Format: Hardback, 660 pages, height x width: 235x155 mm, 50 Illustrations, color; 5 Illustrations,
black and white; XL, 660 p. 55 illus., 50 illus. in color. 2 volume-set., 1 Hardback
Series: Emerging Topics in Statistics and Biostatistics
Pub. Date: 22-Mar-2025
ISBN-13: 9783031915833
In an era defined by the seamless integration of data and sophisticated analytical and modeling techniques, the quest for advanced statistical modeling and methodologies has never been more pertinent. Biostatistics Modeling and Public Health Applications: Study Design and Analysis Methodology in Health Sciences - Volume 1 has eleven chapters, divided in two Parts, with Part I comprising five chapters dealing with the application of Multivariate Analysis techniques and multivariate distributions to a set of different situations, and Part II consisting of six chapters which address the modeling of several interesting phenomena through the use of Heavy-Tailed, Skewed, Circular-Linear and Mixture Distributions, as well as Neural Networks. Statistical Modeling and Applications: Multivariate, Heavy-Tailed, Skewed Distributions, Mixture and Neural-Network Modeling, Volume 2, represents a concerted effort to bridge the gap between theoretical advancements and practical applications in the realm of Statistical Science, namely in the area of Statistical Modeling. It also aims to present a wide range of emerging topics in mathematical and statistical modeling written by a group of distinguished researchers from top-tier universities and research institutes to offer broader opportunities in stimulating further collaborations in the areas of mathematics and statistics.
.- Bootstrap calibrated tests for average bioequivalence and scaled average bioequivalence.
.- Hypothesis Testing within Bayesian Inference "Regression Estimation for Length-Biased Data:A Review and Comparative Study".
.- "Nonparametric Methods for Incomplete Multivariate Data: Applications to Quality of Life Outcomes".
.- Geostatistical Analysis of Under-Five Children Mortality and Associated Factors Across Sub-Saharan African Countries.
.- SEIRD Mathematical Modelling of Malaria Transmission Dynamics in Ethiopia.
.- Robust Principal Component Analysis for Retinal Image Enhancement.
.- "Estimating Average and Individual Treatment Effects in the Presence of Time-Dependent Covariates".
.- "Detection of Quadratic Interactions in Brain Functional Connectivity".
.- "Variable selection in the generalized semiparametric longitudinal model and HIV analysis".
.- Survey design effect in the prediction of events for categorical health outcomes through regression methods: Evidence from Malawi under-five
mortality survey data; 2000-2016.
.- Survey design effect in the prediction of events for categorical health outcomes through regression methods: Evidence from Malawi under-five
mortality survey data; 2000-2016.
.- Issues in Multivariate Spatial Analysis of Multiple Diseases Using Complex Health Survey Data.
Format: Hardback, 616 pages, height x width: 235x155 mm, XVII, 616 p., 1 Hardback
Series: Springer Monographs in Mathematics
Pub. Date: 27-May-2025
ISBN-13: 9783031917516
This book, now in a revised and extended third edition, provides a comprehensive and accessible introduction to modern axiomatic set theory.
After an overview of basic notions in combinatorics and first-order logic, and discussing in great detail the axioms of set theory, the author outlines in the second part the main topics of classical set theory, including Ramsey theory and the axiom of choice. As an application of the axiom of choice, a complete proof of Robinson's construction for doubling a ball by dividing it into only five parts is given. For the new edition, the chapter on permutation models has been extended, and recent results in set theory without the axiom of choice and about cardinal characteristics have been added. The third part explains the sophisticated technique of forcing from scratch, now including more details about iterated forcing. The technique is then used to show that certain statements are neither provable nor disprovable from the axioms of set theory. In particular, it is shown that both Martin's Axiom and Suslin's Hypothesis are independent of the axioms of set theory. The final part, with a new chapter on Laver forcing, is mainly concerned with consistency results obtained by iterations of forcing notions such as Cohen forcing, Sacks forcing, and Mathias forcing. The part begins with an extended chapter on countable support iterations of proper forcing notions, now also including proofs of some preservation theorems such as preservation of properness and of certain ultrafilters. In the following chapters, various consistency results concerning possible relations between cardinal characteristics and the existence of Ramsey ultrafilters are presented. For example, a detailed proof of Shelahs astonishing construction of a model with finitely many Ramsey ultrafilters is given.
Written for graduate students in axiomatic set theory, Combinatorial Set
Theory will appeal to all researchers interested in the foundations of
mathematics. With extensive reference lists, historical remarks, and related
results at the end of the chapters, this book is also suitable for self-study
Part I: Preliminary.- 1 The Setting.- 2 First-Order Logic in a
Nutshell.- 3 Axioms of Set Theory.- Part II: Topics in Combinatorial Set
Theory.- 4 Overture: Ramsey's Theorem.- 5 Cardinal Relations in ZF Only.- 6
Forms of Choice.- 7 How to Make Two Balls from One.- 8 Models of Set Theory
with Atoms.- 9 Thirteen Cardinals and Their Relations.- 10 The Shattering
Number Revisited.- 11 Happy Families and Their Relatives.- 12 Coda: A Dual
Form of Ramseys Theorem.- Part III: From Martins Axiom to Cohens Forcing.-
13 The Idea of Forcing.- 14 Martin's Axiom.- 15 The Notion of Forcing.- 16
Proving Unprovability.- 17 Models in Which AC Fails.- 18 Combining Forcing
Notions.- 19 Models in Which p=c.- 20 Suslins Problem.- Part IV:
Combinatorics of Forcing Extensions.- 21 Properties of Forcing Extensions.-
22 Cohen Forcing Revisited.- 23 Sacks Forcing.- 24 Silver-Like Forcing
Notions.- 25 Miller Forcing.- 26 Mathias Forcing.- 27 Laver Forcing.- 28 How
Many Ramsey Ultrafilters Exist?.- 29 Suite.
Format: Hardback, 567 pages, height x width: 235x155 mm, 88 Illustrations, color;
39 Illustrations, black and white; XXXVI, 567 p. 127 illus., 88 illus. in color., 1 Hardback
Series: Contributions to Statistics
Pub. Date: 24-May-2025
ISBN-13: 9783031923821
This volume gathers peer-reviewed contributions presented at the 6th International Workshop on Functional and Operatorial Statistics, IWFOS 2025, held in Novara, Italy, June 25-27, 2025.
Covering a broad spectrum of topics in functional and operatorial statistics and related fields, including high-dimensional statistics and machine learning, the contributions tackle both fundamental theoretical challenges and practical applications. A variety of features of statistics for functional data are addressed, such as estimation of functional features, exploration and pre-processing of functional data, methodologies for functional regression and forecasting problems, unsupervised and supervised classification, and testing procedures. Nonstandard functional data and situations which go beyond the pattern of samples of independent variables are investigated, and a link to the field of artificial intelligence is presented. Interesting real data applications to medicine, health, economics and the natural, environmental and social sciences are featured throughout.
Initiated at the University of Toulouse in 2008, the series of IWFOS workshops fosters discussion and international collaboration on theoretical advancements, methodological innovations, and applications in functional and operatorial statistics and related fields.
Chapter 42 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.
1 Germ?n Aneiros, Enea G. Bongiorno, Aldo Goia and Marie Hukov?, An
Introduction to the 6th Edition of the International Workshop on Functional
and Operatorial Statistics.- 2 Nihan Acar-Denizli and Pedro Delicado, Local
Constant Likelihood Estimation for Beta Distribution with Time Varying
Parameters.- 3 Mohamed Alahiane, Mustapha Rachdi, Idir Ouassou and Philippe
Vieu, An Expansion of the Functional Projection Pursuit Regression to
Generalized Partially Linear Single Index Models.- 4 Alexander Aue, Sebastian
Kuhnert and Gregory Rice, On the Estimation of Invertible Functional Time
Series.- 5 Patrick Bastian, Rupsa Basu and Holger Dette, Uniform Confidence
Bands for Joint Angles Across Different Fatigue Phases.- 6 Sayan Bhadra and
Anuj Srivastava, Scalar on Shape Regression Using Function Data.- 7 Filip
Boinec, Erik Mendro and Stanislav Nagy, A Comparison of Band-based
Approaches to Functional Depth.- 8 Enea G. Bongiorno, Lax Chan and Aldo Goia,
Analysing the Complexity Mixture Structure of Daily Probability Densities of
Bitcoin Returns.- 9 Teresa Bortolotti, Roberta Troilo, Alessandra Menafoglio
and Simone Vantini, Regularized Nonparametric Estimation of Covariance
Kernels for High-Dimensional Interferometric Data.- 10 Alain Boudou and
Sylvie Viguier-Pla, Statistical Properties of a Random Series Transmitted by
Filtering.- 11 Robert Cantwell and John Aston for the Alzheimers Disease
Neuroimaging Initiative, Multi-Object Regression: A Linear Framework via
Partial Least Squares.- 12 Christian Capezza, Davide Forcina, Antonio Lepore,
Biagio Palumbo, Monitoring the Covariance of Multichannel Profiles.- 13 Herve
Cardot and Caroline Peltier, Statistical Modeling of Categorical Trajectories
with Multivariate Functional Data Approaches.- 14 Roberto Casarin, Radu Craiu
and Qing Wang, Markov Switching Tensor Regressions.- 15 Michele Cavazzutti,
Eleonora Arnone, Ying Sun, Marc G. Genton and Laura M. Sangalli, Functional
Data Depth for the Analysis of Earth Surface Temperatures.- 16 Lax Chan,
Laurent Delsol and Aldo Goia, Improving Finite Samples Performances in
Nonparametric Functional Regression by Using Weighted Pseudo-Metrics.- 17
Aldo Clemente, Alessandro Palummo, Eleonora Arnone and Laura M. Sangalli,
Smoothing with Nonlinear Partial Differential Equation Regularization.- 18
Adela Czolkov?, Karel Hron and Sonja Greven, Functional Principal Component
Analysis for Bivariate Densities and their Orthogonal Decomposition.- 19
Marco F. De Sanctis, Ilenia Di Battista, Eleonora Arnone, Cristian
Castiglione, Mauro Bernardi, Francesca Ieva and Laura M. Sangalli, Estimating
Multiple Quantile Surfaces: A Penalized Functional Approach.- 20 Simone Di
Gregorio and Francesco Iafrate, Neural Drift Estimation for Ergodic
Diffusions: Nonparametric Analysis and Numerical Exploration.- 21 Jacopo Di
Iorio, Marzia A. Cremona and Francesca Chiaromonte, Amplitude-Invariant
Functional Motif Discovery.- 22 Daniel Diz-Castro, Manuel Febrero-Bande and
Wenceslao Gonz?lez-Manteiga, Testing the Significance of Covariates in
Nonparametric Regression without the Curse of Dimensionality.- 23 Patric
Dolmeta and Matteo Giordano, Gaussian Process Methods for Covariate-Based
Intensity Estimation.- 24 Melanie Dreina, Sylvie Viguier-Pla and Stephane
Abide, Spectral Analysis of Multidimensional Thermal Fields.- 25 Matteo Farn?
and Xuanye Dai, Forecasting Dynamic Factor Scores by UNALSE Spectral Density
Matrix Estimator.- 26 Manuel Febrero-Bande, Pedro Galeano and Wenceslao
Gonz?lez-Manteiga, Testing for Linearity and Independence in
Scalar-on-Function Regression with Responses Missing at Random by Generalized
Distance Covariance.- 27 Antonino Gagliano, Chiara Di Maria, Gianluca
Sottile, Sarah Beutler-Traktovenko, Luigi Augugliaro and Valeria Vitelli, A
Novel Spectral Density Operator Approach to Unveil Dynamic Time Dependencies
in Multivariate Long-Term ECGs.- 28 Nouhaila Goujili, Matthieu Saumard and
Maher Jridi, Comparison of Deep Learning Methods for Functional Data.- 29
Nicol?s Hern?ndez and Stanislav Nagy, The Common Support Function with
Applications.- 30 Karel Hron, Multivariate Densities in Bayes Spaces: The
Novel Concept of Marginals and Its Implications.- 31 ?rka Hudecov?, Daniel
Hlubinka and Zdenk Hl?vka, Functional ?? Sample Problem via
Multivariate Optimal Measure Transport-Based Permutation Test.- 32 Marie
Hukov? and Charl Pretorius, Sequential Monitoring for Detection of Breaks in
Panel Data.- 33 Ioannis Kalogridis and Stefan Van Aelst, Robust Penalized
Splines for Location Estimation from Discretely Sampled Functional Data.- 34
Yuwei Jiang and Natalya Pya Arnqvist, Functional Regression with Shape
Constraints.- 35 Francesca Ieva, Nicole Fontana, Carlo Andrea Pivato,
Emanuele Di Angelantonio and Piercesare Secchi, Enhancing Causal Inference in
Functional Data: a Method for Estimating Time-Varying Causal Treatment
Effects.- 36 Luigi Ippoliti, Tonio Di Battista, Luigi Di Carlo, Stefania
Fensore, Eugenia Nissi, Pasquale Valentini, Carlo Zaccardi, Linear and
Nonlinear Regression Models for Spatial Downscaling of Particulate Matter.-
37 Alessandro Lanteri, Raffaele Argiento, Silvia Montagna, A Bayesian
Non-Parametric Model to Learn Functions with Discontinuties.- 38 Salvatore
Latora, Luigi Augugliaro and Gerda Claeskens, A Novel Approach To Estimate
Functional Gaussian Graphical Model Based On Penalized Multivariate
Functional Regression Model.- 39 Niels Lundtorp Olsen, Alessia Pini and
Simone Vantini, Local Null Hypothesis Significance Testing on Riemaniann
Manifolds.- 40 Hassan Maatouk, Didier Rulli?re and Xavier Bay, Efficient
Bayesian Linear Models for a Large Number of Observations- 41 Jitka Machalov?
and Jana Heckenbergerov?, Innovative Approach to Wind Direction Data
Analyses: A Compositional Periodic Spline Representation in Bayes Spaces.- 42
Eva-Maria Maier, Alexander Fottner, Almond Stocker and Sonja Greven, Bayes
Hilbert Space Additive Density-on-Scalar Regression Based on Individual
Observations.- 43 Terence Kevin Manfoumbi Djonguet and Guy Martial Nkiet, A
Kernel-Based Approach for Testing Mutual Independence of Several Functional
Variables.- 44 Alejandra Mercedes Mart?nez, Addressing Robustness and
Sparsity in Partially Linear Additive Models.- 45 Valentina Masarotto and
Yiya Chen, Covariance Operators for Phonetics: Revisiting Tonal
Coarticulation.- 46 Caterina May, Theodoros Ladas, Davide Pigoli and Kalliopi
Mylona, A-optimal Designs of Experiments in Linear Models with Dynamic
Factors and Functional Responses.- 47 Alessandra Menafoglio, Moving
Object-Oriented Spatial Statistics Beyond Stationary and Euclidean
Paradigms.- 48 Erik Mendro and Stanislav Nagy, The Spherical Depth for
Functional Data.- 49 Tom? Mrkvika, False Discovery Rate Envelope and its
Performance for Local Testing in Functional Data Analysis.- 50 Stanislav
Nagy, Interpretable Functional Boxplots.- 51 Silvia Novo, Alessandro Palummo
and Laura M. Sangalli, Scalar-on-Function Regression with Partially Observed
Covariate.- 52 Alessandro Palummo, Eleonora Arnone, Letizia Clementi and
Laura M. Sangalli, Efficient Physics-Informed Smoothing of Space-time
Functional Data.- 53 Giulia Patan?, Federica Nicolussi, Alexander Krauth,
Gunter Gauglitz, Bianca Maria Colosimo, Luca Dede and Alessandra Menafoglio,
Ordinal-on-Function Dimensionality Reduction.- 54 Nicola Pronello, Rosaria
Ignaccolo and Luigi Ippoliti, Varying Coefficient Regression Models on
Fluvial Networks.- 55 Hedvika Ranoov? and Daniel Hlubinka, Non-Parametric
Testing of Time Reversibility in Functional Data.- 56 Mar?a D. RuizMedina
and Rosa M. Crujeiras, An LRD Spectral Test for Irregularly Discretely
Observed Functional Time Series in Manifolds.- 57 Diego Serrano and Eduardo
Garc?a-Portugues, Prediction Regions for Functional-Valued Random Forests.-
58 Mohammad Reza Seydi, Johan Strandberg, Todd C. Pataky, and Lina Schelin,
Sample Size Estimation for Two-Sample Functional Hypothesis Test.- 59 Han Lin
Shang, Forecasting Age Distribution of Deaths at Subnational Level.- 60
Stanislav kora and Jitka Machalov?, Statistical Analysis of Bivariate
Densities with Compositional Splines.- 61 Veronika majserov? and Jitka
Machalov?, Prediction with Mixed Effects Smooth Models by using P-Splines.-
62 Marco Stefanucci, Mauro Bernardi and Antonio Canale, Locally Sparse
Estimation for Functional Linear Models with Scalar Response.- 63 Shahin
Tavakoli, Gilles Nisol, and Marc Hallin, Factor Models for High-Dimensional
Functional Time Series.- 64 Romain Valla, Pavlo Mozharovskyi, and Florence
dAlche-Buc, Anomaly-Driven Visualization of Functional Data.- 65 Simone
Vantini, Leveraging Data Exchangeability for a More Reliable and
Interpretable Functional Data Analysis.- 66 Marc Vidal, A Family of Moment
Operators for Functional Data and Its Discriminative Properties.
Format: Paperback / softback, 154 pages, height x width: 235x155 mm, 27 Illustrations, color;
38 Illustrations, black and white; VI, 154 p. 65 illus., 27 illus. in color., 1 Paperback / softback
Pub. Date: 17-Sep-2025
ISBN-13: 9783031814174
In studying mathematics, one can encounter surprises, whether in results that seem astonishing or in proofs that lead down unexpected paths. Over his career, the author came across many such gems. This book includes his favorites and invites readers to experience the same sense of wonder they inspired in him.
Here are results about number theory, Euclidean geometry, trigonometry, algebra, and infinite series, with stories from the history of mathematics added to the mix. There are theorems from such masters as Euler, Newton, and Ramanujan, while some lesser known mathematicians get their moments in the spotlight. Although the author does not avoid technical details, his narrative style makes the book accessible to anyone with an interest in mathematics.
[ This book] is irresistibly fun. It has everything youd expect from Dunham: delightful math and charming stories, all delivered with his trademark clarity, wit, and enthusiasm.
- Steven Strogatz, Cornell University, and author of Infinite Powers
This is a great read, tremendously entertaining (and informative). Dunham is truly the master of us all with regards to writing about fascinating mathematics.
- Clifford Taubes, Harvard University
As always, Dunhams unique gift for mathematical storytelling shines through. [ His] enthusiasm for mathematics jumps off the page and sweeps his reader along on a joyous intellectual ride.
- Linda McGuire, Muhlenberg College
As the books title promises, unexpected results abound. I encourage you to let Dunham be your guide to some of the wonders of mathematics you are in for an intellectual treat.
- Howard Stone, Princeton University
Preface.- Simple Things.- The Infinitude of Primes: Euclid, Ersatz, and Erdos.- On the Twin Prime Conjecture.- On the Twin Composite Conjecture.- A Prime Family Resemblance.- Once More, Into the Primes.- Your Humble Servant Is, Newton.- Newton's (Original) Method.- The First Great Max/Min Problem.- The Roots of All Evil.- What's Your Sign.- The Square Root of Two and a Shaggy-Log Story.- A Ramanujan Morsel. The Biblical Value of Pi, or Not.- The Power of the Powers of Three.- The Demonic Harmonic Series.- Euler Sums a Series.- The Math Matriarchs of Bryn Mawr.- The Triangle Inequality in Euclid.- Isosceleast Triangles.- My Favorite Proof of Morely's Theorem (as of this morning).- Euclid, Euler, and the Regular Solids.- The Magic of Commutativity.- Wondrous Formulae.- Proving is Believing.- Mathematics and the Humanities.- References.
Format: Hardback, 575 pages, height x width: 235x155 mm, XV, 575 p., 1 Hardback
Pub. Date: 27-May-2025
ISBN-13: 9783031897061
Math for Data Science presents the mathematical foundations necessary for studying and working in Data Science. The book is suitable for courses in applied mathematics, business analytics, computer science, data science, and engineering. The text covers the portions of linear algebra, calculus, probability, and statistics prerequisite to Data Science. The highlight of the book is the machine learning chapter, where the results of the previous chapters are applied to neural network training and stochastic gradient descent. Also included in this last chapter are advanced topics such as accelerated gradient descent and logistic regression trainability.
Clear examples are supported with detailed figures and Python code; Jupyter notebooks and supporting files are available on the author's website. More than 380 exercises and nine detailed appendices covering background elementary material are provided to aid understanding. The book begins at a gentle pace, by focusing on two-dimensional datasets. As the text progresses, foundational topics are expanded upon, leading to deeper results at a more advanced level.
Preface.- List of Figures.- Datasets.- Linear Geometry.- Principal
Components.- Calculus.- Probability.- Statistics.- Machine Learning.- A.
Auxiliary Material.- B. Auxiliary Files.- References.- Python Index.- Index.
Format: Hardback, 389 pages, height x width: 235x155 mm, 39 Illustrations, color; 235 Illustrations, black and white; XXI, 389 p. 274 illus., 39 illus. in color., 1 Hardback
Series: The Materials Research Society Series
Pub. Date: 30-Jul-2025
ISBN-13: 9783031907302
This book, now in an expanded second edition, provides a self-contained undergraduate course on quantum computing based on classroom-tested lecture notes. It reviews the fundamentals of quantum mechanics from the double-slit experiment to entanglement, before progressing to the basics of qubits, quantum gates, quantum circuits, quantum key distribution, and some of the famous quantum algorithms. As well as covering quantum gates in depth, it also describes promising platforms for their physical implementation, along with error correction, and topological quantum computing. With quantum computing expanding rapidly in the private sector, understanding quantum computing has never been so important for graduates entering the workplace or PhD programs. Assuming minimal background knowledge, this book is highly accessible, with rigorous step-by-step explanations of the principles behind quantum computation, further reading, and exercises, ensuring that undergraduate students in physics and engineering emerge well prepared for the future. This edition contains new material on quantum metrology, random circuit sampling, electric dipole spin resonance, dilution refrigeration, photon detection, boson sampling, and continuous variable quantum computing. It also features around 50 new exercises, and lecture slides for course instructors.
Chapter 1: Superposition.
Chapter 2: Quantization.
Chapter 3: Spin.-
Chapter 4: Qubits.
Chapter 5: Entanglement.
Chapter 6: Quantum Key Distribution.
Chapter 7: Quantum Gates.
Chapter 8: Teleportation.
Chapter 10: Computational Complexity.
Chapter 11: Deutsch Algorithm.
Chapter 12: Grover Algorithm.
Chapter 13: Shor Algorithm.
Chapter 14: Physical Implementation of Single-Qubit Gates.
Chapter 15: Electron Spin Resonance.-
Chapter 16: Two-state Dynamics.
Chapter 17: Physical Implementation of Two-qubit Gates.
Chapter 18: DiVincenzo Criteria.
Chapter 19: Nuclear Magnetic Resonance.
Chapter 20: Solid-state Spin Qubits.
Chapter 21: Trapped Ion Quantum Computing.
Chapter 22: Superconducting Qubits.
Chapter 23: Adiabatic Quantum Computing.
Chapter 24: Optical Quantum Computing.-
Chapter 25: Quantum Error Correction.
Chapter 26: Topological Quantum Computing.
Format: Hardback, 186 pages, height x width: 235x155 mm, 42 Illustrations, color;
3 Illustrations, black and white; IX, 186 p. 45 illus., 42 illus. in color., 1 Hardback
Pub. Date: 19-Jul-2025
ISBN-13: 9783031890826
This book examines infinite-equilibriums for the switching bifurcations of two 1-dimensional flows in dynamical systems. Quadratic single-linear-bivariate systems are adopted to discuss infinite-equilibriums in dynamical systems. For such quadratic dynamical systems, there are three types of infinite-equilibriums. The inflection-source and sink infinite-equilibriums are for the switching bifurcations of two parabola flows on the two-directions. The parabola-source and sink infinite-equilibriums are for the switching bifurcations of parabola and inflection flows on the two-directions. The inflection upper and lower-saddle infinite-equilibriums are for the switching bifurcation of two inflection flows in two directions. The inflection flows are for appearing bifurcations of two parabola flows on the same direction. Such switching bifurcations for 1-dimensional flow are based on the infinite-equilibriums, which will help one understand global dynamics in nonlinear dynamical systems. This book introduces infinite-equilibrium concepts and such switching bifurcations to nonlinear dynamics.
Single-linear-bivariate Linear systems.- Constant and Linear-bivariate
Quadratic Systems.- Single-linear-bivariate Linear and Quadratic Systems.-
Single-linear-bivariate Quadratic Systems.