Graphs of Linear Functions

Graphs of Linear Functions tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về tất cả các lĩnh vực k...

LINEAR AND NON-LINEAR OPERATORS, AND THE DISTRIBUTION OF ZEROS OF ENTIRE FUNCTIONS A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI‘I AT M ¯ ANOA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS AUGUST 2013 By Rintaro Yoshida Dissertation Committee: George Csordas, Chairperson Thomas Craven Erik Guentner Marcelo Kobayashi Wayne Smith ACKNOWLEDGMENTS I would like to express my deepest gratitude to my advisor Dr. George Csordas for his patience, enthusiasm, and support throughout the years of my studies. The completion of this dissertation is mainly due to the willingness of Dr. Csordas in generously bestowing me countless hours of advice and insight. The members of the dissertation committee, who willingly accepted to take time out of their busy schedules to review, comment, and attend the defence regarding this work, deserve great recognition. I would like to thank Dr. Pavel Guerzhoy for his understanding when I decided to change my area of study. As iron sharpens iron, fellow advisees Dr. Andrzej Piotrowski, Dr. Matthew Chasse, Dr. Lukasz Grabarek, and Mr. Robert Bates have my greatest appreciation in allowing me to take part in honing my ability to understand our beloved theory of distribution of zeros of entire functions. I appreciate everyone in the Mathematics department, but in particular, Austin Anderson, Mike Andonian, John and Tabitha Brown, William DeMeo, Patricia Goldstein, Alex Gottlieb, Zach Kent, Sue Hasegawa, Mike Joyce, Shirley Kikiloi, Troy Ludwick, Alicia Maedo, John Marriot, Chi Mingjing, Paul Nguyen, Geoff Patterson, John Radar, Gretel Sia, Jacob Woolcutt, Diane Yap, and Robert Young for their congeniality. Financial support was received from the University of Hawai‘i Graduate Student Organization for travel expenses to Macau, China, and the American Institute of Mathematics generously provided funding for travel and accommodation expenses in hosting the workshop “Stability, hyperbolicity, and zero location of functions” in Palo Alto, California. My parents have been very patient during my time in graduate school, and I’m glad to have a sister who is a blessing. I am thankful for my closest friends in California, Kristina Aquino, Jeremy Barker, Rod and Aleta Bollins, Jason Chikami, Daniel and Amie Chikami, Jay Cho, Dominic Fiorello, Todd Gilliam, Jim and Betty Griset, Karl and Jane Gudino, Michael Hadj, Steve and Hoan Hensley, Ruslan Janumyan, Kevin Knight, Phuong Le, Lemee Nakamura, Barry and Irene McGeorge, Israel and Yoko Peralta, Shawn Sami, Roger Yang, Henry Yen, my church family in California at Bethany Bible Fellowship, my church ohana at Kapahulu Bible Church, and last but clearly not least, Yahweh. ii ABSTRACT An important chapter in the theory of distribution of zeros of entire functions pertains to the study of linear operators acting on entire functions. This dissertation presents new results involving not only linear, but also some non-linear operators. If {γ k } ∞ k=0 is a sequence of real numbers, and Q = {q k (x)} ∞ k=0 is a sequence of polynomials, where deg q k (x)= k, associate with the sequence {γ k } ∞ k=0 a linear operator T such that T [q k (x)]=γ k q k (x), k = 0, 1, 2, . . . . The sequence {γ k } ∞ k=0 is termed a Q-multiplier sequence if T is a hyperbolicity preserving operator. Some multiplier sequences are characterized Graphs of Linear Functions Graphs of Linear Functions By: OpenStaxCollege Two competing telephone companies offer different payment plans The two plans charge the same rate per long distance minute, but charge a different monthly flat fee A consumer wants to determine whether the two plans will ever cost the same amount for a given number of long distance minutes used The total cost of each payment plan can be represented by a linear function To solve the problem, we will need to compare the functions In this section, we will consider methods of comparing functions using graphs Graphing Linear Functions In Linear Functions, we saw that that the graph of a linear function is a straight line We were also able to see the points of the function as well as the initial value from a graph By graphing two functions, then, we can more easily compare their characteristics There are three basic methods of graphing linear functions The first is by plotting points and then drawing a line through the points The second is by using the y-intercept and slope And the third is by using transformations of the identity function f(x) = x Graphing a Function by Plotting Points To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values The input values and corresponding output values form coordinate pairs We then plot the coordinate pairs on a grid In general, we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph For example, given the function, f(x) = 2x, we might use the input values and Evaluating the function for an input value of yields an output value of 2, which is represented by the point (1, 2) Evaluating the function for an input value of yields an output value of 4, which is represented by the point (2, 4) Choosing three points is often advisable because if all three points not fall on the same line, we know we made an error How To Given a linear function, graph by plotting points 1/55 Graphs of Linear Functions Choose a minimum of two input values Evaluate the function at each input value Use the resulting output values to identify coordinate pairs Plot the coordinate pairs on a grid Draw a line through the points Graphing by Plotting Points Graph f(x) = − x + by plotting points Begin by choosing input values This function includes a fraction with a denominator of 3, so let’s choose multiples of as input values We will choose 0, 3, and Evaluate the function at each input value, and use the output value to identify coordinate pairs x=0 x=3 x=6 ⇒ (0, 5) f(3) = − (3) + = ⇒ (3, 3) f(6) = − (6) + = ⇒ (6, 1) f(0) = − (0) + = Plot the coordinate pairs and draw a line through the points [link] represents the graph of the function f(x) = − x + 2/55 Graphs of Linear Functions The graph of the linear function f(x) = − x + Analysis The graph of the function is a line as expected for a linear function In addition, the graph has a downward slant, which indicates a negative slope This is also expected from the negative constant rate of change in the equation for the function Try It Graph f(x) = − x + by plotting points 3/55 Graphs of Linear Functions Graphing a Function Using y-intercept and Slope Another way to graph linear functions is by using specific characteristics of the function rather than plotting points The first characteristic is its y-intercept, which is the point at which the input value is zero To find the y-intercept, we can set x = in the equation The other characteristic of the linear function is its slope m, which is a measure of its steepness Recall that the slope is the rate of change of the function The slope of a function is equal to the ratio of the change in outputs to the change in inputs Another way to think about the slope is by dividing the vertical difference, or rise, by the horizontal difference, or run We encountered both the y-intercept and the slope in Linear Functions Let’s consider the following function f(x) = x + The slope is Because the slope is positive, we know the graph will slant upward from left to right The y-intercept is the point on the graph when x = The graph crosses the y-axis at (0, 1) Now we know the slope and the y-intercept We can begin graphing rise by plotting the point (0, 1) We know that the slope is rise over run, m = run From our example, we have m = , which means that the rise is and the run is So starting from our y-intercept (0, 1), we can rise and then run 2, or run and then rise We repeat until we have a few points, and then we draw a line through the points as shown in [link] 4/55 Graphs of Linear Functions A General Note Graphical Interpretation of a Linear Function In the equation f(x) = mx + b • b is the y-intercept of the graph and indicates the point (0, b) at which the graph crosses the y-axis • m is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of ...Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 147069, 12 pages doi:10.1155/2009/147069 Research Article Inclusion Properties for Certain Classes of Meromorphic Functions Associated with a Family of Linear Operators Nak Eun Cho Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea Correspondence should be addressed to Nak Eun Cho, necho@pknu.ac.kr Received 3 March 2009; Accepted 1 May 2009 Recommended by Ramm Mohapatra The purpose of the present paper is to investigate some inclusion properties of certain classes of meromorphic functions associated with a family of linear operators, which are defined by means of the Hadamard product or convolution. Some invariant properties under convolution are also considered for the classes presented here. The results presented here include several previous known results as their special cases. Copyright q 2009 Nak Eun Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let A be the class of analytic functions in the open unit disk U  {z ∈ C : |z| < 1} with the usual normalization f0f  0 − 1  0. If f and g are analytic in U, we say that f is subordinate to g, written f ≺ g or fz ≺ gz, if there exists an analytic function w in U with w00and|wz| < 1forz ∈ U such that fzgwz. Let N be the class of all functions φ which are analytic and univalent in U and for which φU is convex with φ01andRe{φz} > 0forz ∈ U. We denote by S ∗ and K the subclasses of A consisting of all analytic functions which are starlike and convex, respectively. Let M denote the class of functions of the form f  z   1 z  ∞  k0 a k z k , 1.1 which are analytic in the punctured open unit disk D  U \{0}. For 0 ≤ η, β < 1, we denote by MSη, MKη and MCη, β the subclasses of M consisting of all meromorphic functions which are, respectively, starlike of order η, convex of order η and colse-to-convex of order β and type η in U see, for details, 1, 2. 2 Journal of Inequalities and Applications Making use of the principle of subordination between analytic functions, we introduce the subclasses MSη, φ, MKη, φ and MCη, β; φ, ψ of the class M for 0 ≤ η, β < 1and φ, ψ ∈N, which are defined by MS  η; φ  :  f ∈M: 1 1 − η  − zf   z  f  z  − η  ≺ φ  z  in U  , MK  η; φ  :  f ∈M: 1 1 − η  −  1  zf   z  f   z   − η  ≺ φ  z  in U  , MC  η, β; φ,ψ  :  f ∈M: ∃g ∈MS  η; φ  s.t. 1 1 − β  − zf   z  g  z  − β  ≺ ψ  z  in U  . 1.2 We note that the classes mentioned above are the familiar classes which have been used widely on the space of analytic and univalent functions in U see 3–5 and for special choices for the functions φ and ψ involved in these definitions, we can obtain the well-known subclasses of M. For examples, we have MS  η; 1  z 1 − z   MS  η  , MK  η; 1  z 1 − z   MK  η  , MC  η, β; 1  z 1 − z , 1  z 1 − z   MC  η, β  . 1.3 Now we define the function φa, c; z by φ  a, c; z  : 1 z  ∞  k0  a  k1  c  k1 z k , 1.4  z ∈ U; a ∈ R; c ∈ R \ Z − 0 ; Z − 0 : { −1, −2, }  , 1.5 where ν k is the Pochhammer symbol or the shifted factorial defined in terms of Original article Linear and non-linear functions of volume index to estimate woody biomass in high density young poplar stands JY Pontailler R Ceulemans J Guittet F Mau 1 Laboratoire d’écophysiologie végétale (CNRS Ura 2154), bâtiment 362, université Paris-XI, 91405 Orsay cedex, France 2 Department of Biology, University of Antwerpen (UIA), Universiteitsplein I, B-2610 Wilrijk, Belgium (Received 3 April 1996; accepted 9 January 1997) Summary - Biomass estimations are very important in short rotation high density stands, but usu- ally require some destructive sampling. This paper discusses the potential use of allometric rela- tionships based on volume index (height x diameter squared) for accurate and non-destructive esti- mations of stem biomass. When using this approach, one implicitly assumes a constant conversion factor from stem volume index to real stem volume as well as a constant wood infradensity (stem dry mass versus fresh volume), both assumptions being questionable. Our results on five different poplar clones grown at two different sites (Afsnee, near Gent, Belgium and Orsay, near Paris, France) and under two different cultural management regimes underscore the following points: i) stem diameter measured at 22 cm aboveground and in two perpendicular directions is a relevant parameter to com- pute volume index in high density poplar stands; ii) power function regression equations fit the stem volume index versus stem dry mass relationship better than simple linear regressions; iii) attention should be paid to variation in wood infradensity, which ranged from 0.35 to 0.44 kg dm-3 in our study. short rotation forestry / high density plantations / Populus / volume index / allometric relationships Résumé - Fonctions linéaires et non linéaires de l’indice de volume pour l’estimation de la biomasse sèche de jeunes plantations de peupliers. L’estimation de la biomasse sur pied de parcelles denses cultivées en courtes rotations est généralement indispensable mais requiert le plus souvent des techniques destructives lourdes. Cet article discute de l’utilisation potentielle des relations allométriques utilisant l’indice de volume (hauteur du brin x carré de son diamètre à la base) pour l’estimation précise de la biomasse sèche de jeunes tiges de peuplier. Par ce type d’approche, on suppose impli- * Correspondence and reprints Tel: (33) 01 69 15 71 37; fax: (33) 01 69 15 72 38; courriel: jean-yves.pontailler@eco.u.psud.fr citement qu’il existe un facteur de conversion constant entre volume vrai et indice de volume, et que l’infradensité du bois est constante. Ces deux hypothèses sont loin d’être rigoureusement véri- fiées. Les résultats présentés ici portent sur cinq clones de peupliers cultivés sur deux sites (Afsnee, près de Gand en Belgique et Orsay, près de Paris) selon deux techniques culturales différentes. Ils met- tent en évidence les points suivants : i) le diamètre de la tige, mesuré à la hauteur de 22 cm selon deux directions perpendiculaires, est un paramètre pertinent pour le calcul de l’indice de volume de jeunes brins de peupliers ; ii) les tarifs utilisant une fonction puissance de l’indice de volume fournissent des estimations plus précises de la masse sèche des brins que ne le font les tarifs linéaires ; iii) les varia- tions de l’infradensité du bois (ici de 0,35 à 0,44 kg dm-3 ) peuvent réduire considérablement la pré- cision de ces estimations. indice de volume / allométrie / Populus / sylviculture en courte rotation INTRODUCTION Within Recognizing circulant graphs of prime order in polynomial time ∗ Mikhail E. Muzychuk Netanya Academic College 42365 Netanya, Israel mikhail@netvision.net.il Gottfried Tinhofer Technical University of Munich 80290 M¨unchen, Germany gottin@mathematik.tu-muenchen.de Submitted: December 19, 1997; Accepted: April 1, 1998 Abstract A circulant graph G of order n is a Cayley graph over the cyclic group Z n . Equivalently, G is circulant iff its vertices can be ordered such that the cor- responding adjacency matrix becomes a circulant matrix. To each circulant graph we may associate a coherent configuration A and, in particular, a Schur ring S isomorphic to A. A can be associated without knowing G to be circu- lant. If n is prime, then by investigating the structure of A either we are able to find an appropriate ordering of the vertices proving that G is circulant or we are able to prove that a certain necessary condition for G being circulant is violated. The algorithm we propose in this paper is a recognition algorithm for cyclic association schemes. It runs in time polynomial in n. MR Subject Number: 05C25, 05C85, 05E30 Keywords: Circulant graph, cyclic association scheme, recognition algorithm ∗ The work reported in this paper has been partially supported by the German Israel Foundation for Scientific Research and Development under contract # I-0333-263.06/93 the electronic journal of combinatorics 3 (1996), #Rxx 2 1 Introduction The graphs considered in this paper are of the form (X, γ), where X is a finite set and γ is a binary relation on X which is not necessarily symmetric. Let G be a group and G =(X, γ) a graph with vertex set X = G and with adjacency relation γ defined with the aid of some subset C ⊂Gby γ = {(g, h):g,h ∈G∧gh −1 ∈ C}. Then G is called Cayley graph over the group G. Let Z n , n ∈ N, stand for a cyclic group of order n written additively. A circulant graph G over Z n is a Cayley graph over this group. In this particular case, the adjacency relation γ has the form γ = n−1  i=0 {i}×{i+γ(0)} where γ(0) is the set of successors of the vertex 0. Evidently, the set of successors γ(i) of an arbitrary vertex i satisfies γ(i)=i+γ(0). The set γ(0) is called the connection set of the circulant graph G. G is a simple undirected graph if 0 ∈ γ(0) and j ∈ γ(0) implies −j ∈ γ(0). There are different equivalent characterizations of circulant graphs. One of them is this: A graph G is a circulant graph iff its vertex set can be numbered in such a way that the resulting adjacency matrix A(G) is a circulant matrix. We call such a numbering a Cayley numbering. Still another characterization is: G is a circulant graph iff a cyclic permutation of its vertices exists which is an automorphism of G. Cayley graphs, and in particular, circulant graphs have been studied intensively in the literature. These graphs are easily seen to be vertex transitive. In the case of a prime vertex number n circulant graphs are known to be the only vertex transitive graphs. Because of their high symmetry, Cayley graphs are ideal models for commu- nication networks. Routing and weight balancing is easily done on such graphs. Assume that a graph G on the set V (G)={0, ,n−1} is given by its diagram or by its adjacency matrix, or by some other data structure commonly used in dealing with graphs. How can we decide whether G is a Cayley graph or not? In such a generality, this decision problem seems to be far from beeing tractable efficiently. A recognition algorithm for Cayley graphs would have to involve implicitly checking all finite groups of order n. In the special case of circulant graphs, or in any other case where the group G is given, we could recognize Cayley graphs by checking all different numberings of the vertex set and comparing [...]... Departments at the offices / branches of the Reserve Bank, while management of public debt including floatation of new loans is done at Public Debt Office at offices / branches of the Reserve Bank and by the Internal Debt Management Department at the Central Office For the final compilation of the Government accounts, both of the centre and states, the Nagpur office of the Reserve Bank has a Central... Government of India, and the various high-level committees constituted at the industry and national levels  20 3 Monetary Management One of the most important functions of central banks is formulation and execution of monetary policy In the Indian context, the basic functions of the Reserve Bank of India as enunciated in the Preamble to the RBI Act, 1934 are: “to regulate the issue of Bank notes and the... Based on its assessment of macroeconomic and financial conditions, the Reserve Bank takes the call on the stance of monetary policy and monetary measures Its monetary policy statements reflect the changing circumstances and priorities of the Reserve Bank and the thrust of policy measures for the future Faced with multiple tasks and a complex mandate, the Reserve Bank emphasises clear and structured communication... There is a network of 4,281 Currency Chests and 4,044 Small Coin Depots with other banks Currency chests are storehouses where bank notes and rupee coins are stocked on behalf of the Reserve Bank The currency chests have been established with State Bank of India, six associate banks, nationalised banks, private sector banks, a foreign bank, a state cooperative bank and a regional rural bank Deposits into... management of the public debt of these two State Governments As a banker to the Government, the Reserve Bank receives and pays money on behalf of the various Government departments As it has offices in only 27 locations, the Reserve Bank appoints other banks to act as its agents for undertaking the banking business on behalf of the governments The Reserve Bank pays agency bank charges to the banks for... Board of Directors The Central Board of Directors is at the top of the Reserve Bank s organisational structure Appointed by the Government under the provisions of the Reserve Graphs of Logarithmic Functions Graphs of Logarithmic Functions By: OpenStaxCollege In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events How logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation In other words, logarithms give the cause for an effect To illustrate, suppose we invest $2500 in an account that offers an annual interest rate of 5%, compounded continuously We already know that the balance in our account for any year t can be found with the equation A = 2500e0.05t But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation By graphing the model, we can see the output (year) for any input (account balance) For instance, what if we wanted to know how many years it would take for our initial investment to double? [link] shows this point on the logarithmic graph 1/45 Graphs of Logarithmic Functions In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions Finding the Domain of a Logarithmic Function Before working with graphs, we will take a look at the domain (the set of input values) for which ... into the slope-intercept form of a line 14/55 Graphs of Linear Functions Matching Linear Functions to Their Graphs Match each equation of the linear functions with one of the lines in [link] f(x)... factor of Then show the vertical shift as in [link] 11/55 Graphs of Linear Functions The function y = x, shifted down units Try It Graph f(x) = + 2x, using transformations 12/55 Graphs of Linear Functions. .. the graph of the function f(x) = − x + 2/55 Graphs of Linear Functions The graph of the linear function f(x) = − x + Analysis The graph of the function is a line as expected for a linear function