scholarly journals I. On a new method of approximation applicable to elliptic and ultra-elliptic functions

1860 ◽  
Vol 10 ◽  
pp. 473-475
Keyword(s):  
Geometric Mean ◽  
Elliptic Functions ◽  
New Method ◽  
Square Root ◽  
The Real ◽  
Rate Of Approximation ◽  
Real Application ◽  
Harmonic Means

I found my method on the known principle, that the geometric mean between two quantities is also a geometric mean between the arithmetic and harmonic means of those quantities. We may therefore approximate to the geometric mean of two quantities in this way:—Take their arithmetic and harmonic means; then take the arithmetic and harmonic means of those means; then of these last means again, and so on, as far as we please. If the ratio of the original quantities lies within the ratio of 1 : 2, the approximation proceeds with extraordinary rapidity, so that, in obtaining a fraction nearly equal to √2 by this method, we obtain a result true to eleven places of decimals at the fourth mean. I name this merely to show the rate of approximation. The real application of the method is to the integration of functions embracing a radical of the square root.

scholarly journals XIII. On a new method of approximation applicable to elliptic and ultra-elliptic functions

1860 ◽  
Vol 150 ◽  
pp. 223-227
Keyword(s):  
Geometric Mean ◽  
Elliptic Functions ◽  
New Method ◽  
Geometric Means ◽  
The Third ◽  
Harmonic Means ◽  
General Method

The difficulty of finding approximate values of elliptic functions of the third kind has led me to consider a general method of approximation, which I believe to be new, at least in its application to the evaluation of integrals of irrational functions. It depends on the known principle that the geometric mean between two quantities is also a geometric mean between their arithmetic and harmonic means. If we take any two positive quantities, we may approximate to their geometric means as follows:— Take the arithmetic and harmonic means of the two quantities, then again take the arithmetic and harmonic means of those means, and so on: the successive means will approximate with great rapidity to the geometric mean.

Radiation synovectomy of the knee joint

2006 ◽  
Vol 45 (01) ◽  
pp. 57-61
Author(s):  
M. Puille ◽  
D. Steiner ◽  
R. Bauer ◽  
R. Klett
Keyword(s):  
Knee Joint ◽  
Lymph Nodes ◽  
Geometric Mean ◽  
Radiation Synovectomy ◽  
Real Activity ◽  
Anterior View ◽  
Inguinal Lymph Nodes ◽  
The Real ◽  
Therapeutic Modalities ◽  
Real Value

Summary Aim: Multiple procedures for the quantification of activity leakage in radiation synovectomy of the knee joint have been described in the literature. We compared these procedures considering the real conditions of dispersion and absorption using a corpse phantom. Methods: We simulated different distributions of the activity in the knee joint and a different extra-articular spread into the inguinal lymph nodes. The activity was measured with a gammacamera. Activity leakage was calculated by measuring the retention in the knee joint only using an anterior view, using the geometric mean of anterior and posterior views, or using the sum of anterior and posterior views. The same procedures were used to quantify the activity leakage by measuring the activity spread into the inguinal lymph nodes. In addition, the influence of scattered rays was evaluated. Results: For several procedures we found an excellent association with the real activity leakage, shown by an r² between 0.97 and 0.98. When the real value of the leakage is needed, e. g. in dosimetric studies, simultaneously measuring of knee activity and activity in the inguinal lymph nodes in anterior and posterior views and calculation of the geometric mean with exclusion of the scatter rays was found to be the procedure of choice. Conclusion: When measuring of activity leakage is used for dosimetric calculations, the above-described procedure should be used. When the real value of the leakage is not necessary, e. g. for comparing different therapeutic modalities, several of the procedures can be considered as being equivalent.

Single-level anterior cervical fusion: a new method to evaluate the real need for plate augmentation

2019 ◽  
Vol 63 (4) ◽  
Author(s):  
Marco Ajello ◽  
Nicola Marengo ◽  
Paolo Pacca ◽  
Federico Pecoraro ◽  
Francesco Zenga ◽  
...  
Keyword(s):  
Anterior Cervical Fusion ◽  
New Method ◽  
Cervical Fusion ◽  
The Real ◽  
Single Level

The digital function filters – algorithms and applications

2013 ◽  
Vol 61 (2) ◽  
pp. 371-377
Author(s):  
M. Siwczyński ◽  
A. Drwal ◽  
S. Żaba
Keyword(s):  
Digital Filters ◽  
Phase Shifters ◽  
Square Root ◽  
Real Power ◽  
The Real ◽  
Distributed Parameters ◽  
Digital Modeling ◽  
Analysis And Synthesis ◽  
Basic Functions ◽  
Recursive Formulas

Abstract The simple digital filters are not sufficient for digital modeling of systems with distributed parameters. It is necessary to apply more complex digital filters. In this work, a set of filters, called the digital function filters, is proposed. It consists of digital filters, which are obtained from causal and stable filters through some function transformation. In this paper, for several basic functions: exponential, logarithm, square root and the real power of input filter, the recursive algorithms of the digital function filters have been determined The digital function filters of exponential type can be obtained from direct recursive formulas. Whereas, the other function filters, such as the logarithm, the square root and the real power, require using the implicit recursive formulas. Some applications of the digital function filters for the analysis and synthesis of systems with lumped and distributed parameters (a long line, phase shifters, infinite ladder circuits) are given as well.

scholarly journals Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 896-909 ◽  
Author(s):  
Dianchen Lu ◽  
Aly R. Seadawy ◽  
Mujahid Iqbal
Keyword(s):  
Solitary Wave ◽  
Nonlinear Partial Differential Equations ◽  
Research Work ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Mapping Method ◽  
New Method ◽  
Auxiliary Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions

AbstractIn this research work, for the first time we introduced and described the new method, which is modified extended auxiliary equation mapping method. We investigated the new exact traveling and families of solitary wave solutions of two well-known nonlinear evaluation equations, which are generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified forms of Camassa-Holm equations. We used a new technique and we successfully obtained the new families of solitary wave solutions. As a result, these new solutions are obtained in the form of elliptic functions, trigonometric functions, kink and antikink solitons, bright and dark solitons, periodic solitary wave and traveling wave solutions. These new solutions show the power and fruitfulness of this new method. We can solve other nonlinear partial differential equations with the use of this method.

A Novel K-Harmonic Means Clustering Based on Multiple Initial Centers

2013 ◽  
Vol 321-324 ◽  
pp. 1947-1950
Author(s):  
Lei Gu ◽  
Xian Ling Lu
Keyword(s):  
New Method ◽  
Final Solution ◽  
Data Set ◽  
Number Of Clusters ◽  
Multiple Groups ◽  
Harmonic Means

In the initialization of the traditional k-harmonic means clustering, the initial centers are generated randomly and its number is equal to the number of clusters. Although the k-harmonic means clustering is insensitive to the initial centers, this initialization method cannot improve clustering performance. In this paper, a novel k-harmonic means clustering based on multiple initial centers is proposed. The number of the initial centers is more than the number of clusters in this new method. The new method with multiple initial centers can divide the whole data set into multiple groups and combine these groups into the final solution. Experiments show that the presented algorithm can increase the better clustering accuracies than the traditional k-means and k-harmonic methods.

scholarly journals Bounding Lagrangian widths via geodesic paths

2014 ◽  
Vol 150 (12) ◽  
pp. 2143-2183 ◽  
Author(s):  
Matthew Strom Borman ◽  
Mark McLean
Keyword(s):  
Sectional Curvature ◽  
New Method ◽  
Finite Width ◽  
Energy Capacity ◽  
Positive Sectional Curvature ◽  
Floer Cohomology ◽  
The Real ◽  
Ambient Manifold ◽  
Open Map ◽  
Geodesic Paths

AbstractThe width of a Lagrangian is the largest capacity of a ball that can be symplectically embedded into the ambient manifold such that the ball intersects the Lagrangian exactly along the real part of the ball. Due to Dimitroglou Rizell, finite width is an obstruction to a Lagrangian admitting an exact Lagrangian cap in the sense of Eliashberg–Murphy. In this paper we introduce a new method for bounding the width of a Lagrangian$Q$by considering the Lagrangian Floer cohomology of an auxiliary Lagrangian$L$with respect to a Hamiltonian whose chords correspond to geodesic paths in$Q$. This is formalized as a wrapped version of the Floer–Hofer–Wysocki capacity and we establish an associated energy–capacity inequality with the help of a closed–open map. For any orientable Lagrangian$Q$admitting a metric of non-positive sectional curvature in a Liouville manifold, we show the width of$Q$is bounded above by four times its displacement energy.

Further Contributions to the Theory of Peripheral Jets in Ground Proximity

1992 ◽  
Vol 36 (01) ◽  
pp. 88-90
Author(s):  
David S. Tselnik
Keyword(s):  
General Solution ◽  
Infinite Series ◽  
Elliptic Functions ◽  
Jet Flow ◽  
New Method ◽  
Asymptotic Formulas ◽  
Flow Problems ◽  
Complicated Task

A number of plane inviscid jet flow problems of interest in hydrodynamics require the use of elliptic functions theory. Generally speaking, finding the general solution to a problem in terms of elliptic functions is not a complicated task. However, finding solutions as rapidly convergent infinite series or as sound asymptotic formulas is often not as easy, and special ways of treatment may prove to be necessary. In parallel with solving the problem of peripheral jets, the author's earlier paper (1985) proposed some such ways of treatment. In the present paper, a new method of treatment is proposed (and used);this approach may be of help in studies where the methods of elliptic functions theory have to be used.

scholarly journals A new method to retrieve the real part of the equivalent refractive index of atmospheric aerosols

2018 ◽  
Vol 117 ◽  
pp. 54-62 ◽  
Author(s):  
S. Vratolis ◽  
P. Fetfatzis ◽  
A. Argyrouli ◽  
A. Papayannis ◽  
D. Müller ◽  
...  
Keyword(s):  
Refractive Index ◽  
Atmospheric Aerosols ◽  
New Method ◽  
The Real

scholarly journals Generalized Linguistic Hesitant Intuitionistic Fuzzy Hybrid Aggregation Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Wei Yang ◽  
Jiarong Shi ◽  
Yongfeng Pang
Keyword(s):  
Decision Making ◽  
Geometric Mean ◽  
Multiple Attribute Decision Making ◽  
Aggregation Operators ◽  
New Method ◽  
Weighted Averaging ◽  
Fuzzy Information ◽  
Intuitionistic Fuzzy ◽  
Multiple Attribute ◽  
Special Cases

Some hybrid aggregation operators have been developed based on linguistic hesitant intuitionistic fuzzy information. The generalized linguistic hesitant intuitionistic fuzzy hybrid weighted averaging (GLHIFHWA) operator and the generalized linguistic hesitant intuitionistic fuzzy hybrid geometric mean (GLHIFHGM) operator are defined. Some special cases of the new aggregation operators are studied and many existing aggregation operators are special cases of the new operators. A new multiple attribute decision making method based on the new aggregation operators is proposed and a practical numerical example is presented to illustrate the feasibility and practical advantages of the new method.

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