Sentence examples for exists an element of from inspiring English sources

In Klein's approach each geometry is a (connected) manifold endowed with a group of automorphisms, that is, a Lie group \(G\) of "motions" that acts transitively on the manifold, such that two figures are regarded as congruent if and only if there exists an element of the appropriate Lie group \(G\) that transforms one of the figures into the other.

In particular, in the Einstein, Podolsky, Rosen paper, we find the following criterion for the existence of physical reality: "If without in any way disturbing a system we can predict with certainty...the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity" (Einstein et al 1935, 778).

A Banach space X is said to have normal structure if for each bounded closed convex subset K of X, which contains at least two points, there exists an element of K which is not a diametral point of K. Let ({x_{n}}) be a bounded sequence in X and (emptysetneq Esubseteq X).

To see this, since S is a proper subset of X, there exists an element x of S c.

Recall that a closed convex subset C of a Banach space E is said to have a normal structure if for each bounded closed convex subset K of C which contains at least two points, there exists an element x of K which is not a diametral point of K, i.e., sup { ∥ x − y ∥ : y ∈ K } < d ( K ), where d ( K ) is the diameter of K. Let C be a nonempty closed convex subset of E. Let T : C → C be a mapping.

In what follows, we always assume that E is Banach space with the dual E ∗. Recall that a closed convex subset C of E is said to have normal structure if for each bounded closed convex subset K of C which contains at least two points, there exists an element x of K which is not a diametral point of K, i.e., sup { ∥ x − y ∥ : y ∈ K } < d ( K ), where d ( K ) is the diameter of K.

Then there exists an element v of H with (|v|< R) such that the mild solution of equation (5.8) with the initial condition (u(0)=v) is of period ξ.

By hypothesis, there exists an element x of G such that | x G | = | G : C G ( x ) | = 3 ⋅ 7.

By hypothesis, there exists an element x of order p in G such that C G ( x ) = 〈 x 〉 and C G ( x ) is a Sylow p-subgroup of G.

Then there exists an element v of H with ∥ v ∥ < R such that the mild solution of equation (3.1) with the initial condition u ( 0 ) = v is periodic of period ξ.

Moreover, R cannot be contained in Z ( G ). Thus, there exists an element y of R which is not contained in Z ( G ) such that 1 < | y G | ⩽ | R | < 3 ⋅ 7, a contradiction.