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The fixed rate making an interest rate swap have zero initial value. For unit accrual periods, unit notional and payments $r_t-s$, the floating-leg value is $1-P_0(T)$ and the fixed-leg annuity is ${\sum }_{t=1}^T{P}_0(t)$. The formula changes when accrual lengths or payment schedules differ.

A hierarchical statistical model that contains an interaction term retains its associated lower-order effects. For two categorical predictors this means keeping both main effects when keeping their interaction. A nonsignificant averaged main effect may coexist with large opposite effects in different groups. Dropping its lower-order term while retaining the interaction can make the model depend on arbitrary factor coding.

A coefficient multiplying an interaction term controls its strength relative to the chosen field normalization. Its engineering dimension determines whether it is dimensionless, relevant or irrelevant. A renormalized coupling constant constant is specified at a renormalization scale; its change with that scale is described by the renormalization-group beta function.

A smooth curve tangent to a regular integrable distribution lies in one maximal leaf of a regular foliation. In a Frobenius theorem chart its transverse coordinates have zero derivative, so each short segment lies in one plaque. A finite chain of such segments covers any compact parameter interval and joins plaques in the same leaf. A single embedded local plaque only gives confinement while the curve remains in its chart. A dense immersed cylinder in a three-dimensional torus shows why global embedded confinement cannot generally replace the immersed statement.

The map descends to an injective immersion from $\mathbb{R}\times (\mathbb{R}/\mathbb{Z})$ into the three-dimensional torus. Injectivity follows because an integer difference in $s$ with an integer difference in $\alpha s$ must be zero. Its image is dense by an irrational rotation of the circle, and is tangent to the commuting independent fields ${\partial }_x+\alpha {\partial }_z$ and ${\partial }_y$. It is a leaf of a regular foliation, but cannot be an embedded submanifold: in an embedding chart the coordinate plane would be locally closed yet contain a dense ambient subset.

Changing the lower limit in $H(a){\int }_{a_i}^{a}d\tilde{a}/(\tilde{a}H{)}^3$ adds a constant multiple of $H(a)$, the independent matter-era decaying solution. Thus a finite lower limit defines a useful independent solution but can include a decaying admixture. A pure growing normalization is selected by an appropriate linear combination.

When $H\to H_{\Lambda }>0$, the growth integral has an integrable $a^{-3}$ tail. Its product with $H$ tends to a constant, with leading correction proportional to $a^{-2}$ in a matter-plus-Lambda background. Both displayed basis functions can tend to constants; a linear combination cancels that constant and isolates the truly decaying late-time $a^{-2}$ behavior.

If $A$ is an integral domain with fraction field $K$ and $L/K$ is finite, a $K$-basis of $L$ can be rescaled to consist of integral elements over $A$. For a monic algebraic equation, choose $a\in A$ clearing the coefficient denominators; replacing $u$ by $au$ multiplies the coefficient of degree $r-j$ by $a^j$, putting it in $A$.

For a normal Noetherian ring $A$ that is an integral domain, the integral closure in a finite separable field extension of its fraction field is finite as an $A$-module. An integral field basis by denominator clearing gives a finite lattice. Integral traces place the closure in its finite trace-dual lattice; Noetherianity makes that submodule finitely generated.

Differentiating the Euclidean classical equation for an instanton gives a zero eigenfunction proportional to its time derivative. Its norm satisfies $\mu \int {\dot{q}}_I^{2}d\tau =S_I$, so replacing the Gaussian zero-mode integral by the center coordinate gives $\sqrt{S_I/(2\pi \hslash )}d{\tau }_0$. The nonzero modes belong in the primed functional determinant. This removes the false divergence without deleting the center's integration volume.

The canonical gradient is the influence-function representer in the statistical tangent space $\mathcal{T}$. It equals the orthogonal projection onto $\mathcal{T}$ of any representer. Every other representer differs from it by an element of ${\mathcal{T}}^{\perp }$, so the Pythagorean theorem in an inner-product space gives it minimum variance. For $\psi (P)=Pa$ in an unrestricted density model with bounded $a$, it is $a-Pa$.

For $a_j\ge 0$, the finite product satisfiesThe lower inequality follows by product expansion; the upper one uses $1+t\le e^t$. Since both the partial sums and partial products are increasing, one converges to a finite limit exactly when the other does. The monotone bounded sequence theorem proves both implications. The product cannot tend to zero in this nonnegative setting.

If a tail of holomorphic factors admits logarithms whose absolute values have a summable bound uniformly on every compact set, their product converges locally uniformly to the exponential of the logarithmic sum. The tail is holomorphic and nowhere zero. Finite initial factors may have zeros; those finite factors determine the locations and orders of all zeros near any given compact set. The scalar sufficient condition ${\sum }_j|u_j|<\infty$ for factors $1+u_j$ follows from $|\log (1+u_j)|\le 2|u_j|$ for $|u_j|\le 1/2$.

The Weierstrass elementary factor has a simple zero at $w=1$ and no other zeros. Its logarithmic series begins at degree $p+1$: $\log E_p(w)=-{\sum }_{\nu \ge p+1}{w}^{\nu }/\nu$ for $|w|<1$. Raising the factor order $p$ makes the tail arbitrarily small on a fixed compact set and enables infinite product convergence from logarithmic tails for unrestricted discrete zero data.

A nonempty path space $[T]$ is perfect if it has no isolated points: for every path and every finite prefix of it there is a different path sharing that prefix. This property of the closed subset of Cantor space does not imply that its particular finite-string presentation has no dead ends. Requiring every finite node to extend to incompatible nodes is the stronger pruned-tree convention for a perfect tree.

The hadronic R ratio normalizes the photon-channel hadronic relativistic cross-section to the leading massless muon-pair value. The free active-quark result is $R=N_c{\sum }_f{Q}_f^2$. It depends on flavor charges as well as the number of active flavors; it is not simply $N_c{N}_f$.

The measure $d\sigma /\sigma$ is invariant under multiplication by every positive constant. In the logarithmic coordinate it is Lebesgue measure. It cannot be normalized over $(0,\infty )$, although a bounded truncation is a log-uniform distribution.

If a linear functional $\ell$ vanishes on the image of a linear map $T$ but $\ell (b)\ne 0$, the equation $Tx=b$ has no solution. In Euclidean coordinates, a vector $n$ with $n^{T}A=0$ provides such an obstruction whenever $n^{T}b\ne 0$. In finite dimensions these annihilating functionals exactly characterize membership in the image: $b\in \mathrm{im}A$ precisely when it is orthogonal to every vector in $\ker A^T$.

For a morphism in an abelian category, its coimage $\mathrm{coker}(\ker f)$ is canonically isomorphic to its image $\ker (\mathrm{coker}f)$. This yields an epimorphism followed by a monomorphism, unique up to unique compatible isomorphism. Every epi-mono factorization has this same middle object.

An idele obtained by mapping the same nonzero element of a number field into every completion. Only finitely many local valuations are nonzero, so this diagonal tuple belongs to the idele group.