Negative Integer Values Research Articles

Variations on a Conjecture of C. C. Yang Concerning Periodicity

The generalized Yang’s Conjecture states that if, given an entire function f(z) and positive integers n and k, f(z)^nf^{(k)}(z) is a periodic function, then f(z) is also a periodic function. In this paper, it is shown that the generalized Yang’s conjecture is true for meromorphic functions in the case k=1. When kge 2 the conjecture is shown to be true under certain conditions even if n is allowed to have negative integer values.

Characteristics of Student Mental Models in Negative Round Numbers

The purpose of this study is to know, to describe, and to analyze students' mental model on negative integers. This research uses a descriptive qualitative approach. The data collection techniques use test, observation, interview, and documentation. The data analysis techniques use data reduction, data presentation, and conclusions. The result shows that there are three characteristics of students' metal model, the characteristics of early mental model on negative integers are only able to understand the minus sign as a count operation, students with formal mental model characteristics on negative integers understand the existence of negative numbers that have values ​​smaller than 0 but do not understand that numbers going left on the number line, it can be smaller values. Students with formal mental model characteristics on negative integers understand the direction on the number line and the value of each number correctly. It can be concluded that in understanding the scale of negative integer values ​​and negative integer places on number line, every student has different characteristics. The characteristics of the early mental model are the characteristics of the lowest mental model, the characteristics of the synthesis mental model are the characteristics of the moderate mental model and the characteristics of the formal mental model are the characteristics of the highest mental model in understanding negative integers.

General solution of the Schrödinger equation for some trigonometric potentials

In this article, we recursively obtain the general solution of the Schrodinger equation $$y_{\nu }''(x;\lambda )+[\lambda -\nu (\nu +1)v(x)]y_{\nu }(x;\lambda )=0$$ for non negative integer values of $$\nu$$ and an arbitrary values of the eigenvalue parameter $$\lambda$$ where v(x) is certain trigonometric potentials. The recursions are obtained from the contour integral solutions of Tanriverdi. By using these contour integral solutions the author have obtained the first few solutions when $$\nu =n$$, a non negative integer, by means of residue calculations which becomes considerably troublesome or almost impossible for larger values of n. Therefore, the recursive procedure of the present article can be seen as a superior alternative to the method of residue calculation for deriving the general solution for arbitrary values of $$\lambda$$ and non negative integer n. Moreover, the eigenpairs with the homogeneous Drichlet and Neumann boundary conditions are also derived from the general solution.

Holomorphic quadratic differentials on graphs and the chromatic polynomial

We study “holomorphic quadratic differentials” on graphs. We relate them to the reactive power in an LC circuit, and also to the chromatic polynomial of a graph. Specifically, we show that the chromatic polynomial χ of a graph G, at negative integer values, can be evaluated as the degree of a certain rational mapping, arising from the defining equations for a holomorphic quadratic differential. This allows us to give an explicit integral expression for χ(−k).

BFKL eigenvalue and maximal alternation of harmonic sums

We analyze the known results for the eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation in the perturbative regime using the analytic continuation of harmonic sums from even positive arguments to the complex plane. The resulting meromorphic functions have poles at negative integer values of the argument. The typical classification of harmonic sums is determined by two major parameters: a) the weight - a sum of inverse powers of the summation indices; b) the depth - a number of nested summations. We introduce the third parameter: the alternation - a number of nested sign-alternating summations in a given harmonic sum. We claim that the maximal alternation of the nested summation in the functions building the BFKL eigenvalue is preserved from loop to loop in the perturbative expansion. The BFKL equation is formulated for arbitrary color configuration of the propagating states in the t-channel. Based on known results one can state that color adjoint BFKL eigenvalue can be written using only harmonic sums with positive indices, maximal alternation zero, and at most depth one, whereas the singlet BFKL eigenvalue is constructed of harmonic sums with maximal sign alternation being equal one. We also note that for maximal alternation being equal unity the harmonic sums can be expressed through alternation zero harmonic sums with half-shifted arguments.

Analytical solutions for three-dimensional Coulomb transition matrix at negative energy and integer values of interaction parameter

With the use of a stereographic projection of momentum space into a four-dimensional sphere of unit radius, the possibility of analytical solution of the three-dimensional two-body Lippmann–Schwinger equation with the Coulomb interaction at negative energy has been studied. Simple analytical expressions for the three-dimensional Coulomb transition matrix in the case of the repulsive Coulomb interaction and positive integer values of the Coulomb parameter have been obtained. The worked out method has also been applied to the generalized three-dimensional Coulomb transition matrix in the case of the attractive Coulomb interaction and negative integer values of the Coulomb parameter.

LO01: Development and validation of an adjustment score for ruling out MI using a single high-sensitivity cardiac troponin T assay in patients with chest pain and kidney dysfunction

Introduction: Very low concentrations of high-sensitivity cardiac troponin can rule-out myocardial infarction (MI) at ED arrival in patients with chest pain. However, this single troponin rule-out strategy works poorly in patients with renal impairment and elevated baseline troponin levels. The objective of this study was to develop and validate a troponin adjustment strategy to accurately rule-out MI with a single hs-cTnTmeasurement in patients with kidney dysfunction. Methods: We used data from three cohorts of ED chest pain patients to develop an adjustment score for a high-sensitivity troponin T (hs-cTnT) assay in patients with kidney dysfunction. The derivation cohort (n = 8846) used administrative and registry data. Two validation cohorts (n = 1187 and 1092) were prospectively-collected. The score assigned points for increasing hs-cTnT levels and subtracted points for lower estimated glomerular filtration rate (eGFR). In the derivation cohort, hs-cTnT concentrations achieving 98.5% sensitivity in of patients with eGFR ≥60, 45-59, 30-44, 15-29 and <15 were assigned ascending positive integer values. Negative integer values were assigned to eGFR values 45-59, 30-44, 15-29 and <15. The scpres for troponin and eGFR were summed for each patient, with scores ranging from −4 to +5. The proportion of patients with 7-day MI ruled out by a score ≤0, sensitivity, NPV, negative likelihood ratio (LR-) and area under the curve (AUC) were quantified in each study cohort. Results: The derivation and validation cohorts had 7-day MI rates of 5.7, 8.6 and 9.1%. In the derivation cohort, a score ≤0 ruled out MI in 35% of patients, with a sensitivity for 7-day MI of 99.5% (95% CI 98-100), NPV of 99.9% (95% CI 98.4-99.9), LR- of 0.02 (95% CI 0.01-0.05) and AUC of 0.88. In the first validation cohort, a score ≤0 ruled out MI in 45% of patients, with a sensitivity for 7-day MI of 97% (95% CI 90-100%), NPV of 99% (95% CI 98-100%), LR- 0.06 (0.02-0.18) and AUC of 0.89. In the second validation cohort, a score ≤0 ruled out MI in 20% of patients, with a sensitivity for 7-day MI of 96% (95% CI 93-99%), NPV of 98% (95% CI 96-100%), LR- of 0.16 (95% CI 0.07-0.39) and AUC of 0.78. Conclusion: We developed and validated a simple scoring system to adjust hs-cTnT concentrations for a patient's kidney function that enables MI to be ruled out in a large proportion of chest pain patients using a single measurement on ED presentation.

Some results on the beta function and the incomplete beta function

The incomplete Beta function [Formula: see text] is defined for [Formula: see text] and [Formula: see text]. This definition was extended to negative integer values of [Formula: see text] and [Formula: see text] by Özçaḡ et al. using neutrix limits. Partial derivatives of the incomplete Beta function [Formula: see text] for negative integer values of [Formula: see text] and [Formula: see text] were then evaluated. In the following, a number of results are obtained for the incomplete Beta function [Formula: see text] and then the corresponding results for the Beta function [Formula: see text] are deduced.

Aggregate directional distance formulation of DEA with integer variables

Conventional data envelopment analysis (DEA) models make the assumption of non-negativity and real values in the input and output of the systems that are under study. This paper combines these two interrelated ideas. One is the non-radial measurement of efficiency by establishing an aggregate directional distance formulation of the DEA model (ADDM). Another is the introduction of an integer directional distance function. Usually, directional distance formulations of DEA and ADDM projections of efficient targets for inefficient decision making units (DMUs) have non-integer values. In this paper directional distance function is modified and a mixed integer directional distance formulation of DEA is proposed. This model guarantees integer targets for inefficient DMUs and is applicable to measure efficiency even when input and output variables are negative integer values.

(The Role of Money and Credit Aggregates in Conduct of Monetary Policy in Korea)

Korean Abstract: 글로벌 금융위기를 계기로 그동안 경시되어 왔던 통화·신용량의 중요성이 학계 및 중앙은행을 중심으로 재인식되고 있다. 그러나 양적지표를 실제 금리중심의 인플레이션 타게팅 운용체계에 어떻게 접목시킬 것인가에 관한 연구는 이제 시작단계이다. 이와 관련하여 본고는 한국은행 금융통화위원회의 월별 의사록 내용을 토대로 금통위원들이 통화·신용량을 언급하며 표명한 금리결정 의사 및 강도를 지수화(indexation)하고 동 지수를 이용하여 우리나라가 인플레이션 타게팅 제도를 도입한 이후 통화/신용량을 금리결정에 어떻게 반영해 왔는지를 분석하였다. 분석결과 금융통화위원회는 통화·신용량의 움직임에도 유의하면서 기준금리를 결정한 것으로 나타났는데, 이는 한국은행이 자산가격과 신용경색 등을 고려하여 인플레이션 타게팅 제도를 신축적으로 운용해 왔음을 시사한다. English Abstract: After having been de-emphasized from around the early 1990s, the importance of financial factors such as the money and credit aggregates in the implementation of monetary policy under inflation targeting regimes has gained renewed attention since the recent global financial crisis that erupted in August 2007. Further discussions and rigorous analysis are however needed, to reach a broadly shared consensus among academia and policymakers on the issue of when and how the central bank should incorporate these quantity variables into its actual conduct of interest rate-focused inflation targeting.Our aim in this paper is to address the issue of how the Monetary Policy Committee (MPC) of the Bank of Korea took into consideration the actual money and credit aggregates in its policy decisions under inflation targeting over the period 2003-2011. A novelty of our empirical analysis is its construction of an index as a measure for capturing, using the monthly Committee meeting minutes for the period concerned, how many MPC members mentioned their intentions with regard to the policy stance by taking the current movements of money and credit aggregates into consideration just prior to their policy decisions. For the analysis we set up a discrete choice framework for the index. We assign integer values of from 4 through -4, depending upon how many MPC members show their intentions that the policy stance be “tight,” “neutral,” or “easy,” in consideration of the movements of money and credit aggregates for a specific month. The integer value 4 denotes that more than five MPC members mentioned favoring a policy rate hike while the value 2 indicates that it was less than two MPC members. The same classification applies to the negative integer values. A zero value indicates that no MPC member has shown any favoring of a “tight” or “easy” policy stance at all, implying an intended “neutral” stance of the MPC members.We conduct OLS estimation of the effects of the index on the actual policy rate decisions by the Committee members, using an equation controlling for two key macroeconomic indicators such as the inflation gap and the output gap that the Committee considers in setting policy. Interestingly, the estimated index coefficient turns out to be positive and statistically significant. The regression suggests that when the MPC members frequently and strongly mentioned the need to take notice of the movements of money and credit aggregates around the time of the Committee meeting, monetary policy was dominated by concerns over the stabilization of asset prices or massive credit squeezes in the financial markets. We, in addition, estimated two different equations, as a way of checking the robustness of the OLS estimates. The first specification is an ordered probit model in which we estimated the effects of the index on the policy changes in discrete increments. The integer values assigned for the policy changes range from -2 through 2. The values of 2(-2) and 1(-1) correspond to 50 bp and 25 bp hikes (cuts) in the policy rate, respectively, while a value of 0 indicates no policy change. It seems more accurate to use this five-stage classification of policy rate changes as the dependent variable, because the evolution of changes in the policy rate does not follow a smooth process but a step function. In a second experiment, we first estimated the standard Taylor rule over the period concerned, and then the effects of the index on its residual estimates. In both cases, the estimated index coefficients are positive and significant, implying that the results support the OLS estimates mentioned above.To summarize, the evidence suggests that the Bank of Korea has been implementing flexible inflation targeting.

Negative Integer Understanding: Characterizing First Graders' Mental Models

This article presents results of a research study with a pretest, instruction, and posttest design that aimed to identify students' mental models of integers and investigate how these models change based on instruction on the unary and/or binary meanings of the minus sign. Sixty-one first graders' responses to interview questions about negative integer values and order and directed magnitudes were examined to characterize the students' mental models. The models reveal that initially, students over-relied on various combinations of whole number principles as they tried to understand negative integers. However, a significant number of students developed formal mental models based on their instruction.

Existence of the neutrix limit of the [formula omitted]-analogue of the incomplete gamma function and its derivatives

The q -analogue of the incomplete gamma function γ q ( α , x ) is defined for α > 0 , x > 0 and 0 < q < 1 . In this work, the neutrix and the neutrix limit are used to generalize the definition of the q -analogue of the incomplete gamma function and its derivatives for zero and negative integer values of α . We prove the existence of the neutrix limit of the incomplete q -gamma function and its derivatives for all values of α . Recurrence formula for the incomplete q -gamma function at negative integer values is investigated.

Minimal unitary representation of SU(2,2) and its deformations as massless conformal fields and their supersymmetric extensions

We study the minimal unitary representation (minrep) of SO(4,2) over a Hilbert space of functions of three variables, obtained by quantizing its quasiconformal action on a five dimensional space. The minrep of SO(4,2), which coincides with the minrep of SU(2,2) similarly constructed, corresponds to a massless conformal scalar in four space-time dimensions. There exists a one-parameter family of deformations of the minrep of SU(2,2). For positive (negative) integer values of the deformation parameter ζ, one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in (0,ζ∕2)((−ζ∕2,0)) representation of the SL(2,C) subgroup. We construct the supersymmetric extensions of the minrep of SU(2,2) and its deformations to those of SU(2,2|N). The minimal unitary supermultiplet of SU(2,2|4), in the undeformed case, simply corresponds to the massless N=4 Yang–Mills supermultiplet in four dimensions. For each given nonzero integer value of ζ, one obtains a unique supermultiplet of massless conformal fields of higher spin. For SU(2,2|4), these supermultiplets are simply the doubleton supermultiplets studied in the work of Gunaydin et al. [Nucl. Phys. B 534, 96 (1998); e-print arXiv:hep-th/9806042].

The neutrix limit of the [formula omitted]-Gamma function and its derivatives

The q -analogue of the gamma function Γ q ( x ) is defined for x > 0 , 0 < q < 1 . In this work the neutrix and neutrix limit are used to define the q -analogue of the gamma function and its derivatives for zero and negative integer values of x . We prove that the neutrix limit of the q -Gamma function and its derivatives exist for all values of x .

An extension of the incomplete beta function for negative integers

The incomplete beta function B x ( a , b ) is defined for a , b > 0 and 0 < x < 1 . Its definition can be extended, by regularization, to negative non-integer values of a and b. In this paper we define the incomplete beta function B x ( a , b ) for negative integer values of a and b. Further we prove that the function ∂ m + n ∂ a m ∂ b n B x ( a , b ) exists for m , n = 0 , 1 , 2 , … and all a and b.

On a certain family of generalized Laguerre polynomials

Following the work of Schur and Coleman, we prove the generalized Laguerre polynomial L n (−3−n)(x)=∑ j=0 n 1 j! (n−j+1)(n−j+2) 2 x j is irreducible over the rationals for all n⩾1 and has Galois group A n if n+1 is an odd square, and S n otherwise. We also show that for certain negative integer values of α and certain congruence classes of n modulo 8, the splitting field of L n ( α) ( x) can be embedded in a double cover.

An Asymptotic Expansion for the First Derivative of the Generalized Riemann Zeta Function

An asymptotic expansion for the partial derivative a( q)/13z of the generalized Riemann zeta function t( q), for all negative integer values of Z, is obtained. The generalized Riemann zeta function t(z, q) is defined by 00

Правила сумм для амплитуд Хури и разложение Вильсона на световом конусе

The convolution theorem is used to obtain general sum rules for the Khuri amplitudes of the current correlation function (familiar from current algebra). The sum rules can be continued analytically in the Khuri λ-plane to negative integer values of λ; at these points the sum rules relate the finite-energy integral to the Fourier transform of distributions that, by the Gel'fand-Shilov method of analytic continuation, can be defined in a natural way as the current commutator and its derivatives on the light-cone. Assuming that the current commutator involved has a light-cone Wilson expansion, one can write the previously mentioned sum rules in terms of scaling amplitudes. Finally one considers the peculiar features of the sum rules obtained in the case of conserved currents.

Inverse Sampling of a Poisson Distribution

SUMMARY Consider a situation where an observation or may result in a number X = 0, 1, 2, ... of events or successes, with X a Poisson variate of mean uz. A series of independent trials is conducted, resulting in Xi, X2, ... successes respectively. The series is terminated as soon as the total number Xi + X2 + * * * + XN of successes has reached or exceeded a preassigned number k. Let Nk be the number of trials required to reach k successes. The moments, E{ Nm }, of Nk are derived for both positive and negative integer values of m. Exact confidence limits for ui, based on Nk, are derived, and the width of the confidence interval in relation to the size of the trial unit is discussed. In applications to counts of particles in suspension, the most suitable dilution and size of the trial unit are investigated to ensure an approximate Poisson distribution.

Singularities of a Relativistic Scattering Amplitude in the Complex Angular Momentum Plane

The analytic properties of the partial-wave amplitude in the complex angular momentum plane are investigated in a relativistic scalar-meson theory. Using a $\frac{N}{D}$ decomposition, the numerator and denominator are calculated by perturbative expansions to fourth order in the coupling constant. Higher order poles at negative integer values of $l$ are found in both the numerator and denominator, leading to a breakdown in their perturbation expansions near these singularities. The same breakdown occurs for all but the leading Regge trajectory. It is further found that, to fourth order, due to inelastic processes, the denominator has branch cuts for values of $l$ near negative integers.