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2. Solving the membership problem for certain subgroups of $SL_2(\mathbb{Z})$

Author

Han, Sandie, Masuda, Ariane M., Singh, Satyanand, and Thiel, Johann

Subjects
Mathematics - Group Theory, Mathematics - Number Theory
Abstract

For positive integers $u$ and $v$, let $L_u=\begin{bmatrix}1 & 0 \\u&1\end{bmatrix}$ and $R_v=\begin{bmatrix}1 & v \\ 0 & 1\end{bmatrix}$. Let $G_{u,v}$ be the group generated by $L_u$ and $R_v$. In a previous paper, the authors determined a characterization of matrices $M=\begin{bmatrix}a & c \\b&d\end{bmatrix}$ in $G_{u,v}$ when $u,v\geq 3$ in terms of the short continued fraction representation of $b/d$. We extend this result to the case where $u+v> 4$. Additionally, we compute $[\mathscr{G}_{u,v}\colon G_{u,v}]$ for $u,v\geq 1$, extending a result of Chorna, Geller, and Shpilrain., Comment: New version extends results to cases when $u=1$ and index computations/proofs are included

Published
2021

3. Subgroups of $SL_2(\mathbb{Z})$ characterized by certain continued fraction representations

Author

Han, Sandie, Masuda, Ariane M., Singh, Satyanand, and Thiel, Johann

Subjects
Mathematics - Group Theory, Mathematics - Number Theory
Abstract

For positive integers $u$ and $v$, let $L_u=\begin{bmatrix} 1 & 0 \\ u & 1 \end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0 & 1 \end{bmatrix}$. Let $S_{u,v}$ be the monoid generated by $L_u$ and $R_v$, and $G_{u,v}$ be the group generated by $L_u$ and $R_v$. In this paper we expand on a characterization of matrices $M=\begin{bmatrix}a & b \\c & d\end{bmatrix}$ in $S_{k,k}$ and $G_{k,k}$ when $k\geq 2$ given by Esbelin and Gutan to $S_{u,v}$ when $u,v\geq 2$ and $G_{u,v}$ when $u,v\geq 3$. We give a simple algorithmic way of determining if $M$ is in $G_{u,v}$ using a recursive function and the short continued fraction representation of $b/d$., Comment: Post-comment edits

Published
2019
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4. Mean Row Values in $(u,v)$-Calkin-Wilf Trees

Author

Han, Sandie, Masuda, Ariane M., Singh, Satyanand, and Thiel, Johann

Subjects
Mathematics - Number Theory
Abstract

We fix integers $u,v \geq 1$, and consider an infinite binary tree $\mathcal{T}^{(u,v)}(z)$ with a root node whose value is a positive rational number $z$. For every vertex $a/b$, we label the left child as $a/(ua+b)$ and right child as $(a+vb)/b$. The resulting tree is known as the $(u,v)$-Calkin-Wilf tree. As $z$ runs over $[1/u,v]\cap \mathbb{Q}$, the vertex sets of $\mathcal{T}^{(u,v)}(z)$ form a partition of $\mathbb{Q}^+$. When $u=v=1$, the mean row value converges to $3/2$ as the row depth increases. Our goal is to extend this result for any $u,v\geq 1$. We show that, when $z\in [1/u,v]\cap \mathbb{Q}$, the mean row value in $\mathcal{T}^{(u,v)}(z)$ converges to a value close to $v+\log 2/u$ uniformly on $z$.

Published
2018
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5. Maximal entries of elements in certain matrix monoids

Author

Han, Sandie, Masuda, Ariane M., Singh, Satyanand, and Thiel, Johann

Subjects
Mathematics - Number Theory, Mathematics - Combinatorics
Abstract

Let $L_u=\begin{bmatrix}1 & 0\\u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix}1 & v\\0 & 1\end{bmatrix}$ be matrices in $SL_2(\mathbb Z)$ with $u, v\geq 1$. Since the monoid generated by $L_u$ and $R_v$ is free, we can associate a depth to each element based on its product representation. In the cases where $u=v=2$ and $u=v=3$, Bromberg, Shpilrain, and Vdovina determined the depth $n$ matrices containing the maximal entry for each $n\geq 1$. By using ideas from our previous work on $(u,v)$-Calkin-Wilf trees, we extend their results for any $u, v\geq 1$ and in the process we recover the Fibonacci and some Lucas sequences. As a consequence we obtain bounds which guarantee collision resistance on a family of hashing functions based on $L_u$ and $R_v$., Comment: Fixed typos and added comment related to BSV, 31 pages

Published
2017

8. Orphans in Forests of Linear Fractional Transformations

Author

Han, Sandie, Masuda, Ariane M., Singh, Satyanand, and Thiel, Johann

Subjects
Mathematics - Number Theory, Mathematics - Combinatorics, 11A99, 11B75, 05C05
Abstract

This paper studies the set of orphans in certain Calkin-Wilf trees generated by linear fractional transformations., Comment: Expanded section on PLFTs, includes several new proofs and results. This version represents a significant reorganization over the previous version

Published
2015

9. Creating Dynamic Documents with R and Python as a Computational and Visualization Tool for Teaching Differential Equations

Author

Kostadinov, Boyan, Thiel, Johann, and Singh, Satyanand

Abstract

The goal of this paper is to promote the use of R and Python as high-level, free, open-source programming environments that can be used as a computational and visualization tool for teaching differential equations. Both R and Python also allow for creating reproducible dynamic documents using Markdown, which combines live code, plain text, and expressions to generate different formats (pdf, html, Word), including the numerical and graphical output from the code, along with the code itself, properly formatted with minimum effort. This technology can be used for creating interactive teaching documents, randomized exams, homework and project reports, and even publication quality papers. We illustrate this technology with several examples implemented in R and Python. This approach has been classroom-tested, with some promising results. This article has online supplemental material posted on the "PRIMUS" website.

Published
2019
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10. The (u,v)-Calkin-Wilf Forest

Author

Han, Sandie, Masuda, Ariane M., Singh, Satyanand, and Thiel, Johann

Subjects
Mathematics - Number Theory, Mathematics - Combinatorics, 11A55, 11B75, 05C05
Abstract

In this paper we consider a refinement, due to Nathanson, of the Calkin-Wilf tree. In particular, we study the properties of such trees associated with the matrices $L_u=\begin{bmatrix} 1 & 0 \\ u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0& 1\end{bmatrix}$, where $u$ and $v$ are nonnegative integers. We extend several known results of the original Calkin-Wilf tree, including the symmetry, numerator-denominator, and successor formulas, to this new setting. Additionally, we study the ancestry of a rational number appearing in a generalized Calkin-Wilf tree., Comment: 18 pages

Published
2014
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12. Curricular and Strategic Changes in mathematics to Enhance Institutional STEM Education.

Author

Han, Sandie, Kostadinov, Boyan, Liou-Mark, Janet, and Thiel, Johann

Subjects
STEM education, COLLABORATIVE learning, MATHEMATICS, DATA visualization, EXPERIENTIAL learning
Abstract

We implemented a project to create transformative changes in the STEM Education at a Hispanic-serving institution by revamping mathematics curriculum and building a system of support in mathematics learning. The project accomplished three main areas of curriculum development: (1) Restructuring the first-year mathematics courses at the college algebra level using a corequisite model; (2) designing and implementing active learning and problem-solving oriented curriculum in Calculus; (3) building a student support system of peer-tutoring and mentoring through peer-led team learning. High impact strategies were used for increasing and sustaining the participation of women and underrepresented minority students in computing. Strategies included hands-on active and collaborative learning pedagogy, experiential learning with real world relevance and applications, and curriculum that incorporates data analysis, data visualization and computational thinking. The support targeted introductory courses to provide early exposure and engagement. This paper describes the curriculum restructuring and development, which led to the institutional implementation and transformation of mathematics curriculum. The paper also reports on the success of the redesigned courses. [ABSTRACT FROM AUTHOR]

Published
2022

14. Subgroups of SL2(Z) characterized by certain continued fraction representations.

Author

Han, Sandie, Masuda, Ariane M., Singh, Satyanand, and Thiel, Johann

Subjects
CONTINUED fractions, FRACTIONS, RECURSIVE functions, INTEGERS, MATRICES (Mathematics)
Abstract

For positive integers u and v, let Lu = [1 & 0 \\ u & 1] and Rv = [1 & v \\ 0 & 1 ]. Let Su,v be the monoid generated by Lu and Rv, and let Gu,v be the group generated by Lu and Rv. In this paper we expand on a characterization of matrices M = [a & b \\ c & d] in Sk,k and Gk,k when k ≥ 2 given by Esbelin and Gutan to Su,v when u,v ≥ 2 and Gu,v when u,v ≥ 3. We give a simple algorithmic way of determining if M is in Gu,v using a recursive function and the short continued fraction representation of b/d. [ABSTRACT FROM AUTHOR]

Published
2020
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15. MAXIMAL ENTRIES OF ELEMENTS IN CERTAIN MATRIX MONOIDS.

Author

Sandie Han, Masuda, Ariane M., Singh, Satyanand, and Thiel, Johann

Subjects
MONOIDS, MATRICES (Mathematics), LINEAR orderings, POLYNOMIALS, LUTETIUM compounds, HASHING, SURETYSHIP & guaranty
Abstract

Let Lu =[¹ 0 u 1] and Rv =[¹ v 0 1]be matrices in SL2(Z) with u,v ≥ 1. Since the monoid generated by Lu and Rv is free, we can associate a depth to each element based on its product representation. In the cases where u = v = 2 and u = v = 3, Bromberg, Shpilrain, and Vdovina found a depth n matrix containing the maximal entry for each n ≥ 1. By using ideas from our previous work on positive linear fractional transformation (u,v)-Calkin-Wilf trees and a polynomial partial ordering, we extend their results for any u,v ≥ 1 and in the process we recover the Fibonacci and some Lucas sequences. As a consequence we obtain bounds which guarantee collision resistance on a family of Cayley hash functions based on Lu and Rv. [ABSTRACT FROM AUTHOR]

Published
2020

19. The -Calkin-Wilf forest.

Author

Han, Sandie, Masuda, Ariane M., Singh, Satyanand, and Thiel, Johann

Subjects
NUMBER theory, TREE graphs, MATRICES (Mathematics), INTEGERS, PARTITIONS (Mathematics), FRACTIONS
Abstract

In this paper, we consider a refinement, due to Nathanson, of the Calkin-Wilf tree. In particular, we study the properties of such trees associated with the matrices and , where and are non-negative integers. We extend several known results of the original Calkin-Wilf tree, including the symmetry, numerator-denominator, and successor formulas, to this new setting. Additionally, we study the ancestry of a rational number appearing in a generalized Calkin-Wilf tree. [ABSTRACT FROM AUTHOR]

Published
2016
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20. Arrangements of Stars on the American Flag.

Author

Koukoulopoulos, Dimitris and Thiel, Johann

Subjects
*FLAGS of the United States, *FLAG manifolds (Mathematics), *MATHEMATICAL symmetry, *U.S. states
Abstract

The article discusses the arrangement of stars on the American flag representing states in the U.S. The 50 star jack arrangement was designed by Robert G. Heft, designer of America's current national flag. A "Nice arrangement" of stars uses one of the six patterns of the Union Jack arrangement and exhibits a lot of symmetry which rarely changes with each adding star to the flag as number of states in the union increases.

Published
2012

23. Impietatem Doctrinae Pontificiae Pio Veritatis Stylo Improbatam, Consensu Maxime-Reverendae Facultatis Theologicae, In Illustri Prussorum Albertina, Praeside ... Domino Godofredo Wegnero, S.S. Theologiae D. & Prof. Publ. Concionatore Aulico Secund. meritissimo, Examini subiicit publico M. Johannes Thiel, Reg. Pruss. Auctor. In Auditorio Maiori, Anno MDCXCVI. die Martii ...

Author

Derschau, Friedrich von, Rock, Andreas, Werner, Georg, Kohlen, Christoph von, Schwenner, Peter, Mühlküntzel, Georg Wilhelm, Willemsen, Johannes, Kantel, Johannes, Bartsch, Henricus, Ditmer, Georgius, Concius, Jacobus, Hoynovius, Michael, Harnack, Johannes, Hoynovius, Daniel, Wegner, Gottfried, Thiel, Johann, Derschau, Friedrich von, Rock, Andreas, Werner, Georg, Kohlen, Christoph von, Schwenner, Peter, Mühlküntzel, Georg Wilhelm, Willemsen, Johannes, Kantel, Johannes, Bartsch, Henricus, Ditmer, Georgius, Concius, Jacobus, Hoynovius, Michael, Harnack, Johannes, Hoynovius, Daniel, Wegner, Gottfried, and Thiel, Johann

24. Impietatem Doctrinae Pontificiae Pio Veritatis Stylo Improbatam, Consensu Maxime-Reverendae Facultatis Theologicae, In Illustri Prussorum Albertina, Praeside ... Domino Godofredo Wegnero, S.S. Theologiae D. & Prof. Publ. Concionatore Aulico Secund. meritissimo, Examini subiicit publico M. Johannes Thiel, Reg. Pruss. Auctor. In Auditorio Maiori, Anno MDCXCVI. die Martii ...

Author

Derschau, Friedrich von, Rock, Andreas, Werner, Georg, Kohlen, Christoph von, Schwenner, Peter, Mühlküntzel, Georg Wilhelm, Willemsen, Johannes, Kantel, Johannes, Bartsch, Henricus, Ditmer, Georgius, Concius, Jacobus, Hoynovius, Michael, Harnack, Johannes, Hoynovius, Daniel, Wegner, Gottfried, Thiel, Johann, Derschau, Friedrich von, Rock, Andreas, Werner, Georg, Kohlen, Christoph von, Schwenner, Peter, Mühlküntzel, Georg Wilhelm, Willemsen, Johannes, Kantel, Johannes, Bartsch, Henricus, Ditmer, Georgius, Concius, Jacobus, Hoynovius, Michael, Harnack, Johannes, Hoynovius, Daniel, Wegner, Gottfried, and Thiel, Johann

25. Impietatem Doctrinae Pontificiae Pio Veritatis Stylo Improbatam, Consensu Maxime-Reverendae Facultatis Theologicae, In Illustri Prussorum Albertina, Praeside ... Domino Godofredo Wegnero, S.S. Theologiae D. & Prof. Publ. Concionatore Aulico Secund. meritissimo, Examini subiicit publico M. Johannes Thiel, Reg. Pruss. Auctor. In Auditorio Maiori, Anno MDCXCVI. die Martii ...

Author

Derschau, Friedrich von, Rock, Andreas, Werner, Georg, Kohlen, Christoph von, Schwenner, Peter, Mühlküntzel, Georg Wilhelm, Willemsen, Johannes, Kantel, Johannes, Bartsch, Henricus, Ditmer, Georgius, Concius, Jacobus, Hoynovius, Michael, Harnack, Johannes, Hoynovius, Daniel, Wegner, Gottfried, Thiel, Johann, Derschau, Friedrich von, Rock, Andreas, Werner, Georg, Kohlen, Christoph von, Schwenner, Peter, Mühlküntzel, Georg Wilhelm, Willemsen, Johannes, Kantel, Johannes, Bartsch, Henricus, Ditmer, Georgius, Concius, Jacobus, Hoynovius, Michael, Harnack, Johannes, Hoynovius, Daniel, Wegner, Gottfried, and Thiel, Johann

26. Impietatem Doctrinae Pontificiae Pio Veritatis Stylo Improbatam, Consensu Maxime-Reverendae Facultatis Theologicae, In Illustri Prussorum Albertina, Praeside ... Domino Godofredo Wegnero, S.S. Theologiae D. & Prof. Publ. Concionatore Aulico Secund. meritissimo, Examini subiicit publico M. Johannes Thiel, Reg. Pruss. Auctor. In Auditorio Maiori, Anno MDCXCVI. die Martii ...

Author

Derschau, Friedrich von, Rock, Andreas, Werner, Georg, Kohlen, Christoph von, Schwenner, Peter, Mühlküntzel, Georg Wilhelm, Willemsen, Johannes, Kantel, Johannes, Bartsch, Henricus, Ditmer, Georgius, Concius, Jacobus, Hoynovius, Michael, Harnack, Johannes, Hoynovius, Daniel, Wegner, Gottfried, Thiel, Johann, Derschau, Friedrich von, Rock, Andreas, Werner, Georg, Kohlen, Christoph von, Schwenner, Peter, Mühlküntzel, Georg Wilhelm, Willemsen, Johannes, Kantel, Johannes, Bartsch, Henricus, Ditmer, Georgius, Concius, Jacobus, Hoynovius, Michael, Harnack, Johannes, Hoynovius, Daniel, Wegner, Gottfried, and Thiel, Johann

27. Impietatem Doctrinae Pontificiae Pio Veritatis Stylo Improbatam, Consensu Maxime-Reverendae Facultatis Theologicae, In Illustri Prussorum Albertina, Praeside ... Domino Godofredo Wegnero, S.S. Theologiae D. & Prof. Publ. Concionatore Aulico Secund. meritissimo, Examini subiicit publico M. Johannes Thiel, Reg. Pruss. Auctor. In Auditorio Maiori, Anno MDCXCVI. die Martii ...

Author

Derschau, Friedrich von, Rock, Andreas, Werner, Georg, Kohlen, Christoph von, Schwenner, Peter, Mühlküntzel, Georg Wilhelm, Willemsen, Johannes, Kantel, Johannes, Bartsch, Henricus, Ditmer, Georgius, Concius, Jacobus, Hoynovius, Michael, Harnack, Johannes, Hoynovius, Daniel, Wegner, Gottfried, Thiel, Johann, Derschau, Friedrich von, Rock, Andreas, Werner, Georg, Kohlen, Christoph von, Schwenner, Peter, Mühlküntzel, Georg Wilhelm, Willemsen, Johannes, Kantel, Johannes, Bartsch, Henricus, Ditmer, Georgius, Concius, Jacobus, Hoynovius, Michael, Harnack, Johannes, Hoynovius, Daniel, Wegner, Gottfried, and Thiel, Johann

28. Honores In Philosophia Summos In Inclyta ad Albim Academia, Rectore ... Domino Michaele Sennerto, Phil. & Med. D. & Prof. Publ. ... a Decano ... Domino Conrado Samuele Schurtzfleisch/ Poeseos Prof. Publ. ... Viro-Iuveni Godofredo Kipffendero, Olsna Silesio. d. 27. April. Anno 1676. publice collatos Votivis Acclamationibus excipiebant Gottfried Heidenreich/ Johan. Carl Thiel/ Christianus Banner/ Christianus Tschirbock Olsna-Siles.

Author

Thiel, Johann Carl, Banner, Christian, Tschirboch, Christian, Kipffender, Gottfried, Heidenreich, Gottfried, Thiel, Johann Carl, Banner, Christian, Tschirboch, Christian, Kipffender, Gottfried, and Heidenreich, Gottfried

Abstract

Schlüsselseiten aus dem Exemplar der UB Erfurt: 13 - B. 2° 00012 (02), Vorlageform des Erscheinungsvermerks: Wittebergae, Typis Johannis Borckardi , Acad. Typ. Excudebat Simon Lieberhirt. - Erscheinungsjahr nach der Datierung im Titel angegeben

29. Homo Oriens & Florens

Author

Wippermann, Engelbert, Thiel, Johann Carl, Wippermann, Engelbert, and Thiel, Johann Carl

Abstract

Quem Pro Licentia Summos in utroque iure Honores Ac Privilegia Doctoralia consequendi ... Sub Praesidio ... Dn. Engelberti Wippermann ... In Illustri Academia Hasso-Schaumb. ad Examen & intuitum ... Eruditorum sistit Ad diem Iovis & Veneris. Johann Carl Thiel, Schlüsselseiten aus dem Exemplar der SLUB Dresden: Diss.jur.civ.239,16

31. Dissertatio Historica, De Genealogia Lutheri

Author

Thiel, Johann, Harnack, Georgius, Thiel, Johann, and Harnack, Georgius

Abstract

Quam ... In Academia Patria, Publico Eruditorum Examini submittet, Praeses M. Johannes Thiel, Regiomont. Prussus, Respondente Georgio Harnack, Hoh. Pr. S.S. Theol. & Phil. Stud. Horis locoque solitis Anno MDCXCV. Die Februar, Schlüsselseiten aus dem Exemplar der SLUB Dresden: Biogr.erud.D.1617,2, Erscheinungsjahr nach der Datierung im Titel

34. Honores In Philosophia Summos In Inclyta ad Albim Academia, Rectore ... Domino Michaele Sennerto, Phil. & Med. D. & Prof. Publ. ... a Decano ... Domino Conrado Samuele Schurtzfleisch/ Poeseos Prof. Publ. ... Viro-Iuveni Godofredo Kipffendero, Olsna Silesio. d. 27. April. Anno 1676. publice collatos Votivis Acclamationibus excipiebant Gottfried Heidenreich/ Johan. Carl Thiel/ Christianus Banner/ Christianus Tschirbock Olsna-Siles.

Author

Thiel, Johann Carl, Banner, Christian, Tschirboch, Christian, Kipffender, Gottfried, Heidenreich, Gottfried, Thiel, Johann Carl, Banner, Christian, Tschirboch, Christian, Kipffender, Gottfried, and Heidenreich, Gottfried

Abstract

Schlüsselseiten aus dem Exemplar der UB Erfurt: 13 - B. 2° 00012 (02), Vorlageform des Erscheinungsvermerks: Wittebergae, Typis Johannis Borckardi , Acad. Typ. Excudebat Simon Lieberhirt. - Erscheinungsjahr nach der Datierung im Titel angegeben

37. Honores In Philosophia Summos In Inclyta ad Albim Academia, Rectore ... Domino Michaele Sennerto, Phil. & Med. D. & Prof. Publ. ... a Decano ... Domino Conrado Samuele Schurtzfleisch/ Poeseos Prof. Publ. ... Viro-Iuveni Godofredo Kipffendero, Olsna Silesio. d. 27. April. Anno 1676. publice collatos Votivis Acclamationibus excipiebant Gottfried Heidenreich/ Johan. Carl Thiel/ Christianus Banner/ Christianus Tschirbock Olsna-Siles.

Author

Thiel, Johann Carl, Banner, Christian, Tschirboch, Christian, Kipffender, Gottfried, Heidenreich, Gottfried, Thiel, Johann Carl, Banner, Christian, Tschirboch, Christian, Kipffender, Gottfried, and Heidenreich, Gottfried

Abstract

Schlüsselseiten aus dem Exemplar der UB Erfurt: 13 - B. 2° 00012 (02), Vorlageform des Erscheinungsvermerks: Wittebergae, Typis Johannis Borckardi , Acad. Typ. Excudebat Simon Lieberhirt. - Erscheinungsjahr nach der Datierung im Titel angegeben

38. Homo Oriens & Florens

Author

Wippermann, Engelbert, Thiel, Johann Carl, Wippermann, Engelbert, and Thiel, Johann Carl

Abstract

Quem Pro Licentia Summos in utroque iure Honores Ac Privilegia Doctoralia consequendi ... Sub Praesidio ... Dn. Engelberti Wippermann ... In Illustri Academia Hasso-Schaumb. ad Examen & intuitum ... Eruditorum sistit Ad diem Iovis & Veneris. Johann Carl Thiel, Schlüsselseiten aus dem Exemplar der SLUB Dresden: Diss.jur.civ.239,16

41. Dissertatio Historica, De Genealogia Lutheri

Author

Thiel, Johann, Harnack, Georgius, Thiel, Johann, and Harnack, Georgius

Abstract

Quam ... In Academia Patria, Publico Eruditorum Examini submittet, Praeses M. Johannes Thiel, Regiomont. Prussus, Respondente Georgio Harnack, Hoh. Pr. S.S. Theol. & Phil. Stud. Horis locoque solitis Anno MDCXCV. Die Februar, Schlüsselseiten aus dem Exemplar der SLUB Dresden: Biogr.erud.D.1617,2, Erscheinungsjahr nach der Datierung im Titel

45. Calculus One

Author

Holowinsky, Roman, author, Thiel, Johann, author, and Lindberg, David, author

Subjects
Mathematics, Calculus, Textbooks
Abstract

Calculus is about the very large, the very small, and how things change—the surprise is that something seemingly so abstract ends up explaining the real world. This course is a first and friendly introduction to calculus, suitable for someone who has never seen the subject before, or for someone who has seen some calculus but wants to review the concepts and practice applying those concepts to solve problems. One learns calculus by doing calculus, and so this course is based around doing practice problems.

Published
2014

46. Landscape - Marshland

Author

Thiel, Johannes and Fine Arts Museums of San Francisco

Subjects
European, Northern European, German, Germany
Published
1889

47. [Landscape]

Author

Thiel, Johannes and Fine Arts Museums of San Francisco

Subjects
European, Northern European, German, Germany
Published
1889

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