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WALKS, TRAILS, PATHS, CYCLES AND CIRCUITS Definition: A Path is defined as an open trail with no repeated vertices. Notice that all paths must therefore be open walks, as a path cannot both start and terminate at the same vertex. For example, the following orange coloured walk is a path. because the walk does not repeat any edges. Now let's look at the next graph with the tealwalk.
THE COMPLEX COSINE AND SINE FUNCTIONS We will now extend the real-valued sine and cosine functions to complex-valued functions. For reference, the graphs of the real-valued cosine (red) and sine (blue) functions are given below: Recall from The Complex Exponential Function page that for any imaginary number we have that: (1) So then since is an even function and is an oddfunction
THE GLUING LEMMA
The Gluing Lemma. Lemma 1: Let and be topological spaces and let be closed subsets of such that . Furthermore, let and be continuous maps such that for all . Then the function given by is continuous. We require that for all so that is well defined! Proof: To show that is continuous we will show that for every closed set we have that isclosed in .
THE LIMIT SUPERIOR AND LIMIT INFERIOR OF SEQUENCES OF REAL Hence the limit of this sequence will be the supremum of this set, that is: We now have two different ways to express the limit superior and limit inferior of a sequence of real numbers. We now establish some important results regarding the general limit of and the limit superior and limit inferior. Lemma 1: Let be a sequence of realnumbers.
EXAMPLES OF APPLYING HENSEL'S LEMMA Examples of Applying Hensel's Lemma. Recall from the Hensel's Lemma page that if is a polynomial, is a prime, is a solution to , and then there exists a unique lift to a solution where . In particular, if is a solution to and then the recursive formula: (1) is a solution to . We will now look at an example of applying Hensel's Lemma to solving . SEQUENTIAL CRITERION FOR THE CONTINUITY OF A FUNCTION Theorem 1 (Sequential Criterion for Continuity): Let be a function. Then is continuous at the point if and only if for all sequences from with then we have that . Proof: Let be continuous at the point and let be any arbitrary sequence from such that . We want to show that . Let be given. Since is continuous at then exists a such that if and then . THE CONNECTEDNESS OF THE CLOSURE OF A SET Theorem 1: Let be a topological space and let . If the subspace is connected then the closure is also connected. Proof: Let be a connected topological subspace and assume that is disconnected. We will show that this leads to a contradiction. If is disconnected then there exists open sets where , and . Since , if we take the closure ofboth
THE OSCULATING CIRCLE AT A POINT ON A CURVE The Osculating Circle at is the circle that best approximates at . The osculating circle at : 1) Contains the point . 2) Has radius . 3) Has curvature . 4) Shares the same tangent at . If generates a curve , then the osculating circle at is also defined to be the circle that best approximates at by having radius , curvature , and shares the PROOF THAT SIN(X) ≤ X FOR ALL POSITIVE REAL NUMBERS Proof that sin (x) ≤ x for All Positive Real Numbers. A very useful inequality that sometimes appears in calculus and analysis is that for any nonnegative real number we have that . We will now prove this result using an elementary result from calculus - the Mean Value theorem. We state this result below and then prove this inequality. DETERMINING A VECTOR GIVEN TWO POINTS Let's say we have two points in 3-space, one of which has its initial point situated at the origin and its terminal point at coordinates . We can draw the vector as follows: In these cases, the direction of the arrow on top of vector corresponds to the initial and terminal points of the vector , that is the arrow indicates the vector goesfrom
WALKS, TRAILS, PATHS, CYCLES AND CIRCUITS Definition: A Path is defined as an open trail with no repeated vertices. Notice that all paths must therefore be open walks, as a path cannot both start and terminate at the same vertex. For example, the following orange coloured walk is a path. because the walk does not repeat any edges. Now let's look at the next graph with the tealwalk.
THE COMPLEX COSINE AND SINE FUNCTIONS We will now extend the real-valued sine and cosine functions to complex-valued functions. For reference, the graphs of the real-valued cosine (red) and sine (blue) functions are given below: Recall from The Complex Exponential Function page that for any imaginary number we have that: (1) So then since is an even function and is an oddfunction
THE GLUING LEMMA
The Gluing Lemma. Lemma 1: Let and be topological spaces and let be closed subsets of such that . Furthermore, let and be continuous maps such that for all . Then the function given by is continuous. We require that for all so that is well defined! Proof: To show that is continuous we will show that for every closed set we have that isclosed in .
THE LIMIT SUPERIOR AND LIMIT INFERIOR OF SEQUENCES OF REAL Hence the limit of this sequence will be the supremum of this set, that is: We now have two different ways to express the limit superior and limit inferior of a sequence of real numbers. We now establish some important results regarding the general limit of and the limit superior and limit inferior. Lemma 1: Let be a sequence of realnumbers.
EXAMPLES OF APPLYING HENSEL'S LEMMA Examples of Applying Hensel's Lemma. Recall from the Hensel's Lemma page that if is a polynomial, is a prime, is a solution to , and then there exists a unique lift to a solution where . In particular, if is a solution to and then the recursive formula: (1) is a solution to . We will now look at an example of applying Hensel's Lemma to solving . SEQUENTIAL CRITERION FOR THE CONTINUITY OF A FUNCTION Theorem 1 (Sequential Criterion for Continuity): Let be a function. Then is continuous at the point if and only if for all sequences from with then we have that . Proof: Let be continuous at the point and let be any arbitrary sequence from such that . We want to show that . Let be given. Since is continuous at then exists a such that if and then . THE CONNECTEDNESS OF THE CLOSURE OF A SET Theorem 1: Let be a topological space and let . If the subspace is connected then the closure is also connected. Proof: Let be a connected topological subspace and assume that is disconnected. We will show that this leads to a contradiction. If is disconnected then there exists open sets where , and . Since , if we take the closure ofboth
THE OSCULATING CIRCLE AT A POINT ON A CURVE The Osculating Circle at is the circle that best approximates at . The osculating circle at : 1) Contains the point . 2) Has radius . 3) Has curvature . 4) Shares the same tangent at . If generates a curve , then the osculating circle at is also defined to be the circle that best approximates at by having radius , curvature , and shares the WALKS, TRAILS, PATHS, CYCLES AND CIRCUITS Definition: A Path is defined as an open trail with no repeated vertices. Notice that all paths must therefore be open walks, as a path cannot both start and terminate at the same vertex. For example, the following orange coloured walk is a path. because the walk does not repeat any edges. Now let's look at the next graph with the tealwalk.
THE COMPLEX COSINE AND SINE FUNCTIONS We will now extend the real-valued sine and cosine functions to complex-valued functions. For reference, the graphs of the real-valued cosine (red) and sine (blue) functions are given below: Recall from The Complex Exponential Function page that for any imaginary number we have that: (1) So then since is an even function and is an oddfunction
ALGEBRAIC STRUCTURES Definition: A field is a set with the two binary operations of addition and multiplication, both of which operations are commutative, associative, contain identity elements, and contain inverse elements. The identity element for addition is 0, and the identity element for multiplication is 1. Given x, the inverse element for addition is -x, and the multiplicative inverse element for OPEN SETS IN THE COMPLEX PLANE Trivially, the empty set and whole set are open sets. With these two notions, it can be shown that is a topological space. Proposition 1: The open sets of satisfy the following properties: a) and are open in . b) If is an arbitrary collection of open sets in then is open in THE CONNECTEDNESS OF THE CLOSURE OF A SET Theorem 1: Let be a topological space and let . If the subspace is connected then the closure is also connected. Proof: Let be a connected topological subspace and assume that is disconnected. We will show that this leads to a contradiction. If is disconnected then there exists open sets where , and . Since , if we take the closure ofboth
THE DIRECT PRODUCT OF TWO RINGS For example, consider the ring of real number $(\mathbb{R}, +_1, *_1)$ where $+_1$ and $*_1$ denote standard addition and standard multiplication. Also consider the ring of $2 \times 2$ matrices with real coefficients $(M_{22}, +_2, *_2)$ where $+_2$ and $*_2$ denote standard matrix addition and standard matrix multiplication. Then the direct product between these two rings is the following AREAS UNDER PARAMETRIC CURVES EXAMPLES 1 \begin{align} A = \int_0^4 t \cdot 2t \: dt \\ A = 2 \int_0^4 t^2 \: dt \\ A = 2 \left ( \frac{t^3}{3} \right ) \bigg |_0^4 \\ A = 2 \left ( \frac{64}{3} \right 6 COLOUR THEOREM FOR PLANAR GRAPHS 6 Colour Theorem for Planar Graphs. Theorem 1: For a connected planar simple graph , the vertices in can be coloured with or fewer colours for a good (or less) colouring of , that is, a function exists, where such that for every , whenever , . Proof: Let be the statement that for a connected planar simple graph , the vertices in can be coloured GROUP SUBREPRESENTATIONS Group Subrepresentations. Definition: Let be a group and let be a group representation of . A Subrepresentation of is a pair consisting of a subspace of that is -invariant, i.e., for all and for all we have that , and formally, the pair is the subrepresentation, where for each, .
LIMIT DIVERGENCE CRITERIA We will now formulate what is known as the Limit Divergence Criteria, which will establish criteria to establish whether the value is not the limit of as . Theorem 1 (Limit Divergence Criteria): Let be a function and let be a cluster point of . Then for , if either: 1. There exists a sequence from where such that but . 2. PROOF THAT SIN(X) ≤ X FOR ALL POSITIVE REAL NUMBERS Proof that sin (x) ≤ x for All Positive Real Numbers. A very useful inequality that sometimes appears in calculus and analysis is that for any nonnegative real number we have that . We will now prove this result using an elementary result from calculus - the Mean Value theorem. We state this result below and then prove this inequality. WALKS, TRAILS, PATHS, CYCLES AND CIRCUITS Definition: A Path is defined as an open trail with no repeated vertices. Notice that all paths must therefore be open walks, as a path cannot both start and terminate at the same vertex. For example, the following orange coloured walk is a path. because the walk does not repeat any edges. Now let's look at the next graph with the tealwalk.
POINT-NORMAL FORM OF A PLANE Point-Normal Form of a Plane. In 2-space, a line can algebraically be expressed by simply knowing a point that the line goes through and its slope. This can be expressed in the form . In 3-space, a plane can be represented differently. We will still need some point that lies on the plane in 3-space, however, we will now use a value called the ALGEBRAIC STRUCTURES PROVING THE EXISTENCE OF LIMITS Proving The Existence of Limits. Recall from the precise definition of a limit on the Introduction to Limits page, we said that the statement $\lim_{x \to a} f(x) = L$ says that for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $0 < \mid x - a \mid < \delta$ then $\mid f(x) - L \mid < \epsilon$.We will now actually look at proving limits exist with the precise definition of the THE DISTANCE BETWEEN TWO VECTORS Given some vectors , we denote the distance between those two points in the following manner. Definition: Let . Then the Distance between and is . We will now look at some properties of the distance between points in . Theorem 1 (Symmetry Property of Distance): If then . Proof: We note that and that . To show these are equal, we must onlyshow
THE OSCULATING CIRCLE AT A POINT ON A CURVE The Osculating Circle at is the circle that best approximates at . The osculating circle at : 1) Contains the point . 2) Has radius . 3) Has curvature . 4) Shares the same tangent at . If generates a curve , then the osculating circle at is also defined to be the circle that best approximates at by having radius , curvature , and shares the COMPOUND INTEREST WITH DIFFERENTIAL EQUATIONS Compound Interest with Differential Equations. Let be an initial sum of money. Let represent an interest rate. We can model the growth of an initial deposit with respect to the interest rate with differential equations. If represents time, then the rate of change of the initial deposit is and assuming that the initial deposit is compounded 6 COLOUR THEOREM FOR PLANAR GRAPHS 6 Colour Theorem for Planar Graphs. Theorem 1: For a connected planar simple graph , the vertices in can be coloured with or fewer colours for a good (or less) colouring of , that is, a function exists, where such that for every , whenever , . Proof: Let be the statement that for a connected planar simple graph , the vertices in can be coloured PROOFS REGARDING THE SUPREMUM OR INFIMUM OF A BOUNDED SETSEE MORE ON MATHONLINE.WIKIDOT.COM PROOF THAT SIN(X) ≤ X FOR ALL POSITIVE REAL NUMBERS Proof that sin (x) ≤ x for All Positive Real Numbers. A very useful inequality that sometimes appears in calculus and analysis is that for any nonnegative real number we have that . We will now prove this result using an elementary result from calculus - the Mean Value theorem. We state this result below and then prove this inequality. WALKS, TRAILS, PATHS, CYCLES AND CIRCUITS Definition: A Path is defined as an open trail with no repeated vertices. Notice that all paths must therefore be open walks, as a path cannot both start and terminate at the same vertex. For example, the following orange coloured walk is a path. because the walk does not repeat any edges. Now let's look at the next graph with the tealwalk.
POINT-NORMAL FORM OF A PLANE Point-Normal Form of a Plane. In 2-space, a line can algebraically be expressed by simply knowing a point that the line goes through and its slope. This can be expressed in the form . In 3-space, a plane can be represented differently. We will still need some point that lies on the plane in 3-space, however, we will now use a value called the ALGEBRAIC STRUCTURES PROVING THE EXISTENCE OF LIMITS Proving The Existence of Limits. Recall from the precise definition of a limit on the Introduction to Limits page, we said that the statement $\lim_{x \to a} f(x) = L$ says that for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $0 < \mid x - a \mid < \delta$ then $\mid f(x) - L \mid < \epsilon$.We will now actually look at proving limits exist with the precise definition of the THE DISTANCE BETWEEN TWO VECTORS Given some vectors , we denote the distance between those two points in the following manner. Definition: Let . Then the Distance between and is . We will now look at some properties of the distance between points in . Theorem 1 (Symmetry Property of Distance): If then . Proof: We note that and that . To show these are equal, we must onlyshow
THE OSCULATING CIRCLE AT A POINT ON A CURVE The Osculating Circle at is the circle that best approximates at . The osculating circle at : 1) Contains the point . 2) Has radius . 3) Has curvature . 4) Shares the same tangent at . If generates a curve , then the osculating circle at is also defined to be the circle that best approximates at by having radius , curvature , and shares the COMPOUND INTEREST WITH DIFFERENTIAL EQUATIONS Compound Interest with Differential Equations. Let be an initial sum of money. Let represent an interest rate. We can model the growth of an initial deposit with respect to the interest rate with differential equations. If represents time, then the rate of change of the initial deposit is and assuming that the initial deposit is compounded 6 COLOUR THEOREM FOR PLANAR GRAPHS 6 Colour Theorem for Planar Graphs. Theorem 1: For a connected planar simple graph , the vertices in can be coloured with or fewer colours for a good (or less) colouring of , that is, a function exists, where such that for every , whenever , . Proof: Let be the statement that for a connected planar simple graph , the vertices in can be coloured PROOFS REGARDING THE SUPREMUM OR INFIMUM OF A BOUNDED SETSEE MORE ON MATHONLINE.WIKIDOT.COM ALGEBRAIC STRUCTURES Definition: A field is a set with the two binary operations of addition and multiplication, both of which operations are commutative, associative, contain identity elements, and contain inverse elements. The identity element for addition is 0, and the identity element for multiplication is 1. Given x, the inverse element for addition is -x, and the multiplicative inverse element for 6 COLOUR THEOREM FOR PLANAR GRAPHS 6 Colour Theorem for Planar Graphs. Theorem 1: For a connected planar simple graph , the vertices in can be coloured with or fewer colours for a good (or less) colouring of , that is, a function exists, where such that for every , whenever , . Proof: Let be the statement that for a connected planar simple graph , the vertices in can be coloured THE DISTANCE BETWEEN TWO VECTORS Given some vectors , we denote the distance between those two points in the following manner. Definition: Let . Then the Distance between and is . We will now look at some properties of the distance between points in . Theorem 1 (Symmetry Property of Distance): If then . Proof: We note that and that . To show these are equal, we must onlyshow
OPEN AND CLOSED SETS IN THE DISCRETE METRIC SPACE Theorem 1: Let be the discrete metric space where for all , . Then every subset is clopen. Proof: We first show that every singleton set is open. Consider the set and consider the ball . Since if and only if , we see that the ball centered at with radius contains only and that: (2) Therefore is open. Now more generally, for any subset where we SEQUENTIAL CRITERION FOR THE CONTINUITY OF A FUNCTION Theorem 1 (Sequential Criterion for Continuity): Let be a function. Then is continuous at the point if and only if for all sequences from with then we have that . Proof: Let be continuous at the point and let be any arbitrary sequence from such that . We want to show that . Let be given. Since is continuous at then exists a such that if and then . THE LIMIT SUPERIOR AND LIMIT INFERIOR OF SEQUENCES OF REAL Hence the limit of this sequence will be the supremum of this set, that is: We now have two different ways to express the limit superior and limit inferior of a sequence of real numbers. We now establish some important results regarding the general limit of and the limit superior and limit inferior. Lemma 1: Let be a sequence of realnumbers.
LIMIT DIVERGENCE CRITERIA We will now formulate what is known as the Limit Divergence Criteria, which will establish criteria to establish whether the value is not the limit of as . Theorem 1 (Limit Divergence Criteria): Let be a function and let be a cluster point of . Then for , if either: 1. There exists a sequence from where such that but . 2. EXAMPLES OF APPLYING HENSEL'S LEMMA Examples of Applying Hensel's Lemma. Recall from the Hensel's Lemma page that if is a polynomial, is a prime, is a solution to , and then there exists a unique lift to a solution where . In particular, if is a solution to and then the recursive formula: (1) is a solution to . We will now look at an example of applying Hensel's Lemma to solving . STABLE, SEMI-STABLE, AND UNSTABLE EQUILIBRIUM SOLUTIONS Definition: An equilibrium solution is said to be Asymptotically Stable if on both sides of this equilibrium solution, there exists other solutions which approach this equilibrium solution. An equilibrium solution is said to be Semi-Stable if one one side of this equilibrium solution there exists other solutions which approach this equilibrium solution, and on the other side of the equilibrium LAGRANGE MULTIPLIERS WITH TWO CONSTRAINTS EXAMPLES 2 Example 2. Find the extreme values of subject to the constraint equations and . Let and let . In computing the appropriate partial derivatives we get that: (4) The third equation immediately gives us that , and so substituting this into the other two equations and we have that: (5) We will then subtract the second equation from thefirst to get
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The Natural Embedding J is an Open Map24 May 2019, 03:04
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The First and Second Arens Products on A**10 May 2019, 01:48
A** (with First Arens Product) Has a Right I. IFF. A Has a B.R.A.I10 May 2019, 01:32
The Operator Algebra of Bounded Linear Operators on X2 May 2019, 17:01
Closed Unit Ball of X Being Weakly Comp. Implies X is a BanachSpace
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The Casorati-Weierstrass Theorem15 Nov 2018, 00:57
The Left Regular Representation22 Oct 2018, 02:17
Class Functions on a Group22 Oct 2018, 02:03
Coro. to the Orthogonality Thm. for Chars. of Irreducible GroupReps.
22 Oct 2018, 01:49
The Orthogonality Theorem for Characters of Irreducible Group Reps.22 Oct 2018, 01:05
The Character of a Group Representation21 Oct 2018, 16:55
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MAY 1ST, 2019: Sorry for being away for over half a year! I have been quite busy with my own personal studies. I plan to continue to add pages on a slightly more consistent basis from now on, but not as frequently as I used to due to time constraints. I have received a few very nice comments from you guys saying that you appreciate the site, and I'm really glad for it! I created this site 6 years ago when I was starting my undergraduate degree as a way for me to keep my notes and solved problems in case I needed or wanted to relearn them down the road. I had no idea that so many people would enjoy reading my notes and problems (although a lot of them are error-riden!) Thank you all for your kind words and I'm glad you're enjoying the site. Also, I have corrected a lot of the errors posted on the errors page. Many reported errors haven't been fixed yet (and I probably won't be able to correct them at this point in time unless I relearn the material from scratch, which isn't an option for me at the present time unfortunately). Regardless, thank you all who have reported errors, and I'm sorry if any reported error hasn't been fixed by me yet. I hope to get around to it… some day. Keep your eyes keen though, as a large part of math is questioning things that don't look quite right=)!
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