Today, we will prove Cheeger’s Inequality. I consider it to be the most important theorem in spectral graph theory. Cheeger’s inequality has many variants, all of which tell us in some way that when λ2 of a graph is small, the graph has a cut of small conductance (or ratio or sparsity). Recall that in Lecture 5 we proved: Theorem 7.1.1. Let G = (V,E) be a graph and let LG be its Laplacian matri...

we define a new function-valued inner product on l2(g), called ?-bracket product, where g is a locally compact abelian group and ? is a topological isomorphism on g. we investigate the notion of ?-orthogonality, bessel's inequality and ?-orthonormal bases with respect to this inner product on l2(g).

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...

It is pointed out that, one of the results in the recently published article, ’On the Iyengar-Madhava Rao-Nanjundiah inequality and it’s hyperbolic version’ [3] by J´ozsef S´andor is logically incorrect and new corrected result with it’s proof is presented.

After an elementary derivation of Bell’ s inequality, classical, quantum mechanical , and stronger-than-quantum correlation functions for 2-particle-systems are discussed. Special functions are investigated which give rise to an extreme violation of Bell’ s inequality by the value of 4. Referring to a specific quantum system it is shown that under certain conditions such an extreme violation wo...

We mainly study Max TSP with two objective functions. We propose an algorithm which returns a single Hamiltonian cycle with performance guarantee on both objectives. The algorithm is analysed in three cases. When both (resp. at least one) objective function(s) fulfill(s) the triangle inequality, the approximation ratio is 5 12 − ε ≈ 0.41 (resp. 3 8 − ε). When the triangle inequality is not assu...

A pair of B 0 ¯ B 0 mesons from Υ(4S) decay exhibit EPR type non-local particle-antiparticle (flavor) correlation. It is possible to write down Bell Inequality (in the CHSH form: S ≤ 2) to test the non-locality assumption of EPR. Using semileptonic B 0 decays of Υ(4S) at Belle experiment, a clear violation of Bell Inequality in particle-antiparticle correlation is observed: S = 2.725 ± 0.167sta...

In this paper, by using the Euler-Maclaurin expansion for Riemann-$zeta$ function, we establish an inequality of a weight coefficient. Using inequality, derive new reverse Hilbert's type inequality. As applications, equivalent form is obtained.

where f : [a, b] → R is a differentiable function such that |f ′ (x) | ≤ M for all x ∈ [a, b] . More about Ostrowski type inequalities and companion inequalities to the Ostrowski type inequality can be found in papers [4, 5] and in monographs [1, 6]. If f is a differentiable function, f ′ is integrable, and γ ≤ f ′ (s) ≤ Γ, for all s ∈ [a, b] , then the following Ostrowski–Grüss inequality can ...

We explore M. Gromov's counterexamples to systolic inequalities. Does the manifold S 2 × S 2 admit metrics of arbitrarily small volume such that every noncontractible surface inside it has at least unit area? This question is still open, but the answer is affirmative for its analogue in the case of S '~ × S n, n > 3. Our point of departure is M. Gromov's metric on S 1 x S 3, and more general ex...