1. ### To Theory One Class of Three-Dimensional Integral Equation with Super-Singular Kernels by Tube Domain

Nusrat Rajabov

Tajik National University, Research Institute, Dushanbe, Tajikistan.

Abstract: In this work, we investigate one class of three-dimensional integral equation by tube domains, are in power basis and lateral surface and way have super-singularity. In depend of the roots of the characteristic equations (2), (3) integral representation manifold solution is obtained in an explicit form. In the case, when parameters present in kernels, such that general solution integral equation contain arbitrary functions, invers formula is found. On basis obtained integral representation and its invers formula , in the case when general solution integral equation contain arbitrary functions, determined correct stand Dirichlet boundary valued problem and found its solution.

Keywords: Integral representation, super-singular kernels, invers formula, three-dimensional integral equations, Dirichlet type boundary value problem.

2. ### Integration of Geogebra in Teaching and Learning Geometric Transformation

Niroj Dahal, Dinesh Shrestha and Binod Prasad Pant

Kathmandu University School of Education, Department of STEAM Education, Hattiban, Lalitpur, Nepal.

Abstract: This paper discusses the use of GeoGebra in teaching geometric transformations. As GeoGebra is an interactive geometry, algebra, statistics and calculus application of mathematical software is very essential from school to university level to foster mathematical experiments and discoveries. Also, the contribution of this paper was a reflection of several specific examples of transformation namely reflection, rotation, translation and dilation for teaching mathematics to sixteen students in one of the secondary schools of Kathmandu Valley, Nepal. Subscribing teaching experiment as qualitative research methodology, this paper deals with the implementation of GeoGebra in six episodes, we have used adequate illustrations, pictures and animations of objects using GeoGebra to make abstract concepts of transformation visible to the students. The result of experiment shows that GeoGebra is helpful in teaching and learning the abstract concepts of transformation. Findings of this study show that if GeoGebra is used in mathematics classroom, students could become an active constructor of knowledge. Similarly, they collaborate with each other, visualize the process of transformation, and enjoy their authority in such classes. It acts as an important educational tool so as to support the traditional lecture-method of teaching mathematics that shifts education system from teacher centered to learner centered.

Keywords: Constructor of knowledge, GeoGebra, teaching of mathematics, transformation, mathematical experiment, teaching experiment.

3. ### Correlation induced by missing spatial covariates: a connection between variance components models and kriging

Jessica Rothman1, Monica C. Jackson2, Kimberly F. Sellers3, Talithia Williams4, Subhash R. Lele5, and Lance A. Waller6

1. Department of Biostatistics, Yale University, New Haven, CT 06510.
2. Department of Mathematics and Statistics, American University, Washington, DC 20016. Corresponding author: monica@american.edu.
3. Department of Mathematics and Statistics, Georgetown University, Washington, DC 20057.
4. Department of Mathematics, Harvey Mudd College, Claremont, CA 91711.
5. Department of Mathematical and Statistical Sciences, University of Alberta, University of Alberta Edmonton, AB, Canada.
6. Department of Biostatistics and Bioinformatics, Emory University, Atlanta, GA 30322.

Abstract: Residual spatial correlation in linear models of environmental data is often attributed to spatial patterns in related covariates omitted from the fitted model. We connect the nonunique decomposition of error in geostatistical models into trend and covariance components to the similarly non-unique decomposition of mixed models into fixed and random effects. We specify spatial correlation induced by missing spatial covariates as a function of the strength of association and (spatial) covariation of the missing covariates. The connection with variance components models provides insight into estimation procedures. We showed how missing covariates in spatial linear models actually induces spatial autocorrelation in the covariates. This finding was confirmed through the use of simulated data and the Binary Steve dataset.

Keywords: geostatistics, spatial regression, variable selection, semi-variogram, spatial auto-correlation.

4. ### A Dynamic Factor Approach for Estimating an Aggregate Multipollutant Air Quality Indicator

Giuliana Passamani1, Paola Masotti2

1. Department of Economics and Management, University of Trento, Trento, Italy. E-mail address: giuliana.passamani@unitn.it.
2. Department of Economics and Management, University of Trento, Trento, Italy. E-mail address: paola.masotti@unitn.it.

Abstract: Measurement of air pollution has become an important issue since it has been established that air quality is closely connected to human health and environment. International organizations as well as local authorities are particularly concerned with air pollution, but, in spite of the huge amount of data on various pollutants recorded frequently at the monitoring sites located in many countries all over the world, the problem of properly synthetizing the available information is still a matter of discussion in the specialized literature. In this paper we present an explicit dynamic time series factor model that implicitly determines a variable which can be thought of as measuring the state of local air pollution. With the suggested operative approach, we aim to contribute to measuring air quality, by proposing a methodological procedure leading to the estimation of a single site indicator determined jointly by present and past pollution as well as by the meteorological conditions. These single indicators are then spatially aggregated using principal component analysis. The advantage of using this dynamic factor model for the empirical analysis is that, besides measuring air pollution, we can use the estimated model for forecasting future air pollution, given the meteorological predictions. The application of the model in the present paper considers a pollution data set collected at different monitoring sites in the alpine province of Trento.

Keywords: Air pollution measurement, Aggregate air quality indicator, Dynamic factor model.