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Differential Equations Activities:     more detail
1. Epileptiform Activity in Differential Equation Models of Neuronal Networks (Berichte Aus Der Physik) by Christian Hauptmann, 2000-10-12
2. Inverse Problems: Activities for Undergraduates (Classroom Resource Materials) by Charles W. Groetsch, 1999-12-02
3. Computing the Electrical Activity in the Heart (Monographs in Computational Science and Engineering) by Joakim Sundnes, Glenn Terje Lines, et all 2010-11-02
4. Computer Science Research Activities In Asia: Software Technology And Patents, Cim, Scientific Computation And Differential Equations, Computer And Mathmatics Modelling, And System Simulation by David K. Kahaner, 1993-09-30
5. Stochastic estimation in combined arms Lanchester modeling of warfare (Technical report / U.S. Army Materiel Systems Analysis Activity) by Herbert E Cohen, 1985
6. Methodology for stochastic modeling (Technical report / U.S. Army Materiel Systems Analysis Activity) by Herbert E Cohen, 1985
7. Ready-To-Use Vocabulary, Word Attack & Comprehension Activities: Fifth Grade Reading Level (Reading Skills Activities Library) by Henriette L. Allen, Walter B. Barbe, et all 1999-09

61. Research Activities Of Prasanth Nair
Meshfree Solution of Partial differential equations More recently, I initiated researchto Details of some of my previous research activities including links
http://www.soton.ac.uk/~pbn/res.html
 home papers research CV software MDO pages jobs Ongoing Research Stochastic Subspace Projection Theory : A significant focus of my current research is on developing numerical algorithms for analysis of systems governed by stochastic partial differential equations. I am particularly interested in cases when the PDE coefficients are random variables or random fields with finite memory. The long-term objective is to develop the theoretical foundations for a general purpose capability, which parallels existing computational methods for high-fidelity analysis of deterministic systems. As a first step towards achieving these goals, I am developing discrete Stochastic Subspace Projection Schemes for the following problems Random linear algebraic equations [ A x b Algebraic random eigenvalue problems [ A x l B x The stochastic Lyapunov equation A X X B C Random Lambda-matrix problems: ( A + p B + p C x The basic idea of our discrete formulations is to use a 'rich' set of stochastic basis vectors to approximate the solution process [ ]. More recently, I extended the discrete formulations to develop a general approach which employs

62. VIII Workshop On Arcial Differential Equations
Many of the activities of the last Workshops (see Wedp01; Wedp99; Wedp97) were focusedon topics of current interest in Partial differential equations, such as.
http://www.fluid.impa.br/wedp03/ingles/historico-i.html
 A Brief History The Workshop on Partial Differential Equations: Theory, Computation and Applications is held every two years, in July, at Instituto Nacional de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil. The first one took place in 1989, and it was focused on Fluid Dynamics. The other events took place from 1991 to 2001 (odd years); they were more general and they gave rise to six issues of Matematica Contemporanea, a journal of the Brazilian Mathematical Society that publishes refereed articles of high quality meetings in Brazil. Many of the activities of the last Workshops (see ) were focused on topics of current interest in Partial Differential Equations, such as theoretical aspects; existence, stability and long time behavior; numerical analysis of solution methods, as well as exploration of related theory via scientific computing; nonlinear conservation laws, with applications in fluid dynamics, elasticity, flow in porous media; mathematical modeling in geophysical fluid dynamics, with emphasis in theoretical aspects of meteorology and physical oceanography.

63. Liste Des Livres (en Anglais)
Limits, Derivatives, Integrals, Sequences and Series, differential equations, MultiVariableCalculus. B. Antinone, L., Brueningson, E. 25 activities Using the
http://www.ti.com/calc/suisse/vpplistelivres.htm
##### Liste des livres (en anglais)
Vous avez droit à un livre gratuit pour 10 calculatrices graphiques achetées. Veuillez s.v.p. cocher le livre souhaité sur la liste ci-après et joindre cette liste à votre commande. Titre Auteur Contenu TI-82, TI-83:
Graphing Calculator activities for enriching middle school mathematics Browning, Ch., Channell, D. 12 activities Statistics Handbook for the TI-83 Morgan, L. Describing One-Variable Data, Describing Bivariate Data, Fitting an Equation to Bivariate Data, Describing Categorical Data, Colleting Data, Demonstrating Probability, Simulations, and probability Distributions, Estimating and Finding Confidence Intervals, Testing Hypotheses (Means, Proportions and Standard Deviations), Testing Goodness-of -Fit and Two-Way Table Chi-Square, Analyzing Variance, Inference of Correlation and Regression, Forecasting Time Value Money: Applications on the TI-83 Hofmann, R., Hofmann, Ch. TI-86: Using the TI-86 in Collegiate Mathematics Dodd, F., Cruthirds, J. Graphing Fundamentals, Advanced Graphing, Parametric and Polar Graphing, Solving Equations, Differential Calculus, Integral Calculus, Matrices and Vectors, Differential Equations, Statistics, Numerical Integration, Probability Distributions Differential equations with the TI-86 Barton, R., Pence, D.

64. MAT 274 Elementary Differential Equations Spring 2003 RUEDEMANN
math.la.asu.edu/~rick Textbook Elementary differential equations and Boundary plans,work schedules, weddings, family reunions, and other such activities.
http://math.la.asu.edu/~rick/274syl.html
 MAT 274 Elementary Differential Equations,  Spring 2003         (Tentative Syllabus: Subject to change by in-class annoucements.) SLN 53700 : 7:40 - 8:30 MWF in PSA 107 Instructor: Dr. Richard Ruedemann                Office: PSA 725 Phone: 965-4240 (do not leave message)     Office Hours: 9:40-10:30 MWF, 11:40-1:30 T    E-Mail: rick@asu.edu     Web Page: http://math.la.asu.edu/~rick Textbook: Elementary Differential Equations and Boundary Value Problems, 7th edition by Content: We will cover most of 2.1-2.9; 3.1-3.9; 4.1-4.4; 5.1-5.4; 6.1-6.6; 7.1-7.7    Prerequisite: MAT271 (Calculus II) or equivalent with a grade of C or better. Unrestricted Withdrawal Deadline: Friday, 2/14 Restricted Withdrawal Deadline:     Friday, 4/4 Restricted Complete Withdrawal Deadline:   Wednesday, 4/30 Unrestricted Withdrawal: A student may withdraw from a course with a grade of W during the unrestricted withdrawal period. The instructor's signature is not required. Restricted Withdrawal: In order to withdraw during this period, the student must obtain the

65. ENC: Web Links: Math Topics: Equations
The activities offer realtime correlations between equations WWW) site providesa college level course in calculus, differential equations, and matrix
Skip Navigation You Are Here ENC Home Web Links Math Topics Advanced ... Frequently Asked Questions Find detailed information about thousands of materials for K-12 math and science. Read articles about inquiry, equity, and other key topics for educators and parents. Create your learning plan, read the standards, and find tips for getting grants.
##### Math Topics
Lists of web sites categorized by subject areas within mathematics.
##### Equations

This Internet site features student algebra lessons on material ranging from preliminary algebra topics, such as absolute value and rounding and significant digits, to advanced algebra topics including complex numbers, logarithms, and exponential functions...
(For more details see Brief ENC Record or Full ENC Record
This web site, maintained by an algebra teacher, has many different resources for students. Tutorials on a wide variety of algebra topics are available; many are interactive and supply problems with step-by-step solutions. Others include movies or other...
(For more details see Brief ENC Record or Full ENC Record
ExploreMath.com

Associates who participate in the teaching activities Jamylle Carter Selim Esedoglu(Ph.D. Courant Institute, partial differential equations, computer vision
FOR ITS MEMBERS AND FRIENDS
NO. 8 UNIVERSITY OF MINNESOTA - NEWSLETTER OF THE SCHOOL OF MATHEMATICS JANUARY2002
INSIDE THIS ISSUE:
Home

Promotions

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2002 Symposiums

Conference
1st Biennial Yamabe
Symposium Conf. for A. Friedman's 70th Birthday Other Notable Activities - Undergraduate Program Communication Skills Thoughts on Writing Skills Senior Project Goals Changing to Semesters ... MCIM Featuring Dr. John Hoffman Mathematics Library AWM Mentor Network - News from the Centers IMA Digital Technology Center ITCEP Contacting Us
Continuing Postdocs and Visiting Faculty Assistant Professors: Associate Professors : James Louisell (University of Southern Colorado, control theory, delay-differential equations) Professors: Yakar Kannai (Weizmann Institute, partial differential equations), Elisabeth Logak (University of Cergy-Pontoise, France, nonlinear partial differential equations, biomathematics), Fedor Nazarov (Michigan State University, harmonic analysis), Gregory Seregin (Steklov Institute St. Petersburg, applications of calculus of variations and general theory of nonlinear partial differential equations), Bevan Thompson (University of Queensland, applications of methods from nonlinear analysis), and Thomas Zink (University of Bielefeld, algebraic geometry, arithmetic geometry).

67. Florida Tech Catalog 2000-2001
Research activities. analysis, qualitative and quantitative properties of nonlinearevolution equations, including differential equations with delay
http://www.fit.edu/catalog/sci-lib/app-math.html
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##### DEPARTMENT OF MATHEMATICAL SCIENCES V. Lakshmikantham, Ph.D., Head
Bachelor of Science Master of Science Doctor of Philosophy Associate Head
Michael D. Shaw, Ph.D. Professors
Jewgeni H. Dshalalow, Dr. Sci., real analysis, operations research, stochastic processes, queueing theory. Charles T. Fulton, Ph.D., ordinary and partial differential equations, spectral theory of differential operators, numerical linear algebra, numerical methods for two-point boundary value problems, parallel processing. V. Lakshmikantham, Ph.D., nonlinear analysis, differential and integral equations, numerical mathematics, evolution operations. Kamel Rekab, Ph.D., sequential analysis, design of experiments, applied data analysis, reliability theory. Associate Professors
Gary Howell, Ph.D.

68. Activities Of Biomechanics Laboratory
activities of the St.FX Biomechanics Laboratory Erik Anderson attended the InternationalConference on differential equations with Applications to Biology
http://www.stfx.ca/people/edemont/activities.html
Activities of the St.F.X. Biomechanics Laboratory
##### Go to: PublicationsProjectsPeople
Scientific Meetings ... Awards/Grants/Student Scholarships
##### Scientific Meetings
Edwin DeMont was invited to present a talk at the Department of Fisheries and Oceans , Moncton, on the Biomechanics of Invertebrates, which he presented on December 10, 1998.
Edwin DeMont was invited by the Department of Fisheries and Oceans to be an external scientific referee for the 1998 assessments of lobster stocks in the Maritimes Region. The purposes of peer review are: to identify important questions that may have been neglected; to examine the scientific approaches of the stock assessments; to identify any weaknesses in methodology; to help improve the clarity of assessments; to make research recommendations; and to develop collaborative research programs. The peer review includes detailed review of stock assessments and Stock Status Reports. The assessment will occur in the Gulf Fisheries Center, Moncton, N.B., July 7-9, 1998.
Edwin DeMont was invited to present a poster at the 1998 Gordon Research Conference on Innovations in College Teaching of Materials Science . He will be presenting the following poster.
STRETCHED TO THE LIMIT
Most material testing apparatus require expensive equipment, thus material testing in undergraduate labs is limited. I have developed two material testing exercices, which do not require expensive equipment. The materials used in the laboratories were inspired by my research interests in comparative biomechanics. The first exercise is used to examine the stress-strain relationships of fresh and collagen-free cow arteries. The most complex apparatus used in the exercise is a slide projector. This exercise is used in a third year

69. \Large\bf ACTIVITIES AT IMPA
Algebra; Analysis/Partial differential equations; Computer Graphics; MathematicalEconomics; differential Geometry; Operations of the activities currently being
http://www.impa.br/English/activities.html
##### Activities
IMPA has been certified by the Federal Board of Education (Conselho Federal de Educação) as a graduate study center of pure and applied mathematics for Master's and Doctoral programs since 1971. In the Master's program, the student acquires basic knowledge that will permit him/her to teach mathematics-related subjects at the graduate level or enter into professional activities which require mathematical applications for technical or scientific problems. The Doctoral program is especially aimed at training researchers. Its fundamental requirement is the elaboration of an original thesis relevant to the progress of mathematics. Special emphasis is given to the Doctoral program at IMPA and, thus, the Master's program also devised as a way of furnishing rapid access to the Doctoral program. In preparation for graduate studies at IMPA, the Young Scientist Program has been designed with the following features:
• to complete basic graduate training so as to prepare students for the Master's program; to give students who are undertaking graduate studies in mathematics, engineering, physics, etc. an opportunity to test their aptitudes and interests in mathematics while at IMPA, pending their possible admittance in the graduate program.
• 70. AAMP - Teaching Activities
activities. The members of the group contribute to undergraduate courses in calculus,analysis, mathematical modelling and ordinary differential equations.
http://www.math.utwente.nl/mpcm/aamp/teaching.html
 Teaching activities The members of the group contribute to undergraduate courses in calculus, analysis, mathematical modelling and ordinary differential equations. Graduate courses for which the group is responsible include partial differential equations, mathematical modelling, applied analytical methods (variational structures and methods, continuous dynamical systems) and bifurcation theory. Contributions to numerical courses are given as well. At all levels of education, we investigate the appropriate use of computer-algebra and numerical tools for visualisation, mathematical modelling and ordinary differential equations. Please consult the dutch and english education pages of the faculty, the TW-onderWWWijs , or the UT-Teletop for the course program.

71. H. Amann's Editorial Activities
Editorial activities. Member of the following editorial boards Journals. Abstractand Applied Analysis (AAA); Advances in differential equations; Advances in
http://www.math.unizh.ch/amann/editorial_activities.html
##### Editorial activities
Member of the following editorial boards:
##### Journals
• Abstract and Applied Analysis (AAA)
• Advances in Mathematical Sciences and Applications
• Cubo Matemática Educacional
• Differential and Integral Equations
• Houston Journal of Mathematics
• Journal of Evolution Equations (JEE)
• Journal of Mathematical Fluid Mechanics
• Nonlinear Differential Equations and Applications (NoDEA)
• Questiones Mathematicae
• Scientiae Mathematicae Japonicae

Retired from
• Proceedings of the Royal Scociety of Edinburgh (1978-1988)
• Mathematische Annalen (1979-1999, managing editor 1993-1999)
• Dynamics Reported (1987-1995)
##### Book Series
• Monographs in Mathematics

Retired from
• Research Notes in Mathematics Series (formerly: Pitman Research Notes in Mathematics)
• Monographs and Surveys in Pure and Applied Mathematics (formerly: Pitman Monographs and Surveys in Pure and Applied Mathematics)

home

72. CNA, Activities
activities. Scientific Program (About CNA). 20012002 21820 Advanced Topicsin Analysis Partial differential equations and Kinetic Theory, Tartar;
http://www.math.cmu.edu/cna/activities.html
People

Publications

Activities
Application

Contact

CNA Home
##### Topics Courses
• Advanced Topics in Analysis: Partial Differential Equations and Kinetic Theory, Tartar
• Advanced Topics in Analysis: Modern Methods in the Calculus of Variations, Fonseca
• Advanced Topics in Analysis: Homogenization, Tartar
• Advanced Topics in Analysis: Fine Regularity Theory for Partial Differential Equations, Fonseca
• Advanced Topics in Plasticity, Owen
• Advanced Topics in Analysis: Multidimensional Problems in the Calculus of Variations, Fonseca
• Advanced Topics in Analysis: PDE Models in Oceanography, Tartar
##### 2001 CNA Summer School

73. Research Activities
Geometric Integration of differential equations. Also visit GI's own homepage. Symmetriesin numerical solution of ordinary differential equations.
http://www.ii.uib.no/forskningsgrupper/num/forskning/index-eng.shtml
U of Bergen Fac. Math and Nat. Sci. Dep. of Informatics Numerical analysis
##### Research activities
Go to: Numerical
analysis
Research
activities
...
analyse
A common theme for most of the research in the numerical analysis group is parallel computing. This research is to a large extent carried out in close collaboration with Parallab Partitioning and scheduling problems are ubiquitous in parallel computing. Problems of this sort are modeled and solved as graph problems, hence we have a close collaborate with the Algorithm Group on these issues. Numerical techniques of special interest to us are:
• Domain Decomposition techniques for elliptic PDE's.
• Geometric Integration of differential equations. Also visit GI's own homepage.
• Multidimensional Numerical Integration.
• Numerical Linear Algebra
For lists of online publications go to Reports in Informatics or the individual homepages
##### Ongoing projects
People from the numerical analysis group are involved in the following projects: Strategic Project in Scientific Computing Symmetries in numerical solution of ordinary differential equations

74. Activites
The central activities of the group will be a series of workshops. the latest developmentsin the theory and algorithms for differential equations; and a
http://www-lmc.imag.fr/CATHODE/activite.html
The central activities of the group will be a series of workshops. These will be of two main types: the CADE series, open to a wider audience and presenting the latest developments in the theory and algorithms for differential equations; and a series of internal meetings aimed at the development of the specifications of solutions and algorithms. In association with the internal workshops, a series of 1 or 2 day meetings for contacts in industry and academic research will be organised. The aim of these meetings is not only to disseminate our results but also to clarify which of our lines of development offers the most immediate benefit to practical users. To advance work on specific topics within the main theme, a programme of short visits between group members will also be organised.
The results of our workshops will be published in the CADE proceedings and as a series of technical reports, and programs will be disseminated by electronic transfer. Work on individual topics will appear as research papers. We will maintain contacts with the ESPRIT projects ALCOM, POSSO and NEURON by interchange of visits to workshops and meetings.
##### WORKSHOP ACTIVITIES :
• International workshop : CADE-92 has been organised in June 92 at the CIRM in Luminy, Marseille, with the American community.
differential equations Exploration Through Technology (lab manual, prepared by Eric MatthiasKawski's Maple Resources and activities (Arizona State University
Department of Mathematics College of Science and Mathematics University of South Carolina
##### Professional Activities and Interests

76. Applied Mathematics At Texas A&M University
inverse problems. For a description of the breadth of these activities,see the differential equations webpage. Numerical Analysis.
http://www.math.tamu.edu/~jay.walton/applied.html
 Applied Mathematics Control Theory Control Theory is a major research area of both applied mathematics and engineering spanning applications as diverse as the control of space vehicles and the active and passive damping of buildings and bridges in order to improve their earthquake tolerance. The principal members of the Control Theory Group here are Goong Chen and Jianxin Zhou Applied Mechanics Applied Mechanics refers to applied contiuum mechanics and encompasses classical fluid dynamics, solid mechanics, thermo-mechanics, etc. Our research activities in applied mechanics are highly interdisciplinary involving mathematicians working in collaboration with colleagues from engineering, material science, geoscience, economics, medicine and the life sciences. Several faculty work in fluid dynamics, including flow through porous media, and viscoelastic flows, ( Prabir Daripa Richard Ewing , Ciprian Foias, K. R. Rajagopal , and John Slattery ), and solid mechanics, including biological materials ( K.R. Rajagopal John Slattery and Jay Walton ). Members of this group have close ties with colleagues in the interdisciplinary

77. Pavel Drabek
elliptic partial differential equations with use of weighted Sobolev spaces (CNRItaly, grant no. 341/91, 19921994). Significant activities outside department
http://deimos.zcu.cz/Drabek/
University of West Bohemia, Faculty of Applied Sciences, Department of Mathematics P AVEL D RABEK PROF., RND R., D R S C. T EL: F AX: E -MAIL: pdrabek@kma.zcu.cz
[Publications]
[...for Students...] [Back to the Staff] ... [Back to the KMA-Homepage]
##### Qualification:
• Prof., University of West Bohemia, Pilsen.
• DrSc., Czechoslovak Academy of Sciences, Prague,
• Doc., Technical University of Pilsen,
• CSc., Czechoslovak Academy of Sciences, Prague,
• RNDr., Charles University, Prague,
• graduated from Charles University, Prague.
##### Teaching activities:
• ordinary differential equations,
• partial differential equations,
• integral equations,
• functional analysis.
##### Scientific activities:
• nonlinear boundary value problems for ordinary and partial differential equations
• existence results,
• multiplicity results,
• bifurcations,
• stability of nonlinear evolution problems.
##### Language knowledge:
• English (active),
• Russian (active),
• Slovak (passive),
• Italian (passive).
##### Visiting professor:
• National Technical University Athens, Greece ( , 2 weeks),
• F.C.F.M. Universidad de Chile, Santiago, Chile (
• 78. Equations Aux Dérivées Partielles
The theme Partial differential equations (PDE) plays a central role in the LaboratoryMathématiques pour l The activities of the group are described below.
http://mip.ups-tlse.fr/equipes/EDP.html
Equipe suivante (Partial Differential Equations) responsable (team leader) : J. Fleckinger
##### Membres permanents (Faculty)
Laure Cardoulis
Youri Egorov

Jacqueline Fleckinger

Jacques Giacomoni
...
Claude Wagschal
##### Doctorants (Ph.D. Students)
Naziha Besbas
Mokhtaria Boutaleb

Rachid Faouzi

Christian Lobert
...
Nasreddine Megrez
##### Post-Doctorants (Ex. Ph.D.)
Kouassi Theodore Boni
Renaud Camales

Laure Cardoulis
Karim Chaib ... Fouzia Nemrouch
##### Research activities
In the first project dealing with the theory of holomorphic PDE 's, C.Wagschal, his collaborators and PhD students have obtained several new results on ramified Cauchy problems. In the second project entitled Spectral Theory and Inequalities Maximum and Antimaximum Principle A seminar at the University Toulouse 1 brings together the members of the PDE group. The group has previously obtained many significant results in the above mentioned topics. In particular let us mention that Y.Egorov and collaborators have written two books: Foundations of the Classical Theory of Partial Differential Equations , (259 pages), Springer, 1998, with M.A. Shubin and

79. Undergraduate Committee Report, Fall 98
The following is a summary of activities for the Undergraduate Committee of the Departmentof Mathematics for the AP calculus/differential equations sequence .
http://hedgehog.math.arizona.edu/~laetsch/MathNews99/UGC984.html
##### SUMMARY OF UGC ACTIVITIES FOR THE FALL 1998
Submitted by David Gay.
The following is a summary of activities for the Undergraduate Committee of the Department of Mathematics for the fall semester of 1998. Minutes for past UGC meetings can be found at the UGC website (http://www.math.arizona.edu/~mcenter/UGCminutes/index.html).
LOWER DIVISION COURSES
Course renumbering for 100 level courses . A memo listing the proposed changes was distributed early November to all mathematics faculty, all University advisors, the Undergraduate Council, and Michael Gottfredson (University Vice President for Undergraduate Education). There being no objections, changes were submitted December 15 to the Curriculum Office. The Department will disseminate information about the changes to all state high schools, universities, and community colleges.
AP calculus/differential equations sequence . For the last couple of years, the Department has been experimenting with a calculus/differential equations sequence for students entering with Advanced Placement calculus experience. The following descriptions for permanent courses were submitted to the Curriculum Office. MATH 250A Calculus and Differential Equations I (3) I Integral calculus with applications, techniques of integration, solving first order differential equations using separation of variables, introduction to autonomous first order differential equations. The sequence 250A-250B substitutes for the pair of courses 129-254 or the pair 129-355; however, 250A alone does not substitute for 129. P, a score of 4 or 5 on the Calculus AB advanced Placement Exam or by permission of the instructor. Credit allowed for only one of the following courses: MATH 250A or MATH 129.

80. Research Activities
Partial differential equations blowup for parabolic and elliptic equations, maximumprinciples for parabolic equations, parabolic equations with singular
http://riemann.unica.it/eng/ricerca.html
Opening Page Faculty Phone Numbers
##### Annual Research Plans for 1998
The research programs proposed by the faculty of the Department of Mathematics for 1998 have been formulated separately for the various research areas. Algebra and symbolic calculus
Polynomial and differential ideals. Inverse Macaulay systems. Primary Decomposition of polynomial modules. Study of linear and nonlinear dynamical systems as a function of apriori" properties. Mathematical analysis.
Numerical analysis and computational mathematics

Spectral factorization of convolution equations, with applications to the solution of integral equations. Numerical solution of nonlinear equations of large dimension. Orthogonalization of sequences of multivariate functions. Global minimization with improvement of local minima" and simulated annealing" methods. Math education and history of mathematics
Birth of analytic geometry and the notion of volume. Study of teaching/learning processes, with particular attention to the concepts of measure and linear transformation and the part played by proofs. Probability and statistics in junior high school. Riemannian Geometry
Minimal surfaces. Harmonic maps and morphisms. Elastic curves and biharmonic curves. Trigonometry on Grasmannians and other symmetric spaces. Geometry of submanifolds of three dimensional space where it is defined a field of symmetries with respect to an axes. Geometry of D'Atri spaces. Geometry of submanifolds of quaternionic manifolds

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