Positive Integer Values Research Articles

Coupled torsion and inflation of functionally graded and residually stressed Mooney–Rivlin hollow cylinders

Functionally graded structures have material properties continuously varying in one or more directions. Examples include human teeth, sea shells, bamboo stems, and mammal organs in which the varying volume fraction and orientation of fibers optimize their functionalities. Here we analytically study large deformations due to combined torsion and inflation of residually stressed and radially graded Mooney–Rivlin hollow circular cylinders to provide insights into how grading their material moduli according to a power law function of the radius can be advantageously used. We simulate residual stresses in a thin wall hollow cylinder by inverting it inside out and in a thick wall cylinder by assuming that a longitudinal wedge opening parallel to the cylinder axis is closed by deforming it axisymmetrically. It is found that positive integer values of the power law exponent are generally advantageous but its negative values can have deleterious effects on stress distributions. Furthermore, a hollow cylinder with residual stresses generated by one of the foregoing methods in the reference configuration should be modeled as orthotropic rather than transversely isotropic for subsequent deformations. The analytical solutions provided here should help numerical analysts verify their algorithms for large deformations of rubber-like materials that are modeled by the Mooney–Rivlin relation, and design engineers for exploiting the radial gradation of the two material moduli to reduce structure’s mass without sacrificing its performance.

First-principles study on rare earth-based equiatomic quaternary Heusler alloys YbCoCrSb and YbCoTiSn: New candidates for spintronics

We have investigated the structural, mechanical, thermal, electronic, magnetic, and half-metallic properties of new Yb-based EQHAs such as YbCoCrSb and YbCoTiSn using first-principles calculations. We computed the ground state properties of different possible crystal structures and magnetic states. We also calculated their elastic constants and derived mechanical parameters, including bulk, shear, and Young's moduli, as well as Poisson's and Pugh ratios. The electronic structure of both EQHAs shows half-metallic behavior with a spin down band gap of 0.39 eV for YbCoCrSb and 0.30 eV for YbCoTiSn in the GGA-PBE scheme. The estimated total spin magnetic moment has a positive integer value, which is well in accordance with the Slater-Pauling rule of 18. Because of the nearly closed f orbital shell of the Yb atom, the total spin magnetic moment is mainly dependent on 3d transition metal atoms. These alloys exhibit TC > 300 K using the mean field approximation.

Near universal property of a quadratic form of Hessian discriminant 4

A quadratic form is universal if it represents all totally positive integers in a number field. In classical number theory over the rational integers there are some results where a quadratic form takes on all positive integer values except for a certain restricted set. Here we show that one of the quaternary quadratic forms proven universal for ℚ(√5) takes on all totally positive values in ℚ(√2) except for a certain infinite set. The exceptional integers have even norm and the highest power of 2 dividing that norm is 2 to the first power.

Intensity variability in stationary solutions of the Fractional Nonlinear Schrödinger Equation

Solitons that propagate in optical fiber with indexes of refraction, dispersion, and diffraction are balanced, making pulses or electromagnetic waves propagate without any distortion. This is closely related to use of nonlinear refractive index in fiber optics. If an optical fiber only uses a nonlinear refractive index, then the partial signal can be lost over time. This study aims to analyze the variability of stationary solutions in multi-solitons formed using Fractional Nonlinear Schrödinger (FNLS). The parameter p indicates energy level of the solution to FNLS equation which has a positive integer value. This study focuses on 3 variations of p values, namely p = 0 which indicates the ground state, p = 1 which indicates the first excited state, and p = 2 which indicates the second excited state. During the first to second excited state, multi soliton peaks are formed with the same amplitude symmetrically. The amplitude experienced by the middle soliton in second excited state is lower which indicates the input signal obtained from the FNLS solution in the ground state in the form of triple-soliton. The polarization mode cause the soliton pulse width to shrink and the consequent amplitude in the first excited state to increase.

Pole skipping in holographic theories with gauge and fermionic fields

Using covariant expansions, recent work showed that pole skipping happens in general holographic theories with bosonic fields at frequencies i(lb − s)2πT, where lb is the highest integer spin in the theory and s takes all positive integer values. We revisit this formalism in theories with gauge symmetry and upgrade the pole-skipping condition so that it works without having to remove the gauge redundancy. We also extend the formalism by incorporating fermions with general spins and interactions and show that their presence generally leads to a separate tower of pole-skipping points at frequencies i(lf − s)2πT, lf being the highest half-integer spin in the theory and s again taking all positive integer values. We also demonstrate the practical value of this formalism using a selection of examples with spins 0,12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}, 1,32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{3}{2} $$\\end{document}, 2.

Validating available Donor-recipient risk assessment tools in a German Heart Transplant cohort

Abstract Objective Several Risk assessment tools have been developed to help predict mortality after heart transplantation. While validated within their regional transplant cohorts, it remains unclear whether they can also be applied to other cohorts. We therefore aimed to test and compare the predictive value of an US-Score validated in the UNOS Database by Joyce et al., a French Score by Jasseron et al., and the BO-Score within the Eurotransplant area developed by Schramm et al. Methods We applied the three scores to our cohort using published model coefficients and variables (see Figure 1 for distribution). The initial BO-Score as well as the French Score were hereby modified to produce positive integer values within our cohort (mod. French Score = [initial French Score + 2.24]x10, mod. BO Score = [initial BO initial score+10]x3). C-statistics using ROC-analysis (Figure 2) were used to compare the three scores regarding their ability to discriminate between 30-day, 1 year and 5-year survivors and non-survivors within our heart transplant cohort. Results The French Score performed best predicting the 1 survival with an AUC of 0.69 [95% CI: 0.59-0.79, p<0.0001], followed by the BO-Score with 0.66 (95% CI: 0.56-0.77, p=0.003) and the US-Score with an AUC of 0.67, CI 0.57-0.76, p=0.02). Regarding 30-day-Survival, only the French Score with an AUC of 0.64 [95% CI: 0.52-0.76, p=0.006] as well as the BO-Score with 0.71 (95% CI: 0.60-0.81, p=0.008) could predict mortality sufficiently, while the US-Score could not discriminate statistically significant between survivors and non-survivors (AUC 0.64, CI 0.52-0.76, p=0.06). All scores performed worst predicting 5-year survival, with only the BO-Score being statistically significant (AUC 0.61, 95% CI: 0.56-0.77, p=0.003). Conclusion In our heart transplant cohort, the available donor recipient risk assessment tools performed best predicting 1 year survival, with worse and heterogenous results for 30 day or 5 year survival. Further developments will have to incorporate risk factors not yet taken into account in order to make even more reliable predictions, especially for the Eurotransplant area.Figure 1 and 2

Quantum interpolating ensemble: Bi-orthogonal polynomials and average entropies

The density matrix formalism is a fundamental tool in studying various problems in quantum information processing. In the space of density matrices, the most well-known measures are the Hilbert–Schmidt and Bures–Hall ensembles. In this work, the averages of quantum purity and von Neumann entropy for an ensemble that interpolates between these two major ensembles are explicitly calculated for finite-dimensional systems. The proposed interpolating ensemble is a specialization of the [Formula: see text]-deformed Cauchy–Laguerre two-matrix model and new results for this latter ensemble are given in full generality, including the recurrence relations satisfied by their associated bi-orthogonal polynomials when [Formula: see text] assumes positive integer values.

Sign changes of coefficients of powers of the infinite Borwein product

We denote by ct(m)(n) the coefficient of qn in the series expansion of (q;q)∞m(qt;qt)∞−m, which is the m-th power of the infinite Borwein product. Let t and m be positive integers with m(t−1)≤24. We provide asymptotic formula for ct(m)(n), and give characterizations of n for which ct(m)(n) is positive, negative or zero. We show that ct(m)(n) is ultimately periodic in sign and conjecture that this is still true for other positive integer values of t and m. Furthermore, we confirm this conjecture in the cases (t,m)=(2,m),(p,1),(p,3) for arbitrary positive integer m and prime p.

Quick counting of probability distributions for the sum of multi-sided dice: a physical and didactic approach

There are many physical systems (with randomness) where the performance of the probability distribution is of great importance [for example the so-called radioactive dice (Murray and Hart 2012 Phys. Educ. 47 197–201)]. A relevant situation is when the value (positive integers) of the upper faces is added when rolling fair dice (unbiased), that is, balanced, regular or unshaven. It has been widely studied in the basic literature for one, two and a maximum of four dice. However, a generalization using these methods for many dice is too cumbersome to carry out. In this paper we explore a method that uses the moment generating function, already studied in Singh et al (2011 UNLV Gaming Res. Rev. J.), for five six-sided fair dice with positive integer values. We carry out the generalization of this method to study the performance of the probability distribution and make a quick counting without complications for the sum of n fair dice with s sides (s ⩾ 6), and real values. We also study the invariance of the distribution when we set the number of sides, that is, for s-fix. Therefore, we study the distributions for the outgoing total sum (random variable) of roll generalized dice, for combinations of n multi-sided dice.

On the behavior of multiple zeta-functions with identical arguments on the real line

We study the behavior of r-fold zeta-functions of Euler-Zagier type with identical arguments ζr(s,s,…,s) on the real line. Our basic tool is an “infinite” version of Newton's classical identities. We carry out numerical computations, and draw graphs of ζr(s,s,…,s) for real s, for several small values of r. Those graphs suggest various properties of ζr(s,s,…,s), some of which we prove rigorously. When s∈[0,1], we show that ζr(s,s,…,s) has r asymptotes at ℜs=1/k (1≤k≤r), and determine the asymptotic behavior of ζr(s,s,…,s) close to those asymptotes. Numerical computations establish the existence of several real zeros for 2≤r≤10 (in which only the case r=2 was previously known). Based on those computations, we raise a conjecture on the number of zeros for general r, and gives a formula for calculating the number of zeros. We also consider the behavior of ζr(s,s,…,s) outside the interval [0,1]. We prove asymptotic formulas for ζr(−k,−k,…,−k), where k takes odd positive integer values and tends to +∞. Moreover, on the number of real zeros of ζr(s,s,…,s), we prove that there are exactly (r−1) real zeros on the interval (−2n,−2(n−1)) for any n≥2.

A new discrete reaching condition and generalized discrete reaching law with different convergence rates

A new discrete reaching condition of quasi-sliding mode (QSM) and generalized discrete reaching law with different convergence rates are proposed in this paper. The superiority is that they can ensure the faster convergence rate and shorten the reaching phase of the sliding variable by adjusting a positive integer value. The relationship between the generalized discrete reaching law parameters and the width of QSM domain are investigated for both positive and negative decay factors, respectively. Under proper choices of the parameters, the generalized discrete reaching law will satisfy the proposed new reaching condition and guarantee the QSM motion. Finally, a simulation example is given to illustrate the usefulness and superiority of the results.

Infinite Series of Ferrimagnetic Phases Emergent from the Gapless Spin Liquid Phase of Mixed Diamond Chains

The ground-state phases of mixed diamond chains with ($S, \tau^{(1)}, \tau^{(2)})=(1/2,1/2,1)$, where $S$ is the magnitude of vertex spins, and $\tau^{(1)}$ and $\tau^{(2)}$ are those of apical spins, are investigated. The apical spins $\tau^{(1)}$ and $\tau^{(2)}$ are connected with each other by an exchange coupling $\lambda$. Other exchange couplings are set equal to unity. This model has an infinite number of local conservation laws. For large $\lambda$, the ground state is equivalent to that of the uniform spin $1/2$ chain. Hence, the ground state is a gapless spin liquid. For $\lambda \leq 0$, the ground state is a Lieb-Mattis ferrimagnetic phase with spontaneous magnetization $m_{\rm sp}=1$ per unit cell. For intermediate $\lambda$, we find a series of ferrimagnetic phases with $m_{\rm sp}=1/p$ where $p$ takes positive integer values. The phases with $p \geq 2$ are accompanied by the spontaneous breakdown of the $p$-fold translational symmetry. It is suggested that the phase with arbitrarily large $p$, namely infinitesimal spontaneous magnetization, is allowed as $\lambda$ approaches the transition point to the gapless spin liquid phase.

On the design of nonparametric runs‐rules schemes using the Markov chain approach

AbstractThe aim of this paper is to highlight some concerns about the partly inaccurate manner in which the currently available2‐of‐2and2‐of‐3simple and improved runs‐rules charts based on the sign and the signed‐rank statistics were designed using Markov chain matrix. Because of the memory‐less property of the Markov chains, the empirical average run‐length (ARL) values were not affected; but the design structure of the matrix makes it difficult to formulate the general expressions of theARLexplicitly. Also, the dimension (and consequently, the simplicity) of the transition probability matrices (TPMs) and the false alarm rates expressions were affected negatively. Thus, we present some zero‐ and steady‐state formulae that make it easier to construct some key elements of the run‐length distribution (including the TPMs) of the simple and improved 2‐of‐(H+ 1) runs‐rules charts based on the sign and the signed‐rank statistics, for any positive integer valueH> 0, not justH= 1 and 2, as currently available for these monitoring schemes, so that any interested reader may use these formulae to obtain the corresponding empirical study.

The anisotropic chiral boson

We construct the theory of a chiral boson with anisotropic scaling, characterized by a dynamical exponent z that takes positive odd integer values. The action reduces to that of Floreanini and Jackiw in the isotropic case (z = 1). The standard free boson with Lifshitz scaling is recovered when both chiralities are nonlocally combined. Its canonical structure and symmetries are also analyzed. As in the isotropic case, the theory is also endowed with a current algebra. Noteworthy, the standard conformal symmetry is shown to be still present, but realized in a nonlocal way. The exact form of the partition function at finite temperature is obtained from the path integral, as well as from the trace over hat{u} (1) descendants. It is essentially given by the generating function of the number of partitions of an integer into z-th powers, being a well-known object in number theory. Thus, the asymptotic growth of the number of states at fixed energy, including subleading correc- tions, can be obtained from the appropriate extension of the renowned result of Hardy and Ramanujan.

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.

An Asymptotic Expansion for the Error Term in the Brent-McMillan Algorithm for Euler’s Constant

The Brent-McMillan algorithm is the fastest known procedure for the high-precision computation of Euler’s constant γ and is based on the modified Bessel functions I_0(2x) and K_0(2x). An error estimate for this algorithm relies on the optimally truncated asymptotic expansion for the product I_0(2x)K_0(2x) when x assumes large positive integer values. An asymptotic expansion for this optimal error term is derived by exploiting the techniques developed in hyperasymptotics, thereby enabling more precise information on the error term than recently obtained bounds and estimates.

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.

One-Sided and Two-Sided w-of-w Runs-Rules Schemes: An Overall Performance Perspective and the Unified Run-Length Derivations

The one-sided and two-sided Shewhart w-of-w standard and improved runs-rules monitoring schemes to monitor the mean of normally distributed observations from independent and identically distributed (iid) samples are investigated from an overall performance perspective, i.e., the expected weighted run-length (EWRL), for every possible positive integer value of w. The main objective of this work is to use the Markov chain methodology to formulate a theoretical unified approach of designing and evaluating Shewhart w-of-w standard and improved runs-rules for one-sided and two-sided X- schemes in both the zero-state and steady-state modes. Consequently, the main findings of this paper are as follows: (i) the zero-state and steady-state ARL and initial probability vectors of some of the one-sided and two-sided Shewhart w-of-w standard and improved runs-rules schemes are theoretically similar in design; however, their empirical performances are different and (ii) unlike previous studies that use ARL only, we base our recommendations using the zero-state and steady-state EWRL metrics and we observe that the steady-state improved runs-rules schemes tend to yield better performance than the other considered competing schemes, separately, for one-sided and two-sided schemes. Finally, the zero-state and steady-state unified approach run-length equations derived here can easily be used to evaluate other monitoring schemes based on a variety of parametric and nonparametric distributions.

The Smallest Nontrivial Solution to and Related Sequences

For two positive integers we consider the equation and we let s > 1 be its smallest nontrivial integer solution. We prove that as n ranges over the positive integers, the ratio takes each positive integer value infinitely often.

The connected metric dimension at a vertex of a graph

The notion of metric dimension, dim⁡(G), of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdimG(v), the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z∈S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdimG(v) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim(G), is min⁡{cdimG(v):v∈V(G)}. Noting that 1≤dim⁡(G)≤cdim(G)≤cdimG(v)≤|V(G)|−1 for any vertex v of G, we show the existence of a pair (G,v) such that cdimG(v) takes all positive integer values from dim⁡(G) to |V(G)|−1, as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdimG(v)∈{1,|V(G)|−1}. We show that cdim(G)=2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim⁡(H)=2. We also characterize trees and unicyclic graphs G satisfying cdim(G)=dim⁡(G). We show that cdim(G)−dim⁡(G) can be arbitrarily large. We determine cdim(G) and cdimG(v) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems.