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 if and only if This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. TheS-arithmetic group г is of typeFn, resp.FPn, if and only if for allp inS thep-adic completionGp of the corresponding algebraic groupG is of typeCn resp.CPn. We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in ?(G) are diagonalizable. Specifically, a subset is complete if and only if it contains infinitely many even-order autocorrelation functions. We next show that this result is best possible by including a result of Kalton: A frame can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis. We conjecture that (Ω, Λ) is a spectral pair if and only if the translates of some set Ω' by the vectors of Λ tile ?d. Particularly, we find that g and γ0 have better than exponential decay in both domains if and only if the associated Zibulski-Zeevi matrix is unimodular, i.e., its determinant is a constant. Second, we prove that a function belongs toQp(?Δ) if and only if it is M?bius bounded in the Sobolev spaceLp2(?Δ). For example, an integrable function $f$ belongs to $H^1(\R)$ if and only if the maximal Fej\'er operator $\sigma^{1,1}_*$ applied to $f$ belongs to $L^1(\R)$. We show that $C(\gamma)\subset S_0$ if and only if $\gamma\geq\frac{1}{4}$ and that $S_0\subset C(\gamma)$ if and only if $\gamma\leq{-}\frac{1}{4}$. We prove that the cyclic groupZn (n ≥ 3) has ak-regular digraph regular representation if and only if 0 >amp;lt;k >amp;lt;n - 1. And the value will be achieved if and only if the task is completed by its deadline. ThenCn(S) is arc-transitive if and only if Xoacts transitively onS. These methods are exponentially fitted at q0 if and only if its fitted function f(q) satisfies f(q0)=0. In particular, it is proved that and λn(LG) =-2 if and only if G is bipartite. Using this result can prove that two large subalgebras are isomorphic if and only if the corresponding nests are similar. It is showed that the LmSP (limit shadowing property) for flows is invariant under topological equivalence, and the suspension flow ?f of a homeomorphism f under a continuous function ?: X → R>amp;gt;0 has LmSP+ if and only if f has LmSP+. In this paper, it is proved that Δ + 1 ≤ χ(G2) ≤ Δ + 2, and χ(G2) = Δ + 2 if and only if G is Q, where Δ represents the maximum degree of G. This paper shows that, for every T∈B(E, F) with ‖T0+ (T-T0)‖>amp;lt;1, B ≡ (I + T0+(T-T0))-1T0+ is a generalized inverse of T if and only if (I-T0+T0)N(T) = N(T0), where N(·) stands for the null space of the operator inside the parenthesis. In this paper, we prove that a Cayley digraph Γ = Cay(G, S) is a nontrivial lexicographical product if and only if there is a nontrivial subgroup H of G such that S?H is a union of some double cosets of H in G.

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