Teaching multivariable calculus and tensor calculus with computer algebra software

2020 ◽  
Vol 54 (4) ◽  
pp. 134-135
Author(s):  
Miao-jung Yvonne Ou
Keyword(s):  
Computer Algebra ◽  
Directional Derivatives ◽  
Tensor Calculus ◽  
New Concepts ◽  
Problem Solve ◽  
The Many ◽  
Computer Algebra Software ◽  
Vector Valued Functions ◽  
Fundamental Theorems ◽  
Vector Valued

To go from calculus of scalar functions of one variable to multivariate calculus of vector-valued functions is a steep learning curve for many students. It takes a lot of practice to get used to the new concepts such as the directional derivatives, the di.erentiability, the many types of first order di.erential operators, parameterization of surfaces and the fundamental theorems of integrals, e.g. the Divergence Theorem and the Stokes Theorem. Along the learning process of mastering the skills, the students often need to check whether the intermediate steps in the tedious calculations are correct. Unfortunately, this is beyond the capability of an ordinary calculator and the answers provided at the end of the books. This is where computer algebra software, such as Mathematica, can come to students' help. With the developed symbolic computation tools, the students can tweak a given problem, solve a new one by hand and then check the answer against the result obtained by using the computer algebra software.

Higher-order conditions for strict local Pareto minima for problems with partial order introduced by a polyhedral cone

2018 ◽  
Vol 24 (1) ◽  
pp. 45-54
Author(s):  
Aleksandra Stasiak
Keyword(s):  
Partial Order ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Polyhedral Cone ◽  
Directional Derivatives ◽  
Pareto Minima ◽  
Necessary And Sufficient ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Derivatives Of

Abstract Using the definitions of μ-th order lower and upper directional derivatives of vector-valued functions, introduced in Rahmo and Studniarski (J. Math. Anal. Appl. 393 (2012), 212–221), we provide some necessary and sufficient conditions for strict local Pareto minimizers of order μ for optimization problems where the partial order is introduced by a pointed polyhedral cone with non-empty interior.

scholarly journals Existence and Uniqueness of Zeros for Vector-Valued Functions with K-Adjustability Convexity and Their Applications

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 809
Author(s):  
Wei-Shih Du
Keyword(s):  
Fixed Point ◽  
Minimization Problem ◽  
Existence And Uniqueness ◽  
Existence Theorems ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Uniqueness Theorems ◽  
New Concepts ◽  
Vector Valued Functions ◽  
Vector Valued

In this paper, we introduce the new concepts of K-adjustability convexity and strictly K-adjustability convexity which respectively generalize and extend the concepts of K-convexity and strictly K-convexity. We establish some new existence and uniqueness theorems of zeros for vector-valued functions with K-adjustability convexity. As their applications, we obtain existence theorems for the minimization problem and fixed point problem which are original and quite different from the known results in the existing literature.

scholarly journals Farkas-Type Results for Vector-Valued Functions with Applications

2017 ◽  
Vol 173 (2) ◽  
pp. 357-390 ◽  
Author(s):  
N. Dinh ◽  
M. A. Goberna ◽  
M. A. López ◽  
T. H. Mo
Keyword(s):  
Vector Valued Functions ◽  
Vector Valued

Glossary of terms used in medicinal chemistry. Part II (IUPAC Recommendations 2013)

2013 ◽  
Vol 85 (8) ◽  
pp. 1725-1758 ◽  
Author(s):  
Derek R. Buckle ◽  
Paul W. Erhardt ◽  
C. Robin Ganellin ◽  
Toshi Kobayashi ◽  
Thomas J. Perun ◽  
...  
Keyword(s):  
Molecular Biology ◽  
Human Genome ◽  
Human Genome Project ◽  
Medicinal Chemistry ◽  
Genome Project ◽  
Scientific Disciplines ◽  
New Concepts ◽  
Essential Components ◽  
The Many ◽  
The Human Genome Project

The evolution that has taken place in medicinal chemistry practice as a result of major advances in genomics and molecular biology arising from the Human Genome Project has carried with it an extensive additional working vocabulary that has become both integrated and essential terminology for the medicinal chemist. Some of this augmented terminology has been adopted from the many related and interlocked scientific disciplines with which the modern medicinal chemist must be conversant, but many other terms have been introduced to define new concepts and ideas as they have arisen. In this supplementary Glossary, we have attempted to collate and define many of the additional terms that are now considered to be essential components of the medicinal chemist’s expanded repertoire.

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

scholarly journals THE LOCAL NON-HOMOGENEOUS Tb THEOREM FOR VECTOR-VALUED FUNCTIONS

2014 ◽  
Vol 57 (1) ◽  
pp. 17-82 ◽  
Author(s):  
TUOMAS P. HYTÖNEN ◽  
ANTTI V. VÄHÄKANGAS
Keyword(s):  
Singular Integrals ◽  
Local Version ◽  
Square Functions ◽  
Property A ◽  
Martingale Differences ◽  
Umd Spaces ◽  
Function Lattice ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Similar Extension

AbstractWe extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, ‘vector-valued’ means ‘taking values in a function lattice with the UMD (unconditional martingale differences) property’. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local version requires several elaborations of the previous techniques, and raises new questions about the limits of the vector-valued theory.

Strict Topologies for Vector-Valued Functions

1974 ◽  
Vol 26 (4) ◽  
pp. 841-853 ◽  
Author(s):  
Robert A. Fontenot
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Measure Theory ◽  
Strict Topology ◽  
Locally Compact ◽  
Topological Measure ◽  
Topological Measure Theory ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Compact Hausdorff Space

This paper is motivated by work in two fields, the theory of strict topologies and topological measure theory. In [1], R. C. Buck began the study of the strict topology for the algebra C*(S) of continuous, bounded real-valued functions on a locally compact Hausdorff space S and showed that the topological vector space C*(S) with the strict topology has many of the same topological vector space properties as C0(S), the sup norm algebra of continuous realvalued functions vanishing at infinity. Buck showed that as a class, the algebras C*(S) for S locally compact and C*(X), for X compact, were very much alike. Many papers on the strict topology for C*(S), where S is locally compact, followed Buck's; e.g., see [2; 3].

Dictatorship and Democracy, 1918–1939

2012 ◽  
Author(s):  
Thomas Mergel
Keyword(s):  
World War I ◽  
Social Commitment ◽  
The People ◽  
New Concepts ◽  
The Social ◽  
Community People ◽  
The Many ◽  
Social Organism ◽  
German Parliament ◽  
Revolution Of 1848

Both dictatorship and democracy were essentially new concepts of political rule in Germany after World War I. It was true that suffrage had been increasingly extended after the revolution of 1848–1849, and more citizens (male citizens, that is) were entitled to vote in Imperial Germany than, for instance, in Great Britain. Dictatorship, too, was a new form of political control, at least in Germany. The term ‘people’ was to become a standard formula for the self-understanding of German politics after 1918. In its shades of meaning, it saw the people as a social organism, rather than as an ethnic community. ‘People’ referred to the many. It described the social commitment with which a good community was supposed to be built. An inquiry into Reichstag, and the German parliament and incidents and rebellions surrounding it concludes this article.

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