosine Fourier coefficients areOne integration by parts gives the formula; the remaining integral is $\log j+O(1)$ by the Dirichlet test. All these coefficients are nonpositive. The de la Vallée Poussin sum $V_n=2{\sigma }_{2n}-{\sigma }_n$ reproduces degree-at-most-$n$ trigonometric polynomials and has operator norm at most three. Hence $L_n(f)=f(0)-V_{n}f(0)$ has norm at most four and annihilates those polynomials. Its weights on coefficients are zero up to $n$, nonnegative thereafter, and one from $2n$ onwards, so $|L_n(h)|\ge {\sum }_{j\ge 2n}|a_j|\gtrsim \log n/n$. Thus $E_n(h)\ge |L_n(h)|/4\gtrsim \log n/n$; the first Jackson theorem for periodic approximation proves the matching upper bound. This cusp and the critical Weierstrass modulus have the same first-modulus order but different best-approximation rates.

The Born approximation for scalar wave scattering relates an incident plane wave and the transverse Fourier transform of its scattered field to samples of the three-dimensional Fourier transform of a scattering potential on a shifted Ewald sphere. One incident direction at one frequency samples a surface, rather than an entire three-dimensional frequency region. Additional illuminations, frequencies or structural constraints are needed for a full reconstruction.

For an outgoing field outside a sphere of radius $R$, write $u_s=k{\sum }_{n,m}{i}^{n+1}{a}_n^m{h}_n^{(1)}(kr)Y_n^m$. Its far-field pattern is $\sum a_n^m{Y}_n^m$. If its trace on that sphere lies in L2 space, Parseval's identity gives $k^2\sum |h_n^{(1)}(kR){|}^2|a_n^m{|}^2<\infty$. The fixed-argument growth of spherical Hankel functions yields the displayed equivalent range condition, with the low-order weights defined finitely. Square summability of the far-field coefficients alone does not ensure continuation to the chosen sphere.

Fix ordered branch positions and conjugacy classes $C_i$ in a finite group $G$. The generating product-one tuples $(g_1,\dots ,g_r)\in \prod C_i$ form the inner Nielsen set. The type is rigid when this set is nonempty and a single simultaneous-conjugation orbit. Over $\mathbb{C}(T)$ this fixes the connected Galois cover up to isomorphism: tuples prescribe the monodromy epimorphism of the punctured sphere, and conjugate epimorphisms give isomorphic covers; the local completions at punctures are uniquely determined by their cyclic monodromy.

Choose canonical spin states by applying the rotationless Lorentz boost $L(p)$ to rest-frame spin states. Spatial inversion $\Pi =\mathrm{diag}(1,-1,-1,-1)$ commutes with rest-frame rotations, so Schur lemma makes its action on one irreducible rest spin multiplet a phase ${\eta }_P$. Since $\Pi L(p)\Pi =L(\Pi p)$, the formula follows. Canonical spin labels are unchanged; helicity reverses. With $P^2=I$, the intrinsic parity is ${\eta }_P=\pm 1$.

// Target: lie-theory.bigb

In the Dirac representation of the gamma matrices, the parity symmetry in quantum field theory action ${\psi }_P(x)={\gamma }^0\psi (x_P)$ and the charge conjugation action ${\psi }_C=i{\gamma }^2{\psi }^{\ast }$ obey $({\psi }_P{)}_C(x)=-{\gamma }^0{\psi }_C(x_P)$, since ${\gamma }^0$ is real and anticommutes with ${\gamma }^2$. At rest, positive-frequency Dirac spinors have nonzero upper components and negative-frequency Dirac spinors have nonzero lower components; they are respectively $+1$ and $-1$ eigenvectors of ${\gamma }^0$. With the common conventional real intrinsic-parity phase, particle and antiparticle therefore have opposite intrinsic parity. The overall sign is conventional, whereas their relative sign is fixed.

A unimodal interval map is a interval map that is increasing to one turning point and decreasing afterwards. For a differentiable map with a maximum at zero, a critical point of even order $d$ means $f(x)=f(0)+c{x}^d+o(x^d)$ with $c<0$. The critical order distinguishes universality classes of period-doubling cascades.

// Target: dynamical-systems.bigb

Interval root exclusion certifies that a function has no zero of a function on $X$ when a valid interval enclosure $F(X)$ excludes zero. For $g(x)=x(1-x)-1/3$ on $[0,1]$, the interval square range yields $G(X)=[-1/3,-1/12]$, which excludes zero. Direct interval arithmetic yields $[-1/3,2/3]$ and is inconclusive, even though the function has no zero of a function there.

Knowing only that a calculated interval enclosure contains zero does not certify a zero of a function: it may contain unattainable values. Even the positive constant function $f(x)=1$ has a valid, loose interval enclosure $[-1,2]$. A separate existence argument, such as reliable opposite endpoint signs together with continuity in the intermediate value theorem, is required; see root bracket and bisection method. Interval subdivision can resolve some inconclusive pieces, but remaining pieces must be reported as unresolved unless an existence argument succeeds. This makes interval root exclusion useful for systematically discarding regions in a search.

// Target: analysis.bigb

Outward rounding replaces an exact lower endpoint by a representable value no larger than it, and an exact upper endpoint by a representable value no smaller than it. This preserves an interval enclosure under floating-point arithmetic. In the binary numeral system, most rational numbers cannot be represented exactly with finitely many digits, so the rounding direction is essential even for elementary calculations. Rump explains directed rounding and its use in interval bounds in §§4 and 5.3 of his survey.^[ref]

To see the issue without long binary digit strings, use an illustrative decimal grid with step $0.001$. Exact interval division gives $[1,1]/[3,3]=[1/3,1/3]$. Rounding both endpoints to the nearest grid value gives $[0.333,0.333]$, which excludes the true answer because $0.333<1/3$. Outward rounding instead givesThis decimal toy illustrates the rule; its displayed endpoints are not the results of binary directed rounding. Real implementations need valid outward rounding throughout the calculation, including input conversion and intermediate interval arithmetic operations; rounding only the final output cannot repair an earlier lost bound. INTLAB's tutorial demonstrates why decimal input strings and previously rounded intermediate values need care.^[ref]

For closed intervals $X=[a,b]$ and $Y=[c,d]$, exact addition, subtraction and multiplication giveThese interval arithmetic operations describe all results $u\circ v$ for $(u,v)$ in the Cartesian product $X\times Y$: the two inputs are allowed to vary independently. Independence here means freedom to choose either input anywhere in its closed interval; it does not assert anything about probability. For example, $[1,2]+[3,5]=[4,7]$ and $[1,2]-[3,5]=[-4,-1]$.

All four endpoint products matter when signs change. For $X=[-2,3]$ and $Y=[-4,1]$, the endpoint products are $8,-2,-12,3$, so $XY=[-12,8]$. Taking just $ac$ and $bd$, even after sorting them, would give $[3,8]$ and lose valid negative products.

For division, if $0\notin Y$, the multiplicative inverse gives $1/Y=[1/d,1/c]$ and $X/Y=X(1/Y)$. Thus $[1,2]/[3,4]=[1/4,2/3]$. This ordinary finite-closed interval rule requires a denominator that excludes zero. When the denominator contains zero, division is undefined for some pairs and can become unbounded near zero; these formulas cannot supply a finite interval enclosure of all quotients. The operations above use exact endpoints; a computer implementation must also apply outward rounding.

An interval enclosure $F(X)=[L,U]$ for a real-valued function $f$ on a closed interval $X$ satisfiesHere $f(X)$ means the actual image of a function, also called its pointwise range on $X$; $F(X)$ denotes a calculated interval enclosure. They need not agree. Sampling a few inputs can suggest a bound, but does not establish an interval enclosure between those inputs. Kearfott and coauthors explicitly distinguish the image of a function from evaluating an expression with interval arithmetic, because different expressions for the same function can give different bounds.^[ref]

A tensor operator transforming in an irreducible representation $R$ between states in $V$ defines an intertwining operator $R\otimes V\to V$. In a basis of these intertwining maps, its scalar coefficients $g_{\nu }$ are reduced matrix elements, independent of the component indices. Schur lemma makes the number of independent coefficients the multiplicity of $V$ in $R\otimes V$. The component dependence is fixed by the associated Clebsch-Gordan coefficients.

In the Deligne-sheaf convention, $I{H}_{\bar{p}}^i(X)={\mathbb{H}}^i(X,I{C}_{\bar{p}})$, with $I{C}_{\bar{p}}$ normalized in degree zero on the regular stratum. For a compact oriented $N$-dimensional pseudomanifold over a field, $I{H}_{\bar{p}}^i(X)\cong I{H}_{N-i}^{\bar{p}}(X)$ and it is dual to $I{H}_i^{\bar{q}}(X)$, where $\bar{p}(k)+\bar{q}(k)=k-2$. Thus the dual of lower-middle singular chains uses the upper-middle Deligne sheaf. The two middle conventions agree on Witt spaces.