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### Article

Vibration of Visco-Elastic Parallelogram Plate with Thickness Variation Linearly in one Direction and Parabolic in another Direction

1Department of Mathematics, M. S. College, Saharanpur, U.P., India

American Journal of Applied Mathematics and Statistics. 2013, 1(5), 103-109
DOI: 10.12691/ajams-1-5-5

Cite this paper:
Arun Kumar Gupta, Anuj Kumar. Vibration of Visco-Elastic Parallelogram Plate with Thickness Variation Linearly in one Direction and Parabolic in another Direction. American Journal of Applied Mathematics and Statistics. 2013; 1(5):103-109. doi: 10.12691/ajams-1-5-5.

Correspondence to: Arun  Kumar Gupta, Department of Mathematics, M. S. College, Saharanpur, U.P., India. Email: gupta_arunnitin@yahoo.co.in

### Abstract

The main objective of the present investigation is to study the vibration of visco-elastic parallelogram plate whose thickness varies bi-directionally. It is assumed that the plate is clamped on all the four edges and that the thickness varies linearly in one direction and parabolically in another direction. Using the separation of variables method and Rayleigh-Ritz technique with a two-term deflection function, the governing differential equation has been solved for vibration of visco-elastic parallelogram plate. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The assumption of small deflection is made. Visco-elastic of the plate is taken of the “Kelvin Type”. Time period and deflection function at different point for the first two modes of vibration are calculated for various values of taper constant, aspect ratio and skew angle and results are presented in tabular form. Alloy “Duralumin” is considered for all the material constants used in numerical calculations.

### References

 [1] Leissa A. W., Vibration of Plate, NASA SP-160, (1969). [2] Leissa A. W., Plate Vibration Research, 1976-1980, Classical Theory, The Shock and Vibration Digest, 13(9), (1981), 11-22. [3] Leissa A. W., Recent Studies in Plate Vibrations: Part I, Classical Theory, The Shock and Vibration Digest, 19(2), (1987),11-18. [4] Sobotka Z., Analysis of thick visco-elastic plates, Theoretical and Applied Mechanics, Proceeding Vol. I. Publishing House of the Bulgarian Academy of Sciences, Sofia., (1971),379-386. [5] Bland D.R., The theory of linear visco-elasticity, Pergamon Press, (1960).
 [6] Ilanko S., Comments on the historical bases of the Rayleigh and Ritz methods, J. Sound and Vibration, 319( 1-2),(2009), 731-733. [7] Eslami M. R., Shakeri M., Ohadi A.R. and Shiari B., Coupled thermo elasticity of shell of revolution effect of normal stress and coupling, AIAA Journal, 37(4), (1999), 496-512. [8] Leissa A. W. and Narita Y., Vibration studies for simply supported symmetrically laminated rectangular plates, Composite Structures, 12,(1989), 113-132. [9] Lekhnitski S.G., Anistropic plates, Ist Ed.English trans., Am.Iron and Steel Inst. (New York N.Y.) ( 1956). [10] Zhang L. and Zu J.W., Non-linear vibrations of visco-elastic moving belts part-I: force vibration analysis, J. Sound and Vibration, 216(1),(1998),75-91. [11] Zhang L and Zu J.W., Non linear vibration of parametrically excited visco-elastic moving belts part-II stability analysis, J. Appl.Mach., Trans. ASME, 66(2), (1999), 403-409. [12] Mivhel R., A periodic problem in visco-elasticity with variable coefficients, Int. J. of Engg. Sci., 19, (1981), 1145-1168. [13] Garrick I.E., Survey of Aero-thermo-elasticity, J. Aerospace Engg.,22,(1963), 140-147. [14] Gnossi R. O. and Laura P. A. A., Transverse vibrations of rectangular orthotropic plates with one or two free edges while the remaining are elastically restrained against rotation, Ocean Engg.,6(5),(1979),527-539. [15] Park J. and Mongeau Luc., Vibration and radiation of visco-elastically supported mindlin plates, J. Sound and Vibration, 318(4-5),(2008),1230-1249. [16] Gupta A.K., Kumar A. and Gupta Y.K., Vibration study of visco-elastic parallelogram plate of linearly varying thickness, International Journal of Engineering and Interdisciplinary Mathematics, 2(1),(2010),1-9. [17] Nagaya K., Vibrations and dynamic response of visco-elastic plates on non-periodic elastic supports, J. Engg. for Industry, 99, (1977),404-409.
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### Article

Robustness and Power of the Kornbrot Rank Difference, Signed Ranks, and Dependent Samples T-test

1Department of Evaluation and Research, Wayne State University, Detroit, USA

American Journal of Applied Mathematics and Statistics. 2013, 1(5), 99-102
DOI: 10.12691/ajams-1-5-4

Cite this paper:
Norman N. Haidous, Shlomo S. Sawilowsky. Robustness and Power of the Kornbrot Rank Difference, Signed Ranks, and Dependent Samples T-test. American Journal of Applied Mathematics and Statistics. 2013; 1(5):99-102. doi: 10.12691/ajams-1-5-4.

Correspondence to: Norman  N. Haidous, Department of Evaluation and Research, Wayne State University, Detroit, USA. Email: ac6702@wayne.edu

### Abstract

The purpose of the study was to compare the power and accuracy of the Kornbrot rank difference test to classical parametric and nonparametric alternatives when the assumption of normality is not met, the data are ordinal, and the sample size is small. Although the procedure is robust, there was no evidence the rank difference test had power advantages over Wilcoxon Signed-Ranks test.

### References

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 [6] Hodges, J. L., & Lehman, E. L. (1956). The efficiency of some nonparametric competitors of the t-test. Annals of Mathematical Statistics, 27(2), 324-335. [7] Sawilowsky, S. & Fahoome, G. (2003). Statistics Through Monte Carlo Simulation with Fortran. Oak Park: JMASM. [8] Kornbrot, D. E. (1990). The rank difference test: A new and meaningful alternative to the Wilcoxon signed ranks test for ordinal data. British Journal of Mathematical and Statistical Psychology, 43, 241-264. [9] Bradley, J. V. (1978). Robustness? British Journal of Mathematical & Statistical Psychology, 31, 144-152. [10] Headrick, T. C., & Sawilowsky, S. S. (2000). Weighted simplex procedures for determining boundary points and constants for the univariate and multivariate power methods. Journal of Educational and Behavioral Statistics, 25, 417-436. [11] Fleishman, A. I. (1978). A method for simulating non-normal distributions. Psychometrika, 43, 521-532. [12] Smith, J. (2009). Intermediate r values for use in the Fleishman power method. Journal of Modern Applied Statistical Methods, 8(2), 610-612. [13] Sawilowsky, S. (2009). New effect size rules of thumb. Theoretical and Behavioral Foundations, 8,2, 597-599. [14] Gibbons, J., & Chakraborti, S. (1991). Comparisons of the Mann-Whitney, Student’s t, and Alternate t Tests for means of normal distribution. The Journal of Experimental Education 59(3), 258-267.
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### Article

An Assessment of the Changing Climate in Northern Nigeria Using Cokriging

1Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria

2Department of Environmental Sciences, Federal University Dutse, Jigawa State, Nigeria

American Journal of Applied Mathematics and Statistics. 2013, 1(5), 90-98
DOI: 10.12691/ajams-1-5-3

Cite this paper:
U. Usman, S. A. Yelwa, S.U. Gulumbe, A. Danbaba. An Assessment of the Changing Climate in Northern Nigeria Using Cokriging. American Journal of Applied Mathematics and Statistics. 2013; 1(5):90-98. doi: 10.12691/ajams-1-5-3.

Correspondence to: S. A. Yelwa, Department of Environmental Sciences, Federal University Dutse, Jigawa State, Nigeria. Email: bubakar9@yahoo.co.uk

### Abstract

The aim of this paper is to test the applicability of Co-Kriging (CK) on the study of the changing climate in Northern Nigeria. Indices were derived from climatic variables (Rainfall and Temperature) obtained from Nigerian Meteorological Agency (NIMET) and remotely sensed data covering the period from 1981 to 2010 in the form of Normalised Difference Vegetation Index (NDVI) data derived from National Oceanic Atmospheric Administration-Advanced Very High Resolution Radiometer (NOAA-AVHRR). Because of the strong relationship between NDVI and Rainfall, CK method of data interpolation was tested with R-Statistical software. A digital elevation model (DEM) of the study area at 90-meter spatial resolution was used as a supplement in an overlay procedure using the IDRISI Remote sensing and GIS software so as to derive the correct altitude values of the Met stations for comparison with the coefficient of variation of the rainfall dataset. Results from the derived CK prediction maps showed that there are high variability in NDVI and rainfall across the time-series. Furthermore, spatial average variability in the growing season rainfall was 60% with a mean temperature of 4% although coefficient of variation in rainfall for the individual climatic station's ranged from 18.15 to 60.98 per cent. While the highest coefficient of variation in temperature for the entire time series (1981-2010) was located around Katsina area, the lowest was located around Minna. From the results of this analysis it is evident that the higher prediction variance values particularly for vegetation NDVI and rainfall are located in the southern part of the study area particularly around Kaduna, Minna, and Jos as compared to the northern part of the study area falling around Maiduguri, Sokoto and Katsina which indicated relatively lower prediction values. However, further studies should also be undertaken using the raster NDVI dataset in a GIS environment to buttress our view that there were changes in the general ecosystems within the study area as result of climatic impact.

### References

 [1] Adejuwon, J., Azar, C., Baethgen, W., Hope, C,. Moss, R., Leary, N., Richels, R., van Ypersele, J.-P., Kuntz-Duriseti, K. and, Jones, R.N. "Overview of Impacts, Adaptation, and Vulnerability to Climate Change". (Editors: McCarthy, J. J., Canziani, O. F., Leary, N. A., Dokken, D. J. and White, K. S.) Contribution of Working Group II to the Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC), Cambridge University Press, 75-103, 2001. http://grida.no/climate/ipcc_tar/wg2/pdf/wg2TARchap1.pdf. [2] Swart, R. L.; Bernstein, M.; Ha-Duong, and Petersen, A. "Agreeing to disagree: Uncertainty management in assessing climate change, impacts and responses by the IPCC. In. Climate Change. 92, 1-29, 2009. [3] Burton, I., Challenger, B., Huq, S., Klein R. J. T., Yohe, G., Adger, N., Downing, T., Harvey, E., Kane, S., Parry, M., Skinner, M., Smith, J., and Wandel, J. "Adaptation to Climate Change in the context of Sustainable Development and Equity". Intergovernmental Panel on Climate Change (IPCC). Cambridge University Press, 778-912, 2001. http://grida.no/climate/ipcc_tar/wg2/pdf/wg2TARchap18.pdf [4] The United Nations Convention to Combat Desertification (UNCCD) 2004: "A carrying pillar in the global combat against land degradation and food insecurity". Background paper for the San Rossore meeting ‘Climate change: a new global vision. Pisa, Italy, 15-16 July, 2004. [5] Hulme, M. and Kelly, M. "Exploring the links between desertification and climate change". Environment, 35(6), 4-45, 1993.
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"Interlinkages of NOAA/AVHRR derived integrated NDVI to seasonal precipitation and transpiration in drylands tropics". International Journal of Remote Sensing, 18, 2931-2952, 1997. [12] Stone, T. A.; Schlesinger, P.; Houton, R. A. and Woodwell, G. M. "A Map of the Vegetation of South America based on Satellite Imagery". Photogrammetric Engineering and Remote Sensing, 60, 541-551, 1994. [13] Tucker, C. J.; Holben, B. N. and Goff, T. E. "Intensive Forest Clearing in Randonia, Brazil as detected by satellite remote sensing". Remote Sensing of Environment, 15, 255-261, 1984. [14] Tucker, C. J.; Newcomb, W. W.; Los, S. O. and Prince, S. D. "Mean and Inter-Year Variation of growing-season Normalised Difference Vegetation Index for the Sahel 1981-1989". International Journal of Remote Sensing, 12, 1133-1155, 1991b. [15] Turner, M. G. "Spatial and Temporal analysis of landscape patterns". Landscape Ecology, 4, 21-30, 1992. [16] Turner II, B. L. 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D., Delrieu, G. and Lebel, T. "Rain measurement by raingauge-radar combination: a geostatistical approach". Journal of Atmospheric Oceanic Tech., 5 (1), 102-115, 1988. [37] Azimi-Zonooz, A., Krajewski, W. F., Bowles, D. S. and Seo, D. J. "Spatial rainfall estimation by linear and non-linear cokriging of radar-rainfall and raingauge data. Stochastic Hydrol. Hydraul, 3, 51-67, 1989. [38] Raspa, G., Tucci, M. and Bruno, R. "Reconstruction of rainfall fields by combining ground raingauges data with radar maps using external drift method". (Editors: Baafi,E. Y., Schofield, N. A.). Geostatistics Wollongong ’96, Kluwer Academic, Dordrecht, 941-950, 1997. [39] Hevesi, J. A., Flint, A. L., Istok, J. D., 1992a. Precipitation estimation in mountainous terrain using multivariate geostatistics. Part I: structural analysis. Journal of Applied Meteorology. 31, 661-676, 1992a. [40] Hevesi, J. A., Flint, A. L. and Istok, J. D. 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### Article

Cost Effectiveness Statistic: A Proposal to Take Into Account the Patient Stratification Factors

1ICT Department, Italian Senate, LUMSA University, Rome, Italy

American Journal of Applied Mathematics and Statistics. 2013, 1(5), 87-89
DOI: 10.12691/ajams-1-5-2

Cite this paper:
Ciro D'Urso. Cost Effectiveness Statistic: A Proposal to Take Into Account the Patient Stratification Factors. American Journal of Applied Mathematics and Statistics. 2013; 1(5):87-89. doi: 10.12691/ajams-1-5-2.

Correspondence to: Ciro D'Urso, ICT Department, Italian Senate, LUMSA University, Rome, Italy. Email: cirodurso@gmail.com

### Abstract

The formula here proposed can be used to conduct economic analysis in randomized clinical trials. It is based on a statistical approach and aims at calculating a revised version of the incremental cost-effective ratio (ICER) in order to take into account the key factors that can influence the choice of therapy causing confounding by indication. Let us take as an example a new therapy to treat cancer being compared to an existing therapy with effectiveness taken as time to death. A challenging problem is that the ICER is defined in terms of means over the entire treatment groups. It makes no provision for stratification by groups of patients with differing risk of death. For example, for a fair and unbiased analysis, one would desire to compare time to death in groups with similar life expectancy which would be impacted by factors such as age, gender, disease severity, etc. The method we decided to apply is borrowed by cluster analysis and aims at (i) discard any outliers in the set under analysis that may arise, (ii) identify groups (i.e. clusters) of patients with "similar" key factors.

### References

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### Article

Wavelet-Galerkin Method and Some Numerical Method for Lane-Emden Type Differential Equation

1Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Guilan, Rasht, Iran

American Journal of Applied Mathematics and Statistics. 2013, 1(5), 83-86
DOI: 10.12691/ajams-1-5-1

Cite this paper:
Jafar Biazar, Fereshteh Goldoust. Wavelet-Galerkin Method and Some Numerical Method for Lane-Emden Type Differential Equation. American Journal of Applied Mathematics and Statistics. 2013; 1(5):83-86. doi: 10.12691/ajams-1-5-1.

Correspondence to: Jafar Biazar, Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Guilan, Rasht, Iran. Email: biazar@guilan.ac.ir

### Abstract

In this paper, we will compare the performance of Adomian decomposition method and the wavelet-Galerkin method applied to the Lane-Emden type differential equation. The Galerkin Wavelet method (GWM), which is known as a numerical approach is used for the Lane- Emden equation, as an initial value problem. This approach consists of using integral operator, to convert the Lane- Emden equation in to an integral equation, then applying Galerkin Wavelet method to solve the resulted integral equation. The properties of Galerkin Wavelet method (GWM) and the Adomian Decomposition Method are also addressed. Although the Adomian decomposition solution required slightly more computational effort than the wavelet-Galerkin solution, it resulted in more accurate results than the wavelet-Galerkin method. To illustrate the methods two examples are provided and the results are in good agreement with exact solution.

### References

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