Sentence examples for be an arbitrary polynomial from inspiring English sources

Let (f(x)) be an arbitrary polynomial.

Let (mathbf{p}_{1}) be an arbitrary polynomial degree distribution satisfies (2.13).

Let (mathbf{p}_{1}) be an arbitrary polynomial degree distribution satisfying (2.13) and (p_{tau}geq2), (forall tauinmathcal{T}).

Proposition 2.2 Let P ( D ) be an arbitrary polynomial in D acting on two differentiable functions f ( x, t, … ) and g ( x, t, … ), then the following equations hold: ( i ) P ( D ) { f ⋅ g } = P ( − D ) { g f }, (4).

This can be seen as follows: Suppose (T in mathbb {Z}[underline {x}]) is an arbitrary polynomial.

The corresponding kernels are given by the formula h ( t ) = P ( ln t ) t - 1 (1.2)where P ( x ) is an arbitrary polynomial.

When (m+n+kappageq0), the general solution of ({P R}_{m,n}) is Phi z)=e^{-frac{2niz}{a}}X (z ) biggl[ frac{1}{2api i}int _{L_{0}}frac{g(t)}{e^{-frac {2nit}{a}}X^(t)}cotfrac{t-z}{a},mathrm{d}t+P_{m+n+kappa } bigl(e^{frac{iz}{a}} bigr) biggr],quad zin mathcal{C}backslash L, (3.39) where (P_{r}) is an arbitrary polynomial of degree not greater than r.

From [7, 8], we know that, when (m+n+kappageq-1), the general solution of (R_{m+n}) problem (3.13) is Phi_{sharp}(w)=X_{sharp}(w) bigl[ Psi_{sharp}(w)+P_{m+n+kappa }(w) bigr],quad winmathcal{C}backslash Gamma, (3.16) where (P_{r}) is an arbitrary polynomial of degree not greater than r ((P_{r}equiv0) if (r<0)), denoted as (P_{r}inPi_{r}).

Theorem 2 Let P ( x ) ∈ C p [ [ x ] q ] be an arbitrary q-polynomial with ∑ a i [ x ] q i.

where δ n, q → 0 and n → ∞ and δ n, q is independent of a, then μ − q is called a weakly fermionic p-adic q-measure on Z p. If δ n, q is replaced by C p − ν p ( 1 − q n ) (C is some constant), then μ − q is called a strongly fermionic p-adic q-measure on Z p. Let P ( x ) ∈ C p [ [ x ] q ] be an arbitrary q-polynomial with ∑ a i [ x ] q i.

Let be an arbitrary function.