Skip to main content
Top

2024 | Book

Mathematical Methods for Engineering Applications

ICMASE 2023, Madrid, Spain, July 12–14

Editors: Víctor Gayoso Martínez, Fatih Yilmaz, Araceli Queiruga-Dios, Deolinda M.L.D. Rasteiro, Jesús Martín-Vaquero, Ion Mierluş-Mazilu

Publisher: Springer Nature Switzerland

Book Series : Springer Proceedings in Mathematics & Statistics

insite
SEARCH

About this book

These proceedings gather selected, peer-reviewed papers presented at the IV International Conference on Mathematics and its Applications in Science and Engineering – ICMASE 2023, held on July 12–14, 2023 by the University Center of Technology and Digital Arts (U-tad) in Madrid, Spain.

Papers in this volume cover new developments in applications of mathematics in science and engineering, with an emphasis on mathematical and computational modeling of real-world problems. Topics range from the use of differential equations to model mechanical structures to the employ of number theory in the development of information security and cryptography. Educational issues specific to the acquisition of mathematical competencies by engineering and science students at all university levels are also touched on.

Researchers, practitioners, and university students can significantly benefit from this volume, especially those seeking advanced methods for applying mathematics to various contexts and fields.

Table of Contents

Frontmatter
Modeling of Nitrogen, Phosphorus, and Potassium Concentrations in Lakes Affected by Soil Fertilization
Abstract
The excessive use of fertilizer on agricultural land has led to a significant build-up of chemical nutrient stocks in the soil, which increases the risk of diffuse pollution of surface waters. The reservoirs to which these waters drain are also polluted by the chemical components of the fertilizer. This is a major environmental problem. Indeed, the fishery products contained in these water reservoirs are directly contaminated, and this remains a major public health problem. Thus, using ordinary partial derivatives, we established budget equations for the mass concentrations of the major components of fertilizer chemicals such as nitrogen (N), phosphorus (P) and potassium (K) found in each natural water reservoir. Using the Euler discretization method, we linearized these budget equations in order to formulate them numerically with AMPL and solve them. The numerical solutions obtained thus make it possible to predict the mass concentrations of the chemical elements (N, P, K) at the level of each water reservoir in time by taking the measures for a substantial reduction of the chemical content of fresh water.
Déthié Dione, Teubé Cyrille Mbainaissem, Bakary Koné, Paul Python Ndekou
p-Numerical Semigroups of the Triples of the Sequence
Abstract
For a non-negative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of numerical semigroups of \((\nu _n,\nu _{n+1},\nu _{n+2})\), where \(\nu _n=(a^n-b^n)/(a-b)\) with \(\gcd (a,b)=1\) and \(a>b>1\). Here, the p-numerical semigroup \(S_p\) is the set of integers whose non-negative integral linear combinations of given positive integers are expressed more than p ways. When \(p=0\), \(S_0\) with the 0-Frobenius number and the 0-genus is the original numerical semigroup \(S_0\) with the Frobenius number and the genus. Symmetric properties of numerical semigroups are important to characterize the numerical semigroup. In recent works, some closed formulas of p-Frobenius numbers have been successfully given, but no symmetric property has been found. We give a symmetric property of this numerical semigroup.
Takao Komatsu, Ruze Yin
Some Identities for Balancing and Lucas-Balancing Numbers in Bidimensional Version
Abstract
Several number sequences have been studied for many researchers, most of times in their unidimensional version. Our main goal is to study the bidimensional version of two sequences of known numbers. At the expense of studying the unidimensional version of the numerical balancing sequence and the numerical Lucas-balancing sequence, in particular on the basis of some of their properties, some identities related to bidimensional extensions of these two number sequences involving sums of terms are introduced: with odd index, with even index and with generic index.
José Chimpanzo, Paula Catarino, M. Victoria Otero-Espinar
Sequences of Uncountable Iterated Function Systems: The Convergence of the Sequences of Fractals and Fractal Measures Associated
Abstract
In this paper, we consider a sequence of uncountable iterated function system (U.I.F.S.). Each term of this sequence is built using an uncountable family of contractions and a linear and continuous operator. For each U.I.F.S. of the sequence we have an associated attractor, a Markov-type operator and a fractal measure.
We study the convergence of the corresponding attractors and fractal measures sequences.
Ion Mierlus-Mazilu, Lucian Nită
Method of Hydrodynamic Images and Quantum Calculus in Fock-Bargmann Representation of Quantum States
Abstract
We propose a new approach to quantum states in Fock space in terms of classical hydrodynamics. By conformal mapping of complex analytic function, representing the wave function of quantum states in Fock-Bargmann representation, we define the complex potential, describing these quantum states by incompressible and irrotational classical hydrodynamic flow. In our approach, zeros of the wave function appear as a set of point vortices (sources) in plane with the same strength, allowing interpretation of them as images in a bounded domain. For the cat states we find fluid representation as descriptive of a point source in the oblique strip domain, with infinite number of periodically distributed images. For the annular domain, the infinite set of images is described by Jackson q-exponential functions. We show that these functions represent the wave functions of quantum coherent states of the q-deformed quantum oscillator in q-Fock-Bargmann representation and describe the infinite set of point vortices, distributed in geometric progression.
Oktay K. Pashaev
On Leonardo Numbers and Fibonacci Fundamental System
Abstract
Our goal is to explore the generalized Leonardo numbers, through the properties of the Fibonacci fundamental systems related to the elements of this sequence. We explicitly describe the closed connection between the sequences of the Fibonacci fundamental system and the generalized Leonardo numbers. Moreover, the matrix approach is considered for studying the combinatorial identities and the generalized Cassini identity for the generalized Leonardo numbers. In addition, the analytical aspect about each sequence of the generalized Leonardo numbers is elaborated. Finally, the combinatorial and the analytical formula of the generalized Cassini identity are investigated.
Elen Viviani Pereira Spreafico, Paula Maria Machado Cruz Catarino
A Quadratic Estimation Approach from Fading Measurements Subject to Deception Attacks
Abstract
In this paper, using covariance information, the least-squares quadratic filtering and fixed-point smoothing problems are addressed under the assumption that the measurements are perturbed by both a multiplicative noise and a time-correlated additive noise. Additionally, they are affected by the fading phenomena and exposed to random deception attacks. In the least-squares quadratic estimation approach, the signal and observation vectors are augmented by combining the original vectors with their second-order Kronecker powers. Then, using the Kronecker algebra rules and under an innovation approach, the linear estimators of the original signal based on the augmented observations are obtained. These linear estimators provide the required quadratic estimators. A simulation example shows the feasibility of the proposed quadratic estimation algorithms; also, the superiority of the quadratic estimators over the conventional linear ones is illustrated and the influence of the deception attack success probabilities on the estimation accuracy is analyzed.
Raquel Caballero-Águila, Josefa Linares-Pérez
SOLO Taxonomy in the Evaluation of Engineering Students: A Case Study in Mathematics
Abstract
This paper presents the results of a study whose main objective was to analyze the assessments by exam in higher education according to the SOLO Taxonomy (Structure of the Observed Learning Outcome) over the last five years, in particular the cognitive complexity required in each question and the “quality” of the exam. The data were collected in the Mathematical Analysis exams of the Degree in Electrical and Computer Engineering. In total, 130 items presented in 10 exams (2 per year) from 2019 to 2023 were surveyed. The SOLO index of these exams was always higher than 16, with the exception of one exam from 2023 with an index of 14.55. The analysis reveals that the exams are composed of items classified at the highest levels of cognitive complexity, according to the SOLO Taxonomy, and associated with the student’s deep learning. Thus the authors were able to study their pedagogical practices in relation to the assessment and, consequently, to consciously prepare the assessment.
Cristina M. R. Caridade, Verónica Pereira
Is Collaborative Learning a Voluntary Process?
Abstract
Mathematics learning process in Engineering courses is heavily studied, and the integration of Information and communication technologies (ICT) has grown significantly in recent years. Due to the pandemic, teaching processes were adjusted, and assessments were conducted considering the prevailing context we all encountered. Thus, students need innovative and stimulating teaching and learning practices that quickly motivate and involve them in the teaching/learning processes. In the last two years, ICT tools and digital platforms were extensively used, prompting questions about their optimal and effective utilization. Simultaneously, face-to-face group work and involvement with colleagues’ needs lost some space for achievement and effectiveness. At the study’s outset, the concern was about a potential preference for individual work and reduced solidarity among colleagues, except for possible friendships from previous school groups. Collaborative learning (CL) has been proposed as one of the solutions, with the Padlet platform (Padlet, 2023. Retrieved from http://​www.​padlet.​com) being a popular tool for facilitating such interactions. In this paper, the authors propose two collaborative learning platforms using Padlet, one for Statistical Methods in Informatics Engineering and other for Mathematical Analysis I, with the aim of creating a space for virtual interaction among students. The study enrols 533 students from Informatics Engineering, many of whom are student workers, and from Mechanical Engineering 20 students, who were divided into small groups to foster teamwork and cooperation. The paper discusses the results such as students’ participation, collaboration with colleagues, and course’ performance, highlighting when it is, or it is not fruitful to use this type of activities.
Deolinda M. L. D. Rasteiro, Cristina M. R. Caridade
Elliptic Biquaternionic Sequence with Vietoris’ Numbers as Its Components
Abstract
In this study, we introduce an elliptic biquaternionic sequence with Vietoris’ numbers as its components and discuss some of its properties. Also, the generating function and some identities in terms of elliptic biquaternionic sequence with Vietoris’ numbers are given. Furthermore, the construction of this elliptic biquaternion sequence is presented using matrices that generate the quaternionic sequence where the components are Vietoris’ number, and also by applying the determinant to a special kind of matrices.
Regina de Almeida, Paula Catarino
Teaching Mathematics in STEM Education
Abstract
Teaching mathematics in STEM education is a vital component of developing students’ analytical thinking, problem-solving skills, and logical reasoning abilities. To effectively teach mathematics in STEM, educators should employ strategies such as active learning, integration of technology, problem-based learning, differentiated instruction, cross-disciplinary connections, visualization, real-world applications, collaborative learning, formative assessment, and cultivating a growth mindset.
The integration of technology plays a significant role in enhancing mathematics instruction. Educators can utilize interactive software, graphing calculators, spreadsheets, online resources, virtual manipulatives, coding and programming, online collaborative platforms, data visualization tools, virtual reality, augmented reality, online tutorials, and videos to engage students and deepen their understanding of mathematical concepts.
By incorporating technology, teachers can provide dynamic visualizations, simulations, and interactive activities that enable students to explore multiple representations of mathematical ideas. Technology tools also support data analysis, graphing, problem-solving, and collaborative learning experiences. Furthermore, technology facilitates real-world connections and applications of mathematics, helping students recognize its relevance in various STEM fields.
Overall, the integration of technology in mathematics instruction empowers students to actively participate in their learning, visualize abstract concepts, solve complex problems, and develop essential mathematical and STEM skills. By incorporating these strategies, educators can create a dynamic and engaging mathematics learning environment that prepares students for future STEM pursuits.
Ion Mierluş-Mazilu, Fatih Yilmaz
Application of Discrete Wavelet Transform and Tree-Based Ensemble Machine Learning for Modeling of Particulate Matter Concentrations
Abstract
The study of air pollution is an extremely important and urgent problem to be solved on a global and local scale. In this field, huge arrays of measurement data are accumulating, for the analysis of which various approaches based on mathematical, statistical, and machine learning (ML) methods are developed. In this paper, we investigate the application of different discrete wavelet transforms (DWT) families, coupled with state-of-the-art ML algorithms to predict concentrations of particulate matter PM10. Average daily data for this pollutant and several meteorological time series for a period of 630 days were used. A hybrid type models with wavelet decomposition of the initial time series and the application of predictive ensembles (Arcing, Arc-x4) were obtained. All models are cross-validated. The models are applied for short-term pollution forecasts.
Maya Stoimenova-Minova, Snezhana Gocheva-Ilieva, Atanas Ivanov
Fixed Point Theorems in Orthogonal F-Metric Spaces
Abstract
F-metric spaces are defined as a generalization of metric spaces with properties of nonnegativity, self-distance, symmetry and generalized triangular inequality. Moreover, notion of orthogonal F-metric spaces are introduced given by establishing a binary relation on the set. In this work, we give some properties of orthogonal relation on a set and some topological properties of orthogonal F-metric spaces. We introduce some rational type contractions in orthogonal F-metric spaces and prove fixed point theorems for this type contraction. Our results generalize the Jaggi and Ciric type contractions.
Vildan Ozturk
New Trends on Malware Propagation: From IoT Environments to Drone Swarms
Abstract
In this work a brief review of theoretical techniques to design mathematical models for malware propagation is introduced. It is shown that the classical methodology, that considers the basic assumptions of the Mathematical Epidemiology and which is wide employed for designing such mathematical models, is not suitable for the special case of malware spreading on IoT networks. Specifically, some recommendations for modeling are introduced and the individual-based paradigm is proposed as the fundamental framework.
A. Martín del Rey
Forms of Assessment in view of Development of Mathematical Competencies
Abstract
During the past twenty years, the need to define outcomes of education as Competences has been recognized in many countries. The European Parliament and the Council of the European Union acknowledged the mathematical competence in 2006 to be a key competence for everyday life (Recommendation of the European Parliament and of the Council of 18 December 2006 on key competences for lifelong learning (2006/962/EC). https://​eur-lex.​europa.​eu/​legal-content/​EN/​TXT/​PDF/​?​uri=​CELEX:​32006H0962&​from=​EN. Last accessed 10 July 2023) and it was incorporated into the national educational programs. The engineering tertiary educational bodies grouped around SEFI, Mathematics Special Interest Group adopted the concept of eight overlaying mathematical competencies introduced by Danish KOM project (Niss M, Mathematical competencies and the learning of mathematics: the Danish KOM project. In: A. Gagatsis, S. Papastravidis (eds.) 3rd Mediterranean conference on mathematics education. Hellenic Mathematical Society and Cyprus Mathematical Society, Athens, Greece, pp 115–124) in 2003 which were linked to the carefully specified core content, and formulated as the core content-related competencies defining learning outcomes in the third edition of “A Framework for Mathematics Curricula in Engineering Education” (Alpers, B. et al., A Framework for Mathematics Curricula in Engineering Education. SEFI, Brussels, (2013)).
The paper deals with appropriate forms of summative assessment, applicable in the higher engineering education. Special attention is paid to mutual communication in a class, a phenomenon that was negatively affected to a large extent by pandemic restrictions. We introduce discussion as a part of summative assessment, not only providing the space for evaluating the mathematical competencies, but more, as an aspect that essentially contributes to their cultivation. The results of a study in which 230 students participated is presented.
The strategy of deploying more assessment forms is discussed, taking into account building of comprehensive Mathematical Competency in gradual continuous and permanent manner.
Daniela Richtarikova
Deep-Control of Memory via Stochastic Optimal Control and Deep Learning
Abstract
In this survey work, we introduce Stochastic Differential Delay Equations and their impacts on Stochastic Optimal Control problems. We observe time delay in the dynamics of a state process that may correspond to inertia or memory in a financial system. For such systems, we demonstrate two special approaches to handle delayed control problems by applying the Dynamic Programming Principle. Moreover, we clarify the technical challenges rising as a consequence of the conflict between the path-dependent, infinite-dimensional nature of the problem and the necessity of the Markov property. Furthermore, we present two different Deep Learning algorithms to solve targeted delayed control tasks and illustrate the results for a complete memory portfolio optimization problem.
Emel Savku
Exponentiated Weibull Mixture Cure Model to Handle Right-Censored Data Set
Abstract
Survival analysis is an important statistical tool for analyzing time-to-event data, such as the time to failure of a product or the time to death in medical research. Often, these data sets are subject to right-censoring, where some individuals are still alive or have not experienced the event of interest at the end of the study period. However, the cure rate models are frequently used to model this type of data. The models provide estimates of the fraction of patients cured of disease as well as the distribution of survival times for uncured patients. In this paper, we propose an exponentiated Weibull mixture cure model, to handle right-censored data sets in survival analysis. The model assumes that the population is a mixture of two sub-populations: a cured population that will never experience the event of interest, and a susceptible population that will experience the event of interest at some point in time. The proposed model applied to the real-life data set and compared its performance with other commonly used survival models. The results show that the exponentiated Weibull mixture cure model provides a better fit to the data and more accurate predictions of survival than other models.
Mohamed A. S. Ishag, Anthony Wanjoya, Aggrey Adem, Ahmed Z. Afify
On Strong Fuzzy Partial Metric Spaces
Abstract
In 2019, Gregori et al. (Int. J. Gen. Syst. 48(3):260–279, 2019), initiated the structure of fuzzy partial metric space (FPMS) based on the residuum operators. Since it is not easy to show different examples when one works with residuum operators, in recent papers, we face to lack of examples of FPMs. Motivated by this fact, in this work, we aim to generate different FPMs from partial metrics (PMs) and vice versa. For this aim, we first give the concept of strong FPMSs by investigating some properties of them and providing some examples by considering different t-norms. In continuation, we give some techniques to construct strong FPMs from existing PMs by means of the pseudo-inverse (p-inverse) of the additive generator (a-generator) of continuous Archimedean t-norms. We also show that FPMs can induce PMs via a-generators. Moreover, we study whether the topologies induced by the presented techniques coincide or not.
Elif Güner, Halis Aygün
An Application of Linear Diophantine Fuzzy Sets to the Edge Detection Techniques
Abstract
The utilization of fuzzy set theory within the domain of image processing provides a lot of advantages such as encompassing the management of uncertainty, adaptability to variations, noise tolerance and adaptive classification compared to the other techniques. These advantages contribute to heightened precision and adaptability in the realm of image processing, enabling more precise and versatile handling of visual data. The process of edge detection performs a pivotal role in the segmentation of foreground objects from the image background. So, it facilitates subsequent analysis and comprehension of the image’s underlying structural properties through complex computational procedures. This complex process can be handled with the notion of fuzzy sets and their generalizations. The concept of linear Diophantine fuzzy sets is a generalization of fuzzy sets where the use of reference parameters corresponds to membership and non-membership grades. The aim of this study is to give an application of linear Diophantine fuzzy sets to edge detection of images. For this aim, we conduct a comprehensive evaluation to ascertain the similarity values using the linear Diophantine fuzzy similarity measure by leveraging the normalized membership values of the gray level associated with fundamental edge detection techniques.
Başak Aldemir, Elif Güner, Halis Aygün
Delamination Resistance of Laminated Glass Plates Having Ethyl Vinyl Acetate, Polyvinyl Butyral and Sentryglas Plus Interlayers
Abstract
Laminated glass is produced by laminating glass layers with an interlayer at specified pressure and temperatures using an autoclave. The characteristic properties of laminated glass can be affected considerably by the used interlayers. Polyvinyl Butyral, Ethyl Vinyl Acetate and SentryGlass are three common interlayers used by glass manufacturers. In order to select the appropriate interlayer parameters such as blast performance, availability, cost, durability, optical property and manufacturing equipment should be considered. Usage of laminated glass is expanding due to its safety and security property as well as comfort and design. As a consequence of possess superiority of laminated glass for structural applications in various industries, they strengthen their position between building materials. Nevertheless delamination which may be result of manufacturing process and service is regarded the primary concern and most undesirable failure mode for the analysis of laminated glass unit. For that cause it is hot research area of composite industry. Delamination strength of laminated glass with the mentioned three types of interlayers are presented in the current research. A mathematical model is presented to analyze delamination behavior of laminated glass plate. Five nonlinear equations are written in matrix form. Solution of matrix system is obtained by using special matrix solvers and successive over relaxation method is used to overcome the convergence problems. Deflection and stress values which represent mechanical behavior of unit are presented in figures. In order to carry out validation assumptions of model the finite element model was constituted. A good fit between the results is observed.
Ebru Dural
Algebraic and Quantum Mechanical Approach to Spinors
Abstract
Clifford algebras and quantum mechanics are closely associated to each other. We are familiar with some popular terms, such as algebra, light polarization, and quantum spin. Dirac spinors, Majorana spinors, and Weyl spinors are discussed in particular as subspaces of Clifford algebras with some remarkable algebraic features. Furthermore, we are interested in demonstrating how the quantum spin state and classical polarization of light waves can be derived from one another along with the Bloch sphere and Poincare sphere representations.
Tahir Manzoor, S. N. Hasan
New G-Closed Sets with Related to an Ideal
Abstract
Ideal is a family which has two properties as known heredity and finite additive. Besides it is a partially ordered collection of sets. This using this concept with a topology was obtained a new topology finer than old. Elements of topology are open sets and their complements are closed sets. It is known that generalized closed set is weaker than closed. Then, several authors studied on this subject both topological spaces and ideal topological spaces. In this paper, we recall some concepts and their basic properties. Then, we introduce new g-closed set called \(*Ig\)-closed set by using ideal and obtained some properties of it. We have obtain a diagram showing the relationships between them and other types of g-closed sets in the literature. Besides, we define a concept of \(*Ig\)-open set and investigate it. We give that a family of consisting of this sets in any I-space forms a minimal structure. Finally, we mention when \(*Ig\)-closed is conserved.
Aynur Keskin Kaymakci
Service-Learning Activity in a Statistics Course
Abstract
Learning through service, a combination of community-based activities with learning objectives, or experiential education, are some features of service-learning (SL) methodology, which improves the realities where the service is performed, and which considers who receives the service as a central element. A SL activity was proposed to undergraduate sophomore students from Industrial Engineering degrees. Students developed a teamwork as part of their statistics course collaborating with a local entity. To make this possible, teachers identified a community need in line with the course contents and after that students were invited to participate. A teamwork was proposed to all students and one group decided to collaborate with an association that helps families in need. The work consisted of creating a database with the data gathered by volunteers and after that present a basic descriptive analysis. The built database and the analysis performed were grounded on the association needs to better help the families that it serves. This SL activity was assessed as part of the course activities.
Araceli Queiruga-Dios, Deolinda M. L. D. Rasteiro, Beatriz Sánchez Barbero, Ángel Martín-del Rey, Ion Mierlus-Mazilu
Fermatean Fuzzy Type a Three-Way Correlation Coefficients
Abstract
In challenging decision-making circumstances, tools like aggregation operators and information measures are routinely used. Using correlation coefficient (KK)’s estimate of the amount of reliance between two sets is another way to decide which choice is better. There are a few methods for figuring out the KK of Fermatean fuzzy sets (FFSs). These previous methodologies, which are specified between \([0, 1]\), simply evaluate the strength of the link between FFSs. This study, using the notions of variance and covariance, respectively, provides a three-way method for computing the KKs between FFSs. The strength of the association between the considered FFSs is demonstrated by this novel approach, which is defined inside the interval \([-1, 1]\) similar to standard statistics, and it identifies whether the FFSs are positively or negatively associated. The potential of inaccuracy owing to information leakage is reasonably mitigated by the suggested technique’s inclusion of the three traditional FFSs parameters. To demonstrate the new technique’s usefulness as an accurate information measure, several theoretical results will be used to validate it. The advantages of the new methods over comparable methods will be demonstrated using a few numerical examples. With the help of the novel approach, several decision-making issues involving pattern recognition and diagnostic medicine will be handled. Multi-attribute decision-making issues can be resolved using the three-way technique for determining the correlation coefficient between FFSs.
Murat Kirişci
On Some Gaussian Oresme Numbers
Abstract
Sequences with recurrence relations are used in many branches of science such as mathematics, physics and engineering. The most well-known and studied sequence among these is Horadam sequence. This sequence is a generalization of many sequences and has an important place in the literature. New sequences can also be obtained from this sequence by changing the initial conditions. The most common sequences obtained by this method in the literature are Fibonacci sequence, Lucas sequence and Jacobstal sequence. The Fibonacci sequence, which has an important place among these sequences, has been studied by many authors. Horadam, who also worked on Fibonacci sequences, defined these sequences in complex space and discussed Gaussian Fibonacci sequences. A new sequence, which was defined by Nicole Oresme in the fourteenth century and called the Oresme sequence, is a special case of the Horadam sequence whose initial conditions are rational numbers. Subsequently, this was reviewed by many authors. Based on these studies, Gaussian Oresme number sequences were studied.
In this study, we obtained some identities of Gaussian Oresme sequences. We have obtained some sum formulas of Gaussian Oresme numbers. We obtained the matrix for the Gaussian Oresme sequences and calculated the nth power.
Serpil Halici, Elifcan Sayin
Metadata
Title
Mathematical Methods for Engineering Applications
Editors
Víctor Gayoso Martínez
Fatih Yilmaz
Araceli Queiruga-Dios
Deolinda M.L.D. Rasteiro
Jesús Martín-Vaquero
Ion Mierluş-Mazilu
Copyright Year
2024
Electronic ISBN
978-3-031-49218-1
Print ISBN
978-3-031-49217-4
DOI
https://doi.org/10.1007/978-3-031-49218-1

Premium Partners