A polynomial generalization of the Euler characteristic for algebraic sets
Miguel A. Marco-Buzunáriz, with an appendix by J. V. Rennemo

Abstract:
We present a method to compute the Euler characteristic of an algebraic subset of C^n. This method relies on classical tools such as Gröbner basis and primary decomposition. The existence of this method allows us to define a new invariant for such varieties. This invariant is related to the problem of counting rational points over finite fields. In an appendix, Jørgen Vold Rennemo proves the relation between this invariant and the Chern-Schwartz-MacPherson class of the variety.

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About the Journal of Singularities:
The Journal of Singularities is an online, freely accessible, refereed journal, which publishes only the highest-quality research articles in all areas of singularity theory, including, but not limited to, the areas of real and complex analytic spaces and maps, subanalytic spaces, stratifications, resolutions of singularities, hyperplane arrangements, mixed Hodge theory, knot theory and Milnor fibrations, metric properties, singularities in characteristic p, and applications of singularity theory. In addition, papers in related areas of differential geometry, algebraic geometry, commutative algebra, and other fields are welcomed.

View Volume 3, 2011
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Future articles will be published immediately upon acceptance.

	Math Ref Free 2.4: This app seems to have a little bit of everything. Users can search through the content by using the topic list or search bar. One of the best features we found was the ability to share equations and formulas through e-mail or printing directly from the app. (Free, also available for Android)
	DocAS Lite: This app is great for note taking. It allows users to type or use their finger to write formulas, notes, or sketches. The background of the notepad can be formatted to appear plain, lined, or as graph paper. Multiple folders on the home screen are available to help organize notes of different classes or topics. This app makes it easy to import PDFs from the web that can be annotated in the app. (Free)
	CalcMadeEasy Free: This app includes a calculator and notepad. Everything typed into the calculator is recorded and can be edited in the notepad. The calculator includes a list of universal constants and the app changes annotation with tablet movement. (Free)
	Equation Solver: The Equation Solver app solves 1st, 2nd, and 3rd degree equations as well as systems of equations with 2 or 3 unknowns. We were disappointed to find that this app does not rotate with the tablet. (Free)
	Math Quizzer: This app is great for testing addition, subtraction, multiplication, division, or any combination of the four. You are presented with a digital chalkboard where equations are shown with four possible solutions in a multiple-choice format. Choose the correct solution and it keeps track of your score. Unfortunately, it seems they are still having development issues when it comes to scoring, but it is still a great quizzing tool. (Free)
	Wow math.org: Wow math.org has created multiple apps for calculus and algebra. All of the apps have the same format; a list of lessons that each contain a video showing an example as well as practice problems with answers. We found the apps both simple and effective. ($0.99)

The Center of Math has created three affordable study guide apps for Differential (Calc I), Integral (Calc II), and AP Calculus. Our apps have full-length video lectures on all topics covered in the given course followed by written examples and/or video solutions. All of our apps are available through iTunes. We also have a new app coming soon called Worldwide AB Calculus powered by LearnBop via edmodo.

Please share with us any apps you have found useful for mathematics by commenting on this post or sharing it on our Facebook page!

A Volume-Preserving Normal Form for a Reduced Normal Crossing Function Germ
Adrian Szawlowski

Abstract:
We will derive a volume-preserving normal form for holomorphic function germs that are right-equivalent to the product of all coordinates.

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About the Journal of Singularities:
The Journal of Singularities is an online, freely accessible, refereed journal, which publishes only the highest-quality research articles in all areas of singularity theory, including, but not limited to, the areas of real and complex analytic spaces and maps, subanalytic spaces, stratifications, resolutions of singularities, hyperplane arrangements, mixed Hodge theory, knot theory and Milnor fibrations, metric properties, singularities in characteristic p, and applications of singularity theory. In addition, papers in related areas of differential geometry, algebraic geometry, commutative algebra, and other fields are welcomed.

View Volume 3, 2011
View Previous Volumes
Future articles will be published immediately upon acceptance.

We have open source lecture videos on a variety of Calculus topics:

Worldwide Lecture Videos by Worldwide Center of Mathematics, LLC are licensed under a Creative Commons Attribution 3.0 Unported License.

	Differential Calculus (28 videos) Rates of Change and the Derivative Basic Rules for Calculuating Derivatives Applications of Differentiation Introduction to Differential Equations
	Integral Calculus (31 videos) Anti-differentiation: the definite Integral Continuous Sums: The Definite Integral Applications of Integration Understanding Functions via Polynomials and Power Series Theorems on Sequences and Series
	AP Calculus (52 videos) Rates of Change and the Derivative Basic Rules for Calculating Derivatives Applications of Differentiation Introduction to Differential Equations Continuous Sums: The Definite Integral Applications of Integration Understanding Functions via Polynomials and Power Series Theorems on Sequences and Series
	Multivariable Calculus (28 videos) An assortment of lectures from Worldwide Multivariable Calculus chapters 2-4. Comprehensive list coming soon.

We plan on continuing to expand the content available into different mathematics area in 2012.

We also have open research talks by world-renowned mathematicians:

Worldwide Research Lecture Videos by Worldwide Center of Mathematics, LLC are licensed under a Creative Commons Attribution 3.0 Unported License.

See the complete list of research talks.

We are excited to announce our first open licensed textbook!

A Term of Commutative Algebra by Worldwide Center of Mathematics, LLC is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

A Term of Commutative Algebra By Allen Altman and Steven Kleiman Coming Fall 2012 From the Preface: There is no shortage of books on Commutative Algebra, but the present book is different. Most books are monographs, with extensive coverage. There is one notable exception: Atiyah and Macdonald’s 1969 classic. It is a clear, concise, and efficient textbook, aimed at beginners, with a good selection of topics. So it has remained popular. However, its age and flaws do show. So there is need for an updated and improved version, which the present book aims to be. Contact us to receive a free digital copy as soon as it is complete. We are also looking for faculty that may be willing to use or review the beta version this summer in order to provide feedback to the authors.

Inflection Points of Real and Tropical Plane Curves
Erwan Brugallé and Lucia López de Medrano

Abstract:
We prove that Viro’s patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic $M$-curves realize many isotopy types. The strategy we adopt in this paper is tropical: we study tropical limits of inflection points of classical plane algebraic curves. The main tropical tool we use to understand these tropical inflection points are tropical modifications.

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About the Journal of Singularities:
The Journal of Singularities is an online, freely accessible, refereed journal, which publishes only the highest-quality research articles in all areas of singularity theory, including, but not limited to, the areas of real and complex analytic spaces and maps, subanalytic spaces, stratifications, resolutions of singularities, hyperplane arrangements, mixed Hodge theory, knot theory and Milnor fibrations, metric properties, singularities in characteristic p, and applications of singularity theory. In addition, papers in related areas of differential geometry, algebraic geometry, commutative algebra, and other fields are welcomed.

View Volume 3, 2011
View Previous Volumes
Future articles will be published immediately upon acceptance.

Abstract: In a 2007 paper entitled “On the Centralizer of K in U(g)”, Bertram Kostant gave an algorithm for computing the generators of the centralizer of a maximal compact group K in the enveloping algebra of a complex semisimple Lie algebra U(g). His motivation is to rescue an old program that sought to determine an irreducible Harish-Chandra module H of U(g) via the finite- dimensional action of the above centralizer on a primary k-component in H. “This original approach of Harish-Chandra to a determination of all H has largely been abandoned because one knows very little about generators of that centralizer”. However, Kostant’s approach produces an algorithm of very high time and space complexity. We present a different algorithm of a lower complexity and compute some previously unknown examples. This is joint work with Steven Jackson.

The Worldwide Center of Mathematics (www.centerofmath.org) is located midway between Harvard and Central Squares, at 929 Massachusetts Avenue, Cambridge, MA, in Suite #102. Travel to the Center by public transportation is easy via the #1 bus, or by taking the subway (the T) to Central Square, and walking for 10 minutes. Suite #102 is located on floor 01.

All attendees will need to sign a release form, as the lecture will be recorded for distribution on the Web.

Abstract: We demonstrate how topological singularity invariants are the key to “seeing” geometry inside of fundamental groups of function fields of surfaces, and explain the relation to Grothendieck’s program of Anabelian Geometry.

The Worldwide Center of Mathematics (www.centerofmath.org) is located midway between Harvard and Central Squares, at 929 Massachusetts Avenue, Cambridge, MA, in Suite #102. Travel to the Center by public transportation is easy via the #1 bus, or by taking the subway (the T) to Central Square, and walking for 10 minutes. Suite #102 is located on floor 01.

All attendees will need to sign a release form, as the lecture will be recorded for distribution on the Web.

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Last week the team from the Worldwide Center of Mathematics attended the 2012 Joint Mathematics Meeting (JMM) at Hynes Convention Center in Boston. The JMM is the largest annual mathematics meeting in the world with participants ranging from students to leading researchers in mathematics as well as innovators in the field of mathematical education programs. This year the JMM had approximately 7,000 attendees; a new record. We received a larger response than ever by faculty and students looking for affordable and open source textbooks, as well as free online resources.