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10: Some Prerequisite Topics

10: Some Prerequisite Topics


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The topics presented in this section are important concepts in mathematics and therefore should be examined.

  • 10.1: Sets and Set Notation
    A set is a collection of things called elements. For example {1,2,3,8} would be a set consisting of the elements 1,2,3, and 8. To indicate that 3 is an element of {1,2,3,8}, it is customary to write 3∈{1,2,3,8}. We can also indicate when an element is not in a set, by writing 9∉{1,2,3,8} which says that 9 is not an element of {1,2,3,8}. Sometimes a rule specifies a set.
  • 10.2: Well Ordering and Induction

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Overview

Why Prerequisites at Hopkins?

  • Top-ranked Nursing School 
  • Easy Application Process
  • Affordable
  • No transcripts required 
  • An introduction to the excellence of a Johns Hopkins education
  • Health-focused perspective delivered through a facilitated teaching approach (participatory, interactive, engaging)
  • Instructor led, online convenience
  • 10 week format

All students pursuing a health-based education can now take the following instructor-led prerequisites all online and get a taste of Hopkins Nursing.

  • Nutrition
  • Human Growth and Development Through the Lifespan
  • Biostatistics
  • Microbiology with virtual lab*
  • Anatomy with virtual lab*
  • Physiology with virtual lab*
  • Chemistry with virtual lab§
  • Biochemistry with virtual lab§

For a full list of required courses visit our pre-licensure Master of Science in Nursing: Entry in Nursing program webpage.

*Virtual labs are accepted at Hopkins Nursing, but not everywhere. Check your university and state licensure requirements for prerequisite courses. 

§Offered but not required for the Johns Hopkins School of Nursing MSN Entry into Nursing program.

Registration & Tuition

All prerequisite courses are available in the summer, fall, and spring semesters.

  • Summer 2021: May 25- August 2
  • Fall 2021: August 31 - November 8
  • Spring 2022: January 25 - April 4

Registration Questions? Contact [email protected] or look in our FAQS.  Returning students register through the Johns Hopkins University Student Information System.

Students may enroll in up to three courses per semester, but should not enroll in more than two lab courses at the same time. Anatomy is to be taken prior to Physiology and may not be taken in the same semester. 

Tuition

  • $350 per credit
  • $1050 per 3 credit course
  • $1400 per 4 credit course

Tuition is due at time of registration.

Course Description

This course will cover the science and fundamentals of human nutrition. Topics covered include
nutritional requirements related to changing individual and family needs, food choices, health
behaviors, food safety, prevention of chronic disease and nutrition-related public health in the
United States and globally. (3 credits)

SAMPLE Course Syllabus (pdf download)
Note: This is a SAMPLE course syllabus and does not necessarily reflect the most recent version of the course syllabus. Do not order the textbook found on this sample syllabus. For further information regarding materials for this course, please visit the Prerequisite Textbook Information page

Course Outcomes

  1. Know the six classes of nutrients and explain their role as it relates to promoting optimal health, information on food labels, and the accuracy of statements made in popular media about nutrition.
  2. Relate the importance of good nutrition to different stages in human development and the promotion of a healthy lifestyle.
  3. Determine, compare and contrast the nutritional value of current eating habits to current recommendations and propose modifications to reduce the risk for developing chronic diseases.
  4. Identify strategies to eating a healthy diet in different cultural and environmental settings.

Required Textbook

For further information regarding materials for this course, please visit the Prerequisite Textbook
Information page

Course Description

This course provides an overview of major concepts, theories, and research related to human
development through the lifespan from the prenatal period to the end of life. Significant factors that
influence individual functioning are explored. (3 credits)

SAMPLE Course Syllabus (pdf download)
Note: This is a SAMPLE course syllabus and does not necessarily reflect the most recent version of the course syllabus. Do not order the textbook found on this sample syllabus. For further information regarding materials for this course, please visit the Prerequisite Textbook Information page

Course Outcomes

Analyze theoretical and conceptual frameworks and research findings related to human development through the lifespan.

Apply theoretical models and research findings of human development and functioning to health and illness behaviors through the lifespan and within a variety of biological, environmental, social and cultural contexts.

Required Textbook

For further information regarding materials for this course, please visit the Prerequisite Textbook
Information page.

Course Description

This course provides an introduction to the basic concepts of statistical ideas and methods that aims
to equip students to carry out common statistical procedures and to follow statistical reasoning in
their fields of study. Principles of measurement, data summarization, and univariate and bivariate
statistics are examined. Emphasis is placed on the application of fundamental concepts to real
world situations. (3 credits)

SAMPLE Course Syllabus (pdf download)
Note: This is a SAMPLE course syllabus and does not necessarily reflect the most recent version of the course syllabus. Do not order the textbook found on this sample syllabus. For further information regarding materials for this course, please visit the Prerequisite Textbook Information page

Course Outcomes

  1. Summarize and interpret data visually through appropriate statistical graphs.
  2. Describe density curves and the properties of the normal distributions.
  3. Examine correlations and linear relationships of explanatory and response variables.
  4. Describe sampling distributions and the central limit theorem.
  5. Discuss statistical inference using confidence intervals and tests of significance.
  6. Explain the differences among various statistical techniques and identify an appropriate technique for a given set of variables and research questions.

Required Textbook

For further information regarding materials for this course, please visit the Prerequisite Textbook
Information page.

Course Description

This course introduces the core concepts and basic principles in microbiology, examining
microorganisms and how they interact with humans and the environment. Information regarding
classification of microorganisms, characteristics of different cell types and processes critical for cell
survival is presented. Topics such as bacterial metabolism, microbial nutrition, genetics, antimicrobial
approaches and interaction of pathogenic bacteria with humans are discussed. The course
includes a virtual laboratory component designed to complement lecture topics. The course
content provides the foundation of general microbiology necessary for students who are interested
in applying to health profession programs. (4 credits)

SAMPLE Course Syllabus (pdf download)
Note: This is a SAMPLE course syllabus and does not necessarily reflect the most recent version of the course syllabus. Do not order the textbook found on this sample syllabus. For further information regarding materials for this course, please visit the Prerequisite Textbook Information page

Course Outcomes

  1. Describe and differentiate among the broad classes of microorganisms, including bacteria, protozoa, fungi, helminthes, and viruses.
  2. Describe in appropriate terminology the structure, function and characteristics of prokaryotes, eukaryotes and viruses.
  3. Explain the metabolic processes necessary for microbe survival, focusing on the different methods of energy acquisition.
  4. Describe ways microbes can cause infection and pathology in humans and apply this understanding to infection prevention and control in healthcare settings.
  5. Identify strategies employed by antimicrobial drugs and how they specifically target certain pathogens and apply this understanding to antimicrobial treatment, drug resistance and interaction with host.
  6. Demonstrate knowledge and skills in common laboratory procedures.

Required Textbook and Course Materials

The custom bundle required for this course includes an electronic copy of the textbook and access to resources within McGraw-Hill Connect.

For further information regarding materials for this course, please visit the Prerequisite Textbook
Information page.

Course Description

This course will introduce components and structures of the human body at the level of gross and microscopic anatomy.  Students will learn organ localization in the body and structural features comprising the different body systems.  The body systems covered will include the skin, heart, lungs, and brain, among others.  Upon completion, students will have an understanding of normal healthy anatomy that will prepare them for professional health programs. This course includes a virtual laboratory component designed to complement lecture topics. (4 credits). 

SAMPLE Course Syllabus (pdf download)
Note: This is a SAMPLE course syllabus and does not necessarily reflect the most recent version of the course syllabus. Do not order the textbook found on this sample syllabus. For further information regarding materials for this course, please visit the Prerequisite Textbook Information page

Course Outcomes

  1. Define the body orientation terms, including planes of section, directional terms, body regions, pleura and pericardium and organ systems.
  2. Identify human body systems and major organs located in each system.
  3. Describe the general anatomical structures and their locations associated with each body system.
  4. Recognize the various layers and normal histology of the integumentary system.
  5. List key components of the skeletal and muscular systems.
  6. Describe the anatomical features of the cardiovascular and respiratory systems.
  7. Detail the important anatomy of the brain and head.
  8. Identify the gross anatomical structures of the digestive, urinary, and reproductive systems.

Required Textbooks and Course Materials

The textbook for this course will be used for both the Anatomy and the Physiology courses. This custom bundle includes an electronic copy of the textbook, access to resources and assignments within McGraw-Hill Connect, and access to Anatomy and Physiology Revealed 3.2.

For further information regarding materials for this course, please visit the Prerequisite Textbook
Information page.

Course Description

This course will introduce the functions of several human body systems.  Students will learn how each part within a body system works together to seamlessly accomplish tasks.  We will also discuss regulation of organ function, a critical component of physiology.  After an introduction on electrolytes, the physiologic processes we will cover include cardio vasculature, lymphatics, and digestion among others.  Upon completion, students will have an understanding of normal healthy anatomical function that will prepare them for professional health programs. This course includes a virtual laboratory component designed to complement lecture topics. (4 credits). 

SAMPLE Course Syllabus (pdf download)
Note: This is a SAMPLE course syllabus and does not necessarily reflect the most recent version of the course syllabus. Do not order the textbook found on this sample syllabus. For further information regarding materials for this course, please visit the Prerequisite Textbook Information page

Course Outcomes

  1. Describe the functions and interactions of major organ systems in human body.
  2. Explain the mechanism and importance of maintaining water, electrolytes, acid-base balance.
  3. Discuss the function of the cardiovascular system.
  4. Describe the key purpose of the lymphatic and specific cranial elements.
  5. List the functions of the components necessary for respiration, digestion, and urination.
  6. Identify the functions of the regulatory elements, i.e. hormones, of the reproductive systems.

Required Textbooks and Course Materials

The textbook for this course will be used for both the Anatomy and the Physiology courses. This custom bundle includes an electronic copy of the textbook, access to resources and assignments within McGraw-Hill Connect, and access to Anatomy and Physiology Revealed 3.2.

For further information regarding materials for this course, please visit the Prerequisite Textbook
Information page.

Course Description

This course introduces the core concepts of matter and energy, atomic structure, the periodic system, chemical bonding, nomenclature, stoichiometry, weight relationships, gases, solutions, chemical reactions, thermodynamics and equilibrium. The course includes a virtual laboratory component designed to enhance lecture topics. The course content provides the foundation of general chemistry necessary for students who are interested in applying to health profession programs. (4 credits)

SAMPLE Course Syllabus (pdf download)
Note: This is a SAMPLE course syllabus and does not necessarily reflect the most recent version of the course syllabus. Do not order the textbook found on this sample syllabus. For further information regarding materials for this course, please visit the Prerequisite Textbook Information page

* Offered, but not required for the Johns Hopkins School of Nursing MSN Entry into Nursing Program.

Course Outcomes

  1. Interconvert amount of substance between moles, mass and molecular weight
  2. Use conversion factors in calculations involving solids, liquids, gases, solutions, heat and energy.
  3. Calculate and express solution concentrations in various ways, such as mass percent, parts per million, mole fraction, molality, and molarity.
  4. Write balanced chemical equations and distinguish between different types of chemical reactions.
  5. Describe the major components of an atom, write symbols for isotopes and calculate the average masses of elements.
  6. Predict direction of change in reactions at equilibrium and measure reaction rates.
  7. Predict the types of intermolecular forces within a compound.
  8. Describe the geometry and polarity of molecules and predict their physical properties.
  9. Describe the properties of acids and bases and measure their concentrations in solutions.

Required Textbook and Course Materials

The custom bundle required for this course includes an electronic copy of the textbook and access to resources within Cengage OWL. Students will also need to purchase access to Labster to complete course lab content.

For further information regarding materials for this course, please visit the Prerequisite Textbook
Information page.

Prerequisite

NR.110.206 Chemistry with Lab or the equivalent

Course Description

Biochemistry is a natural science that investigates life processes at the molecular level. This course begins with an introduction to the structure and function of the four classes of biomolecules: proteins, nucleic acids, carbohydrates, and lipids. In the second half of the course, glycolysis, the citric acid cycle, and oxidative phosphorylation will provide a context for an introduction to the fundamentals of enzyme catalysis, kinetics, bioenergetics, and metabolic regulation. The virtual lab promotes mastery of the lecture content while exploring lab techniques used in biochemical research. Upon completion, students will have a solid background in the science that provides the foundation of the biomedical sciences.

SAMPLE Course Syllabus (pdf download)
Note: This is a SAMPLE course syllabus and does not necessarily reflect the most recent version of the course syllabus. Do not order the textbook found on this sample syllabus. For further information regarding materials for this course, please visit the Prerequisite Textbook Information page

* Offered, but not required for the Johns Hopkins School of Nursing MSN Entry into Nursing Program.

Course Outcomes

  1. Describe the affect of pH on chemical structure and the relevance to biochemistry.
  2. Describe the structure and function of the four classes of biomolecules: proteins, nucleic acids, carbohydrates, and lipids.
  3. Explain how enzymes increase the rate of biochemical reactions.
  4. Explain how animal cells extract and use energy from food.
  5. Explain how enzymes are used to regulate metabolic pathways.
  6. Describe the lab techniques used to study biomolecules.

Required Textbook and Course Materials

Pratt, C. & Cornely, K. (2018). Essential Biochemistry, (4th ed.). Hoboken, NJ: Wiley.

WileyPLUS with ORION adaptive learning primer. Students will also need to purchase access to Labster to complete course lab content.

All prerequisite courses are available in the summer, fall, and spring semesters. First-time students need to submit a simple online application form.

Office of Admissions

Join our mailing list and receive updated information about our prereq courses.

Spring 2022
January 25 – April 4

Summer 2021
May 25 – August 2

Fall 2021
August 31 – November 8

Please note that Blackboard access will be available one week before course start date.

Recorded Events

Prerequisite Info Session and Virtual Classroom Tour, May 17, 2021.


How to start a business: 10 steps to starting a business

Starting a business is exciting&mdashbut also demanding. This guide addresses some of the most common startup steps to ensure your company is ready for success.

Prepare a business plan and materials

  1. An important first step is preparing a business plan to define your business, products and services, and outline your goals, operating procedures and competition. If your company needs funding from a traditional loan or venture capitalists, a business plan will be required. Make sure your plan includes a marketing approach, so people are aware of what you're selling and how to find you.
  2. Create a business logo, cards and stationery. These items establish your company&rsquos identity and help potential customers find and remember you. Meet legal requirements
  3. Of course, incorporating your business or forming an LLC with the state is important because it protects your personal assets from business debts and liabilities. Other benefits of forming a corporation or LLC include tax advantages and greater credibility with customers, vendors and business partners.
  4. Select an accountant and attorney. Many small business owners seek advice from accountants and attorneys. As you search for an accountant and attorney, get referrals from friends or family, and look for professionals who have worked with other small business owners or companies in your specific industry.
  5. Get necessary tax identification numbers, licenses and permits. A federal tax identification number, or employer identification number (EIN), acts like a social security number and is required for corporations and LLCs that will have employees. Contact your state's taxation department to learn if a state tax identification number is required in your state. Also keep in mind that most businesses need licenses and/or permits to operate&mdashin your city, municipality, county and/or state.
  6. Insure your business and investigate other requirements. Some industries have specific insurance requirements. Discuss your needs with your insurance agent to get the right type and amount of insurance. Remember to look into any other government tax and insurance requirements that might apply to your business, particularly if you have employees. For example:
    • Unemployment insurance
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    • OSHA requirements
    • Federal tax
    • State and local tax
    • Self-employment tax
    • Payroll tax requirements (such as FICA, federal unemployment tax, and state unemployment tax)
    • Sales and use tax


Mathematics (MATH)

Basic equations and inequalities, linear and quadratic functions, and systems of simultaneous equations.

Grading Option: Workload Credit Letter Grade with P/NP.

MATHفB. Pre-Calculus. 4 Units.

Preparation for calculus and other mathematics courses. Exponentials, logarithms, trigonometry, polynomials, and rational functions. Satisfies no requirements other than contribution to the 180 units required for graduation.

Prerequisite: MATHفA. Placement into MATHفB via the Calculus Placement exam, or a score of 450 or higher on the Mathematics section of the SAT Reasoning Test.

Restriction: MATHفB may not be taken for credit if taken after MATHقA.

MATHقA. Single-Variable Calculus. 4 Units.

Introduction to derivatives, calculation of derivatives of algebraic and trigonometric functions applications including curve sketching, related rates, and optimization. Exponential and logarithm functions.

Prerequisite: MATHفB or AP Calculus AB or SAT Mathematics or ACT Mathematics. MATHفB with a grade of C or better. AP Calculus AB with a minimum score of 3. SAT Mathematics with a minimum score of 650. ACT Mathematics with a minimum score of 29. Placement via the Calculus Placement exam (fee required) is also accepted.

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment. School of Info & Computer Sci students have first consideration for enrollment.

MATHقB. Single-Variable Calculus. 4 Units.

Definite integrals the fundamental theorem of calculus. Applications of integration including finding areas and volumes. Techniques of integration. Infinite sequences and series.

Prerequisite: MATHقA or MATHمA or AP Calculus AB or AP Calculus BC. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 3

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment. School of Info & Computer Sci students have first consideration for enrollment.

MATHقD. Multivariable Calculus. 4 Units.

Differential and integral calculus of real-valued functions of several real variables, including applications. Polar coordinates.

Prerequisite: MATHقB or MATHمB or AP Calculus BC. AP Calculus BC with a minimum score of 4

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment. School of Info & Computer Sci students have first consideration for enrollment. Undeclared Majors have first consideration for enrollment.

MATHقE. Multivariable Calculus. 4 Units.

The differential and integral calculus of vector-valued functions. Implicit and inverse function theorems. Line and surface integrals, divergence and curl, theorems of Greens, Gauss, and Stokes.

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment.

MATH H2D. Honors Multivariable Calculus. 4 Units.

Differential and integral calculus of real-valued functions of several real variables, including applications. Polar coordinates. Covers the same material as MATHقD-E, but with a greater emphasis on the theoretical structure of the subject matter.

Prerequisite: MATHقB or MATHمB or (AP Calculus BC and (MATH H3A) or (MATHكA and MATH㺍)). MATHقB with a grade of A or better. MATHمB with a grade of A or better. AP Calculus BC with a minimum score of 5. MATH H3A with a grade of B- or better. MATHكA with a grade of A or better. MATH㺍 with a grade of A or better

MATH H2E. Honors Multivariable Calculus. 4 Units.

Differential and integral calculus of real-valued functions of several real variables, including applications. Polar coordinates. Covers the same material as MATHقD-E, but with a greater emphasis on the theoretical structure of the subject matter.

Prerequisite: MATH H2D. MATH H2D with a grade of B- or better

MATHكA. Introduction to Linear Algebra. 4 Units.

Systems of linear equations, matrix operations, determinants, eigenvalues and eigenvectors, vector spaces, subspaces, and dimension.

Prerequisite: MATHقB or MATHمB or AP Calculus BC. AP Calculus BC with a minimum score of 4

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment. Undeclared Majors have first consideration for enrollment.

MATHكD. Elementary Differential Equations. 4 Units.

Linear differential equations, variation of parameters, constant coefficient cookbook, systems of equations, Laplace tranforms, series solutions.

Prerequisite: (MATHكA or MATH H3A) and (MATHقD or MATH H2D) and (MATHقB or AP Calculus BC). AP Calculus BC with a minimum score of 4

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment.

MATH H3A. Honors Introduction to Linear Algebra. 4 Units.

Systems of linear equations, matrix operations, determinants, eigenvalues, eigenvectors, vector spaces, subspaces, and dimension.

Prerequisite: MATHقB or MATHمB or AP Calculus BC. MATHقB with a grade of A or better. MATHمB with a grade of A or better. AP Calculus BC with a minimum score of 5

Restriction: School of Physical Sciences students only. School of Engineering students only. Mathematics Majors only. Undeclared Majors only.

MATHمA. Calculus for Life Sciences. 4 Units.

Differential calculus with applications to life sciences. Exponential, logarithmic, and trigonometric functions. Limits, differentiation techniques, optimization and difference equations.

Prerequisite: MATHفB or AP Calculus AB or SAT Mathematics or ACT Mathematics. MATHفB with a grade of C or better. AP Calculus AB with a minimum score of 3. SAT Mathematics with a minimum score of 650. ACT Mathematics with a minimum score of 29. Placement via the Calculus Placement exam (fee required) is also accepted.

Restriction: School of Biological Sciences students have first consideration for enrollment.

MATHمB. Calculus for Life Sciences. 4 Units.

Integral calculus and multivariable calculus with applications to life sciences. Integration techniques, applications of the integral, phase plane methods and basic modeling, basic multivariable methods.

Prerequisite: MATHمA or MATHقA or AP Calculus AB or AP Calculus BC. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 3

Restriction: School of Biological Sciences students have first consideration for enrollment. Cannot be taken for credit after MATHقB.

MATHهA. Single-Variable Calculus I. 4 Units.

Introduction to derivatives, calculation of derivatives of algebraic and trigonometric functions applications including curve sketching, related rates, and optimization. Exponential and logarithm functions.

Prerequisite: MATHفB or AP Calculus AB or SAT Mathematics or ACT Mathematics. MATHفB with a grade of C or better. AP Calculus AB with a minimum score of 3. SAT Mathematics with a minimum score of 650. ACT Mathematics with a minimum score of 29. Placement via the Calculus Placement exam (fee required) is also accepted.

Restriction: Mathematics Majors only.

MATHهB. Single-Variable Calculus II. 4 Units.

Definite integrals the fundamental theorem of calculus. Applications of integration including finding areas and volumes. Techniques of integration. Infinite sequences and series.

Prerequisite: MATHقA or MATHمA or AP Calculus AB or AP Calculus BC or MATHهA. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 3

Restriction: Mathematics Majors only.

MATHو. Explorations in Functions and Modeling. 4 Units.

Explorations of applications and connections in topics in algebra, geometry, calculus, and statistics for future secondary math educators. Emphasis on nonstandard modeling problems.

Prerequisite or corequisite: MATHقA or AP Calculus AB or AP Calculus BC. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 3

MATHى. Introduction to Programming for Numerical Analysis. 4 Units.

Introduction to computers and programming using Matlab and Mathematica. Representation of numbers and precision, input/output, functions, custom data types, testing/debugging, reading exceptions, plotting data, numerical differentiation, basics of algorithms. Analysis of random processes using computer simulations.

Restriction: Mathematics Majors have first consideration for enrollment.

MATH㺊. Introduction to Programming for Data Science. 4 Units.

Intro to algorithms in data science using Python and R. Basic concepts of Python, store, access, and manipulate data in lists functions, methods, and packages NumPy, Numerical stability, and accuracy. Gradient descent and Newton’s method. Basics of R Programming.

Restriction: Mathematics Majors have first consideration for enrollment.

MATH㺍. Introduction to Abstract Mathematics. 4 Units.

Introduction to formal definition and rigorous proof writing in mathematics. Topics include basic logic, set theory, equivalence relations, and various proof techniques such as direct, induction, contradiction, contrapositive, and exhaustion.

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧅩A. Numerical Analysis I. 4 Units.

Introduction to the theory and practice of numerical computation with an emphasis on solving equations. Solving transcendental equations linear systems, Gaussian elimination, QR factorization, iterative methods, eigenvalue computation, power method.

Corequisite: MATH𧅩LA
Prerequisite: MATHكA or MATH H3A. Familiarity with computer programming is required.

MATH𧅩B. Numerical Analysis II. 4 Units.

Introduction to the theory and practice of numerical computation with an emphasis on topics from calculus and approximation theory. Lagrange interpolation Gaussian quadrature Fourier series and transforms Methods from data science including least squares and L1 regression.

MATH𧅩LA. Numerical Analysis Laboratory. 1 Unit.

Provides practical experience to complement the theory developed in Mathematics 105A.

MATH𧅩LB. Numerical Analysis Laboratory. 1 Unit.

Provides practical experience to complement the theory developed in Mathematics 105B.

MATH𧅫. Numerical Differential Equations. 4 Units.

Theory and applications of numerical methods to initial and boundary-value problems for ordinary and partial differential equations.

MATH𧅫L. Numerical Differential Equations Laboratory. 1 Unit.

Provides practical experience to complement the theory developed in Mathematics 107.

MATH𧅮A. Optimization I. 4 Units.

Introduction to optimization, linear search method, trust region method, Newton method, linear programming, linear, and non-linear least square methods.

MATH𧅮B. Optimization II. 4 Units.

The simplex method, interior point method, penalty barrier method, primal dual method, augmented Lagrangian method, and stochastic gradient method.

Prerequisite: MATH𧅮A. MATH𧅮A with a grade of C or better

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧅰A. Introduction to Partial Differential Equations and Applications. 4 Units.

Introduction to ordinary and partial differential equations and their applications in engineering and science. Basic methods for classical PDEs (potential, heat, and wave equations). Classification of PDEs, separation of variables and series expansions, special functions, eigenvalue problems.

MATH𧅰B. Introduction to Partial Differential Equations and Applications. 4 Units.

Introduction to partial differential equations and their applications in engineering and science. Basic methods for classical PDEs (potential, heat, and wave equations). Green functions and integral representations, method of characteristics.

MATH𧅰C. Introduction to Partial Differential Equations and Applications. 4 Units.

Nonhomogeneous problems and Green's functions, Sturm-Liouville theory, general Fourier expansions, applications of partial differential equations in different areas of science.

MATH𧅱A. Mathematical Modeling in Biology. 4 Units.

Discrete mathematical and statistical models difference equations, population dynamics, Markov chains, and statistical models in biology.

MATH𧅱B. Mathematical Modeling in Biology. 4 Units.

Linear algebra differential equations models dynamical systems stability hysteresis phase plane analysis applications to cell biology, viral dynamics, and infectious diseases.

Prerequisite: MATHقB or AP Calculus BC or MATHمB. AP Calculus BC with a minimum score of 4

MATH𧅳. Mathematical Modeling. 4 Units.

Mathematical modeling and analysis of phenomena that arise in engineering physical sciences, biology, economics, or social sciences.

MATH𧅵. Dynamical Systems. 4 Units.

Introduction to the modern theory of dynamical systems including contraction mapping principle, fractals and chaos, conservative systems, Kepler problem, billiard models, expanding maps, Smale's horseshoe, topological entropy.

MATH𧅶. The Theory of Differential Equations. 4 Units.

Existence and uniqueness of solutions, continuous dependence of solutions on initial conditions and parameteres, Lyapunov and asymptotic stability, Floquet theory, nonlinear systems, and bifurcations.

MATH𧅸A. Introduction to Abstract Algebra: Groups. 4 Units.

Axioms for group theory permutation groups, matrix groups. Isomorphisms, homomorphisms, quotient groups. Advanced topics as time permits. Special emphasis on doing proofs.

Prerequisite: (MATHكA or MATH H3A) and MATH㺍. MATH㺍 with a grade of C or better

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧅸B. Introduction to Abstract Algebra: Rings and Fields. 4 Units.

Basic properties of rings ideals, quotient rings polynomial and matrix rings. Elements of field theory.

Prerequisite: MATH𧅸A. MATH𧅸A with a grade of C- or better

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧅸C. Introduction to Abstract Algebra: Galois Theory. 4 Units.

Galois Theory: proof of the impossibility of certain ruler-and-compass constructions (squaring the circle, trisecting angles) nonexistence of analogues to the "quadratic formula" for polynomial equations of degree 5 or higher.

Restriction: Mathematics Majors have first consideration for enrollment.

MATH H120A. Honors Introduction to Graduate Algebra I. 5 Units.

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, and symmetric operators. Introduction to groups, rings, and fields, including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

Prerequisite: (MATHكA or MATH H3A) and MATH㺍 and (MATH𧅸A or MATH𧅹A). MATH㺍 with a grade of A or better. MATH𧅸A with a grade of A or better. MATH𧅹A with a grade of A or better

Restriction: Mathematics Honors students only.

MATH H120B. Honors Introduction to Graduate Algebra II. 5 Units.

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, and symmetric operators. Introduction to groups, rings, and fields, including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

Restriction: Mathematics Honors students only.

MATH H120C. Honors Introduction to Graduate Algebra III. 5 Units.

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, and symmetric operators. Introduction to groups, rings, and fields, including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

Restriction: Mathematics Honors students only.

MATH𧅹A. Linear Algebra. 4 Units.

Introduction to modern abstract linear algebra. Special emphasis on students doing proofs. Vector spaces, linear independence, bases, dimension. Linear transformations and their matrix representations. Theory of determinants.

Prerequisite: (MATHكA or MATH H3A) and MATH㺍. MATH㺍 with a grade of C or better

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧅹B. Linear Algebra. 4 Units.

Introduction to modern abstract linear algebra. Special emphasis on students doing proofs. Canonical forms inner products similarity of matrices.

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧆂A. Probability I. 4 Units.

Combinatorial probability, conditional probabilities, independence, discrete and continuous random variables, expectation and variance, common probability distributions.

Prerequisite: (MATHقA or AP Calculus BC or AP Calculus AB) and (MATHقB or AP Calculus BC) and (MATHكA or MATH H3A). AP Calculus BC with a minimum score of 3. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 4

MATH𧆂B. Probability II. 4 Units.

Joint distributions, sums of independent random variables, conditional distributions and conditional expectation, covariances, moment generating functions, limit theorems.

MATH𧆂C. Stochastic Processes. 4 Units.

Markov chains, Brownian motion, Gaussian processes, applications to option pricing and Markov chain Monte Carlo methods.

MATH𧆅A. Statistical Methods with Applications to Finance. 4 Units.

Overview of probability, statistics, and financial concepts: distribution, point estimation, confidence interval, linear regression, hypothesis testing, principal component analysis, financial applications.

MATH𧆅B. Statistical Methods with Applications to Finance. 4 Units.

Overview of markets and options: asset modeling, Brownian motion, risk neutrality, option pricing, value at risk, MC simulations.

MATH𧆅C. Statistical Methods with Applications to Finance. 4 Units.

Overview of interest theory, time value of money, annuities/cash flows with payments that are not contingent, loans, sinking funds, bonds, general cash flow and portfolios, immunization, duration and convexity, swaps.

MATH𧆆A. Fixed Income. 4 Units.

Overview of interest theory, time value of money, annuities/cash flows with payments that are not contingent, loans, sinking funds, bonds, general cash flow and portfolios, immunization, duration and convexity, swaps.

MATH𧆆B. Mathematics of Financial Derivatives. 4 Units.

General derivatives call/put options hedging and investment strategies: spreads and collars risk management forwards and futures bonds.

MATH𧆆C. Mathematical Models for Finance. 4 Units.

General properties of options: option contracts (call and put options, European, American and exotic options) binomial option pricing model, Black-Scholes option pricing model risk-neutral pricing formula using Monte-Carlo simulation option greeks and risk management interest rate derivatives, Markowitz portfolio theory.

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧆌A. Elementary Analysis. 4 Units.

Introduction to real analysis, including convergence of sequence, infinite series, differentiation and integration, and sequences of functions. Students are expected to do proofs.

Prerequisite: (MATHقB or AP Calculus BC) and (MATHقD or MATH H2D) and (MATHكA or MATH H3A) and MATH㺍. AP Calculus BC with a minimum score of 4. MATH㺍 with a grade of C or better

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧆌B. Elementary Analysis. 4 Units.

Introduction to real analysis including convergence of sequences, infinite series, differentiation and integration, and sequences of functions. Students are expected to do proofs.

Prerequisite: MATH𧆌A. MATH𧆌A with a grade of C- or better

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧆌C. Analysis in Several Variables . 4 Units.

Rigorous treatment of multivariable differential calculus. Jacobians, Inverse and Implicit Function theorems.

MATH H140A. Honors Introduction to Graduate Analysis I. 5 Units.

Construction of the real number system, topology of the real line, concepts of continuity, differential and integral calculus, sequences and series of functions, equicontinuity, metric spaces, multivariable differential and integral calculus, implicit functions, curves and surfaces.

Prerequisite: (MATHقE or MATH H2E) and (MATHكA or MATH H3A) and MATH㺍 and MATH𧅹A and MATH𧆌A and MATH𧆌B. MATHقE with a grade of A or better. MATH H2E with a grade of A or better. MATH㺍 with a grade of A or better. MATH𧆌A with a grade of A or better. MATH𧆌B with a grade of A or better

Restriction: Mathematics Honors students only.

MATH H140B. Honors Introduction to Graduate Analysis II. 5 Units.

Construction of the real number system, topology of the real line, concepts of continuity, differential and integral calculus, sequences and series of functions, equicontinuity, metric spaces, multivariable differential and integral calculus, implicit functions, curves and surfaces.

Restriction: Mathematics Honors students only.

MATH H140C. Honors Introduction to Graduate Analysis III. 5 Units.

Construction of the real number system topology of the real line concepts of continuity, differential, and integral calculus sequences and series of functions, equicontinuity, metric spaces, multivariable differential, and integral calculus implicit functions, curves and surfaces.

Restriction: Mathematics Honors students only.

MATH𧆍. Introduction to Topology. 4 Units.

The elements of naive set theory and the basic properties of metric spaces. Introduction to topological properties.

MATH𧆓. Complex Analysis. 4 Units.

Rigorous treatment of basic complex analysis: analytic functions, Cauchy integral theory and its consequences, power series, residue calculus, harmonic functions, conformal mapping. Students are expected to do proofs.

Prerequisite or corequisite: MATH𧆌A and MATH𧆌B

Restriction: MATH 114A may not be taken for credit after MATH𧆓.

MATH𧆖. Introduction to Mathematical Logic. 4 Units.

First order logic through the Completeness Theorem for predicate logic.

Prerequisite: MATH㺍 or (ICSنB and ICSنD). MATH㺍 with a grade of C- or better

MATH𧆡. Modern Geometry. 4 Units.

Euclidean Geometry Hilbert's Axioms Absolute Geometry Hyperbolic Geometry the Poincare Models and Geometric Transformations.

Prerequisite: MATH㺍 or (ICSنB and ICSنD). MATH㺍 with a grade of C- or better

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧆢A. Introduction to Differential Geometry. 4 Units.

Applications of advanced calculus and linear algebra to the geometry of curves and surfaces in space.

MATH𧆢B. Introduction to Differential Geometry. 4 Units.

Applications of advanced calculus and linear algebra to the geometry of curves and surfaces in space.

MATH𧆭A. Introduction to Cryptology. 4 Units.

Introduction to some of the mathematics used in the making and breaking of codes, with applications to classical ciphers and public key systems. Includes topics from number theory, probability, and abstract algebra.

Prerequisite: (MATHقB or AP Calculus BC) and (MATHكA or MATH H3A) and MATH㺍 or (ICSنB and ICSنD). AP Calculus BC with a minimum score of 4. MATH㺍 with a grade of C or better

MATH𧆭B. Introduction to Cryptology. 4 Units.

Introduction to some of the mathematics used in the making and breaking of codes, with applications to classical ciphers and public key systems. The mathematics which is covered includes topics from number theory, probability, and abstract algebra.

MATH𧆯. Combinatorics . 4 Units.

Introduction to combinatorics including basic counting principles, permutations, combinations, binomial coefficients, inclusion-exclusion, derangements, ordinary and exponential generating functions, recurrence relations, Catalan numbers, Stirling numbers, and partition numbers.

Prerequisite: (MATHقB or AP Calculus BC) and MATH㺍. AP Calculus BC with a minimum score of 4. MATH㺍 with a grade of C or better

MATH𧆰. Mathematics of Finance. 4 Units.

After reviewing tools from probability, statistics, and elementary differential and partial differential equations, concepts such as hedging, arbitrage, Puts, Calls, the design of portfolios, the derivation and solution of the Blac-Scholes, and other equations are discussed.

Restriction: Business Economics Majors have first consideration for enrollment. Economics Majors have first consideration for enrollment. Quantitative Economics Majors have first consideration for enrollment. Mathematics Majors have first consideration for enrollment.

MATH𧆴A. Number Theory. 4 Units.

Introduction to number theory and applications. Divisibility, prime numbers, factorization. Arithmetic functions. Congruences. Quadratic residue. Diophantine equations. Introduction to cryptography.

Prerequisite: (MATHكA or MATH H3A) and MATH㺍. MATH㺍 with a grade of C or better

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧆴B. Number Theory. 4 Units.

Introduction to number theory and applications. Analytic number theory, character sums, finite fields, discrete logarithm, computational complexity. Introduction to coding theory. Other topics as time permits.

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧆸. History of Mathematics. 4 Units.

Topics vary from year to year. Some possible topics: mathematics in ancient times the development of modern analysis the evolution of geometric ideas. Students will be assigned individual topics for term papers.

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧆸L. History of Mathematics Lesson Lab. 1 Unit.

Aspiring math teachers research, design, present, and peer review middle school or high school math lessons that draw from history of mathematics topics.

MATH𧇀. Studies in the Learning and Teaching of Secondary Mathematics. 2 Units.

Focus is on historic and current mathematical concepts related to student learning and effective math pedagogy, with fieldwork in grades 6-14.

Grading Option: Pass/no pass only.

Repeatability: May be taken for credit 2 times.

MATH𧇂. Problem Solving Seminar. 2 Units.

Develops ability in analytical thinking and problem solving, using problems of the type found in the Mathematics Olympiad and the Putnam Mathematical Competition. Students taking the course in fall will prepare for and take the Putnam examination in December.

Grading Option: Pass/no pass only.

Repeatability: May be taken for credit 2 times.

MATH𧇃W. Mathematical Writing. 4 Units.

Techniques of mathematical writing and communication. Covers effectively writing mathematical papers, creating effective presentations, and communicating mathematics in a variety of media. Focuses on utilizing LaTeX for typesetting mathematics.

Prerequisite: MATH𧅸A or MATH𧅹A or MATH𧆌A. MATH𧅸A with a grade of C or better. MATH𧅹A with a grade of C or better. MATH𧆌A with a grade of C or better. Satisfactory completion of the Lower-Division Writing requirement.

Restriction: Mathematics Majors have first consideration for enrollment.

MATH𧇇A. Special Studies in Mathematics. 2-4 Units.

Supervised reading. For outstanding undergraduate Mathematics majors in supervised but independent reading or research of mathematical topics.

Repeatability: Unlimited as topics vary.

MATH𧇇B. Special Studies in Mathematics. 2-4 Units.

Supervised reading. For outstanding undergraduate Mathematics majors in supervised but independent reading or research of mathematical topics.

Repeatability: Unlimited as topics vary.

MATH𧇇C. Special Studies in Mathematics. 2-4 Units.

Supervised reading. For outstanding undergraduate Mathematics majors in supervised but independent reading or research of mathematical topics.

Repeatability: Unlimited as topics vary.

MATH𧇍A. Introduction to Graduate Analysis. 5 Units.

Construction of the real number system, topology of the real line, concepts of continuity, differential and integral calculus, sequences and series of functions, equicontinuity, metric spaces, multivariable differential and integral calculus, implicit functions, curves and surfaces.

Prerequisite: (MATHقE or MATH H2E) and (MATHكA MATH H3A) and MATH㺍. MATHقE with a grade of A or better. MATH H2E with a grade of A or better. MATH㺍 with a grade of C or better

MATH𧇍B. Introduction to Graduate Analysis. 5 Units.

Construction of the real number system, topology of the real line, concepts of continuity, differential and integral calculus, sequences and series of functions, equicontinuity, metric spaces, multivariable differential and integral calculus, implicit functions, curves and surfaces.

MATH𧇍C. Introduction to Graduate Analysis. 5 Units.

Construction of the real number system, topology of the real line, concepts of continuity, differential and integral calculus, sequences and series of functions, equicontinuity, metric spaces, multivariable differential and integral calculus, implicit functions, curves and surfaces.

MATH𧇎A. Introduction to Graduate Algebra. 5 Units.

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, symmetric operators. Introduction to groups, rings, and fields including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

MATH𧇎B. Introduction to Graduate Algebra. 5 Units.

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, symmetric operators. Introduction to groups, rings, and fields including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

MATH𧇎C. Introduction to Graduate Algebra. 5 Units.

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, symmetric operators. Introduction to groups, rings, and fields including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

MATH𧇒A. Real Analysis. 4 Units.

Measure theory, Lebesgue integral, signed measures, Radon-Nikodym theorem, functions of bounded variation and absolutely continuous functions, classical Banach spaces, Lp spaces, integration on locally compact spaces and the Riesz-Markov theorem, measure and outer measure, product measure spaces.

MATH𧇒B. Real Analysis. 4 Units.

Measure theory, Lebesgue integral, signed measures, Radon-Nikodym theorem, functions of bounded variation and absolutely continuous functions, classical Banach spaces, Lp spaces, integration on locally compact spaces and the Riesz-Markov theorem, measure and outer measure, product measure spaces.

MATH𧇒C. Real Analysis. 4 Units.

Measure theory, Lebesgue integral, signed measures, Radon-Nikodym theorem, functions of bounded variation and absolutely continuous functions, classical Banach spaces, Lp spaces, integration on locally compact spaces and the Riesz-Markov theorem, measure and outer measure, product measure spaces.

MATH𧇚A. Introduction to Manifolds and Geometry. 4 Units.

General topology and fundamental groups, covering space Stokes theorem on manifolds, selected topics on abstract manifold theory.

MATH𧇚B. Introduction to Manifolds and Geometry. 4 Units.

General topology and fundamental groups, covering space Stokes theorem on manifolds, selected topics on abstract manifold theory.

MATH𧇚C. Introduction to Manifolds and Geometry. 4 Units.

General topology and fundamental groups, covering space Stokes theorem on manifolds, selected topics on abstract manifold theory.

MATH𧇜A. Analytic Function Theory. 4 Units.

Standard theorems about analytic functions. Harmonic functions. Normal families. Conformal mapping.

MATH𧇜B. Analytic Function Theory. 4 Units.

Standard theorems about analytic functions. Harmonic functions. Normal families. Conformal mapping.

MATH𧇜C. Analytic Function Theory. 4 Units.

Standard theorems about analytic functions. Harmonic functions. Normal families. Conformal mapping.

MATH𧇞A. Several Complex Variables and Complex Geometry. 4 Units.

Several Complex variables, d-bar problems, mappings, Kaehler geometry, de Rham and Dolbeault Theorems, Chern Classes, Hodge Theorems, Calabi conjecture, Kahler-Einstein geometry, Monge-Ampere.

MATH𧇡A. Introduction to Numerical Analysis and Scientific Computing. 4 Units.

Introduction to fundamentals of numerical analysis from an advanced viewpoint. Error analysis, approximation of functions, nonlinear equations.

Restriction: Graduate students only.

MATH𧇡B. Introduction to Numerical Analysis and Scientific Computing. 4 Units.

Introduction to fundamentals of numerical analysis from an advanced viewpoint. Numerical linear algebra, numerical solutions of differential equations stability.

Restriction: Graduate students only.

MATH𧇡C. Introduction to Numerical Analysis and Scientific Computing. 4 Units.

Introduction to fundamentals of numerical analysis from an advanced viewpoint. Numerical linear algebra, numerical solutions of differential equations stability.

Restriction: Graduate students only.

MATH𧇢A. Computational Differential Equations. 4 Units.

Finite difference and finite element methods. Quick treatment of functional and nonlinear analysis background: weak solution, Lp spaces, Sobolev spaces. Approximation theory. Fourier and Petrov-Galerkin methods mesh generation. Elliptic, parabolic, hyperbolic cases in 226A-B-C, respectively.

MATH𧇢B. Computational Differential Equations. 4 Units.

Finite difference and finite element methods. Quick treatment of functional and nonlinear analysis background: weak solution, Lp spaces, Sobolev spaces. Approximation theory. Fourier and Petrov-Galerkin methods mesh generation. Elliptic, parabolic, hyperbolic cases in 226A-B-C, respectively.

MATH𧇢C. Computational Differential Equations. 4 Units.

Finite difference and finite element methods. Quick treatment of functional and nonlinear analysis background: weak solution, Lp spaces, Sobolev spaces. Approximation theory. Fourier and Petrov-Galerkin methods mesh generation. Elliptic, parabolic, hyperbolic cases in 226A-B-C, respectively.

MATH𧇣A. Mathematical and Computational Biology. 4 Units.

Analytical and numerical methods for dynamical systems, temporal-spatial dynamics, steady state, stability, stochasticity. Application to life sciences: genetics, tissue growth and patterning, cancers, ion channels gating, signaling networks, morphogen gradients. Analytical methods.

Prerequisite: (MATHقA or AP Calculus BC or AP Calculus AB) and (MATHقB or AP Calculus BC or MATHمB) and (MATHكA or MATH H3A). AP Calculus BC with a minimum score of 3. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 4

MATH𧇣B. Mathematical and Computational Biology. 4 Units.

Analytical and numerical methods for dynamical systems, temporal-spatial dynamics, steady state, stability, stochasticity. Application to life sciences: genetics, tissue growth and patterning, cancers, ion channels gating, signaling networks, morphogen gradients. Numerical simulations.

MATH𧇣C. Mathematical and Computational Biology . 4 Units.

Analytical and numerical methods for dynamical systems, temporal-spatial dynamics, steady state, stability, stochasticity. Application to life sciences: genetics, tissue growth and patterning, cancers, ion channels gating, signaling networks, morphogen gradients. Probabilistic methods.

MATH𧇦A. Algebra. 4 Units.

Elements of the theories of groups, rings, fields, modules. Galois theory. Modules over principal ideal domains. Artinian, Noetherian, and semisimple rings and modules.

MATH𧇦B. Algebra. 4 Units.

Elements of the theories of groups, rings, fields, modules. Galois theory. Modules over principal ideal domains. Artinian, Noetherian, and semisimple rings and modules.

MATH𧇦C. Algebra. 4 Units.

Elements of the theories of groups, rings, fields, modules. Galois theory. Modules over principal ideal domains. Artinian, Noetherian, and semisimple rings and modules.

MATH𧇨A. Algebraic Number Theory. 4 Units.

Algebraic integers, prime ideals, class groups, Dirichlet unit theorem, localization, completion, Cebotarev density theorem, L-functions, Gauss sums, diophantine equations, zeta functions over finite fields. Introduction to class field theory.

MATH𧇨B. Algebraic Number Theory. 4 Units.

Algebraic integers, prime ideals, class groups, Dirichlet unit theorem, localization, completion, Cebotarev density theorem, L-functions, Gauss sums, diophantine equations, zeta functions over finite fields. Introduction to class field theory.

MATH𧇨C. Algebraic Number Theory. 4 Units.

Algebraic integers, prime ideals, class groups, Dirichlet unit theorem, localization, completion, Cebotarev density theorem, L-functions, Gauss sums, diophantine equations, zeta functions over finite fields. Introduction to class field theory.

MATH𧇩A. Algebraic Geometry. 4 Units.

Basic commutative algebra and classical algebraic geometry. Algebraic varieties, morphisms, rational maps, blow ups. Theory of schemes, sheaves, divisors, cohomology. Algebraic curves and surfaces, Riemann-Roch theorem, Jacobians, classification of curves and surfaces.

MATH𧇩B. Algebraic Geometry. 4 Units.

Basic commutative algebra and classical algebraic geometry. Algebraic varieties, morphisms, rational maps, blow ups. Theory of schemes, sheaves, divisors, cohomology. Algebraic curves and surfaces, Riemann-Roch theorem, Jacobians, classification of curves and surfaces.

MATH𧇩C. Algebraic Geometry. 4 Units.

Basic commutative algebra and classical algebraic geometry. Algebraic varieties, morphisms, rational maps, blow ups. Theory of schemes, sheaves, divisors, cohomology. Algebraic curves and surfaces, Riemann-Roch theorem, Jacobians, classification of curves and surfaces.

MATH𧇫A. Mathematics of Cryptography. 4 Units.

Mathematics of public key cryptography: encryption and signature schemes RSA factoring primality testing discrete log based cryptosystems, elliptic and hyperelliptic curve cryptography and additional topics as determined by the instructor.

MATH𧇯A. Analytic Methods in Arithmetic Geometry. 4 Units.

Riemann zeta function, Dirichlet L-functions, prime number theorem, zeta functions over finite fields, sieve methods, zeta functions of algebraic curves, algebraic coding theory, L-Functions over number fields, L-functions of modular forms, Eisenstein series.

MATH𧇯B. Analytic Methods in Arithmetic Geometry. 4 Units.

Riemann zeta function, Dirichlet L-functions, prime number theorem, zeta functions over finite fields, sieve methods, zeta functions of algebraic curves, algebraic coding theory, L-Functions over number fields, L-functions of modular forms, Eisenstein series.

MATH𧇯C. Analytic Methods in Arithmetic Geometry. 4 Units.

Riemann zeta function, Dirichlet L-functions, prime number theorem, zeta functions over finite fields, sieve methods, zeta functions of algebraic curves, algebraic coding theory, L-Functions over number fields, L-functions of modular forms, Eisenstein series.

MATH𧇰A. Differential Geometry. 4 Units.

Riemannian manifolds, connections, curvature and torsion. Submanifolds, mean curvature, Gauss curvature equation. Geodesics, minimal submanifolds, first and second fundamental forms, variational formulas. Comparison theorems and their geometric applications. Hodge theory applications to geometry and topology.

MATH𧇰B. Differential Geometry. 4 Units.

Riemannian manifolds, connections, curvature and torsion. Submanifolds, mean curvature, Gauss curvature equation. Geodesics, minimal submanifolds, first and second fundamental forms, variational formulas. Comparison theorems and their geometric applications. Hodge theory applications to geometry and topology.

MATH𧇰C. Differential Geometry. 4 Units.

Riemannian manifolds, connections, curvature and torsion. Submanifolds, mean curvature, Gauss curvature equation. Geodesics, minimal submanifolds, first and second fundamental forms, variational formulas. Comparison theorems and their geometric applications. Hodge theory applications to geometry and topology.

MATH𧇵A. Topics in Differential Geometry. 4 Units.

Studies in selected areas of differential geometry, a continuation of MATH𧇰A-MATH𧇰B-MATH𧇰C. Topics addressed vary each quarter.

Repeatability: Unlimited as topics vary.

MATH𧇵B. Topics in Differential Geometry. 4 Units.

Studies in selected areas of differential geometry, a continuation of MATH𧇰A-MATH𧇰B-MATH𧇰C. Topics addressed vary each quarter.

Repeatability: Unlimited as topics vary.

MATH𧇵C. Topics in Differential Geometry. 4 Units.

Studies in selected areas of differential geometry, a continuation of MATH𧇰A-MATH𧇰B-MATH𧇰C. Topics addressed vary each quarter.

Repeatability: Unlimited as topics vary.

MATH𧇺A. Algebraic Topology. 4 Units.

Provides fundamental materials in algebraic topology: fundamental group and covering space, homology and cohomology theory, and homotopy group.

MATH𧇺B. Algebraic Topology. 4 Units.

Provides fundamental materials in algebraic topology: fundamental group and covering space, homology and cohomology theory, and homotopy group.

MATH𧇺C. Algebraic Topology. 4 Units.

Provides fundamental materials in algebraic topology: fundamental group and covering space, homology and cohomology theory, and homotopy group.

MATH𧈄A. Functional Analysis. 4 Units.

Normed linear spaces, Hilbert spaces, Banach spaces, Stone-Weierstrass Theorem, locally convex spaces, bounded operators on Banach and Hilbert spaces, the Gelfand-Neumark Theorem for commutative C*-algebras, the spectral theorem for bounded self-adjoint operators, unbounded operators on Hilbert spaces.

MATH𧈄B. Functional Analysis. 4 Units.

Normed linear spaces, Hilbert spaces, Banach spaces, Stone-Weierstrass Theorem, locally convex spaces, bounded operators on Banach and Hilbert spaces, the Gelfand-Neumark Theorem for commutative C*-algebras, the spectral theorem for bounded self-adjoint operators, unbounded operators on Hilbert spaces.

MATH𧈄C. Functional Analysis. 4 Units.

Normed linear spaces, Hilbert spaces, Banach spaces, Stone-Weierstrass Theorem, locally convex spaces, bounded operators on Banach and Hilbert spaces, the Gelfand-Neumark Theorem for commutative C*-algebras, the spectral theorem for bounded self-adjoint operators, unbounded operators on Hilbert spaces.

MATH𧈎A. Probability. 4 Units.

Probability spaces, distribution and characteristic functions. Strong limit theorems. Limit distributions for sums of independent random variables. Conditional expectation and martingale theory. Stochastic processes.

MATH𧈎B. Probability. 4 Units.

Probability spaces, distribution and characteristic functions. Strong limit theorems. Limit distributions for sums of independent random variables. Conditional expectation and martingale theory. Stochastic processes.

MATH𧈎C. Probability. 4 Units.

Probability spaces, distribution and characteristic functions. Strong limit theorems. Limit distributions for sums of independent random variables. Conditional expectation and martingale theory. Stochastic processes.

MATH𧈏A. Stochastic Processes. 4 Units.

Processes with independent increments, Wiener and Gaussian processes, function space integrals, stationary processes, Markov processes.

MATH𧈏B. Stochastic Processes. 4 Units.

Processes with independent increments, Wiener and Gaussian processes, function space integrals, stationary processes, Markov processes.

MATH𧈏C. Stochastic Processes. 4 Units.

Processes with independent increments, Wiener and Gaussian processes, function space integrals, stationary processes, Markov processes.

MATH𧈘A. Mathematical Logic. 4 Units.

Basic set theory models, compactness, and completeness basic model theory Incompleteness and Gödel's Theorems basic recursion theory constructible sets.

MATH𧈘B. Mathematical Logic. 4 Units.

Basic set theory models, compactness, and completeness basic model theory Incompleteness and Gödel's Theorems basic recursion theory constructible sets.

MATH𧈘C. Mathematical Logic. 4 Units.

Basic set theory models, compactness, and completeness basic model theory Incompleteness and Gödel's Theorems basic recursion theory constructible sets.

MATH𧈙A. Set Theory. 4 Units.

Ordinals, cardinals, cardinal arithmetic, combinatorial set theory, models of set theory, Gödel's constructible universe, forcing, large cardinals, iterate forcing, inner model theory, fine structure.

MATH𧈙B. Set Theory. 4 Units.

Ordinals, cardinals, cardinal arithmetic, combinatorial set theory, models of set theory, Gödel's constructible universe, forcing, large cardinals, iterate forcing, inner model theory, fine structure.

MATH𧈙C. Set Theory. 4 Units.

Ordinals, cardinals, cardinal arithmetic, combinatorial set theory, models of set theory, Gödel's constructible universe, forcing, large cardinals, iterate forcing, inner model theory, fine structure.

MATH𧈚A. Model Theory. 4 Units.

Languages, structures, compactness and completeness. Model-theoretic constructions. Omitting types theorems. Morley's theorem. Ranks, forking. Model completeness. O-minimality. Applications to algebra.

MATH𧈚B. Model Theory. 4 Units.

Languages, structures, compactness and completeness. Model-theoretic constructions. Omitting types theorems. Morley's theorem. Ranks, forking. Model completeness. O-minimality. Applications to algebra.

MATH𧈚C. Model Theory. 4 Units.

Languages, structures, compactness and completeness. Model-theoretic constructions. Omitting types theorems. Morley's theorem. Ranks, forking. Model completeness. O-minimality. Applications to algebra.

MATH𧈝A. Topics in Mathematical Logic. 4 Units.

Studies in selected areas of mathematical logic, a continuation of MATH𧈘A-MATH𧈘B-MATH𧈘C. Topics addressed vary each quarter.

Repeatability: Unlimited as topics vary.

MATH𧈢A. Methods in Applied Mathematics. 4 Units.

Introduction to ODEs and dynamical systems: existence and uniqueness. Equilibria and periodic solutions. Bifurcation theory. Perturbation methods: approximate solution of differential equations. Multiple scales and WKB. Matched asymptotic. Calculus of variations: direct methods, Euler-Lagrange equation. Second variation and Legendre condition.

MATH𧈢B. Methods in Applied Mathematics. 4 Units.

Introduction to ODEs and dynamical systems: existence and uniqueness. Equilibria and periodic solutions. Bifurcation theory. Perturbation methods: approximate solution of differential equations. Multiple scales and WKB. Matched asymptotic. Calculus of variations: direct methods, Euler-Lagrange equation. Second variation and Legendre condition.

MATH𧈢C. Methods in Applied Mathematics. 4 Units.

Introduction to ODEs and dynamical systems: existence and uniqueness. Equilibria and periodic solutions. Bifurcation theory. Perturbation methods: approximate solution of differential equations. Multiple scales and WKB. Matched asymptotic. Calculus of variations: direct methods, Euler-Lagrange equation. Second variation and Legendre condition.

MATH𧈧A. Partial Differential Equations. 4 Units.

Theory and techniques for linear and nonlinear partial differential equations. Local and global theory of partial differential equations: analytic, geometric, and functional analytic methods.

MATH𧈧B. Partial Differential Equations. 4 Units.

Theory and techniques for linear and nonlinear partial differential equations. Local and global theory of partial differential equations: analytic, geometric, and functional analytic methods.

MATH𧈧C. Partial Differential Equations. 4 Units.

Theory and techniques for linear and nonlinear partial differential equations. Local and global theory of partial differential equations: analytic, geometric, and functional analytic methods.

MATH𧈨. Topics in Partial Differential Equations. 4 Units.

Studies in selected areas of partial differential equations, a continuation of MATH𧈧A-MATH𧈧B-MATH𧈧C. Topics addressed vary each quarter.

Repeatability: Unlimited as topics vary.

Restriction: Graduate students only.

MATH𧈩. Mathematics Colloquium. 1 Unit.

Weekly colloquia on topics of current interest in mathematics.

Grading Option: Satisfactory/unsatisfactory only.

Repeatability: May be repeated for credit unlimited times.

MATH𧈪A. Seminar . 2 Units.

Seminars organized for detailed discussion of research problems of current interest in the Department. The format, content, frequency, and course value are variable.

Grading Option: Satisfactory/unsatisfactory only.

Repeatability: Unlimited as topics vary.

MATH𧈪B. Seminar . 2 Units.

Seminars organized for detailed discussion of research problems of current interest in the Department. The format, content, frequency, and course value are variable.

Grading Option: Satisfactory/unsatisfactory only.

Repeatability: Unlimited as topics vary.

MATH𧈪C. Seminar . 2 Units.

Seminars organized for detailed discussion of research problems of current interest in the Department. The format, content, frequency, and course value are variable.

Grading Option: Satisfactory/unsatisfactory only.

Repeatability: Unlimited as topics vary.

MATH𧈫A. Supervised Reading and Research. 1-12 Units.

Supervised reading and research with Mathematics faculty.

Repeatability: May be repeated for credit unlimited times.

MATH𧈫B. Supervised Reading and Research. 1-12 Units.

Supervised reading and research with Mathematics faculty.

Repeatability: May be repeated for credit unlimited times.

MATH𧈫C. Supervised Reading and Research. 1-12 Units.


Academics

Readiness to Learn in the Classroom

A successful early intervention program prepares a learner for continued learning in a classroom environment by teaching her the prerequisite skills that are needed for success in that environment. Many children with ASD, even after great success in home-based behavioral intervention programs, still do not develop many skills that are fundamental to success in the school environment. Some basic skills to consider targeting include:

Responding to group instructions

Completing assignments independently

Understanding the meaning of grades

Raising one’s hand to get the teacher’s attention (rather than speaking out)

Responding to instructions delivered at a distance (e.g., a teacher standing at the front of class)

Standing in line for recess or to participate in sports

Responding to group instructions from the teacher (even though the teacher is not speaking directly to the learner)

Looking to peers as cues for what to do in the moment when the learner misses the teacher’s request

As with any other skill taught in an ABA program, the skills listed above can be taught through prompting, reinforcement, prompt fading and generalization training. Some creativity is needed to simulate classroom settings, and role-play may be needed to simulate situations involving teachers or peers.

In addition to teaching fundamental school-readiness skills, make sure to set the learner up for success by consulting with the school to determine whether it uses particular language arts or math procedures and incorporating those procedures into the learner’s ABA program. By incorporating these variables into the learner’s home-based therapy before she enters school, you can help ensure that the stimuli in the school environment will already cue the desired behavior when the learner arrives in the classroom.


Topics

Below are a variety of topics handled by the Department of Homeland Security.

Border Security

Protecting our borders from the illegal movement of weapons, drugs, contraband, and people, while promoting lawful trade and travel, is essential to homeland security, economic prosperity, and national sovereignty.

Citizenship and Immigration

Managed by U.S. Citizenship and Immigration Services (USCIS), the United States’ lawful immigration system is one of the most generous in the world.

Cybersecurity

Our daily life, economic vitality, and national security depend on a stable, safe, and resilient cyberspace.

Disasters

Whatever the disaster, the Federal Emergency Management Agency, or FEMA, leads the federal government’s response as part of a team of responders.

Economic Security

America’s and the world’s economic prosperity increasingly depends on the uninterrupted flow of goods and services, people and capital, and information and technology across our borders.

Election Security

A secure and resilient electoral process is a vital national interest and one of our highest priorities at the Department of Homeland Security.

Homeland Security Enterprise

Since the Department's creation, the goal is simple: one DHS, with integrated, results-based operations.

Human Trafficking

Human trafficking is a crime and a form of modern-day slavery. DHS is working to end it.

Immigration and Customs Enforcement

The mission of U.S. Immigration and Customs Enforcement (ICE) is to protect America from the cross-border crime and illegal immigration that threaten national security and public safety.

Preventing Terrorism

Protecting the American people from terrorist threats is the reason DHS was created, and remains our highest priority.

Resilience

DHS works with all levels of government, the private and nonprofit sectors, and individual citizens to make our nation more resilient to acts of terrorism, cyber attacks, pandemics, and catastrophic natural disasters.

Science & Technology

The DHS Science and Technology Directorate (S&T) is the Department’s primary research and development arm and manages science and technology research, from development through transition, for the Department's operational components and first responders.


Windows Server 2016 prerequisites for Exchange 2016

The prerequisites that are needed to install Exchange 2016 on computers running Windows Server 2016 depends on which Exchange role you want to install. Read the section below that matches the role you want to install.

Windows Server 2016 requires Exchange 2016 Cumulative Update 3 or later.

Exchange 2016 Mailbox servers on Windows Server 2016

Run the following command in Windows PowerShell to install the required Windows components:

Install the following software in order:

You can only install this update if your Windows Server 2016 version is 14393.576 or earlier (circa December, 2016). You can check your Windows Server version by running the winver command. If your Windows Server 2016 version is greater than 14393.576, you don't need this update or its replacement KB3213522, which was released one week later. Exchange 2016 Setup looks for the installation of this update, won't allow you to continue if this update is missing, and will clearly inform you if you need it.

The system requirements for the Visual C++ redistributable package do not mention support for Windows Server 2016 or Windows Server 2019, but the redistributable package is safe to install on these versions of Windows.

An overview of the latest supported versions is available at: Visual C++ Redistributable versions.

Only the Mailbox role requires the Visual C++ Redistributable Packages for Visual Studio 2013. Other Exchange installations (management tools and Edge Transport) only require the Visual C++ Redistributable Packages for Visual Studio 2012.

Exchange 2016 Edge Transport servers on Windows Server 2016

Run the following command in Windows PowerShell to install the required Windows components:

Install the following software in order:

The system requirements for the Visual C++ redistributable package do not mention support for Windows Server 2016 or Windows Server 2019, but the redistributable package is safe to install on these versions of Windows.

An overview of the latest supported versions is available at: Visual C++ Redistributable versions.


Pathway to Success

There are several ways to ensure success in research. When in graduate school, students need to undertake several measures to identify a compelling research topic. Although conducting a thorough literature survey certainly facilitates this process, it is virtually impossible to choose the right research topic solely based on literature surveys. Students and early-stage researchers, therefore, need to brainstorm thoroughly with their advisor, talk to experts, and attend research seminars/conferences to listen to (and network with) established researchers. Quite often, taking up the relevant coursework (especially for interdisciplinary research areas) simplifies the process of research topic selection.

Choosing the right research question helps researchers stay focused and motivated throughout their career. Meaningful research questions eventually lead to meaningful discoveries and inventions. Robert Smith presented in Graduate Research: A Guide for Students in the Sciences (ISI Press, 1984) a list of 11 research questions to consider:

  1. Can you enthusiastically pursue it?
  2. Can you sustain your interest while pursuing it?
  3. Is the problem solvable?
  4. Is it worth pursuing?
  5. Will it lead to other research problems?
  6. Is it manageable in size?
  7. What is the potential for making an original contribution to the literature in the field?
  8. Will the scholars in your field receive the results well if you solve the problem?
  9. Are you (or will you become) competent to solve it?
  10. By solving it, will you have demonstrated independent skills in your discipline?
  11. Will the necessary research prepare you in an area of demand or promise for the future?

Keeping these questions in mind while developing a research question can set the stage for a productive and fulfilling career.


Discussion

This study revealed an Age × Learning Strategy Type interaction for the effectiveness of GLSs. As hypothesized, children remembered more facts after generating predictions than after generating examples, and this difference was significantly greater than in adults. Furthermore, compared to generating examples, generating predictions was associated with a larger pupillary surprise response upon seeing the correct result. This pupillary response was a better predictor of children’s learning in the prediction condition than in the example condition, and this effect correlated with the behavioral performance difference. Taken together, these findings suggest that prediction-induced surprise promoted children’s learning. Also in line with our hypotheses, we found that children’s reasoning abilities were positively related to their performance in the example condition and negatively with the performance difference between the prediction and example condition. That is, the better (i.e., more mature) their reasoning abilities, the more do children resemble adults in that generating examples is similarly effective than generating predictions. In summary, the results support our hypothesis that there are distinct cognitive prerequisites for generating predictions and generating examples, which result in different degrees of effectiveness of these strategies for different age groups.

Our findings provide a proof of concept for the importance of cognitive prerequisites for understanding individual differences as well as age-related trends in the effectiveness of GLSs. Our study bridges the gap between research comparing the effectiveness of different GLSs within the same age group (e.g., Brod et al., 2018 Ritchie & Volkl, 2010 Yeo & Fazio, 2019 ) and research comparing the effectiveness of the same GLS between different age groups (e.g., Gurlitt & Renkl, 2008 ). The former approach has identified specific mechanisms underlying different GLSs (e.g., Brod et al., 2018 ), whereas the latter approach has revealed the need for instructional adaptations to support younger learners (Gurlitt & Renkl, 2008 ). Only the combination of the two approaches, however, allows to investigate why different GLSs might be differentially effective for learners of various ages and with varying cognitive abilities.

Our findings regarding the mechanisms underlying the effectiveness of generating predictions in children correlate well with previous research in adults. In a recent study, Brod et al. ( 2018 ) used pupillometry data to demonstrate that predictions trigger surprise for expectancy-violating events and that, on a between-subject level, the strength of this surprise effect relates to better learning. Our pupillary results are consistent with these findings in that pupil dilations upon seeing the correct results were larger after generating predictions than after generating examples. Our findings extend the ones by Brod et al. ( 2018 ) by demonstrating a similar pupillary surprise response in children, which supports the notion that surprise is an age-invariant mechanism (Schützwohl & Reisenzein, 1999 ) that can be triggered by generating predictions. Moreover, the link between children’s surprise response and subsequent memory predicted children’s performance difference. These results are in line with research that suggests that surprise increases attention to task-relevant information (Fazio & Marsh, 2009 Stahl & Feigenson, 2019 ), thereby enhancing learning. In a different experiment (Brod & Breitwieser, 2019 ), we were able to show that generating a prediction further increases attention to one’s knowledge gap, which leads to increased curiosity for the correct answer. Taken together, the results of our experiments suggest that the effectiveness of generating predictions is at least partially mediated by enhanced attention to the new information.

Generating predictions is arguably one of the simplest possible GLSs. The results of the present study attest to the notion that it is this simplicity that makes generating predictions particularly effective in children, whose limited attentional and cognitive control functions impede the effectiveness of more complex strategies. To improve our understanding of this strategy’s underlying mechanisms, future studies should address the link between prediction errors, surprise, and learning more explicitly as generating predictions does not necessarily require prior knowledge activation. Its effectiveness likely relies on where on the continuum from mere guessing to effortful prior knowledge retrieval the generated prediction lies, as prediction errors made with greater confidence should increase the experience of surprise, which should in turn enhance learning. A previous study by Brod, Breitwieser, Hasselhorn, and Bunge ( 2019 ) suggests that children are not always able to leverage the benefit of surprise for learning, however. Metacognitive skills might need to be in place to override a previous held belief with the correct information (Brod et al., 2019 ). In the current study, the accuracy of children’s retrospective judgments of task performance was not significantly above chance, in contrast with the adults’. This finding fits our knowledge of the protracted development of procedural metamemory (i.e., monitoring and regulation of memory performance Fritz, Howie, & Kleitman, 2010 Schneider, 2008 ). Future studies should look into the mediating role of procedural metamemory for the effect of surprise on learning more closely by measuring metacognitive judgments during task performance.

We also found evidence that generating examples can be an effective learning strategy. Adults utilized examples almost as effectively as predictions. The effectiveness of generating examples in children was linked to their analogical reasoning abilities: Children with better reasoning abilities showed a smaller performance decrease in the example condition (i.e., an adult-like performance pattern). Our results, thus, suggest that analogical reasoning abilities are an important and distinct prerequisite for the benefits of example generation to occur. The late development of analogical reasoning abilities (e.g., Richland et al., 2006 ) can then explain why, on average, children did not meet the cognitive requirements to utilize examples as effectively as adults. Although it has been argued before that analogical reasoning might underlie the successful use of examples (Zamary & Rawson, 2018 ), to our knowledge, this is the first study that explicitly tested the moderating effect of analogical reasoning abilities for the effectiveness of example-based learning. While the correlational results clearly do not allow to infer a causal relation between analogical reasoning and example-based learning, they are a good starting point for future research to test their causal relation via intervention studies.

Unlike previous research, the present study tested example generation not as a means of learning declarative concepts (e.g., Rawson & Dunlosky, 2016 Zamary & Rawson, 2018 ), but as a means of learning isolated facts. Thus, in our study, the primary purpose of the examples was not to make sense of an abstract concept but to associate the new information with a self-generated retrieval cue, which has been shown to aid recall (Greenwald & Banaji, 1989 ). It proved similarly effective to generating predictions in the university students. Nevertheless, one might argue that the task design was not ideal for the benefits of example generation to occur. Specifically, unlike the predictions, the examples did not directly relate to the to-be-learned information (i.e., the numbers). While example generation still required elaboration of task-relevant information, the appropriateness of the examples was not determined by the correct number. It remains an open question to what extent this has dampened the effectiveness of generating examples.

Since the present study did not include a control condition without any learning strategy prompt, we cannot make inferences about how much performance was boosted by generating predictions or generating examples compared to “baseline” performance. Such a baseline condition would be difficult to implement because university students can be expected to use effective learning strategies without being prompted to—in accordance with their knowledge of strategy effectiveness (Justice & Weaver-McDougall, 1989 ). Elementary school children, in contrast, are unlikely to spontaneously use effective learning strategies, and large interindividual differences are to be expected due to the ongoing development of metamemory abilities (Bjorklund, 2010 Bjorklund & Coyle, 1995 ). We, thus, deemed baseline performance to be difficult to compare between age groups and decided to not include a baseline condition in our within-subjects design. Having said that, however, an interesting question for future research could be to elucidate which strategies children and adults spontaneously use to learn facts, whether there are interactions between spontaneous and instructed strategy use (i.e., switching of strategies during the experiment), and how this impacts learning success.

Another limitation of the current study is the sample size, which was determined a priori to be sufficient for testing the hypothesized Age × GLS interaction effect as well as pupillary within-subject condition differences. These preregistered parts of the study were based on a pilot study with 26 children and 18 adults and can, thus, be considered a successful replication. The sample size is too small, however, to ensure reliable estimates of the between-subject correlational analyses. We see these exploratory analyses as a first step toward exploring the specific mechanisms and cognitive requirements of different GLSs to explain age-related performance differences. Conceptual replications are required to test the generalizability of the results to other populations, strategies, and task types. Including additional measures of cognitive abilities would further allow to test the specificity of the effects found in the present study. Finally, the time span between the study and test phases was rather short. We are therefore unable to draw conclusions about the long term effects of generating predictions and generating examples.

In closing, the present study demonstrates the importance of considering learners’ cognitive prerequisites for selecting the most effective GLS. Leveraging an age-group comparison and knowledge of typical developmental trajectories of specific cognitive abilities, this study suggests that different GLS can differ strongly in effectiveness depending on maturity of these abilities. Thus, to achieve the goal of selecting optimal learning strategies for a particular group of learners or even for individual learners, knowledge of their cognitive abilities as well as knowledge of the cognitive prerequisites of a specific GLS are needed. While this goal seems distant still, it is becoming clear that, rather than proclaiming the most effective learning strategy for all learners, the effectiveness of any learning strategy will critically depend on the fit between its cognitive requirements and the cognitive prerequisites of the learner.

Figure S1. Pupillary Response to Seeing the Correct Result, Separately for Later Remembered and Forgotten Facts

Table S1. Sample of Items Used in the Numerical Facts Learning Task (Translated From German)