Fractional Colorings and Zykov Products of graphs
|
|
|
- Ἀκύλας Παππάς
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011
2 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is a set of unordered pairs of vertices
3 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is a set of unordered pairs of vertices Example Figure: Peterson graph
4 Subgraphs A subgraph H of a graph G is a graph such that V (H) V (G) and E(H) E(G).
5 Subgraphs Example A subgraph H of a graph G is a graph such that V (H) V (G) and E(H) E(G). Figure: Subgraph of the Peterson graph
6 Subgraphs An induced subgraph, H, of G is a subgraph with property that any two vertices are adjacent in H if and only if they are adjacent in G.
7 Subgraphs Example An induced subgraph, H, of G is a subgraph with property that any two vertices are adjacent in H if and only if they are adjacent in G. Figure: Induced Subgraph of Peterson graph
8 Independent Sets A set of vertices, S, is said to be independent if those vertices induce a graph with no edges.
9 Example Independent Sets A set of vertices, S, is said to be independent if those vertices induce a graph with no edges. Figure: Independent set
10 Example Independent Sets A set of vertices, S, is said to be independent if those vertices induce a graph with no edges. Figure: Independent set The set of all independent sets of a graph G is denoted I (G).
11 Weighting I (S) A weighting of I (G) is a function w : I (G) R 0.
12 Example Weighting I (S) A weighting of I (G) is a function w : I (G) R 0. e d a c b S w(s) {a} 1/3 {b} 1/3 {c} 1/3 {d} 1/3 {e} 1/3 {a,c} 1/3 {a,d} 1/3 {b,d} 1/3 {b,e} 1/3 {e,c} 1/3 Figure: C 5 and a corresponding weighting
13 Fractional k-coloring A fractional k-coloring of a graph, G, is a weighting of I (G) such that S I (G) w(s) = k; and
14 Fractional k-coloring A fractional k-coloring of a graph, G, is a weighting of I (G) such that S I (G) w(s) = k; and For every v V (G), S I (G) v S w(s) = w[v] 1
15 Example Fractional k-coloring e d a c b S w(s) {a} 1/3 {b} 1/3 {c} 1/3 {d} 1/3 {e} 1/3 {a,c} 1/3 {a,d} 1/3 {b,d} 1/3 {b,e} 1/3 {e,c} 1/3 Figure: A fractional coloring of C 5 with weight 10/3
16 Example Fractional k-coloring e d a c b S w(s) {a} 1/3 {b} 1/3 {c} 1/3 {d} 1/3 {e} 1/3 {a,c} 1/3 {a,d} 1/3 {b,d} 1/3 {b,e} 1/3 {e,c} 1/3 Figure: A fractional coloring of C 5 with weight 10/3 S I (G) w(s) = 10/3
17 Example Fractional k-coloring e d a c b S w(s) {a} 1/3 {b} 1/3 {c} 1/3 {d} 1/3 {e} 1/3 {a,c} 1/3 {a,d} 1/3 {b,d} 1/3 {b,e} 1/3 {e,c} 1/3 Figure: A fractional coloring of C 5 with weight 10/3 S I (G) w(s) = 10/3 w[v] = 1 for every v V (G)
18 Fractional Chromatic Number The fractional chromatic number, χ f (G), is the minimum possible weight of a fractional coloring.
19 Example Fractional Chromatic Number The fractional chromatic number, χ f (G), is the minimum possible weight of a fractional coloring. e d a c b S w(s) {a} 0 {b} 0 {c} 0 {d} 0 {e} 0 {a,c} 1/2 {a,d} 1/2 {b,d} 1/2 {b,e} 1/2 {e,c} 1/2 Figure: A weighting C 5
20 Example Fractional Chromatic Number e d a c b S w(s) {a} 0 {b} 0 {c} 0 {d} 0 {e} 0 {a,c} 1/2 {a,d} 1/2 {b,d} 1/2 {b,e} 1/2 {e,c} 1/2 Figure: A fractional 5/2-coloring of C 5
21 Example Fractional Chromatic Number e d a c b S w(s) {a} 0 {b} 0 {c} 0 {d} 0 {e} 0 {a,c} 1/2 {a,d} 1/2 {b,d} 1/2 {b,e} 1/2 {e,c} 1/2 Figure: A fractional 5/2-coloring of C 5 S I (G) w(s) = 5/2
22 Example Fractional Chromatic Number e d a c b S w(s) {a} 0 {b} 0 {c} 0 {d} 0 {e} 0 {a,c} 1/2 {a,d} 1/2 {b,d} 1/2 {b,e} 1/2 {e,c} 1/2 Figure: A fractional 5/2-coloring of C 5 S I (G) w(s) = 5/2 w[v] = 1 for every v V (G)
23 Fractional Chromatic Number How do we know what the minimum is?
24 Fractional Chromatic Number How do we know what the minimum is? Linear Programming
25 Fractional Chromatic Number How do we know what the minimum is? Linear Programming Formulas
26 Zykov Product of Graphs
27 Zykov Product of Graphs The Zykov product Z(G 1, G 2,..., G n ) of simple graphs G 1, G 2,..., G n is formed as follows.
28 Zykov Product of Graphs The Zykov product Z(G 1, G 2,..., G n ) of simple graphs G 1, G 2,..., G n is formed as follows. Take the disjoint union of G i
29 Zykov Product of Graphs The Zykov product Z(G 1, G 2,..., G n ) of simple graphs G 1, G 2,..., G n is formed as follows. Take the disjoint union of G i Example Figure: Drawings of P 2 and P 3
30 Zykov Product of Graphs The Zykov product Z(G 1, G 2,..., G n ) of simple graphs G 1, G 2,..., G n is formed as follows. Take the disjoint union of G i For each (x 1,..., x n ) V (G 1 ) V (G 2 )... V (G n ) add a new vertex adjacent to the vertices {x 1,..., x n }
31 Zykov Product of Graphs The Zykov product Z(G 1, G 2,..., G n ) of simple graphs G 1, G 2,..., G n is formed as follows. Take the disjoint union of G i For each (x 1,..., x n ) V (G 1 ) V (G 2 )... V (G n ) add a new vertex adjacent to the vertices {x 1,..., x n } Example Figure: Constructing Z(P 2, P 3 )
32 Zykov Product of Graphs The Zykov product Z(G 1, G 2,..., G n ) of simple graphs G 1, G 2,..., G n is formed as follows. Take the disjoint union of G i For each (x 1,..., x n ) V (G 1 ) V (G 2 )... V (G n ) add a new vertex adjacent to the vertices {x 1,..., x n } Example Figure: Z(P 2, P 3 )
33 Zykov Graphs
34 Zykov Graphs The Zykov graphs, Z n, are formed as follows: Set Z 1 as a single vertex
35 Zykov Graphs The Zykov graphs, Z n, are formed as follows: Set Z 1 as a single vertex Define Z n := Z(Z 1,..., Z n 1 ) for all n 2 Figure: Drawing of Z 1
36 Zykov Graphs The Zykov graphs, Z n, are formed as follows: Set Z 1 as a single vertex Define Z n := Z(Z 1,..., Z n 1 ) for all n 2 Figure: Drawings of Z 1 and Z 2
37 Zykov Graphs The Zykov graphs, Z n, are formed as follows: Set Z 1 as a single vertex Define Z n := Z(Z 1,..., Z n 1 ) for all n 2 Figure: Drawings of Z 1, Z 2, and Z 3
38 Zykov Graphs The Zykov graphs, Z n, are formed as follows: Set Z 1 as a single vertex Define Z n := Z(Z 1,..., Z n 1 ) for all n 2 Figure: Drawing of Z 4
39 Jacobs Conjecture
40 Jacobs Conjecture Corollary For n 1, χ f (Z n+1 ) = χ f (Z n ) + 1 χ f (Z n )
41 Jacobs Conjecture Corollary For n 1, χ f (Z n+1 ) = χ f (Z n ) + 1 χ f (Z n ) Example χ f (Z 1 ) = 1
42 Jacobs Conjecture Corollary For n 1, χ f (Z n+1 ) = χ f (Z n ) + 1 χ f (Z n ) Example χ f (Z 1 ) = 1 χ f (Z 2 ) = = 2
43 Jacobs Conjecture Corollary For n 1, χ f (Z n+1 ) = χ f (Z n ) + 1 χ f (Z n ) Example χ f (Z 1 ) = 1 χ f (Z 2 ) = = 2 χ f (Z 3 ) = = 5 2
44 Verifying χ f (C 5 )
45 Verifying χ f (C 5 ) Notice that Figure: Z 3 and C 5
46 Verifying χ f (C 5 ) Notice that Figure: Z 3 and C 5 So, χ f (Z 3 ) = χ f (C 5 ) = 5/2
47 The Main Result: Theorem 1
48 Theorem The Main Result: Theorem 1 For n 2, let G 1,..., G n be graphs. Set G := Z(G 1,..., G n ) and χ i = χ f (G i ). Suppose also that the graphs G i are numbered such that χ i χ i+1. Then ( n n ) χ f (G) = max χ n, 2 + (1 ) 1χk i=2 k=i
49 Theorem Example The Main Result: Theorem 1 For n 2, let G 1,..., G n be graphs. Set G := Z(G 1,..., G n ) and χ i = χ f (G i ). Suppose also that the graphs G i are numbered such that χ i χ i+1. Then ( n n ) χ f (G) = max χ n, 2 + (1 ) 1χk i=2 k=i ( ( χ f (Z(P 2, P 3 )) = max 2, )) 2 = max(2, 5 2 ) = 5 2.
50 Lower Bound: χ f (G) max (χ n, f (n))
51 Lower Bound: χ f (G) max (χ n, f (n)) Lemma The fractional chromatic number of a subgraph, H, is at most equal to the fractional chromatic number of a graph, G.
52 Lower Bound: χ f (G) max (χ n, f (n)) Lemma Conclusion The fractional chromatic number of a subgraph, H, is at most equal to the fractional chromatic number of a graph, G. χ f (G) χ n
53 Lower Bound: χ f (G) max (χ n, f (n)) Lemma Let G be a graph and w a weighting of X I (G). Then, for every induced subgraph H of G, there exists x V (H) such that w[x] 1 χ f (H) w(s). S X
54 Lower Bound: χ f (G) max (χ n, f (n)) Start with w, a χ f -coloring of G and x 1 V (G 1 ).
55 Lower Bound: χ f (G) max (χ n, f (n)) Start with w, a χ f -coloring of G and x 1 V (G 1 ). Construct F 1 = {S I (G) : x 1 S} with the property S F 1 w(s) = w[x 1 ] 1.
56 Lower Bound: χ f (G) max (χ n, f (n)) Start with w, a χ f -coloring of G and x 1 V (G 1 ). Construct F 1 = {S I (G) : x 1 S} with the property S F 1 w(s) = w[x 1 ] 1. Construct F 2 = {S I (G) : S {x 1, x 2 } } with the property ( ) S F 2 w(s) χ 2 S F 1 w(s).
57 Lower Bound: χ f (G) max (χ n, f (n)) Continue this process so that for all k {1,..., n},
58 Lower Bound: χ f (G) max (χ n, f (n)) Continue this process so that for all k {1,..., n}, F k = {S I (G) : S {x 1,..., x k } } with the property ( ) S F k w(s) χ k S F k 1 w(s)
59 Lower Bound: χ f (G) max (χ n, f (n)) Continue this process so that for all k {1,..., n}, F k = {S I (G) : S {x 1,..., x k } } with the property ( ) S F k w(s) χ k S F k 1 w(s) It follows that w(s) 1 + S F n n n ) (1 1χk = f (n) 1. i=2 k=i
60 Lower Bound: χ f (G) max (χ n, f (n)) Conclusion χ f (G) f (n)
61 Upper Bound: χ f (G) max (χ n, f (n)) Special Sets and Cool Weightings
62 Upper Bound: χ f (G) max (χ n, f (n)) Special Sets and Cool Weightings Special Sets Let M (G) I (G) be the set of all maximal independent sets of G
63 Upper Bound: χ f (G) max (χ n, f (n)) Special Sets and Cool Weightings Special Sets Let M (G) I (G) be the set of all maximal independent sets of G and for each i {1,..., n}, F i := {S M (G) S V (G j ) = if and only if j < i}
64 Upper Bound: χ f (G) max (χ n, f (n)) Special Sets and Cool Weightings Special Sets Let M (G) I (G) be the set of all maximal independent sets of G and for each i {1,..., n}, F i := {S M (G) S V (G j ) = if and only if j < i} Weightings w i : I (G i ) R 0, a χ f (G i )-coloring of each G i
65 Upper Bound: χ f (G) max (χ n, f (n)) Special Sets and Cool Weightings Special Sets Let M (G) I (G) be the set of all maximal independent sets of G and for each i {1,..., n}, F i := {S M (G) S V (G j ) = if and only if j < i} Weightings w i : I (G i ) R 0, a χ f (G i )-coloring of each G i p i : I (G i ) R 0 where p i := w i (S)/χ i
66 Upper Bound: χ f (G) max (χ n, f (n)) Special Sets and Cool Weightings Special Sets Let M (G) I (G) be the set of all maximal independent sets of G and for each i {1,..., n}, F i := {S M (G) S V (G j ) = if and only if j < i} Weightings w i : I (G i ) R 0, a χ f (G i )-coloring of each G i p i : I (G i ) R 0 where p i := w i (S)/χ i p : n i=1 F i R 0 where p(s) := n i=1 p i(s V (G i ))
67 Upper Bound: χ f (G) max (χ n, f (n)) The Final Weighting Final Weighting We construct a fractional max(χ n, f (n))-coloring of G defined by the weighting (χ i χ i 1 )p(s), S F i w(s) = max(0, f (n) χ n ), S = V 0 0, otherwise
68 Upper Bound: χ f (G) max (χ n, f (n)) The Final Weighting works! We can show, S I (G) w(s) = max(χ n, f (n)) w[x] 1 for all x V (G) So, w is a fractional max(χ n, f (n))-coloring of G. Conclusion χ f (G) max (χ n, f (n))
69 Results Theorem For n 2, let G 1,..., G n be graph. Suppose also that the graphs G i are numbered such that χ i χ i+1. Then ( n n ) χ f (Z(G 1,..., G n )) = max χ n, 2 + (1 ) 1χk i=2 k=i Corollary For every n 2, χ f (Z n+1 ) = χ f (Z n ) + 1 χ f (Z n )
70 Jacobs Conjecture - Proved! Corollary For every n 2, χ f (Z n+1 ) = χ f (Z n ) + 1 χ f (Z n )
71 Jacobs Conjecture - Proved! Corollary For every n 2, χ f (Z n+1 ) = χ f (Z n ) + 1 χ f (Z n ) Proof. By induction on n 2, we prove χ n+1 = f (n) = χ n + χ 1 n.
72 Jacobs Conjecture - Proved! Corollary For every n 2, χ f (Z n+1 ) = χ f (Z n ) + 1 χ f (Z n ) Proof. By induction on n 2, we prove χ n+1 = f (n) = χ n + χ 1 n. Base Case: χ f (Z 1 ) = 1 and f (1) = 2 = χ 2
73 Jacobs Conjecture - Proved! Corollary For every n 2, χ f (Z n+1 ) = χ f (Z n ) + 1 χ f (Z n ) Proof. By induction on n 2, we prove χ n+1 = f (n) = χ n + χ 1 n. Base Case: χ f (Z 1 ) = 1 and f (1) = 2 = χ 2 Inductive Hypothesis: Suppose χ n = f (n 1).
74 Jacobs Conjecture - Proved! Proof. Inductive Hypothesis: Suppose χ n = f (n 1).
75 Jacobs Conjecture - Proved! Proof. Inductive Hypothesis: Suppose χ n = f (n 1). Then f (n) = 2 + = 2 + n ) (1 1χk i=2 k i (1 1χn ) (f (n 1) 1) = χ n + 1 χ n
76 Jacobs Conjecture - Proved! Proof. Inductive Hypothesis: Suppose χ n = f (n 1). Then f (n) = 2 + = 2 + n ) (1 1χk i=2 k i (1 1χn ) (f (n 1) 1) = χ n + 1 χ n Since χ n+1 = max(χ n, f (n)), we have χ n+1 = χ n + 1 χ n.
77 Questions?
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
New bounds for spherical two-distance sets and equiangular lines
New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a
Reminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Every set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Lecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
A Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
The Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
EE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
C.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Uniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS
MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS FUMIE NAKAOKA AND NOBUYUKI ODA Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006 Some properties of minimal closed sets and maximal closed
The Probabilistic Method - Probabilistic Techniques. Lecture 7: The Janson Inequality
The Probabilistic Method - Probabilistic Techniques Lecture 7: The Janson Inequality Sotiris Nikoletseas Associate Professor Computer Engineering and Informatics Department 2014-2015 Sotiris Nikoletseas,
4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
1. Introduction and Preliminaries.
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 22:1 (2008), 97 106 ON δ SETS IN γ SPACES V. Renuka Devi and D. Sivaraj Abstract We
3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Homomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
Finite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Θεωρία Γραφημάτων Θεμελιώσεις-Αλγόριθμοι-Εφαρμογές
Θεωρία Γραφημάτων Θεμελιώσεις-Αλγόριθμοι-Εφαρμογές Ενότητα 8 ΤΕΛΕΙΑ ΓΡΑΦΗΜΑΤΑ Σταύρος Δ. Νικολόπουλος 2017-18 www.cs.uoi.gr/~stavros Εισαγωγή Βασικοί Αλγόριθμοι Γραφημάτων Πολυπλοκότητα χώρου και χρόνου:
5. Choice under Uncertainty
5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation
Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Commutators of cycles in permutation groups
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 67 77 Commutators of cycles in permutation groups Aleš Vavpetič
Other Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Cyclic or elementary abelian Covers of K 4
Cyclic or elementary abelian Covers of K 4 Yan-Quan Feng Mathematics, Beijing Jiaotong University Beijing 100044, P.R. China Summer School, Rogla, Slovenian 2011-06 Outline 1 Question 2 Main results 3
Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
SOME PROPERTIES OF FUZZY REAL NUMBERS
Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical
Sequent Calculi for the Modal µ-calculus over S5. Luca Alberucci, University of Berne. Logic Colloquium Berne, July 4th 2008
Sequent Calculi for the Modal µ-calculus over S5 Luca Alberucci, University of Berne Logic Colloquium Berne, July 4th 2008 Introduction Koz: Axiomatisation for the modal µ-calculus over K Axioms: All classical
Solution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
ST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Some new generalized topologies via hereditary classes. Key Words:hereditary generalized topological space, A κ(h,µ)-sets, κµ -topology.
Bol. Soc. Paran. Mat. (3s.) v. 30 2 (2012): 71 77. c SPM ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v30i2.13793 Some new generalized topologies via hereditary
Commutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
On a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ
ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΗΜΥ 311: Διακριτή Ανάλυση και Δομές Χειμερινό Εξάμηνο 016 Σειρά Ασκήσεων 5: Απαρίθμηση, Αρχή της Θυρίδας, Συνδυασμοί και Μεταθέσεις, Γραφήματα και
Chapter 3: Ordinal Numbers
Chapter 3: Ordinal Numbers There are two kinds of number.. Ordinal numbers (0th), st, 2nd, 3rd, 4th, 5th,..., ω, ω +,... ω2, ω2+,... ω 2... answers to the question What position is... in a sequence? What
Concrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Tridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
F A S C I C U L I M A T H E M A T I C I
F A S C I C U L I M A T H E M A T I C I Nr 46 2011 C. Carpintero, N. Rajesh and E. Rosas ON A CLASS OF (γ, γ )-PREOPEN SETS IN A TOPOLOGICAL SPACE Abstract. In this paper we have introduced the concept
Partial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)
![2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)](/thumbs/40/21364883.jpg "2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)")
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Abstract Storage Devices
Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD
Homework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
de Rham Theorem May 10, 2016
de Rham Theorem May 10, 2016 Stokes formula and the integration morphism: Let M = σ Σ σ be a smooth triangulated manifold. Fact: Stokes formula σ ω = σ dω holds, e.g. for simplices. It can be used to define
The challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
Chap. 6 Pushdown Automata
Chap. 6 Pushdown Automata 6.1 Definition of Pushdown Automata Example 6.1 L = {wcw R w (0+1) * } P c 0P0 1P1 1. Start at state q 0, push input symbol onto stack, and stay in q 0. 2. If input symbol is
w o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Knaster-Reichbach Theorem for 2 κ
Knaster-Reichbach Theorem for 2 κ Micha l Korch February 9, 2018 In the recent years the theory of the generalized Cantor and Baire spaces was extensively developed (see, e.g. [1], [2], [6], [4] and many
b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Models for Probabilistic Programs with an Adversary
Models for Probabilistic Programs with an Adversary Robert Rand, Steve Zdancewic University of Pennsylvania Probabilistic Programming Semantics 2016 Interactive Proofs 2/47 Interactive Proofs 2/47 Interactive
F19MC2 Solutions 9 Complex Analysis
F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
12. Radon-Nikodym Theorem
Tutorial 12: Radon-Nikodym Theorem 1 12. Radon-Nikodym Theorem In the following, (Ω, F) is an arbitrary measurable space. Definition 96 Let μ and ν be two (possibly complex) measures on (Ω, F). We say
Statistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:
TMA4115 Matematikk 3
TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet
Distances in Sierpiński Triangle Graphs
Distances in Sierpiński Triangle Graphs Sara Sabrina Zemljič joint work with Andreas M. Hinz June 18th 2015 Motivation Sierpiński triangle introduced by Wac law Sierpiński in 1915. S. S. Zemljič 1 Motivation
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.
Generating Set of the Complete Semigroups of Binary Relations
Applied Mathematics 06 7 98-07 Published Online January 06 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/036/am067009 Generating Set of the Complete Semigroups of Binary Relations Yasha iasamidze
Solutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Problem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Section 8.2 Graphs of Polar Equations
Section 8. Graphs of Polar Equations Graphing Polar Equations The graph of a polar equation r = f(θ), or more generally F(r,θ) = 0, consists of all points P that have at least one polar representation
Section 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a
THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS. Daniel A. Romano
235 Kragujevac J. Math. 30 (2007) 235 242. THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS Daniel A. Romano Department of Mathematics and Informatics, Banja Luka University, Mladena Stojanovića
Σχέσεις, Ιδιότητες, Κλειστότητες
Σχέσεις, Ιδιότητες, Κλειστότητες Ορέστης Τελέλης telelis@unipi.gr Τµήµα Ψηφιακών Συστηµάτων, Πανεπιστήµιο Πειραιώς Ο. Τελέλης Πανεπιστήµιο Πειραιώς Σχέσεις 1 / 26 Εισαγωγή & Ορισµοί ιµελής Σχέση R από
Overview. Transition Semantics. Configurations and the transition relation. Executions and computation
Overview Transition Semantics Configurations and the transition relation Executions and computation Inference rules for small-step structural operational semantics for the simple imperative language Transition
Partial Trace and Partial Transpose
Partial Trace and Partial Transpose by José Luis Gómez-Muñoz http://homepage.cem.itesm.mx/lgomez/quantum/ jose.luis.gomez@itesm.mx This document is based on suggestions by Anirban Das Introduction This
Lecture 21: Properties and robustness of LSE
Lecture 21: Properties and robustness of LSE BLUE: Robustness of LSE against normality We now study properties of l τ β and σ 2 under assumption A2, i.e., without the normality assumption on ε. From Theorem
CRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Lecture 13 - Root Space Decomposition II
Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).
ΑΛΓΟΡΙΘΜΟΙ Άνοιξη I. ΜΗΛΗΣ
ΑΛΓΟΡΙΘΜΟΙ http://eclass.aueb.gr/courses/inf6/ Άνοιξη 26 - I. ΜΗΛΗΣ NP-complete προβλήματα ΑΛΓΟΡΙΘΜΟΙ - ΑΝΟΙΞΗ 26 - Ι. ΜΗΛΗΣ 6 NP-COMPLETENESS II Tree of reductions (partial) Cook s Th. Π NP SAT 3-SAT
Capacitors - Capacitance, Charge and Potential Difference
Capacitors - Capacitance, Charge and Potential Difference Capacitors store electric charge. This ability to store electric charge is known as capacitance. A simple capacitor consists of 2 parallel metal
Arithmetical applications of lagrangian interpolation. Tanguy Rivoal. Institut Fourier CNRS and Université de Grenoble 1
Arithmetical applications of lagrangian interpolation Tanguy Rivoal Institut Fourier CNRS and Université de Grenoble Conference Diophantine and Analytic Problems in Number Theory, The 00th anniversary
2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Mean-Variance Analysis
Mean-Variance Analysis Jan Schneider McCombs School of Business University of Texas at Austin Jan Schneider Mean-Variance Analysis Beta Representation of the Risk Premium risk premium E t [Rt t+τ ] R1
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Τέλος Ενότητας Χρηματοδότηση Το παρόν εκπαιδευτικό υλικό έχει αναπτυχθεί
Elements of Information Theory
Elements of Information Theory Model of Digital Communications System A Logarithmic Measure for Information Mutual Information Units of Information Self-Information News... Example Information Measure
Matrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Heisenberg Uniqueness pairs
Heisenberg Uniqueness pairs Philippe Jaming Bordeaux Fourier Workshop 2013, Renyi Institute Joint work with K. Kellay Heisenberg Uniqueness Pairs µ : finite measure on R 2 µ(x, y) = R 2 e i(sx+ty) dµ(s,
From root systems to Dynkin diagrams
From root systems to Dynkin diagrams Heiko Dietrich Abstract. We describe root systems and their associated Dynkin diagrams; these notes follow closely the book of Erdman & Wildon ( Introduction to Lie
k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +](/thumbs/73/69566903.jpg "k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +")
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions
International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for
A Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics
A Bonus-Malus System as a Markov Set-Chain Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics Contents 1. Markov set-chain 2. Model of bonus-malus system 3. Example 4. Conclusions
Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp
Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp.115-126. α, β, γ ORTHOGONALITY ABDALLA TALLAFHA Abstract. Orthogonality in inner product spaces can be expresed using the notion of norms.
Μηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
PARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Congruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Approximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth