We give deterministic versions of randomized approximation algorithms for several ranking and clustering problems that were proposed by Ailon, Charikar and Newman[1]. We show that under a reasonable extension of the triangle inequality in clustering problems, we can resolve Ailon et al.’s open question whether there is an approximation algorithm for weighted correlation clustering with weights ...

1 Proof of the Proposition 1 We now prove the Proposition 1 that gives the condition of compactness of sublevel set. Proof. Let B(r) and S(r) denote the ball and sphere of radius r, centered at the origin. By affine transformation, we can assume that X∗ contains the origin O, X∗ ⊂ B(1), and X∗ ∩ S(1) = φ. Then, we have that for ∀x ∈ S(1), (∇f(x), x) ≥ f(x)− f(O) > 0, where we use convexity for ...

This paper considers the robust strict positive real (SPR) problem for a family of plants of the form G(s; q) = N(s; q n)D ?1 (s; q d)?, where N(s; q n) and D(s; q d) are multiaane in uncertain parameters q n and q d , respectively , and > 0. In the discrete-time setting, this problem plays an important role in digital quantiza-tion. Several results are presented. First, we prove that this plan...

In this note, we give some estimate of the generalized quadrature formula of Gauss-Jacobi$$underset{a}{overset{a+eta left( b,aright) }{int }}left( x-aright)^{p}left( a+eta left( b,aright) -xright) ^{q}fleft( xright) dx$$in the cases where $f$ and $left| fright| ^{lambda }$ for $lambda >1$, are $s$-preinvex functions in the second sense.

QCD Laplace Sum-Rules must satisfy a fundamental Hölder inequality if they are to consistently represent an integrated hadronic spectral function. The Laplace sum-rules of pion currents is shown to violate this inequality unless the u and d quark masses are sufficiently large, placing a lower bound on mu + m d , the SU (2)-invariant combination of the light-quark masses. In this paper we briefl...

Background and Aim: In 2020 the rank of Iran in happiness index and gender inequality were reported to be rather low, they being, among 153 countries, 118 and 113, respectively. This study aimed to investigate the relationship between gender inequality and happiness in Iran's provinces. Materials and Methods: This ecological study was conducted based on secondary analysis of data on happiness ...

Given a graph G, we consider the model where G is given a random orientation by giving each edge a random direction. It is proven that for a, b, s ∈ V (G), the events {s → a} and {s → b} are positively correlated. This correlation persists, perhaps unexpectedly, also if we first condition on {s 9 t} for any vertex t 6= s. With this conditioning it is also true that {s → b} and {a → t} are negat...

In the problem of the month 1999 : 106], one was to prove that p a + b ; c + p b + c ; a + p c + a ; b p a + p b + p c , where a, b, c are sides of a triangle. It is to be noted that this inequality will follow immediately from the Majorization Inequality 1]. Here, if A and B are vectors (a 1 a 2 : : : a n), we say that A majorizes B and write it as A B. Then, if F is a convex function, If F is...

In this paper we study and characterize those Diophantine inequalities axmod b ≤ x whose set of solutions is a symmetric numerical semigroup. Given two integers a and b with b = 0 we write a mod b to denote the remainder of the division of a by b. Following the notation used in [8], a modular Diophantine inequality is an expression of the form ax mod b ≤ x. The set S(a, b) of integer solutions ...