Graphs of Logarithmic Functions

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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 the logarithmic function is defined Recall that the exponential function is defined as y = bx for any real number x and constant b > 0, b ≠ 1, where • The domain of y is ( − ∞, ∞) • The range of y is (0, ∞) In the last section we learned that the logarithmic function y = logb(x) is the inverse of the exponential function y = bx So, as inverse functions: 2/45 Graphs of Logarithmic Functions • The domain of y = logb(x) is the range of y = bx : (0, ∞) • The range of y = logb(x) is the domain of y = bx : ( − ∞, ∞) Transformations of the parent function y = logb(x) behave similarly to those of other functions Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections—to the parent function without loss of shape In Graphs of Exponential Functions we saw that certain transformations can change the range of y = bx Similarly, applying transformations to the parent function y = logb(x) can change the domain When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers That is, the argument of the logarithmic function must be greater than zero For example, consider f(x) = log4(2x − 3) This function is defined for any values of x such that the argument, in this case 2x − 3, is greater than zero To find the domain, we set up an inequality and solve for x : 2x − > 2x > x > 1.5 Show the argument greater than zero Add Divide by In interval notation, the domain of f(x) = log4(2x − 3) is (1.5, ∞) How To Given a logarithmic function, identify the domain Set up an inequality showing the argument greater than zero Solve for x Write the domain in interval notation Identifying the Domain of a Logarithmic Shift What is the domain of f(x) = log2(x + 3) ? The logarithmic function is defined only when the input is positive, so this function is defined when x + > Solving this inequality, x+3>0 x> −3 The input must be positive Subtract 3/45 Graphs of Logarithmic Functions The domain of f(x) = log2(x + 3) is ( − 3, ∞) Try It What is the domain of f(x) = log5(x − 2) + ? (2, ∞) Identifying the Domain of a Logarithmic Shift and Reflection What is the domain of f(x) = log(5 − 2x) ? The logarithmic function is defined only when the input is positive, so this function is defined when – 2x > Solving this inequality, − 2x > − 2x > − x< The input must be positive Subtract Divide by − and switch the inequality ( The domain of f(x) = log(5 − 2x) is – ∞, ) Try It What is the domain of f(x) = log(x − 5) + ? (5, ∞) Graphing Logarithmic Functions Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions The family of logarithmic functions includes the parent function y = logb(x) along with all its transformations: shifts, stretches, compressions, and reflections We begin with the parent function y = logb(x) Because every logarithmic function of this form is the inverse of an exponential function with the form y = bx, their graphs will be reflections of each other across the line y = x To illustrate this, we can observe the relationship between the input and output values of y = 2x and its equivalent x = log2(y) in [link] 4/45 Graphs of Logarithmic Functions x −3 −2 −1 2x = y 2 log2(y) = x − − − 1 Using the inputs and outputs from [link], we can build ...Annals of Mathematics Norm preserving extensions of holomorphic functions from subvarieties of the bidisk By Jim Agler and John E. McCarthy* Annals of Mathematics, 157 (2003), 289–312 Norm preserving extensions of holomorphic functions from subvarieties of the bidisk By Jim Agler and John E. M c Carthy* 1. Introduction A basic result in the theory of holomorphic functions of several complex variables is the following special case of the work of H. Cartan on the sheaf cohomology on Stein domains ([10], or see [14] or [16] for more modern treat- ments). Theorem 1.1. If V is an analytic variety in a domain of holomorphy Ω and if f is a holomorphic function on V , then there is a holomorphic function g in Ω such that g = f on V . The subject of this paper concerns an add-on to the structure considered in Theorem 1.1 which arose in the authors’ recent investigations of Nevanlinna- Pick interpolation on the bidisk. The definition for a general pair (Ω,V)isas follows. Definition 1.2. Let V be an analytic variety in a domain of holomor- phy Ω. Say V has the extension property if whenever f is a bounded holo- morphic function on V , there is a bounded holomorphic function g on Ω such that (1.3) g| V = f and sup Ω |g| = sup V |f|. More generally, if Hol ∞ (V ) denotes the bounded holomorphic functions on V and A ⊆ Hol ∞ (V ), then we say V has the A-extension property if there is a bounded holomorphic function g on Ω such that (1.3) holds whenever f ∈ A. Before continuing we remark that in Definition 1.2 it is not essential that V beavariety: interpret f to be holomorphic on V if f has a holomorphic extension to a neighborhood of V . Also, in this paper we shall restrict our attention to the case where Ω = 2 . The authors intend to publish their ∗ The first author was partially supported by the National Science Foundation. The second author was partially supported by National Science Foundation grant DMS-0070639. 290 JIM AGLER AND JOHN E. M C CARTHY results on more general cases in a subsequent paper. Finally, we point out that the notion in Definition 1.2 is different but closely related to extension problems studied by the group that worked out the theory of function algebras in the 60’s and early 70’s (see e.g. [19] and [4]). We now describe in some detail how we were led to formulate the notions in Definition 1.2. The classical Nevanlinna-Pick Theory gives an exhaustive analysis of the following extremal problem on the disk. For data λ 1 , ,λ n ∈ and z 1 , ,z n ∈ , consider (1.4) ρ = inf {sup λ∈ |ϕ(λ)| : ϕ : holo −→ ,ϕ(λ i )=z i }. Functions ψ for which (1.4) is attained are referred to as extremal and the most important fact in the whole theory is that there is only one extremal for given data. Once this fact is realized it comes as no surprise that there is a finite algebraic procedure for creating a formula for the extremal in terms of the data and the critical value ρ (as an eigenvalue problem) an important result, not only in function theory [13], but in the model theory for Hilbert space contractions [12] and in the mathematical theory of control [15]. Now, let us consider the associated extremal problem on the bidisk. For data λ i =(λ 1 i ,λ 2 i ) ∈ 2 , 1 ≤ i ≤ n, and z i ∈ , 1 ≤ i ≤ n, let (1.5) ρ = inf { sup λ∈ 2 |ϕ(λ)| : ϕ : 2 holo −→ ,ϕ(λ i )=z i }. Unlike the case of the disk, extremals for (1.5) are not unique. The authors however have discovered the interesting fact that there is a polynomial variety in the LOCAL POLYNOMIAL CONVEXITY OF GRAPHS OF FUNCTIONS IN SEVERAL VARIABLES KIEU PHUONG CHI Abstract. In that paper, we investigate the locally polynomial convexity of graphs of smooth functions in several variables. We also give a sufficient condition for real analytic function g defined near 0 in C which behaves like z n near the origin so that the algebra generated by z m and g is dense in the space of continuous functions on D for all disks D close enough to the origin in C. 1. Introduction ˆ we denote the polynomial convex We recall that for a given compact K in Cn , by K hull of K i.e., ˆ = {z ∈ Cn : |p(z)| ≤ p K K for every polynomial p in Cn }. ˆ = K. A compact K is called locally We say that K is polynomially convex if K polynomially convex at a ∈ K if there exists the closed ball B(a) centered at a such that B(a) ∩ K is polynomially convex. The interest for studying polynomial convexity stems from the celebrated Oka-Weil approximation theorem (see [1], page 36) which states that holomorphic functions near a compact polynomially convex subset of Cn can be uniformly approximated by polynomials in Cn . A compact K ⊂ C is polynomially convex if is C \ K connected. In higher dimensions, there is no such topological characterization of polynomially convex sets, and it is usual difficult to determine whether a given compact subset is polynomially convex. By a well-known result of Wermer ([19]; see also [1], Theorem. 17.1), every totally real manifold is locally polynomially convex. Recall that a C 1 smooth real manifold M is called totally real at p ∈ M if the real tangent space Tp M contains no complex line. In this paper, we are concerned with local polynomial convexity at the origin of the graph Γf of a C 2 smooth function f near 0 ∈ Cn such that f (0) = 0. By the theorem of Wermer just cited, we ∂f know that if (0) = 0 for all i = 1, 2, ..., n then Γf is locally polynomially convex at ∂z i ∂f the origin of Cn+1 . Thus it remains to consider the case where (0) = 0 for some i. ∂z i Our study is motivated by a similar problem in one complex variable. More precisely, let f be a C 2 smooth function near 0 ∈ C such that f (0) = 0. Under certain condition of f , one can show that Γf is locally polynomial convex at the origin of C2 . The work 2010 Mathematics Subject Classification. 46J10, 46J15, 47H10. Key words and phrases. polynomially convex, plurisubharmonic, totally real . 1 associated with these direction of research is too numerous to list here; instead, the reader is referred to [2, 3, 20] and the references given therein. In the section 3, we will refine the technical from [6] to attack the problem in several variables. For the readers convenience, we repeat a reasoning due to [6]. First, we construct nonnegative smooth functions vanishing exactly on Γf . These functions are, in general, plurisubharmonic only on open sets whose boundaries contain the origin. Secondly,under some technical assumptions, we may add small strictly plurisubharmonic functions to obtain plurisubharmonic functions on certain open sets containing the (local) polynomially convex hull of Γf . Finally, by invoking the nontrivial fact of about equivalence of plurisubharmonic hull and polynomial hulls, we can conclude that Γf is locally polynomially convex at the origin. In this vein, we obtain some known results in one variable. We also give some examples to show that our results are effective. In section 4, we shall present some results about locally uniform approximation of continuous function. Let D be a small closed disk in the complex plane, centered at the origin and g be a C 2 function on D which behaves like z n near the origin. By [z m , g; D] we denote the function algebra consisting of uniform limits on D of all polynomials in z m and g. Our goal finding conditions on g Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics, pages 703–711, Portland, Oregon, June 19-24, 2011. c 2011 Association for Computational Linguistics The impact of language models and loss functions on repair disfluency detection Simon Zwarts and Mark Johnson Centre for Language Technology Macquarie University {simon.zwarts|mark.johnson|}@mq.edu.au Abstract Unrehearsed spoken language often contains disfluencies. In order to correctly inter- pret a spoken utterance, any such disfluen- cies must be identified and removed or other- wise dealt with. Operating on transcripts of speech which contain disfluencies, we study the effect of language model and loss func- tion on the performance of a linear reranker that rescores the 25-best output of a noisy- channel model. We show that language mod- els trained on large amounts of non-speech data improve performance more than a lan- guage model trained on a more modest amount of speech data, and that optimising f-score rather than log loss improves disfluency detec- tion performance. Our approach uses a log-linear reranker, oper- ating on the top n analyses of a noisy chan- nel model. We use large language models, introduce new features into this reranker and examine different optimisation strategies. We obtain a disfluency detection f-scores of 0.838 which improves upon the current state-of-the- art. 1 Introduction Most spontaneous speech contains disfluencies such as partial words, filled pauses (e.g., “uh”, “um”, “huh”), explicit editing terms (e.g., “I mean”), par- enthetical asides and repairs. Of these, repairs pose particularly difficult problems for parsing and related Natural Language Processing (NLP) tasks. This paper presents a model of disfluency detec- tion based on the noisy channel framework, which specifically targets the repair disfluencies. By com- bining language models and using an appropriate loss function in a log-linear reranker we are able to achieve f-scores which are higher than previously re- ported. Often in natural language processing algorithms, more data is more important than better algorithms (Brill and Banko, 2001). It is this insight that drives the first part of the work described in this paper. This paper investigates how we can use language models trained on large corpora to increase repair detection accuracy performance. There are three main innovations in this paper. First, we investigate the use of a variety of language models trained from text or speech corpora of vari- ous genres and sizes. The largest available language models are based on written text: we investigate the effect of written text language models as opposed to language models based on speech transcripts. Sec- ond, we develop a new set of reranker features ex- plicitly designed to capture important properties of speech repairs. Many of these features are lexically grounded and provide a large performance increase. Third, we utilise a loss function, approximate ex- pected f-score, that explicitly targets the asymmetric evaluation metrics used in the disfluency detection task. We explain how to optimise this loss func- tion, and show that this leads to a marked improve- ment in disfluency detection. This is consistent with Jansche (2005) and Smith and Eisner (2006), who observed similar Graphs of the Sine and Cosine Functions Graphs of the Sine and Cosine Functions By: OpenStaxCollege Light can be separated into colors because of its wavelike properties (credit: "wonderferret"/ Flickr) White light, such as the light from the sun, is 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 ... functions: 2/45 Graphs of Logarithmic Functions • The domain of y = logb(x) is the range of y = bx : (0, ∞) • The range of y = logb(x) is the domain of y = bx : ( − ∞, ∞) Transformations of the parent.. .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. .. Subtract 3/45 Graphs of Logarithmic Functions The domain of f(x) = log2(x + 3) is ( − 3, ∞) Try It What is the domain of f(x) = log5(x − 2) + ? (2, ∞) Identifying the Domain of a Logarithmic Shift