# Math 4600 Homework 1

## Math 4600, Spring 2012

**Lecturer:**Alla Borisyuk

**Office:**LCB 303,

**Phone:**(801) 585-1639

**E-Mail:**borisyuk at math.utah.edu

**Office Hours:**Th 11-12 in LCB 303

**TA:**Christopher Remien (remien at math.utah.edu)

**Office:**LCB 318

**Office hours:**Thursday 1:45-2:45 in LCB 115

**The goals of the class**are (i) to introduce the students to a range of modern mathematical tools (from dynamical systems, stochastic processes and partial differential equations); (ii) to teach the students the skill of building tractable mathematical models of biological processes; (iii) to show how to combine the mathematical knowledge, the numerical simulations (in Matlab) and biological intuition to derive new insights into the functioning of living systems.

Mathematical topics will include introduction to linear algebra, complex numbers, geometric dynamical systems, partial differential equations

Biological topics will include modeling heart and circulation, brain rhythms, HIV, insulin-secreting cells, antibiotic resistance in bacteria, regulation of gene expression, biological pattern formation

**Class time:**TTh 12:25-1:45 in LCB 215,

**Required computer lab:**Friday 11:50 or 12:55 in LCB 115

**Prerequisites:**Math 2250 or Math 2270-2280 sequence

**4 credit hours**(3 lecture hours + a computer lab a week)

**Take-home final, due Tuesday May 1 at 12:30pm**

**In class midterm is March 1**

Review session February 28

**Midterm practice, Part 1**

**Midterm practice, Part 2**

**Homeworks/Labs**

**Lecture Material**

**List of books**

**With any questions, please, contact me by email**: borisyuk at math.utah.edu

**Text:**there is no required text. Students are expected to take notes in class. Supplementary material in the form of book excerpts, and research papers will be provided as necessary.

**Homework:**Included with weekly lab assignments, contains both mathematical exercises and simulations in Matlab.

**Tests:**In class midterm February 23 (preliminary date), take-home final exam

**Grade:**labs (40%) + midterm (30%) + final (30%)

**List of Topics:**

I. Introduction to mathematical models, review of ideas from calculus and ODEs

- Heart and circulation. Dynamics of the pulse

II. Dynamical systems: geometric analysis, phase planes. Introduction to bifurcation theory. Phase oscillators, phase response curves (Supplementary material, Strogatz).

- Law of mass action, Hill's function

- Gene activation model. Bistability and biological switches.

- Hodgkin-Huxley equations

- Pancreatic beta-cells

- Epidemics

- HIV modeling

~ 5 weeks.

III. Probability, random variables, Markov chains.

- Natural selection

- Plasmids (antibiotic resistance in bacteria)

- Regulation of gene expression

~ 4 weeks.

IV. Introduction to partial differential equations

- Models for cancer-immune system interaction and for cancer-growth inhibitor interaction

~ 2 weeks.

**ADA statement.**The University of Utah seeks to provide equal access to its programs, services and activities for people with disabilities. If you will need accommodations in the class, reasonable prior notice needs to be given to the Center for Disability Services, 162 Olpin Union Building, 581-5020 (V/TDD). CDS will work with you and the instructor to make arrangements for accommodations

Here is a list of suggested homework problems from the new Kendall-Hunt text.

The problems with numbers in parentheses are somewhat similar to other problems in the list. However, this is a subject that rewards practice, so solving all of the listed problems is a wise use of time.

The final exam will assume familiarity with the material covered by these problems.

The exercises are listed by section; see the syllabus to determine which sections go with which lecture.

The answers (not solutions) to the odd-numbered problems in this list are in the back of the textbook. Here is a link to the answers (prepared by Math 135 instructor M. Lieberman) to the even-numbered problems starting with Chapter 2 in this list. But be sure to work on the problems yourself before you check your work by looking up the answers.

SECTION | PROBLEMS |
---|---|

1.2 | 2, 4, 10, 24, 28, 36. |

1.3 | 2, 12, 13, 20, 40, 42, (45). |

1.4 | 10, 14, 21, 24, 25b, 29, 37, 38, 48, 51. |

2.1 | 1, 2, (3), 4, (5), 6, 13, 29, 49. |

2.2 | 6, 8, 12, 14, 16, 18, 22, 23, 26, 38, 39, 41, 49, 52, 55. |

2.3 | 16, 21, 25, 30, 35, 39, 45, 46, 54, 43, 44. |

2.4 | 6, 7, 10, 12, 21, 27, 29, 32, 33, 35, 36, 44, (47), 49, 50. |

3.1 | 6, 8, 10, 12, 14, 22, 23, 24, 26, 32, 33, 38, 41, 42, 43. |

3.2 | 8, 9, 12, 15, 18, 24, 29, 36, 41. |

3.3 | 1, 4, 11, 17, (18), 29, 37, 41, 52. |

3.4 | 3, 5, (7), 12, (13), 16, (19), 22, 34, 39, 48, 49, (53). |

3.5 | 6, 8, 12, 21, 24, 31, 34, 38, 42, 46, 47, 48. |

3.6 | 1, 4, (5), 7, 8, 9, 11, 14, 26, 27, 31, 35, 36, 38, 43, 45, 51, 52. |

3.7 | 8, 14, 21, 23, 26, (28), 29, (30), 31, 35, (36), 37, 38, 39, 40, (41), (46), 53. |

3.8 | 3, 4, (8), 13, 19, 20, 23, 25, 28, 29, 31, 33, 35, (40), 42, (44), 45. |

4.1 | 4, 5, 11, (12), 17, 25, 27, 32, 36, 42, 50. |

4.2 | 7, 10, 21, 22, 27, 30. |

4.3 | 5, 6, 11, 25, 34, 36, 40, 42, 45. |

4.4 | 10, 11, 12, 15, 20, 23, 27, 29, 33, 38, 42, 47, 48. |

4.5 | 1, 3, 6, 7, 11, 12, 13, 17, 21, 23, 30, 37, 38, 39. Also: the Section 4.3 problems 17, 19, 27 |

4.6 | 7, 8, 16, 27, 28, 34, 35, 39, 42. |

4.7 | 6, 13, 14, 15, 16, 18, 20, 21, 25, 26, 28, 38. |

5.1 | 7, 8, 9, 10, 11, 17, 23, 26, 40, 41, 43, 44, 46. |

5.2 | 4, 8, 25, 28. |

5.3 | 3, 4, 6, 35, 38, 44, 45, 46. |

5.4 | 2, 7, 10, 14, 17, 23, 29, 32, 33, 35, 40, 45, 51. |

5.5 | 1, 3, 6, 9, 10, 13, 15, 16, 21, 27, 30, 40, 44, 49, 51. |

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