Arithmetic Geometric Mean Research Articles

Dosing Time of Insulin Tregopil, a Novel Oral Insulin, in Type 2 Diabetes Mellitus Patients and Its Pharmacodynamic Effect

Our aim was to evaluate pharmacodynamics (PD) of insulin tregopil (IN-105) in relation to pre-meal dosing time in type 2 diabetes mellitus (T2DM) patients in a randomized, placebo-controlled, crossover study. Fifteen patients were randomised to 5 sequences (3 patients each) and administered 4 treatments (Insulin tregopil 30 mg given 30, 20 and 10 minutes before American diabetes association (ADA) meal and placebo given 20 minutes before ADA meal) in a partial replicate crossover fashion with a washout period of 1-2 days between periods. PD sampling was done from dosing to 180 min post-meal. AUC0-180min (glucose exposure), Cmin (minimum glucose concentration observed in the sampling period) and tmin (indicating time of peak PD effect) are presented in Table 1. Maximal reduction in plasma glucose (difference between baseline [meal administration] and minimum post-meal plasma glucose concentration) was 15.7 mg/dL, 9.5 mg/dL and 9.7 mg/dL for 30, 20 and 10 minutes groups, respectively. This peak PD effect was observed 16 to 37 minutes post-meal, indicating ultra-short action profile. Insulin tregopil given orally 10-20 minutes before meals has rapid onset of action. It results in effective lowering of post-meal glucose exposure with peak PD effect early in the post-meal period, hence minimizing chances of hypoglycaemia.Table 1. Pharmacodynamic Parameters for Plasma Insulin TregopilTreatment groupsBaseline-Corrected Plasma Glucose (Relative to time of Meal Administration)AUC0-180 min (min*mg/mL) ± SDCmin (mg/mL) ± SDTmin (min) ± SDInsulin tregopil 10 min217.39 (31.58)0.86 (0.11)37.33 (59.09)Insulin tregopil 20 min227.04 (67.89)0.81 (0.10)26.00 (43.22)Insulin tregopil 30 min267.67 (46.81)0.81 (0.08)16.66 (7.23)Placebo251.96 (39.11)0.98 (0.03)33.66 (66.72)ParametersComparisonGeometric mean ratio (CI [%])AUC0-180 min (min*mg/mL)Insulin tregopil 30 min vs. placebo106.4 (87.2, 129.8)Insulin tregopil 20 min vs. placebo80.0 (65.6, 97.6)Insulin tregopil 10 min vs. placebo86.6 (71.0, 105.7)Cmin (mg/mL)Insulin tregopil 30 min vs. placebo82.7 (78.3, 87.4)Insulin tregopil 20 min vs. placebo83.0 (78.5, 87.7)Insulin tregopil 10 min vs. placebo87.8 (83.1, 92.8)Insulin tregopil was administered at 30, 20 or 10 min before the ADA meal and placebo was administered at 20 min before the ADA meal. CI-90% confidence interval; Min-minutes. Baseline-corrected arithmetic mean and geometric mean values are presented. Disclosure A.P. Khedkar: None. V. Jose: None. H.E. Lebovitz: Stock/Shareholder; Self; Abbott, AbbVie Inc.. Advisory Panel; Self; Biocon. Stock/Shareholder; Self; General Electric, Gilead Sciences, Inc., IBM. Advisory Panel; Self; Intarcia Therapeutics, Inc.. Stock/Shareholder; Self; Nestlé, Novartis AG. Advisory Panel; Self; Poxel SA. A.D. Cherrington: Advisory Panel; Self; Biocon. Consultant; Self; Boston Scientific Corporation. Research Support; Self; Boston Scientific Corporation. Consultant; Self; Eli Lilly and Company. Advisory Panel; Self; Fractyl Laboratories, Inc.. Stock/Shareholder; Self; Fractyl Laboratories, Inc.. Consultant; Self; Galvani Bioelectronics Limited. Research Support; Self; Galvani Bioelectronics Limited. Consultant; Self; MedImmune. Advisory Panel; Self; Metavention. Stock/Shareholder; Self; Metavention. Consultant; Self; Novo Nordisk Inc.. Research Support; Self; Novo Nordisk Inc.. Advisory Panel; Self; NuSirt Biopharma, Inc., Sensulin Labs, LLC.. Other Relationship; Self; Sensulin Labs, LLC.. Consultant; Self; Silver Lake. Research Support; Self; Silver Lake. Consultant; Self; Thermalin Diabetes, LLC., Thetis Pharmaceuticals LLC.. Stock/Shareholder; Self; Thetis Pharmaceuticals LLC.. Advisory Panel; Self; VTV Therapeutics. Consultant; Self; VTV Therapeutics. Advisory Panel; Self; Zafgen. Research Support; Self; Zafgen. Stock/Shareholder; Self; Zafgen. Consultant; Self; Abvance. Other Relationship; Self; Abvance. Consultant; Self; California Institute for Biomedical Research (Calibr). Advisory Panel; Self; These Three Medical, Inc (T3M). G. Fleming: Consultant; Self; Arisaph Pharmaceuticals, Inc., ADOCIA. Board Member; Self; Biocon. Consultant; Self; CocoaVia. Board Member; Self; Diasome Pharmaceuticals, Inc.. Consultant; Self; Dompé Pharmaceutical, InsuLine Medical Ltd., Intarcia Therapeutics, Inc., Ipsen Biopharmaceuticals, Inc., Janssen Pharmaceuticals, Inc., Mylan, Merck KGaA, Oramed Pharmaceuticals, Inc.. Board Member; Self; Orgenesis Inc., NuSirt Biopharma, Inc.. Other Relationship; Self; ProSciento. Consultant; Self; QuintilesIMS, Rhythm Pharmaceuticals Inc., Sanofi, Skyepharma PLC., Thermalin Diabetes, LLC.. Stock/Shareholder; Self; Thetis Pharmaceuticals LLC.. Consultant; Self; Xeris Pharmaceuticals, Inc., Veracyte, Inc., Versartis, Inc., VeroScience, LLC. S.N. Athalye: Employee; Self; Biocon. A. Vishweswaramurthy: Employee; Self; Biocon. Other Relationship; Self; JDRF.

Dispersion Theory of Risk: Characteristics and a Comparative Study of Five Models

Risk and uncertainty in decisions play an important role in many fields including engineering, economics and social sciences. In order to represent the decision problem mathematically or to compare prospects, a scalar either as dispersion measure or risk measure is used. Adequately measuring dispersion or risk is one of the most challenging topics in decision theory. According to well-known definitions in the literature, a dispersion or risk measure should satisfy several good properties; non-negativity, monotonicity, and convexity. In this paper, we investigate properties of two dispersion measures of Geometric Dispersion Theory (GDT) which are based on arithmetic-geometric means and show that they are nonnegative and convex. Also, these new dispersion measures of GDT are less sensitive to noise in comparison to standard deviation and its generalizations. Furthermore, we show that these new measures can present riskiness of alternatives better than other measures; thus enabling GDT to explain several well-known decision-making paradoxes.

Complete p-elliptic integrals and a computation formula of $$\pi _p$$ π p for $$p=4$$ p = 4

The complete $p$-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function $\sin_p{\theta}$ and its half-period $\pi_p$. It is shown, only for $p=4$, that the generalized $p$-elliptic integrals yield a computation formula of $\pi_p$ in terms of the arithmetic-geometric mean. This is a $\pi_p$-version of the celebrated formula of $\pi$, independently proved by Salamin and Brent in 1976.

An algorithm for numerically inverting the modular j-function

The modular j-function is a bijective map from $$X_0(1) \setminus \{\infty \}$$ to $$\mathbb {C}$$ . A natural question is to describe the inverse map. Gauss offered a solution to the inverse problem in terms of the arithmetic–geometric mean. This method relies on an elliptic curve model and the Gaussian hypergeometric series. Here we use the theory of polar harmonic Maass forms to solve the inverse problem by directly examining the Fourier expansion of the weight 2 polar harmonic Maass form obtained by specializing the logarithmic derivative of the denominator formula for the Monster Lie algebra.

Norm inequalities related to the arithmetic–geometric mean inequalities for positive semidefinite matrices

In this paper, we propose three new matrix versions of the arithmetic–geometric mean inequality for unitarily invariant norms, which stem from the fact that the Heinz mean of two positive real numbers interpolates between the geometric and arithmetic means of these numbers. Related trace inequalities are also presented.

On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind

In the article, we prove that the double inequalities1+(6p−7)r′p+(5p−6)r′πtanh−1⁡(r)2r<K(r)<1+(6q−7)r′q+(5q−6)r′πtanh−1⁡(r)2r,qA(1,r)+(5q−6)G(1,r)A(1,r)+(6q−7)G(1,r)L(1,r)<AGM(1,r)<pA(1,r)+(5p−6)G(1,r)A(1,r)+(6p−7)G(1,r)L(1,r) hold for all r∈(0,1) if and only if p≥π/2=1.570796⋯ and q≤89/69=1.289855⋯, where K(r)=∫0π/2(1−r2sin2⁡t)−1/2dt is the complete elliptic integral of the first kind, tanh−1⁡(r)=log⁡[(1+r)/(1−r)]/2 is the inverse hyperbolic tangent function, r′=1−r2, and A(1,r)=(1+r)/2, G(1,r)=r, L(1,r)=(r−1)/log⁡r and AGM(1,r) are the arithmetic, geometric, logarithmic and Gaussian arithmetic-geometric means of 1 and r, respectively.

A note on the matrix arithmetic-geometric mean inequality

This note proves the following inequality: If $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $\bA_1,\bA_2,\dots,\bA_n$, the following inequality holds: \begin{equation*}\label{eq:main} \frac{1}{n^3} \, \Big\|\sum_{j_1,j_2,j_3=1}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\| \,\geq\, \frac{(n-3)!}{n!} \, \Big\|\sum_{\substack{j_1,j_2,j_3=1,\\\text{$j_1$, $j_2$, $j_3$ all distinct}}}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\|, \end{equation*} where $\|\cdot\|$ represents the operator norm. This inequality is a special case of a recent conjecture proposed by Recht and R\'{e} (2012).

Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates

In this paper we revisit the anisotropic isoperimetric and the Brunn−Minkowski inequalities for convex sets. The best known constantC(n) =Cn7depending on the space dimensionnin both inequalities is due to Segal [A. Segal,Lect. Notes Math., Springer, Heidelberg2050(2012) 381–391]. We improve that constant toCn6for convex sets and toCn5for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the formCn2,i.e., quadratic inn. The tools are the Brenier’s mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.

Feature extraction using Pythagorean means for classification of epileptic EEG signals

Electroencephalogram (EEG) is a widely used tool for the study and diagnosis of epilepsy. The patients subjected to epilepsy require long term monitoring of EEG. Automatic seizure detection will eliminate chances of missing any seizure, make detection easy and reduce burden on physicians. In this work, different combination of Pythagorean means (time domain features) namely arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM) of energy per epoch are used as features to classify EEG data into normal, seizure free and seizure classes by using a linear classifier. The classification accuracy of 100% is achieved in two and three class problem with a single feature/epoch and in five class problem with two features/epoch. The novelty of this work is use of new and simple features (in epileptic EEG signal classification), reduced complexity and high performance.

Generalized Irreducible α-Matrices and Its Applications

The class of generalized α-matrices is presented by Cvetković, L. (2006), and proved to be a subclass of H-matrices. In this paper, we present a new class of matrices-generalized irreducible α-matrices, and prove that a generalized irreducible α-matrix is an H-matrix. Furthermore, using the generalized arithmetic-geometric mean inequality, we obtain two new classes of H-matrices. As applications of the obtained results, three regions including all the eigenvalues of a matrix are given.

Distant Decimals of $$\pi $$ π : Formal Proofs of Some Algorithms Computing Them and Guarantees of Exact Computation

We describe how to compute very far decimals of $$\pi $$ and how to provide formal guarantees that the decimals we compute are correct. In particular, we report on an experiment where 1 million decimals of $$\pi $$ and the billionth hexadecimal (without the preceding ones) have been computed in a formally verified way. Three methods have been studied, the first one relying on a spigot formula to obtain at a reasonable cost only one distant digit (more precisely a hexadecimal digit, because the numeration basis is 16) and the other two relying on arithmetic–geometric means. All proofs and computations can be made inside the Coq system. We detail the new formalized material that was necessary for this achievement and the techniques employed to guarantee the accuracy of the computed digits, in spite of the necessity to work with fixed precision numerical computation.

Quadratic refinements of matrix means

The main target of this article is to present refinements of the matrix arithmetic-geometric mean inequality. The main difference between these refinements and the ones in the literature is the quadratic behavior of the refining terms. These refinements include the L\{o}ewner partial ordering, determinants, trace and unitarily invariant norms refinements.

An improved Bennett's inequality

ABSTRACTThis paper gives an improvement to Bennett's inequality for tail probability of sum of independent random variables, without imposing any additional condition. The improved version has a closed form expression. Using a refined arithmetic-geometric mean inequality, we further improve the obtained inequality. Numerical comparisons show that the proposed inequalities often improve the upper bound significantly in the far tail area, and these improvements get more prominent for larger sample size.

Extensions of interpolation between the arithmetic-geometric mean inequality for matrices

In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if A, B, X are ntimes n matrices, then ∥AXB∗∥2≤∥f1(A∗A)Xg1(B∗B)∥∥f2(A∗A)Xg2(B∗B)∥,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\bigl\\Vert AXB^{*} \\bigr\\Vert ^{2}\\leq \\bigl\\Vert f_{1} \\bigl(A^{*}A\\bigr)Xg_{1}\\bigl(B^{*}B\\bigr) \\bigr\\Vert \\bigl\\Vert f_{2}\\bigl(A^{*}A\\bigr)Xg_{2}\\bigl(B^{*}B\\bigr) \\bigr\\Vert , \\end{aligned}$$ \\end{document} where f_{1}, f_{2}, g_{1}, g_{2} are non-negative continuous functions such that f_{1}(t)f_{2}(t)=t and g_{1}(t)g_{2}(t)=t (tgeq0). We also obtain the inequality |||AB∗|||2≤|||p(A∗A)mp+(1−p)(B∗B)s1−p||||||(1−p)(A∗A)n1−p+p(B∗B)tp|||,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\bigl\\vert \\!\\bigl\\vert \\!\\bigl\\vert AB^{*} \\bigr\\vert \\!\\bigr\\vert \\!\\bigr\\vert ^{2} &\\leq \\bigl\\vert \\!\\bigl\\vert \\!\\bigl\\vert p \\bigl(A^{*}A\\bigr)^{\\frac{m}{p}}+ (1-p) \\bigl(B^{*}B\\bigr)^{\\frac {s}{1-p}} \\bigr\\vert \\!\\bigr\\vert \\!\\bigr\\vert \\bigl\\vert \\!\\bigl\\vert \\!\\bigl\\vert (1-p) \\bigl(A^{*}A\\bigr)^{\\frac{n}{1-p}}+ p\\bigl(B^{*}B \\bigr)^{\\frac{t}{p}} \\bigr\\vert \\!\\bigr\\vert \\!\\bigr\\vert , \\end{aligned}$$ \\end{document} in which m, n, s, t are real numbers such that m+n=s+t=1, vert !vert !vert cdot vert !vert !vert is an arbitrary unitarily invariant norm and pin[0,1].

A refined Hoeffding’s upper tail probability bound for sum of independent random variables

Using a refined arithmetic–geometric mean inequality, this paper gives an improved version of Hoeffding’s inequality which has a closed form and is easy to evaluate. Numerical simulation comparing the performance of the proposed inequality to the original version of Hoeffding’s inequality is also presented.

Statistics and characterization of matrices by determinant and trace

Answering a question of Erdos, Komlos proved in 1968 that almost all $$n\times n$$ Bernoulli matrices are nonsingular as $$n\rightarrow \infty $$ . In this paper, we offer a new perspective on the question of Erdos by studying $$n\times n$$ matrices with prime number entries in an almost all sense. Precisely, it is shown that, as $$x\rightarrow \infty $$ , the probability of randomly choosing a nonsingular $$n\times n$$ matrix among all $$n\times n$$ matrices with prime number entries that are $${\le }x$$ is 1. If A is a unitary matrix, then it is well known that $$|{\det } A|=1$$ . However, the converse is far from being true. As a remedy of this defect, we search for necessary and sufficient conditions for being a unitary matrix by teaming up determinant with trace. In this way, we are led to simple characterizations of unitary matrices in the set of normal matrices. The question of which nonsingular commuting complex matrices with real eigenvalues have the same characteristic polynomial is formulated via determinant and trace conditions. Finally, through a study of eigenvectors, we obtain new characterizations of Hermitian and normal matrices. Our approach to proving these results benefits from a modular interpretation of nonsingularity and the spectral theorem for normal operators together with equality cases of classical inequalities such as the arithmetic–geometric mean inequality and the Cauchy–Schwarz inequality.

Graph indices via the AM–GM inequality

Numerical values associated with chemical graphs have their role in describing the graph. Examples of these values are the Zagreb and Wiener indices. The main goal of this article is to present new relations between the multiplicative and additive versions of these indices, where refinements and reverses of some well known relations will be presented.The main tool in our discussion is the arithmetic–geometric mean inequality (AM–GM), which will be refined and reversed in a new form that allows us to obtain the aimed results about the graph indices.

Modeling circular inductors coupled to a semi-infinite magnetic medium considering the proximity effect

This paper presents a simple 2D electromagnetic model that solves the high-frequency current distribution, considering the proximity effect, in a planar spiral coil above a linear conductive or non-conductive semi-infinite magnetic medium. The conductive regions are divided into axisymmetric elements in which the current is assumed constant. The current flowing in each element depends on its complex impedance and is computed by Kirchhoff’s circuit law. To take the effect of the magnetic medium into account, a new set of mutual inductance formulas are presented. Those formulas are expressed in terms of elliptic integrals and the fast-converging arithmetic–geometric mean iteration of Gauss. The geometric mean distance method is used to deal with elements of arbitrarily shaped cross-section. Elliptic integrals are also used to express the magnetic flux density. The current distribution, the magnetic field and the equivalent impedance computed with the multifilament model agree well with the results obtained using commercial finite element software.

Comparison of different ways of computing grades in continuous assessment into the final grade

&lt;p&gt;We present the results of comparing various ways of calculating students' final grades from continuous assessment grades. Traditionally the weighted arithmetic mean has been used and we compare this method with other alternatives: arithmetic mean, geometric mean, harmonic mean and multiplication of the percentage of overcoming of each activi-ty. Our objective is to verify, if any of the alternative methods, agree with the student’s performance proposed by the teacher of the subject, further discriminating the grade be-tween high and low learning outcomes and reducing the number of approved opportunists.&lt;/p&gt;&lt;p&gt; &lt;/p&gt;&lt;p&gt;[Comparación del efecto de diferentes modos de agregar las califica-ciones de evaluación continua en la nota final]&lt;/p&gt;

Optimal bounds for arithmetic-geometric and Toader means in terms of generalized logarithmic mean

In this paper, we find the greatest values alpha_{1},alpha_{2} and the smallest values beta_{1},beta_{2} such that the double inequalities L_{alpha_{1}}(a,b)<operatorname{AG}(a,b)<L_{beta_{1}}(a,b) and L_{alpha_{2}}(a,b)< T(a,b)< L_{beta_{2}}(a,b) hold for all a, b>0 with aneq b, where operatorname{AG}(a,b), T(a,b) and L_{p}(a,b) are the arithmetic-geometric, Toader and generalized logarithmic means of two positive numbers a and b, respectively.