On a theorem of Graham

AbstractA strengthened form of Gurevich's conjecture was proved by R. L. Graham, which says that for any α > 0 and any pair of non-parallel lines L1 and L2, in any partition of the plane into finitely many classes, some class contains the vertices of a triangle which has area α and two sides parallel to the lines Li. In this note, using the main idea of Graham, we present a shorter proof of the result

On weighted zero-sum sequences

Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest positive integer $m$, denoted by s_A(G), such that any sequence {c_i}_{i=1}^m with terms from G has a length n=exp(G) subsequence {c_{i_j}}_{j=1}^n for which there are a_1,...,a_n in A such that sum_{j=1}^na_ic_{i_j}=0. When G is a p-group, A contains no multiples of p and any two distinct elements of A are incongruent mod p, we show that s_A(G) is at most $\lceil D(G)/|A|\rceil+exp(G)-1$ if |A| is at least (D(G)-1)/(exp(G)-1), where D(G) is the Davenport constant of G and this upper bound for s_A(G)in terms of |A| is essentially best possible. In the case A={1,-1}, we determine the asymptotic behavior of s_{{1,-1}}(G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank, s_{{1,-1}}(G)=exp(G)+log_2|G|+O(log_2log_2|G|) as exp(G) tends to the infinity. Combined with a lower bound of $exp(G)+sum{i=1}{r}\lfloor\log_2 n_i\rfloor$, where $G=\Z_{n_1}\oplus...\oplus \Z_{n_r}$ with 1<n_1|... |n_r, this determines s_{{1,-1}}(G), for even exponent groups, up to a small order error term. Our method makes use of the theory of L-intersecting set systems. Some additional more specific values and results related to s_{{1,-1}}(G) are also computed.Comment: 24 pages. Accepted version for publication in Adv. in Appl. Mat

On an error term related to the Jordan totient function Jk(n)

AbstractWe investigate the error terms Ek(x)=∑n⩽xJk(n)−xk+1(k+1)ζ(k+1) for k⩾2, where Jk(n) = nkΠp|n(1 − 1pk) for k ≥ 1. For k ≥ 2, we prove ∑n⩽xEk(n)∼xk+12(k+1)ζ(k+1). Also, lim infn→∞Ek(x)xk⩽Dζ(k+1), where D = .7159 when k = 2, .6063 when k ≥ 3. On the other hand, even though lim infn→∞Ek(x)xk⩽−12ζ(k+1), Ek(n) > 0 for integers n sufficiently large

User Information needs and Information Seeking Behaviour of Physics Department at the University of Burdwan

Purpose – The study found on the University of Burdwan physics department faculties and research scholars’ information needs and information seeking behaviour. The purpose is to find out the information needs, source use, satisfaction level, and improvement needs for library infrastructure, sources and services. Methodology – In this information needs and seeking behaviour study, two sets online questionnaire has been developed to achieve the objective. Only physics faculties and regular registered research scholars (session 2011 – 2020) in the University of Burdwan have been selected as main respondents. Data has been collected using Google form. Online questionnaire have been served through email to all the physics faculty members and one hundred five registered research scholars in the University of Burdwan. The response rates from respondents are 47.62% (faculty) and 64.86% (scholars). The ratio of male – female respondents’ of physics faculty members and scholars are 9:1 and 38:10 respectively. Findings – The study depicted the physics department faculty members and research scholars’ information needs and information seeking behaviour at the University of Burdwan. Information need and information seeking behaviour of physics department faculty members and regular research scholars was in-depth understanding of the major roles on overall university library development in the angle from the students; purview on Burdwan university library. Originality/value – The current finding are original and reflected latest observation on the physics department faculties and research scholars’ information-seeking behaviour in the University of Burdwan. The study benefits information Seekers and as well as library policy makers who provide the library collection development and services to the users of the University

On a question regarding visibility of lattice points—III

AbstractFor a positive integer m, let ω(m) denote the number of distinct prime factors of m. Let h(n) be a function defined on the set of positive integers such that h(n)→∞ as n→∞ and let En(h)={d:disapositiveinteger,d⩽n,ω(d)⩾h(n)}. Writing Δn={(x,y):x,yareintegers,1⩽x,y⩽n}, in the present paper we show that one can give explicit description of a set Xn⊂Δn such that Δn is visible from Xn with at most 100|En(h)|2 exceptional points and for all sufficiently large n, one has|Xn|⩽800h(n)loglogh(n).As a corollary it follows that one can give explicit description of a set Yn⊂Δn such that for large n's, Δn is visible except for at most 100n2/(loglogn)2 exceptional points from Yn where Yn satisfies|Yn|=O((loglogn)(loglogloglogn))

Prediction of fabric hand characteristics using extraction principle

Prediction of fabric handle characteristics using extraction principle has been studied. An instrument for objective measurement of fabric handle characteristics has been developed using nozzle extraction method. This instrument measures the force exerted on the periphery of the nozzle by the fabric being drawn out of the nozzle on the periphery of the nozzle. This force, called the radial force, is a measure of the certain low stress mechanical characteristics of the fabric that determine handle. The instrument also measures the force required to extract the fabric through the nozzle. Woven fabric samples have been sourced from industry and categorized into suiting and shirting fabrics. The fabric samples were also tested in KES-F system. An attempt has been made to predict the shear force and bending rigidity by using artificial neural network. It has been observed that there are very good correlations between the extraction force values and the various KES-F parameters. The fabric extraction force obtained through nozzle extraction instrument is found to be well enough to predict fabric handle/feel value

The Kadison-Singer problem

In mathematics, the Kadison-Singer problem, posed in 1959, was a problem in C ∗ - algebra about whether certain extensions of certain linear functionals on certain C ∗ - algebras were unique. The uniqueness was proven in 2013. The statement arose from work on the foundations of quantum mechanics done by Paul Dirac in the 1940s and was formalized in 1959 by Richard Kadison and Isadore Singer. The problem was subsequently shown to be equivalent to numerous open problems in pure mathematics, applied mathematics, engineering and computer science. Kadison, Singer, and most later authors believed the statement to be false, but, in 2013, it was proven true by Adam Marcus, Daniel Spielman and Nikhil Srivastava, who received the 2014 Polya Prize for the achievement. We will discuss about the Kadison-Singer problem for a separable Hilbert space. First of all we will characterize all functions that can possibly have the Kadison-Singer property and then among these which class of functions fail to have the Kadison-Singer property and also finally which class will have the Kadison-Singer property

Remarks on monochromatic configurations for finite colorings of the plane

Gurevich had conjectured that for any finite coloring of the Euclidean plane, there always exists a triangle of unit area with monochromatic vertices. Graham ([5], [6]) gave the first proof of this conjecture; a much shorter proof has been obtained recently by Dumitrescu and Jiang [4]. A similar result in the case of a trapezium, claimed by the present authors in [3] does not hold due to an error and a weaker result is recovered for quadrilaterals in this paper. We also take up the original question of triangle

A Rare Case of Childhood Hypertension and Hyponatremia

Hypertension and hyponatremia together, is an uncommon entity in children. We here described a 10-year-old boy presented with hypertensive emergency and altered sensorium with hyponatremia. After initial stabilization USG (ultrasonography) Doppler showed shrunken right kidney with absence of flow in the right renal artery. Right Renal Resistive Index was 0.9. Therefore, patient underwent right total nephrectomy and blood pressure ultimately came under control.Keywords: Hypertension; Hyponatremia; Child