according to a class of constrained‎ ‎minimization problems‎, ‎the schwartz symmetrization process and the‎ ‎compactness lemma of strauss‎, ‎we prove that there is a‎ ‎nontrivial ground state solution for a class of $p$-laplace‎ ‎equations without the ambrosetti-rabinowitz condition‎.

The Johnson-Lindenstrauss lemma and its proof This lecture aims to prove the Johnson–Lindenstrauss lemma. Since the lemma is proved easily with another interesting lemma, a part of this lecture is focused on the proof of this second lemma. At the end, the optimality of the Johnson–Lindenstrauss lemma is discussed. Lemma 1 (Johnson-Lindenstrauss). Given the initial space X ⊆ R n s.t. |X| = N , <...

B Properties of the Declarative System 23 1 Lemma (Declarative Weakening) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Lemma (Declarative Term Substitution) . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Lemma (Reflexivity of Declarative Subtyping) . . . . . . . . . . . . . . . . . . . . . 23 4 Lemma (Subtyping Inversion) . . . . . . . . . . . . . . . . . . . . . . . . . ....

In the above paper, Lemma (4), on page 2659 play the key role for deriving the main results in the paper. The statement as well as the proof of Lemma (4), page 2659, [1] is not correct. Here mainly two serious errors in this Lemma, one is in the statement of lemma and the other one in the proof of Lemma (4), on page 2666, [1]. In the Lemma (4), page 2659, [1], Wan and Yi, states the lemma as fo...

we show that in {bf zf} set theory without choice, the ultrafilter mbox{principle} ({bf up}) is equivalent to several compactness theorems for alexandroff discrete spaces and to rudin&apos;s lemma, a basic tool in topology and the theory of quasi-continuous domains. important consequences of rudin&apos;s lemma are various lift lemmas, saying that certain properties of posets are inherited by th...

We give further results for Perron-Frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. We indicate two techniques for establishing the main theorem ofPerron and Frobenius on the numerical range. In the rst method, we use acorresponding version of Wielandt's lemma. The second technique involves graphtheory.

This study concerns the existence and multiplicity of positive weak solutions for a class of semilinear elliptic systems with nonlinear boundary conditions. Our results is depending on the local minimization method on the Nehari manifold and some variational techniques. Also, by using Mountain Pass Lemma, we establish the existence of at least one solution with positive energy.

We consider the extended trust region subproblem (eTRS) as the minimization of an indefinite quadratic function subject to the intersection of unit ball with a single linear inequality constraint. Using a variation of the S-Lemma, we derive the necessary and sufficient optimality conditions for eTRS. Then, an OCP/SDP formulation is introduced for the problem. Finally, several illustrative examp...