Sentence examples for m is finite from inspiring English sources

In the first part we show that a Sobolev Orlicz imbedding implies that the bottom of the L2-spectrum of H is an eigenvalue (i.e. the existence of the ground state) with finite multiplicity, provided m is finite.

In addition, we compute the asymptotic distribution of total winding numbers in the double-scaling regime in which the expected number of walkers with winding number not equal to m is finite.

In the second part we prove that for a large class of operators, namely those for which Persson's characterization of the bottom of the essential spectrum holds true, a Sobolev Orlicz imbedding always implies the discreteness of the L2-spectrum of H, provided m is finite.

It follows from (1.1) and (4.1) that m is finite.

Assume that M is finite.

On the contrary, assume that M is finite.

From (H), we infer that p i * ; i = 0, …, m are finite.

It should be noted that the vector field Φ m is finite-dimensional and, thus, the degree involved in (55) is the Brower degree.

Before we state our main results, the following assumptions will be made: (A1):  The parametric space ϒ is compact with (Upsilon={alpha: deltaleqalpha_{0}leq M, 0leq alpha_{1}+cdots +alpha_{p}leq M^{ast}<1, alpha_{i}geq0, i=1, 2, ldots, p }), where δ and M are finite positive constants, and the true parameter value (alpha^{0}) is an interior point in ϒ.

The parametric space ϒ is compact with (Upsilon={alpha: deltaleqalpha_{0}leq M, 0leq alpha_{1}+cdots +alpha_{p}leq M^{ast}<1, alpha_{i}geq0, i=1, 2, ldots, p }), where δ and M are finite positive constants, and the true parameter value (alpha^{0}) is an interior point in ϒ.

This means that M i is finite; thus, K is finite as N→∞.