Author Archives: mathtuition88

About mathtuition88

http://mathtuition88.com

Cast Iron Pan Singapore Review

Recently bought a cast iron pan/skillet for home cooking. Cast iron is an ancient technology that has several benefits over the more modern non-stick technology. It is supposed to be cheap (just US$10 in America), but in Singapore it is … Continue reading

Posted in math | Tagged , , | Leave a comment

Inspirational Scientist: Dan Shechtman

Source: https://www.theguardian.com/science/2013/jan/06/dan-shechtman-nobel-prize-chemistry-interview To stand your ground in the face of relentless criticism from a double Nobel prize-winning scientist takes a lot of guts. For engineer and materials scientist Dan Shechtman, however, years of self-belief in the face of the eminent Linus … Continue reading

Posted in math | Tagged | 1 Comment

Structure Theorem for finitely generated (graded) modules over a PID

If is a PID, then every finitely generated module over is isomorphic to a direct sum of cyclic -modules. That is, there is a unique decreasing sequence of proper ideals such that where , and . Similarly, every graded module … Continue reading

Posted in math | Tagged , | Leave a comment

Secondary Level Chinese Tuition

Looking for O Level / IP / JC Chinese Tuition? Ms Gao specializes in teaching secondary level chinese (CL/HCL) tuition in Singapore. Ms Gao has taught students from various schools, including RI (Raffles Institution IP Programme). Teaches West / Central … Continue reading

Posted in chinese tuition | Tagged , , | Leave a comment

Persistence Interval

Next, we want to parametrize the isomorphism classes of the -modules by suitable objects. A -interval is an ordered pair with . We may associate a graded -module to a set of -intervals via a bijection . We define for … Continue reading

Posted in math | Tagged | Leave a comment

The Map of Mathematics (YouTube)

A nicely done video on how the various branches of mathematics fit together. It is amazing that he has managed to list all the major branches on one page. Also see: Beautiful Map of Mathematics.

Posted in math | Tagged , | Leave a comment

Homogenous / Graded Ideal

Let be a graded ring. An ideal is homogenous (also called graded) if for every element , its homogenous components also belong to . An ideal in a graded ring is homogenous if and only if it is a graded … Continue reading

Posted in math | Tagged , | Leave a comment

Donate to help Stray Dogs in Singapore

URL: https://give.asia/movement/run_for_exclusively_mongrels 3 Singaporeans – Dr Gan, A Dentist, Dr Herman, A Doctor, and Mr Ariffin, a Law Undergraduate will be taking on the Borneo Ultra Trail Marathon on Feb 18th 2017 to raise 30k for Exclusively Mongrels Ltd; a welfare … Continue reading

Posted in math | Tagged , , | Leave a comment

Water cuts through rock, not because of its strength, but because of its persistence.

Water cuts through rock, not because of its strength, but because of its persistence.

Image | Posted on | Tagged | Leave a comment

Smooth/Differentiable Manifold

Smooth Manifold A smooth manifold is a pair , where is a topological manifold and is a smooth structure on . Topological Manifold A topological -manifold is a topological space such that: 1)  is Hausdorff: For every distinct pair of … Continue reading

Posted in math | Tagged | Leave a comment

Persistence module and Graded Module

We show that the persistent homology of a filtered simplicial complex is the standard homology of a particular graded module over a polynomial ring. First we review some definitions. A graded ring is a ring (a direct sum of abelian … Continue reading

Posted in math | Tagged | Leave a comment

GEP Selection Test Review and Experience

The following is a parent’s review and experience of the GEP Selection Test (2016). Original text (in Chinese) at: http://mp.weixin.qq.com/s/xQpLynFWpZ6QNpI_vlw4cw Interested readers may also want to check out Recommended Books for GEP Selection Test. Translation: One day in September 2016 afternoon, read … Continue reading

Posted in math | Tagged , | Leave a comment

Persistence module and Finite type

A persistence module is a family of -modules , together with homomorphisms . For example, the homology of a persistence complex is a persistence module, where maps a homology class to the one that contains it. A persistence complex (resp.\ … Continue reading

Posted in math | Tagged | Leave a comment

To Live Your Best Life, Do Mathematics

This article is a very good read. 100% Recommended to anyone interested in math. The ancient Greeks argued that the best life was filled with beauty, truth, justice, play and love. The mathematician Francis Su knows just where to find … Continue reading

Posted in math | Tagged | 2 Comments

Homotopy for Maps vs Paths

Homotopy (of maps) A homotopy is a family of maps , , such that the associated map given by is continuous. Two maps are called homotopic, denoted , if there exists a homotopy connecting them. Homotopy of paths A homotopy … Continue reading

Posted in math | Tagged | Leave a comment

Universal Property of Quotient Groups (Hungerford)

If is a homomorphism and is a normal subgroup of contained in the kernel of , then “factors through” the quotient uniquely. This can be used to prove the following proposition: A chain map between chain complexes and induces homomorphisms … Continue reading

Posted in math | Tagged , , | Leave a comment

Some Homology Definitions

Chain Complex A sequence of homomorphisms of abelian groups with for each . th Homology Group is the free abelian group with basis the open -simplices of . -chains Elements of , called -chains, can be written as finite formal … Continue reading

Posted in math | Tagged , | Leave a comment

RP^n Projective n-space

Define an equivalence relation on by writing if and only if . The quotient space is called projective -space. (This is one of the ways that we defined the projective plane .) The canonical projection is just . Define , … Continue reading

Posted in math | Tagged | Leave a comment

Introduction to Persistent Homology (Cech and Vietoris-Rips complex)

Motivation Data is commonly represented as an unordered sequence of points in the Euclidean space . The global `shape’ of the data may provide important information about the underlying phenomena of the data. For data points in , determining the … Continue reading

Posted in math | Tagged | 2 Comments

Equivalence of C^infinity atlases

Equivalence of atlases is an equivalence relation. Each atlas on is equivalent to a unique maximal atlas on . Proof: Reflexive: If is a atlas, then is also a atlas. Symmetry: Let and be two atlases such that is also … Continue reading

Posted in math | Tagged | Leave a comment

Natural Equivalence relating Suspension and Loop Space

Theorem: If , , , , Hausdorff and locally compact, then there is a natural equivalence defined by , where if is a map then is given by . We need the following two propositions in order to prove the … Continue reading

Posted in math | Tagged | Leave a comment

Fundamental Group of S^n is trivial if n>=2

if We need the following lemma: If a space is the union of a collection of path-connected open sets each containing the basepoint and if each intersection is path-connected, then every loop in at is homotopic to a product of … Continue reading

Posted in math | Tagged | Leave a comment

Tangent Space is Vector Space

Prove that the operation of linear combination, as in Definition 2.2.7, makes into an -dimensional vector space over . The zero vector is the infinitesimal curve represented by the constant . If , then where , defined for all sufficiently … Continue reading

Posted in math | Tagged | Leave a comment

Balance Quote

Never let success go to your head, and never let failure go to your heart.

Posted in math | Tagged | Leave a comment

Analysis: 97 marks not enough for Higher Chinese cut-off point for Pri 1 pupils

Quite tough to be a primary school kid nowadays, even 97 marks is not enough to be admitted for Higher Chinese classes. From experience, the main underlying reasons behind this scenario could be: Due to intensive tuition starting from preschool, students … Continue reading

Posted in math | Tagged | Leave a comment

Functors, Homotopy Sets and Groups

Functors Definition: A functor from a category to a category is a function which – For each object , we have an object . – For each , we have a morphism Furthermore, is required to satisfy the two axioms: – For … Continue reading

Posted in math | Tagged | 2 Comments

H2 Maths Tuition by Ex-RI, NUS 1st Class Honours (Mathematics)

Junior College H2 Maths Tuition About Tutor (Mr Wu): https://mathtuition88.com/singapore-math-tutor/ – Raffles Alumni – NUS 1st Class Honours in Mathematics Experience: More than 10 years experience, has taught students from RJC, NJC, ACJC and many other JCs. Personality: Friendly, patient and … Continue reading

Posted in math | Tagged , | Leave a comment

viXra vs arXiv

viXra (http://vixra.org/) is the cousin of arXiv (http://arxiv.org/) which are electronic archives where researchers can submit their research before being published on a journal. The difference is that viXra allows anyone to submit their article, whereas arXiv requires an academic … Continue reading

Posted in math | Tagged , | Leave a comment

Algebraic Topology: Fundamental Group

Homotopy of paths A homotopy of paths in a space is a family , , such that (i) The endpoints and are independent of . (ii) The associated map defined by is continuous. When two paths and are connected in this way … Continue reading

Posted in math | Tagged | Leave a comment

Multivariable Derivative and Partial Derivatives

If is a derivative of at , then . In particular, if is differentiable at , these partial derivatives exist and the derivative is unique. Proof: Let , then becomes since . By choosing (all zeroes except in th position), … Continue reading

Posted in math | Tagged | Leave a comment

Existence and properties of normal closure

If is an algebraic extension field of , then there exists an extension field of (called the normal closure of over ) such that (i)  is normal over ; (ii) no proper subfield of containing is normal over ; (iii) if is … Continue reading

Posted in math | Tagged | Leave a comment

Happy New Year to Readers of Mathtuition88.com

Wishing all readers of Mathtuition88.com a happy new year, and may 2017 bring you peace and joy in your life. No matter which stage of life you are in (student/career/parent/retiree), here is my sincere wishes that you will achieve your … Continue reading

Posted in math | Tagged | Leave a comment

Printable Calendar 2017

2017 Calendar Printable Calendar 2017, with (Singapore) holidays. Generated by http://www.calendarlabs.com/customize/pdf-calendar/monthly-calendar-01.

Posted in math | Tagged | Leave a comment

A little more perseverance, maybe success is near

Wishing all readers a happy new year ahead. May your dreams and wishes come true! 再努力一下,或许就是成功!

Posted in math | Tagged | Leave a comment

A finitely generated torsion-free module A over a PID R is free

A finitely generated torsion-free module over a PID is free. Proof (Hungerford 221) If , then is free of rank 0. Now assume . Let be a finite set of nonzero generators of . If , then () if and … Continue reading

Posted in math | Tagged | Leave a comment

Tensor is a right exact functor Elementary Proof

This is a relatively elementary proof (compared to others out there) of the fact that tensor is a right exact functor. Proof is taken from Hungerford, and reworded slightly. The key prerequisites needed are the universal property of quotient and … Continue reading

Posted in math | Tagged | Leave a comment

Note on Finitely Generated Abelian Groups

We state and prove a sufficient condition for finitely generated Abelian Groups to be the direct product of its generators, and state a counterexample to the conclusion when the condition is not satisfied. Theorem Let be an abelian group and … Continue reading

Posted in math | Tagged | Leave a comment

Locally Lipschitz implies Lipschitz on Compact Set Proof

Assume is locally Lipschitz on , that is, for any , there exists (depending on ) such that for all . Then, for any compact set , there exists a constant (depending on ) such that for all . That … Continue reading

Posted in math | Tagged | Leave a comment

Gauss Lemma Proof

There are two related results that are commonly called “Gauss Lemma”. The first is that the product of primitive polynomial is still primitive. The second result is that a primitive polynomial is irreducible over a UFD (Unique Factorization Domain) D, … Continue reading

Posted in math | Tagged , | Leave a comment

Free Math Notes by AMS

Just learnt from Professor Terence Tao’s blog that there is a new series of free math notes by the American Mathematical Society: http://www.ams.org/open-math-notes. Many of the notes there are of exceptionally high quality (check out “A singular mathematical promenade”, by Étienne Ghys). … Continue reading

Posted in math | Tagged | Leave a comment

Non-trivial submodules of direct sum of simple modules

Suppose and are two non-isomorphic simple, nonzero -modules. Determine all non-trivial submodules of . Let be a non-trivial submodule of . Note that is a composition series. By Jordan-Holder theorem, all composition series are equivalent and have the same length. … Continue reading

Posted in math | Tagged | Leave a comment

Recommended Educational Toys

Does your child complain that science is “boring”? This may be because science is often taught in a boring manner. The solution may be to supplement teaching with hands-on experiments that develop the inner curiosity of the child. For parents … Continue reading

Posted in math | Tagged , | 1 Comment

Kids with tuition fare worse?

Article: http://www.straitstimes.com/opinion/kids-with-tuition-fare-worse Those who read the news, either online or in print, would probably have seen this article: “Kids with tuition fare worse”. In the article, it is claimed that: “In fact, children who received tuition actually scored about 0.256 standard … Continue reading

Posted in math | Tagged , | 1 Comment

Real Life Tortoise vs. Hare

Quite motivational indeed!

Posted in math | Tagged | 1 Comment

Commutator subgroup G’ is the unique smallest normal subgroup N such that G/N is abelian.

Commutator subgroup is the unique smallest normal subgroup such that is abelian. If is a group, then is a normal subgroup of and is abelian. If is a normal subgroup of , then is abelian iff contains . Proof Let … Continue reading

Posted in math | Tagged | Leave a comment

Ascending Central Series and Nilpotent Groups

Ascending Central Series of Let be a group. The center of is a normal subgroup. Let be the inverse image of under the canonical projection . By Correspondence Theorem, is normal in and contains . Continue this process by defining … Continue reading

Posted in math | Tagged | Leave a comment

Singaporean Student Wins US$250K Scholarship

Congratulations to Ms See for the win. The amount of the scholarship is no joke, it is the price of a HDB flat (a home in Singapore). If you are interested to take part, here are the details: The Breakthrough Junior … Continue reading

Posted in math | Tagged | Leave a comment

Existence of Splitting Field with degree less than n!

If is a field and has degree , then there exists a splitting field of with . Proof: We use induction on . Base case: If , or if splits over , then is a splitting field with . Induction … Continue reading

Posted in math | Tagged | Leave a comment

Fate: A Hebrew Folktale

Source: https://www.storyarts.org/library/nutshell/stories/fate.html A Hebrew Folktale King Solomon’s servant came breathlessly into the court, “Please! Let me borrow your fastest horse!” he said to the King. “I must be in a town ten miles south of here by nightfall!” “Why?” asked King … Continue reading

Posted in math | Tagged | 1 Comment

Transitivity of Algebraic Extensions

Let be a tower of fields. If and are algebraic, then is algebraic. Proof: (Hungerford pg 237, reworded) Let . Since is algebraic over , there exists some () such that Let , then is algebraic over . Hence is … Continue reading

Posted in math | Tagged | Leave a comment