I have long been fascinated by group theory. I was initially interested only in pure group theory... more I have long been fascinated by group theory. I was initially interested only in pure group theory, but I am now fascinated by representation theory as well. Group theory is one of the areas where I would like to pursue research. Here, I describe my progress in group theory so far. 1. First brushes with group theory 1.1. An old book by Shanti Narayan. Time period: 2002 03 When in eleventh standard, I chanced upon an old algebra book by Shanti Narayan. The rudimentary introduction to groups was my first exposure to these algebraic structures that I have explored so much in the last few years. 1.2. The first Olympiad experience. Time period: May-June 2003 At the International Mathematical Olympiad Training Camp(camp name) (IMOTC) that I attended in twelfth standard, I learnt of the following applications related to group theory: • “Transformation geometry”: The camp was my first systematic introduction to transformations. The transformations (automorphisms) of any structure naturally f...
I’ve always been fascinated by the problems of invariant theory. I’m currently reading Kostant’s ... more I’ve always been fascinated by the problems of invariant theory. I’m currently reading Kostant’s paper for VSRP under Professor Dipendra Prasad. I also stumbled on a book by Igor V. Dolgachev on Invariant Theory. Based on these, and my own understanding gathered from other sources, I document what I think invariant theory is and could be about. 1. Groups acting on sets 1.1. A basic Galois correspondence. Let A and B be two sets, with a relation R between A and B. That is, R is a map A × B → {0, 1}, with a related to b if and only if R(a, b) = 1. Then, define f : 2 → 2 as follows. For S ⊆ A, f(S) is the set of all elements b ∈ B such that R(a, b) = 1 for everya ∈ S. Similarly g : 2 → 2 takes T ⊆ B to the set of all elements a ∈ A such that R(a, b) = 1 for every b ∈ T . The following are true: • If S1 ⊆ S2 ⊆ A, then f(S2) ⊆ f(S1). Analogously for B and g. • For any S ⊆ A, S ⊆ g(f(S)).Analogously for B and g. • Combining the above two, f(g(f(S))) = f(S). Subsets in B of the form f(S) a...
These are lecture notes for the course titled Lie-theoretic methods in analysis taught by Profess... more These are lecture notes for the course titled Lie-theoretic methods in analysis taught by Professor Alladi Sitaram. In these lecture notes, Professor Sitaram defines a linear Lie group and the exponential map and discusses preliminary results on these. 1. Linear Lie groups 1.1. What is a linear Lie group? Definition. A linear Lie group(defined) is a group along with an embedding as a closed subgroup of GL(n, R). Some comments: (1) GL(n, C) can be embedded as a closed subgroup of GL(2n, R) by treating C as a two-dimensional vector space over R. (2) GL(n, R) is an open subset in Rn2 and the multiplication and inverse maps are smooth maps. (3) GL(n, R) is not connected. It has two connected components: the subgroup comprising matrices of positive determinant (denoted as GL(n, R)) and the subgroup comprising matrices of negative determinant 1.2. Examples of linear Lie groups. The following are linear Lie groups, either directly viewed as closed subgroups of GL(n, R), or viewed as closed...
Corresponding material in the book: Section 12.1, 12.2, 12.3. What students should definitely get... more Corresponding material in the book: Section 12.1, 12.2, 12.3. What students should definitely get: The summation notation and how it works, series, concepts of convergence. The use of telescoping and forward difference operator ideas to sum up series. The use of the integral test and other tests to determine whether a series converges and obtain numerical estimates. Convergence rules for rational functions. What students should hopefully get: How the summation notation is similar to the integral notation, how the parallels can be worked out better. 1. The summation notation Suppose we want to write: 1 + 2 + 3 + · · ·+ n The “...” in betwee in somewhat ambiguous. Since we’re good mind readers, we know what is meant. However, it would be better to have a notation that allows us to compactify this while removing the ambiguity. More generally, for a function f defined on {1, 2, 3, . . . , n}, we want a shorthand notation for: f(1) + f(2) + · · ·+ f(n) The shorthand notation is:
The Whitney embedding theorem states that any smooth compact manifold of dimension n can be embed... more The Whitney embedding theorem states that any smooth compact manifold of dimension n can be embedded as a closed submanifold of R2n. In this article, I prove a weaker version of the thoerem, which states that any smooth compact manifold of dimension n can be embedded as a closed submanifold of R2n+1. I do this in two steps: first, embed it in RN for some finite N , and then, through iterated projections, push the dimension to 2n + 1. 1. The weak form 1.1. Statement. Theorem 1 (Whitney embedding: weak form). A smooth compact real manifold of real dimension n can be embedded as a closed submanifold of R for some N ≥ n. Some remarks: • The closed part arises automatically because any compact subset of a Hausdorff space is closed. • It suffices to restrict our attention to connected components. This is because every compact space has only finitely many connected components, and if we obtain embeddings of each, then we can obtain embeddings of their union by simply placing the images suf...
This is a short note intended to explore the applications of duality theory to the study of manif... more This is a short note intended to explore the applications of duality theory to the study of manifolds. I discuss Alexander duality, Lefschetz duality and Poincare duality, along with applications to the study of compact connected orientable manifolds. 1. Manifolds and points 1.1. The core question. One of the questions we shall be interested in is: Given two manifolds M and N , what are the ways in which N embeds as a submanifold of M? In other words, what are the submanifolds of M homeomorphic to N? Roughly speaking, we want to know how N “sits inside” M merely from the data of what M and N look like abstractly. First, we need to define what it means for “ways in which N embeds”. Definition (Equivalently embedded subsets). Give a topological space X and subspaces Y1 and Y2, we say that Y1 and Y2 are i equivalently embedded subsets(defined)f there is a homeomorphism of X under which Y1 maps homeomorphically to Y2, or equivalently, there is a homeomorphism of the pair (X, Y1) and (X,...
It is easy to see that an inner automorphism of a group can always be extended to an inner automo... more It is easy to see that an inner automorphism of a group can always be extended to an inner automorphism of any group containing it. Are inner automorphisms the only such automorphisms? This is the problem of extensible automorphisms. Here, I discuss the problem of extensible automorphisms and its generalizations, as well as the progress that has been made on the problem via correspondence between Dr. Isaacs and me. 1. The problem statement 1.1. A special property of inner automorphisms. Before beginning with the main content of this article, we make an important observation about inner automorphisms. An inner automorphism of a group is a map of the form x 7→ gxg−1 where g is a fixed element of the group. The inner automorphism corresponding to g is also called the conjugation by g or the transform by g. An automorphism on a group is said to lift or extend an automorphism on a subgroup if its restriction to the subgroup is that automorphism on the subgroup. Given a group, an embeddin...
In this journey, we are going to explore the so called “separation axioms” in greater detail. We ... more In this journey, we are going to explore the so called “separation axioms” in greater detail. We shall try to understand how these axioms are affected on going to subspaces, taking products, and looking at small open neighbourhoods. 1. What this journey entails 1.1. Prerequisites. Familiarity with definitions of these basic terms is expected: • Topological space • Open sets, closed sets, and limit points • Basis and subbasis • Continuous function and homeomorphism • Product topology • Subspace topology The target audience for this article are students doing a first course in topology. 1.2. The explicit promise. At the end of this journey, the learner should be able to: • Define the following: T0, T1, T2 (Hausdorff), T3 (regular), T4 (normal) • Understand how these properties are affected on taking subspaces, products and other similar constructions 2. What are separation axioms? 2.1. The idea behind separation. The defining attributes of a topological space (in terms of a family of ...
This is a write-up of Artin’s theorem for the student seminar in the course on Representation The... more This is a write-up of Artin’s theorem for the student seminar in the course on Representation Theory of Finite Groups. Much of the material is taken from Serre’s book on representation theory. 1. Notation and motivation 1.1. Ring of functions on a finite group. First, some basic observations: • Given a set and a ring, the set of functions from the set to the ring forms a ring forms a ring under pointwise addition and pointwise multiplication. • Given a set and an equivalence relation on it, the set of functions constant on the equivalence classes forms a subring of this ring. We are interested in a setup where the set in question is a finite group, the equivalence relation isthat of beig conjugate, and functions are being taken to the complex number C. By the natural ring structure on C, the set of all functions from G to C has a ring structure. Further, the ring of functions constant on the cnojguacy classes (that we shall call class functions) is a subring of this. We shall denote...
These are lecture notes for the course titled Lie-theoretic methods in analysis taught by Profess... more These are lecture notes for the course titled Lie-theoretic methods in analysis taught by Professor Alladi Sitaram. In these lecture notes, Professor Sitaram discusses the Peter-Weyl theorem which is the analogue in compact groups of the fact that representative functions generate all functions on a finite group. 1. The Peter-Weyl theorem 1.1. Haar measure. Let G be a compact linear Lie group. Then, there exists a unique volume element dg on G with the following properties: (1) The integral ∫ G dg = 1 (2) The integral is invariant under left translation, viz: ∫ G f(gh)dh = ∫ G f(h)dh (3) The integral is invariant under right translation, viz: ∫ G f(hg)dh = ∫ G f(h)dh 1.2. Extending results from finite to compact groups. Recall that if (π, V ) is a finite-dimensional representation of a finite group, it can be conjugated to within a unitary representation. The same is in fact true for any compact group. More specifically, as for finite groups, we can say the following: (1) If (π, V )...
We’re not interested in a function here or a function there. We’re interested in the collection o... more We’re not interested in a function here or a function there. We’re interested in the collection of all functions to R or to C. Why are we interested in these? Functions describe lots of things. For example, a function on a “physical body” could be used to describe the temperature at every point on the body. Functions to R or C are scalar-valued: they have their image in a field. We are often interested in vector-valued functions, for instance, vector fields on open sets in Euclidean space. However, the theory of vector-valued functions, in the finite-dimensional case, isn’t very different from that of scalar-valued functions, because, after all, the vector-valued function can be described by its scalar components. This raises a number of interesting points that we shall explore as we proceed. Another important point is that for most practical purposes, theorems which work for functions to R also work for functions to C. The main differences are: (1) For functions to R we use the abs...
Group theory is an important topic in most undergraduate math curricula, and forms the basis for ... more Group theory is an important topic in most undergraduate math curricula, and forms the basis for higher studies both in math and in areas where math is applied. This article is a lightweight, hands on treatment of group theory from a refreshing perspective. The central theme is that of properties. However, prior knowledge or understanding of property theory is not a prerequisite to understanding the article. It is suitable for beginning college students or for others with a hobby interest in group theory. Its insights may also be valuable to more advanced students. 1. The promise – a great journey 1.1. It is a crucial journey. Group theory is one of the first topics taught in undergraduate math curricula, and is the first plunge of college students into the realm of abstract algebra in particular, and possibly college mathematics in general. Groups are a great excuse to introduce a diverse range of concepts that are crucial to the language and toolkit of the whole of mathematics. Th...
Permutation-Extensible: A Route to Extensible Automorphisms
This article describes the use of permutation representations to obtain a partial solution to the... more This article describes the use of permutation representations to obtain a partial solution to the Extensible Automorphisms Problem. It introduces a notion of permutation-extensibility and proves the equivalence of this notion with the notion of being subgroup-conjugating. 1. The problem we want to solve 1.1. Three big problems. A quick recall of three definitions: Definition. (1) An automorphism σ of a group G is termed extensible(defined) if for any embedding G ≤ H, there is an automorphism σ′ of H such that σ|G = σ. For a particular embedding G ≤ H, the group of automorphisms of G that can be extended to automorphisms of H is denoted as Extensible ( G ; H). The group of extensible automorphisms of G is thus given by:
How much do these laws matter? Look at the black market prices poor migrants eagerly pay to hop t... more How much do these laws matter? Look at the black market prices poor migrants eagerly pay to hop to the border. Smuggler fees from Mexico to the U.S. are now about $4000four years' income for a typical farm laborer in Mexico. 9 Prices for more distant countries are predictably higher. A median-income Indian would need to save all his income for over a decade to pay the $60,000 smugglers currently charge for illegal transport to the United States. 10 High though they are, fees underestimate foreigners' commitment to moving. Migrants along the Mexico-U.S. border brave a difficult trek through a hot desert, migrants from Africa to Europe sail on rickety boats, and bordercrossers to South Africa risk getting eaten by lions. 11 After full deregulation, smuggling fees and the attendant dangers would all but disappear. How many people would choose to relocate? Gallup has conducted worldwide polls since 2010 asking adults whether they would move to another country immediately if allowed. Over 600 million adults-14% of the world adult populationwish to permanently move to another country. Over a billion want to seek temporary work abroad. 12 For comparison, 232 million people currently live outside their country of birth. 13 The United States is the first-choice destination for over 100 million adults. 14 Gallup has used these polls to estimate population gain and loss for each country if everyone migrated to their first-choice destination. The effects are huge: Haiti would lose half its population. Australia, Singapore, and New Zealand's would more than double. Even the United States, the world's third most populous country, would see population increase by 60%. 15 This does not mean that 200 million immigrants would arrive tomorrow if the United
Proving concurrence of lines, collinearity of points and concyclicity of points are important cla... more Proving concurrence of lines, collinearity of points and concyclicity of points are important class of problems in elementary geometry. In this article, we use an abstraction from higher geometry to unify these classes of problems. We build upon the abstraction to develop strategies suitable for solving problems of collinearity, concurrence and concyclicity. The article is suited for high school students interested in Olympiad math. The first section is meant as a warmup for junior students before plunging into the main problem.
A Boolean function on N variables is called evasive if its decision-tree complexity is N . A sequ... more A Boolean function on N variables is called evasive if its decision-tree complexity is N . A sequence Bn of Boolean functions is eventually evasive if Bn is evasive for all sufficiently large n.
A Boolean function on N variables is called evasive if its decision-tree complexity is N . A sequ... more A Boolean function on N variables is called evasive if its decision-tree complexity is N . A sequence Bn of Boolean functions is eventually evasive if Bn is evasive for all sufficiently large n.
I have long been fascinated by group theory. I was initially interested only in pure group theory... more I have long been fascinated by group theory. I was initially interested only in pure group theory, but I am now fascinated by representation theory as well. Group theory is one of the areas where I would like to pursue research. Here, I describe my progress in group theory so far. 1. First brushes with group theory 1.1. An old book by Shanti Narayan. Time period: 2002 03 When in eleventh standard, I chanced upon an old algebra book by Shanti Narayan. The rudimentary introduction to groups was my first exposure to these algebraic structures that I have explored so much in the last few years. 1.2. The first Olympiad experience. Time period: May-June 2003 At the International Mathematical Olympiad Training Camp(camp name) (IMOTC) that I attended in twelfth standard, I learnt of the following applications related to group theory: • “Transformation geometry”: The camp was my first systematic introduction to transformations. The transformations (automorphisms) of any structure naturally f...
I’ve always been fascinated by the problems of invariant theory. I’m currently reading Kostant’s ... more I’ve always been fascinated by the problems of invariant theory. I’m currently reading Kostant’s paper for VSRP under Professor Dipendra Prasad. I also stumbled on a book by Igor V. Dolgachev on Invariant Theory. Based on these, and my own understanding gathered from other sources, I document what I think invariant theory is and could be about. 1. Groups acting on sets 1.1. A basic Galois correspondence. Let A and B be two sets, with a relation R between A and B. That is, R is a map A × B → {0, 1}, with a related to b if and only if R(a, b) = 1. Then, define f : 2 → 2 as follows. For S ⊆ A, f(S) is the set of all elements b ∈ B such that R(a, b) = 1 for everya ∈ S. Similarly g : 2 → 2 takes T ⊆ B to the set of all elements a ∈ A such that R(a, b) = 1 for every b ∈ T . The following are true: • If S1 ⊆ S2 ⊆ A, then f(S2) ⊆ f(S1). Analogously for B and g. • For any S ⊆ A, S ⊆ g(f(S)).Analogously for B and g. • Combining the above two, f(g(f(S))) = f(S). Subsets in B of the form f(S) a...
These are lecture notes for the course titled Lie-theoretic methods in analysis taught by Profess... more These are lecture notes for the course titled Lie-theoretic methods in analysis taught by Professor Alladi Sitaram. In these lecture notes, Professor Sitaram defines a linear Lie group and the exponential map and discusses preliminary results on these. 1. Linear Lie groups 1.1. What is a linear Lie group? Definition. A linear Lie group(defined) is a group along with an embedding as a closed subgroup of GL(n, R). Some comments: (1) GL(n, C) can be embedded as a closed subgroup of GL(2n, R) by treating C as a two-dimensional vector space over R. (2) GL(n, R) is an open subset in Rn2 and the multiplication and inverse maps are smooth maps. (3) GL(n, R) is not connected. It has two connected components: the subgroup comprising matrices of positive determinant (denoted as GL(n, R)) and the subgroup comprising matrices of negative determinant 1.2. Examples of linear Lie groups. The following are linear Lie groups, either directly viewed as closed subgroups of GL(n, R), or viewed as closed...
Corresponding material in the book: Section 12.1, 12.2, 12.3. What students should definitely get... more Corresponding material in the book: Section 12.1, 12.2, 12.3. What students should definitely get: The summation notation and how it works, series, concepts of convergence. The use of telescoping and forward difference operator ideas to sum up series. The use of the integral test and other tests to determine whether a series converges and obtain numerical estimates. Convergence rules for rational functions. What students should hopefully get: How the summation notation is similar to the integral notation, how the parallels can be worked out better. 1. The summation notation Suppose we want to write: 1 + 2 + 3 + · · ·+ n The “...” in betwee in somewhat ambiguous. Since we’re good mind readers, we know what is meant. However, it would be better to have a notation that allows us to compactify this while removing the ambiguity. More generally, for a function f defined on {1, 2, 3, . . . , n}, we want a shorthand notation for: f(1) + f(2) + · · ·+ f(n) The shorthand notation is:
The Whitney embedding theorem states that any smooth compact manifold of dimension n can be embed... more The Whitney embedding theorem states that any smooth compact manifold of dimension n can be embedded as a closed submanifold of R2n. In this article, I prove a weaker version of the thoerem, which states that any smooth compact manifold of dimension n can be embedded as a closed submanifold of R2n+1. I do this in two steps: first, embed it in RN for some finite N , and then, through iterated projections, push the dimension to 2n + 1. 1. The weak form 1.1. Statement. Theorem 1 (Whitney embedding: weak form). A smooth compact real manifold of real dimension n can be embedded as a closed submanifold of R for some N ≥ n. Some remarks: • The closed part arises automatically because any compact subset of a Hausdorff space is closed. • It suffices to restrict our attention to connected components. This is because every compact space has only finitely many connected components, and if we obtain embeddings of each, then we can obtain embeddings of their union by simply placing the images suf...
This is a short note intended to explore the applications of duality theory to the study of manif... more This is a short note intended to explore the applications of duality theory to the study of manifolds. I discuss Alexander duality, Lefschetz duality and Poincare duality, along with applications to the study of compact connected orientable manifolds. 1. Manifolds and points 1.1. The core question. One of the questions we shall be interested in is: Given two manifolds M and N , what are the ways in which N embeds as a submanifold of M? In other words, what are the submanifolds of M homeomorphic to N? Roughly speaking, we want to know how N “sits inside” M merely from the data of what M and N look like abstractly. First, we need to define what it means for “ways in which N embeds”. Definition (Equivalently embedded subsets). Give a topological space X and subspaces Y1 and Y2, we say that Y1 and Y2 are i equivalently embedded subsets(defined)f there is a homeomorphism of X under which Y1 maps homeomorphically to Y2, or equivalently, there is a homeomorphism of the pair (X, Y1) and (X,...
It is easy to see that an inner automorphism of a group can always be extended to an inner automo... more It is easy to see that an inner automorphism of a group can always be extended to an inner automorphism of any group containing it. Are inner automorphisms the only such automorphisms? This is the problem of extensible automorphisms. Here, I discuss the problem of extensible automorphisms and its generalizations, as well as the progress that has been made on the problem via correspondence between Dr. Isaacs and me. 1. The problem statement 1.1. A special property of inner automorphisms. Before beginning with the main content of this article, we make an important observation about inner automorphisms. An inner automorphism of a group is a map of the form x 7→ gxg−1 where g is a fixed element of the group. The inner automorphism corresponding to g is also called the conjugation by g or the transform by g. An automorphism on a group is said to lift or extend an automorphism on a subgroup if its restriction to the subgroup is that automorphism on the subgroup. Given a group, an embeddin...
In this journey, we are going to explore the so called “separation axioms” in greater detail. We ... more In this journey, we are going to explore the so called “separation axioms” in greater detail. We shall try to understand how these axioms are affected on going to subspaces, taking products, and looking at small open neighbourhoods. 1. What this journey entails 1.1. Prerequisites. Familiarity with definitions of these basic terms is expected: • Topological space • Open sets, closed sets, and limit points • Basis and subbasis • Continuous function and homeomorphism • Product topology • Subspace topology The target audience for this article are students doing a first course in topology. 1.2. The explicit promise. At the end of this journey, the learner should be able to: • Define the following: T0, T1, T2 (Hausdorff), T3 (regular), T4 (normal) • Understand how these properties are affected on taking subspaces, products and other similar constructions 2. What are separation axioms? 2.1. The idea behind separation. The defining attributes of a topological space (in terms of a family of ...
This is a write-up of Artin’s theorem for the student seminar in the course on Representation The... more This is a write-up of Artin’s theorem for the student seminar in the course on Representation Theory of Finite Groups. Much of the material is taken from Serre’s book on representation theory. 1. Notation and motivation 1.1. Ring of functions on a finite group. First, some basic observations: • Given a set and a ring, the set of functions from the set to the ring forms a ring forms a ring under pointwise addition and pointwise multiplication. • Given a set and an equivalence relation on it, the set of functions constant on the equivalence classes forms a subring of this ring. We are interested in a setup where the set in question is a finite group, the equivalence relation isthat of beig conjugate, and functions are being taken to the complex number C. By the natural ring structure on C, the set of all functions from G to C has a ring structure. Further, the ring of functions constant on the cnojguacy classes (that we shall call class functions) is a subring of this. We shall denote...
These are lecture notes for the course titled Lie-theoretic methods in analysis taught by Profess... more These are lecture notes for the course titled Lie-theoretic methods in analysis taught by Professor Alladi Sitaram. In these lecture notes, Professor Sitaram discusses the Peter-Weyl theorem which is the analogue in compact groups of the fact that representative functions generate all functions on a finite group. 1. The Peter-Weyl theorem 1.1. Haar measure. Let G be a compact linear Lie group. Then, there exists a unique volume element dg on G with the following properties: (1) The integral ∫ G dg = 1 (2) The integral is invariant under left translation, viz: ∫ G f(gh)dh = ∫ G f(h)dh (3) The integral is invariant under right translation, viz: ∫ G f(hg)dh = ∫ G f(h)dh 1.2. Extending results from finite to compact groups. Recall that if (π, V ) is a finite-dimensional representation of a finite group, it can be conjugated to within a unitary representation. The same is in fact true for any compact group. More specifically, as for finite groups, we can say the following: (1) If (π, V )...
We’re not interested in a function here or a function there. We’re interested in the collection o... more We’re not interested in a function here or a function there. We’re interested in the collection of all functions to R or to C. Why are we interested in these? Functions describe lots of things. For example, a function on a “physical body” could be used to describe the temperature at every point on the body. Functions to R or C are scalar-valued: they have their image in a field. We are often interested in vector-valued functions, for instance, vector fields on open sets in Euclidean space. However, the theory of vector-valued functions, in the finite-dimensional case, isn’t very different from that of scalar-valued functions, because, after all, the vector-valued function can be described by its scalar components. This raises a number of interesting points that we shall explore as we proceed. Another important point is that for most practical purposes, theorems which work for functions to R also work for functions to C. The main differences are: (1) For functions to R we use the abs...
Group theory is an important topic in most undergraduate math curricula, and forms the basis for ... more Group theory is an important topic in most undergraduate math curricula, and forms the basis for higher studies both in math and in areas where math is applied. This article is a lightweight, hands on treatment of group theory from a refreshing perspective. The central theme is that of properties. However, prior knowledge or understanding of property theory is not a prerequisite to understanding the article. It is suitable for beginning college students or for others with a hobby interest in group theory. Its insights may also be valuable to more advanced students. 1. The promise – a great journey 1.1. It is a crucial journey. Group theory is one of the first topics taught in undergraduate math curricula, and is the first plunge of college students into the realm of abstract algebra in particular, and possibly college mathematics in general. Groups are a great excuse to introduce a diverse range of concepts that are crucial to the language and toolkit of the whole of mathematics. Th...
Permutation-Extensible: A Route to Extensible Automorphisms
This article describes the use of permutation representations to obtain a partial solution to the... more This article describes the use of permutation representations to obtain a partial solution to the Extensible Automorphisms Problem. It introduces a notion of permutation-extensibility and proves the equivalence of this notion with the notion of being subgroup-conjugating. 1. The problem we want to solve 1.1. Three big problems. A quick recall of three definitions: Definition. (1) An automorphism σ of a group G is termed extensible(defined) if for any embedding G ≤ H, there is an automorphism σ′ of H such that σ|G = σ. For a particular embedding G ≤ H, the group of automorphisms of G that can be extended to automorphisms of H is denoted as Extensible ( G ; H). The group of extensible automorphisms of G is thus given by:
How much do these laws matter? Look at the black market prices poor migrants eagerly pay to hop t... more How much do these laws matter? Look at the black market prices poor migrants eagerly pay to hop to the border. Smuggler fees from Mexico to the U.S. are now about $4000four years' income for a typical farm laborer in Mexico. 9 Prices for more distant countries are predictably higher. A median-income Indian would need to save all his income for over a decade to pay the $60,000 smugglers currently charge for illegal transport to the United States. 10 High though they are, fees underestimate foreigners' commitment to moving. Migrants along the Mexico-U.S. border brave a difficult trek through a hot desert, migrants from Africa to Europe sail on rickety boats, and bordercrossers to South Africa risk getting eaten by lions. 11 After full deregulation, smuggling fees and the attendant dangers would all but disappear. How many people would choose to relocate? Gallup has conducted worldwide polls since 2010 asking adults whether they would move to another country immediately if allowed. Over 600 million adults-14% of the world adult populationwish to permanently move to another country. Over a billion want to seek temporary work abroad. 12 For comparison, 232 million people currently live outside their country of birth. 13 The United States is the first-choice destination for over 100 million adults. 14 Gallup has used these polls to estimate population gain and loss for each country if everyone migrated to their first-choice destination. The effects are huge: Haiti would lose half its population. Australia, Singapore, and New Zealand's would more than double. Even the United States, the world's third most populous country, would see population increase by 60%. 15 This does not mean that 200 million immigrants would arrive tomorrow if the United
Proving concurrence of lines, collinearity of points and concyclicity of points are important cla... more Proving concurrence of lines, collinearity of points and concyclicity of points are important class of problems in elementary geometry. In this article, we use an abstraction from higher geometry to unify these classes of problems. We build upon the abstraction to develop strategies suitable for solving problems of collinearity, concurrence and concyclicity. The article is suited for high school students interested in Olympiad math. The first section is meant as a warmup for junior students before plunging into the main problem.
A Boolean function on N variables is called evasive if its decision-tree complexity is N . A sequ... more A Boolean function on N variables is called evasive if its decision-tree complexity is N . A sequence Bn of Boolean functions is eventually evasive if Bn is evasive for all sufficiently large n.
A Boolean function on N variables is called evasive if its decision-tree complexity is N . A sequ... more A Boolean function on N variables is called evasive if its decision-tree complexity is N . A sequence Bn of Boolean functions is eventually evasive if Bn is evasive for all sufficiently large n.
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