Graphs of the Other Trigonometric Functions

bernhard waldenfels 3 Levinas and the face of the other The human face we encounter first of all as the other’s face strikes us as a highly ambiguous phenomenon. It arises here and now without finding its place within the world. Being neither something real in- side, nor something ideal outside the world, the face announces the corporeal absence (leibhaftige Abwesenheit) of the other. In Merleau- Ponty’s terms we may call it the corporeal emblem of the other’s otherness. 1 But we do not thereby resolve the enigma of the other’s face. This enigma may be approached in different ways. In contrast to the later Merleau-Ponty, who tries to deepen our experience more and more, looking for the invisible within the visible, the untouch- able within the touchable, Levinas prefers a kind of thinking and writing which may be called eruptive. Many sentences, especially in his last writings, look like blocks of lava spat out by a hidden vulcan. Words like ‘evasion’, ‘rupture’, ‘interruption’ or ‘invasion’ indicate a thinking which is obsessed by the provocative otherness of the other. They suggest a special sort of immediacy. In contrast to Hegel’s im- mediacy, which is only the beginning of a long process of mediation, Levinas’s immediacy breaks through all kinds of mediations, be it laws, rules, codes, rituals, social roles or any other kind of order. The otherness or strangeness of the other manifests itself as the extra- ordinary par excellence: not as something given or intended, but as a certain disquietude, as a d ´ erangement which puts us out of our common tracks. The human face is just the foyer of such bewilder- ments, lurking at the borderlines which separate the normal from the anomalous. The bewildering effects lose their stimulating force if the face is taken either as something too real or as something too sublime. Although Levinas explicitly repudiates both possibilities, we will see that he has more problems avoiding the latter. He pays 63 64 the cambridge companion to levinas much more attention to the breaking of orders than to the orders themselves. But phenomenologically orientated ethics, approaching the demand of the other, turns into moralism when starting imme- diately from the other, instead of trying to show that it has always already done so. Similar to Merleau-Ponty’s claim that ontology can approach Being only in terms of an indirect ontology, we may as- sume that ethics can approach the other only in terms of an indirect ethics. What deviates from certain orders and exceeds them will turn to nothing unless supported by something which it exceeds and devi- ates from. Otherwise the extra-ordinary will turn into another order, and we are still there where we began. So we must be careful not to get into such traps, and Levinas would be the last to deny that. the common face Close to certain theological traditions, Levinas initially approaches the face of the other by the double way of via negationis and of via eminentiae. In his view the human face is not simply what it seems to be, and it is much more than that. So it may be useful to give a first idea of that manifold pre-understanding which gets transformed by Levinas’s philosophy of the other. What is called ‘face’ in English is less common than it Graphs of the Other Trigonometric Functions Graphs of the Other Trigonometric Functions By: OpenStaxCollege We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole But what if we want to measure repeated occurrences of distance? Imagine, for example, a police car parked next to a warehouse The rotating light from the police car would travel across the wall of the warehouse in regular intervals If the input is time, the output would be the distance the beam of light travels The beam of light would repeat the distance at regular intervals The tangent function can be used to approximate this distance Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever The graph of the tangent function would clearly illustrate the repeated intervals In this section, we will explore the graphs of the tangent and other trigonometric functions Analyzing the Graph of y = tan x We will begin with the graph of the tangent function, plotting points as we did for the sine and cosine functions Recall that tan x = sin x cos x The period of the tangent function is π because the graph repeats itself on intervals of π π kπ where k is a constant If we graph the tangent function on − to , we can see the behavior of the graph on one complete cycle If we look at any larger interval, we will see that the characteristics of the graph repeat We can determine whether tangent is an odd or even function by using the definition of tangent 1/49 Graphs of the Other Trigonometric Functions tan(−x) = = sin(−x) cos(−x) Definition of tangent − sin x cos x = − Sine is an odd function, cosine is even sin x cos x The quotient of an odd and an even function is odd = − tan x Definition of tangent Therefore, tangent is an odd function We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in [link] x − π − π − π tan(x) undefined − √3 –1 − π π π − √3 √3 √3 undefined 3 π π These points will help us draw our graph, but we need to determine how the graph π π behaves where it is undefined If we look more closely at values when < x < , we π can use a table to look for a trend Because ≈ 1.05 and radian measures 1.05 < x < 1.57 as shown in [link] x 1.3 1.5 tan π ≈ 1.57, we will evaluate x at 1.55 1.56 x 3.6 14.1 48.1 92.6 π As x approaches , the outputs of the function get larger and larger Because y = tan x is an odd function, we see the corresponding table of negative values in [link] x −1.3 −1.5 −1.55 −1.56 tan x −3.6 −14.1 −48.1 −92.6 π We can see that, as x approaches − , the outputs get smaller and smaller Remember that there are some values of x for which cos x = For example, cos cos ( π2 ) = and ( 3π2 ) = At these values, the tangent function is undefined, so the graph of y = tan x has discontinuities at x = π and 3π At these values, the graph of the tangent has vertical asymptotes [link] represents the graph of y = tan x The tangent is positive from to and from π to 3π 2, π corresponding to quadrants I and III of the unit circle 2/49 Graphs of the Other Trigonometric Functions Graph of the tangent function Graphing Variations of y = tan x As with the sine and cosine functions, the tangent function can be described by a general equation y = Atan(Bx) We can identify horizontal and vertical stretches and compressions using values of A and B The horizontal stretch can typically be determined from the period of the graph With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions Instead, we will use the phrase stretching/compressing factor when referring to the constant A A General note label Features of the Graph of y = Atan(Bx) • The stretching factor is |A| π • The period is P = |B| π π • The domain is all real numbers x, where x ≠ 2|B| + |B| k such that k is an integer • The range is (−∞, ∞) π π • The asymptotes occur at x = 2|B| + |B| k, where k is an integer • y = Atan(Bx) is an odd function 3/49 Graphs of the Other Trigonometric Functions Graphing One Period of a Stretched or Compressed Tangent Function We can use what we know about the properties of the tangent function to quickly sketch a graph of any stretched and/or compressed tangent function of the form f(x) = Atan(Bx) We focus on a single period of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function’s P P domain if we wish Our limited domain is then the interval − , and the graph ( ( ) ) P π π π has vertical asymptotes at ± where P = B On − , , the graph will come up from π the left asymptote at x = − , cross through the origin, and continue to increase as it π approaches the right ...RESEARCH Open Access Some certain properties of the generalized hypercubical functions Duško Letić 1 , Nenad Cakić 2 , Branko Davidović 3* , Ivana Berković 1 and Eleonora Desnica 1 * Correspondence: iwtbg@beotel. net 3 Technical High School, Kragujevac, Serbia Full list of author information is available at the end of the article Abstract In this article, the results of theoretical research of the generalized hypercube function by generalizing two known functions referring to the cube hypervolume and hypersurface and the recurrent relation between them have been presented. By introducing two degrees of freedom k and n (and the third half-edge r), we are able to develop the derivative functions for all three arguments and discuss the possibilities of their use. The symbolic evaluation, numerical experiment, and graphic presentation of the functions are realized using Mathcad Professional and Mathematica. MSC 2010: 33E30; 33E50; 33E99; 52B11. Keywords: special functions, hypercube function, derivate 1. Introduction The hypercube function (HC) is a hypothetical function connected with multidimen- sional space. I t belongs to the group of special functions, so its testing is being per- formed on the basis of known functions of the type: Γ–gamma, ψ–psi, ln–logarithm, exp–exponential function, and so on. By introducing two degrees of freedom k and n, we generalize it from discrete to continual [1,2]. In addition, we can advance from the field of the natura l integers o f the dimensions–degrees of freedom of cube geometry, to the field of real and non-integer values, where all the conditions concur for a more condense mathematical analysis of the function HC(k, n, r). In this article, the analysis is focused on the infinitesimal calculus application of the HC which is given in the generalized form. For research papers on the development of multidimensional func- tion theory, see Bowen [3], Conway [4], Coxeter [5], Dewdney [6], Hinton [7], Hocking and Young [8], Gardner [9], Manning [10], Maunder [11], Neville [12], Rucker [13], Skiena [14], Sloane [15], Sommerville [16], Wilker [17], and others and for its testing, see Letić et al. [18]. Today the results of the HC researc h are represented both in geo- metry and topology and in other branches of mathematics and physics, such as Boole’s algebra, operational researches, theory of algorithms and graphs, combinatorial analy- sis, nuclear and astrophysics, molecular dynamics, and so on. 2. The derivative HCs 2.1. The hypercube functional matrix The former results [2], as it is known, give the functions of the hypercube surface (n = 2), i.e., volume (n = 3), therefore, we have, respectively Letić et al . Advances in Difference Equations 2011, 2011:60 http://www.advancesindifferenceequations.com/content/2011/1/60 © 2011 Letićć et al; li censee Springer. This is a n Open Acce ss article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrest ricted use, distribution, and reproduction in any medium, provided the original work is properly cited. HC(k,2,r)=2kr k−1 = ∂ ∂r HC(k,3,r)orHC(k,3,r)= r  0 HC(k,2,r) dr =(2r) k . On the basis of the above recurrent relations, we formulated the general form of the HC [1]. Definition 2.1. The generalized HC is defined by equality HC(k, n, r)= 2 k r k+n−3 (k +1) (k + n − 2) (k, n ∈, r ∈ N). (2:1) where r is the half-edge of the On certain integral Schreier graphs of the symmetric group Paul E. Gunnells ∗ Department of Mathematics and Statistics University of Massachusetts Amherst, Massachusetts, USA gunnells@math.umass.edu Richard A. Scott † Department of Mathematics and Computer Science Santa Clara University Santa Clara, California, USA rscott@math.scu.edu Byron L. Walden Department of Mathematics and Computer Science Santa Clara University Santa Clara, California, USA bwalden@math.scu.edu Submitted: Feb 17, 2006; Accepted: May 3, 2007; Published: May 31, 2007 Mathematics Subject Classification: 05C25, 05C50 Abstract We compute the spectrum of the Schreier graph of the symmetric group S n corresponding to the Young subgroup S 2 × S n−2 and the generating set consisting of initial reversals. In particular, we show that this spectrum is integral and for n ≥ 8 consists precisely of the integers {0, 1, . . . , n}. A consequence is that the first positive eigenvalue of the Laplacian is always 1 for this family of graphs. ∗ Supported in part by NSF grant DMS 0401525. † Supported in part by a Presidential Research Grant from Santa Clara University. the electronic journal of combinatorics 14 (2007), #R43 1 1 Introduction Given a group G, a subgroup H ⊂ G, and a generating set T ⊂ G, we let X(G/H, T) denote the associated Schreier graph: the vertices of X(G/H, T) are the cosets G/H and two cosets aH and bH are connected by an edge whenever aH = tbH and t ∈ T . We shall assume that T satisfies t ∈ T ⇔ t −1 ∈ T so that X(G/H, T) can be regarded as an undirected graph (with loops). The main result of this article is the following. Theorem 1.1. Let S n be the symmetric group on n letters, let H n be the Young subgroup S 2 × S n−2 ⊂ S n , and let T n = {w 1 , . . . , w n } where w k denotes the involution that reverses the initial interval 1, 2, . . . , k and fixes k + 1, . . . , n. Then for n ≥ 8, the spectrum of the Schreier graph X(S n /H n , T n ) consists precisely of the integers 0, 1, . . . , n. The full spectrum, complete with multiplicities, is given in Theorem 7.2 and seems interesting in its own right. There are, however, some connections with results in the literature that are worth mentioning. Given a graph X, let λ = λ(X) denote the difference between the largest and second largest eigenvalue of the adjacency matrix. For a connected graph, λ coincides with the first positive eigenvalue of the Laplacian and is closely related to certain expansion coefficients for X. In particular, one way to verify that a given family of graphs has good expansion properties is to show that λ is uniformly bounded away from zero (see, e.g., [Lu2]). Given a group G and generating set T ⊂ G, we denote by X(G, T ) the corre- sponding Cayley graph. Several papers in the literature address spectral properties of X(S n , T n ) for certain classes of subsets T n . In the case where T n is the set of transposi- tions {(1, 2), (2, 3), . . . , (n −1, n)}, i.e., the Coxeter generators for S n , the entire spectrum is computed by Bacher [Ba]. Here λ = 2 − 2 cos(π/n), which approaches zero as n gets large. On the other hand, in the case where T n is the more symmetric generating set {(1, 2), (1, 3), . . . , (1, n)}, the eigenvalue gap λ is always 1 ([FOW, FH]). In [Fr], it is shown that among Cayley graphs of S n generated by transpositions, this family is optimal in the sense that λ ≤ 1 for any set T n consisting of n − 1 transpositions. In applications, one typically wants an expanding family with bounded degree, meaning there exists some k and some  > 0 such that every graph in the family has λ ≥  and vertex degrees ≤ k. In [Lu1] Lubotzky poses the question as to whether Cayley graphs of the symmetric group can contain such a family. When restricting T n to transpositions, this is impossible, since one needs at least n − 1 transpositions to generate S n . In [Na] the case where T n is a set of “reversals” (permutations that reverse the order of an entire subinterval of {1, 2, . . 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 Another .: môt .nào đó số ít ,dùng khi nói đến một đối tượng nào đó không xác định This book is boring .Give me another quyển sách này chán quá đưa tôi quyển khác xem => quyển nào cũng được ,không xác định Others : những khác Số nhiều ,dùng khi nói đến những đối tượng nào đó không xác định These books are boring .Give me others : những quyển sách này chán quá ,đưa tôi những quyển khác xem => tương tự câu trên nhưng số nhiều the other : .còn lại Xác định ,số ít I have two brothers .One is a dotor ; the other is a teacher Tôi có 2 ngừoi anh .Một ngừoi là bác sĩ ngừoi còn lại là giáo viên. the others : những .còn lại Xác định ,số nhiều I have three brothers .One is a dotor ; the others are teachers Tôi có 3 ngừoi anh .Một ngừoi là bác sĩ những ngừoi còn lại là giáo viên. The others = The other + N số nhiều There are 5 books on the table .I don't like this book .I like the others = ( I like the other books ) http://www.tienganh.com.vn/showthread.php?p=144147#post144147 The Other Trigonometric Functions The Other Trigonometric Functions By: OpenStaxCollege A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is 12 or less, regardless of its length A tangent represents a ratio, so this means that for every inch of rise, the ramp must have 12 inches of run Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions We have already defined the sine and cosine functions of an angle Though sine and cosine are the trigonometric functions most often used, there are four others Together they make up the set of six trigonometric functions In this section, we will investigate the remaining functions Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent To define the remaining functions, we will once again draw a unit circle with a point (x, y) corresponding to an angle of t, as shown in [link] As with the sine and cosine, we can use the (x, y) coordinates to find the other functions The first function we will define is the tangent The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle In [link], y the tangent of angle t is equal to x , x≠0 Because the y-value is equal to the sine of t, 1/28 The Other Trigonometric Functions and the x-value is equal to the cosine of t, the tangent of angle t can also be defined as sin t cos t , cos t ≠ 0.The tangent function is abbreviated as tan The remaining three functions can all be expressed as reciprocals of functions we have already defined • The secant function is the reciprocal of the cosine function In [link], the secant 1 of angle t is equal to cos t = x , x ≠ The secant function is abbreviated as sec • The cotangent function is the reciprocal of the tangent function In [link], the cos t x cotangent of angle t is equal to sin t = y , y ≠ The cotangent function is abbreviated as cot • The cosecant function is the reciprocal of the sine function In [link], the 1 cosecant of angle t is equal to sin t = y , y ≠ The cosecant function is abbreviated as csc A General Note Tangent, Secant, Cosecant, and Cotangent Functions If t is a real number and (x, y) is a point where the terminal side of an angle of t radians intercepts the unit circle, then y tan t = , x ≠ x sec t = , ... Graphs of the Other Trigonometric Functions decreases, the graph of the cosecant function increases Where the graph of the sine function increases, the graph of the cosecant function decreases The. .. and the local minimum at ( 3π8 , 3).[link] 19/49 Graphs of the Other Trigonometric Functions try it feature Graph one period of f(x) = 0.5csc(2x) 20/49 Graphs of the Other Trigonometric Functions. .. I and III of the unit circle 2/49 Graphs of the Other Trigonometric Functions Graph of the tangent function Graphing Variations of y = tan x As with the sine and cosine functions, the tangent