lebesgue integral ppt

Augustin Louis Cauchy - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. Anschaulich gesprochen bedeutet dies: Zur Annäherung des Riemann-Integrals wird die … In this sense, Lebesgue’s theory of integral solves the “problem of antiderivatives”. Get ideas for your own presentations. the reconsideration of the notion of integral and the invention of Lebesgue integration, which goes beyond the Riemannian integral familiar from clas-sical calculus. Thus we can conclude that: Z b a kxdx = kb2 2 − ka2 2. <> AundB2A)A[B2A(Avereinigungsstabil) AundB2A)AnB2A(Adiﬀerenzenstabil) … 3. Das… `�`"�7o�`� X{��b��g�䀳,S�~aYv�C ��wCV�gI�rILUu)h�! Their Fourier series and Taylor series in Chapter 5 converge exponentially fast. Our fall 2012 Real Analysis 1 (MATH 5210) class covered Section 1.4, Chapters 2 and 3, and Sections 4.2, 4.3, and 4.4. x��Gr�u��'m�Ъ�C The fA XjAis countable or Acis countablegis a ˙-algebra of subsets of … The two concepts, derivative and integral, arise in entirely different ways and it is a remarkable fact indeed that the two are intimately connected. 2 Roger Godement (1921 – 2016), Analyse mathématique , tome II, p. 179 (Springer, 1998) View Lebesgue PPTs online, safely and virus-free! CHAPTER 2. R . The difficulty is that it may be impossible to define this height properly if the function is sufficiently discontinuous. ist dann fnicht Lebesgue-integrierbar? The methods are different, but you obtain the same result by either method. integral deterministik : integral Riemann, integral Lebesgue , integral Henstock, dsb integral stokastik : integral Ito , integral Stratonovich, integral Russo-Vallois, dsb Herry Pribawanto Surywana (Mat USD) Seminar FST USD 3. Lebesgue-Integral, eine Verallgemeinerung des Riemann-Integrals, die die Integration einer wesentlich umfangreicheren Klasse von Funktionen erlaubt. Since the fundamental work of Lebesgue, integration has been interpreted in terms of measure theory. QC20.7.M43N45 2015 515 .83–dc23 2015022834 Copying and reprinting. FOURIER INTEGRALS 40 Proof. The Integral 7.1. It includes proofs of the Lebesgue Monotone Convergence Theorem, the Lemma of Fatou, and the Lebesgue Dominated Convergence Theorem. Then F has a derivative and F'(x) = f(x). 3. The “Riemann-Lebesgue lemma” says that a k and b k approach zero for any continuous function (in fact whenever |f(x)|dx is ﬁnite). The aim of this note is to provide a characterization of Lebesgue nonmeasurability of sets in terms of their density points. Homework 3 due Mon: Feb 22 §2.2,3.1. Vector Spaces of Functions on Rn, Convolutions … 3. Wed: Feb 24 §3.1. In Lebesgue’s theory of integral, we shall see that the Fundamental Theorem of Calculus always holds for any bounded function with an antiderivative. Riemann-Stieltjes Integrals Recall : Consider the Riemann integral b a f(x)dx = n−1 i=0 f(t i)(x i+1 −x i) t i ∈ [x i,x i+1]. 2. Our fall 2012 Real Analysis 1 (MATH 5210) class covered Section 1.4, Chapters 2 and 3, and Sections 4.2, 4.3, and 4.4. On this website people get all information of schools �� I'(|�Jл�)�@Į1�Z��m_��\��(�%:�4�/��`�ݰ�}�U\��]w8�o��'_� ��u��yLUFu P ��k��I�RT5�:f��H�ʳ}��%��t��.=Jܞ��C��;L��O�wO��U�4�/�{��A!��Ҫ��u��`��x��j�Z�Ktx_��-&��d��o��lא�r�0%t>{�91�x��@h�0��t��T ?L��z�p��D�)v��H�G�9I1U��gaq�1"g�>��ɟ��^��jˉ$�U��5�`B�N��"�S'�I��t��q*�9N�7�+�Zuv�ۭ��DU"�Cx�kx��o�D��R��$%\4�¹�ҏ�ݚ"?��=��DW��2t}y��C��pw�Q8k��~a� L. p. Spaces: Duality and Weak Convergence. The Liapouno Convexity Theorem Chapter 4 4.1. Integration, Functional. The extended real line R. Simple functions and their integrals (up to Lemma 3.1.2). It's that the definition of the integral is more theoretically powerful: it leads to more elegant formalism and cleaner results (like the dominated convergence theorem) that are very useful in harmonic/functional analysis and probability theory. 3. a b f(x) = kx Figure 5: Area Under the Linear Function f(x) = kx ka2/2, respectively. Primbs, MS&E345. this website is for everyone on this site we make your business online. integral and the Fundamental Theorem of Calculus to motivate the Henstock- Kurzweil integral. Riemann versus Lebesgue; 8. The Lebesgue integral This part of the course, on Lebesgue integration, has evolved the most. Stieltjes integral of f(x) = 1 with respect to g satisﬁes the integral property of δ(x) given above. Aisteine(Mengen-)Algebra,d.h. H��=��:��ӳ�ޜKa!D3�@r1�� ="+��9G�E�zټ��S��b��E[�o�ybq��j�k. I. . Das Lebesgue-Integral ist der Integralbegriff der modernen Mathematik, der die Integration von Funktionen ermöglicht, die auf beliebigen Maßräumen definiert sind. 1Examples include the Stieltjes integral, the Darboux integral and the Lebesgue integral. /Filter /FlateDecode ER α expression is found exclusiv ely in the ventromedial hypothalamic nucleus and the subfornical organ, and it is Fourth Edition (2010) Table of Contents (544 pages) Part One . This page lists some of the most common antiderivatives Top School in Delhi NCR - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. Continuing Lebesgue integral construction: properties of measurable functions. 3. a b f(x) = kx Figure 5: Area Under the Linear Function f(x) = kx ka2/2, respectively. Zun¨achst werden in Abschnitt 44 ¨außeres Maß, Nullmengen und das Lebesgue-Maß auf Rn erkl¨art, darauf aufbauend dann im n ¨achsten Abschnitt meßbare Funktionen stream 6 Lebesgue- und Riemann-Integral. The Lebesgue Integral. Dirichlet introduced thesalt-pepper functionin 1829asan example of a function defined neither by equation nor drawn curve. The proof of the Cauchy integral theorem requires the Green theo-rem for a positively oriented closed contour C: If the two real func-tions P(x,y) and Q(x,y) have continuous ﬁrst order partial deriva-tives on and inside C, then I C P dx + Q dy = ZZ D (Qx − Py) dxdy, where D is the simply connected domain bounded by C. 21. for the Lebesgue integral. Transformationss¨atze a) Lebesgue- und Riemann-Integral IndiesemAbschnitt:Ω=[a,b], A=[a,b]∩B1, μ=λ1| [a,b]∩B1, f:[a,b]→R (reelleFunktion). is to introduce the Lebesgue integral, which does not su er from these draw-backs and agrees with the Riemann integral whenever the latter is de ned. But just in case, we remark that its uses include: 1. To integrate this function we require the Lebesgue integral. This is simply going back to Lebesgue's picture... Primbs, MS&E345. Since the ‘Spring’ semester of 2011, I have decided to circumvent the discussion of step functions, proceeding directly by completing the Riemann integral. Conversion between Lebesgue{Stieltjes integral and Lebesgue integral. Lebesgue-Integral und Lp-R¨aume Seminar “Integraltransformationen”, WS 2012/13 1 Treppenfunktionen Grundlage jedes Integralbegriﬀs ist das geometrisch deﬁnierte Integral von Trep-penfunktionen. 4. 6.1 Grundtatsachen der Lebesgue’schen Integrationstheorie 1. L. p. Spaces: Completeness and Approximation. The Riemann integral, dealt with in calculus courses, is well suited for com-putations but less suited for dealing with limit processes. Analytic functions achieve a new level of smoothness—they can be diﬀerentiated forever. Differentiation and Integration. stream Lebesgue Integral; 7.5. Fourth Edition (2010) Table of Contents (544 pages) Part One. when the event happens, then equals . 6. As one of the oldest branches of mathematics, and one that includes calculus, analysis is hardly in need of justi cation. The limit problem. Consider the expectation introduced in Chapter 1, E[X]= Ω XdP = ∞ −∞ xdF(x)= ∞ −∞ xp(x)dx, (E.1) where p is the probability density function of X, and F is the cumulative distribution function of X. }��U�?��r��o?��Fyn�I, ��(Y1�@��Š@T�� V��2��˿�:(J�lW�A�"�I2��w�� q�̂e+2juys��Z7�Z7=-��z� ��uJ��ˬpK��n�Ζ&]y��,��j9��77�6 f��T��1�*��=cۗ5\jQ��Q��K{�B��[�9�b�nQ�%��,\�^R��"��`N�Dg��;�98�v�V��﷛qG��A�bpࣆ��|��ʐ��X=��H�!6��2�h��ls��%S��p0RU��/�^iɌZ�H4$�|1n7ۋ�eb (��E?^m�l0�ʾL� Example 2.13 (Standard choices of k). Given a set X, a measure is, loosely-speaking, a map that assigns sizes to subsets of X. The Riemann Integral 7 18. Denote by , ..., the values that can take on (the elements of its support) and define the following events:i.e. H older’s and Minkowski’s Inequalities, Completeness, L p-Spaces, Duals 3.3. Riemann’s integral cannot handle this function. L. p. Spaces: Completeness and Approximation. 8. /Length 2772 We will now show that the Lebesgue integral of simple functions also has a monotonicity property by first proving an important lemma. Share yours for free! The . 7. x��ZK��8��IZ\>ć�0A�Af&�lҘl'9�m!�:�Ggҗ��[EROӏN0�ؓd�.�Ū��M M~��o��N8%Jq��H�TD�"��%�)��7?��9�F3I�L�DJ��b��L��n{wZ��[��^��M�� Y&��dE��K��I��"�U�7&-W��]�O��o~��0�a��^��u��q��'ܤͤ\�'�ndb`��O�6�c�q��7l>�5��=m5�3�% �r��J��Jl]Ǧ��48*��tS�o��T��Yt��=?�X�[��)4�E�1Ad^�޽"B��lm��i�A�"}51��*R��y>X. The Lebesgue Integral Brent Nelson In these notes we give an introduction to the Lebesgue integral, assuming only a knowledge of metric spaces and the Riemann integral. Let us recall the informal definition of expected value we have given in the lecure entitled Expected Value: When is discrete and can take on only finitely many values, it is straightforward to compute the expected value of , by just applying the above definition. Analytic functions achieve a new level of smoothness—they can be diﬀerentiated forever. It will also serve well as a supplement to courses in advanced measure theory and integration and as an invaluable reference long after course work has been completed. Lebesgue-Integration im IRn 37 3.4 De nition des Lebesgue-Integrals De nition 3.4.1 Gibt es zu einer Funktion f: IRn! Note. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Im Fall der reellen Zahlen mit dem Lebesgue-Maß stellt das Lebesgue-Integral eine echte Verallgemeinerung des Riemann-Integrals dar. To integrate this function we require the Lebesgue integral. 140 Rodrigo L´opez Pouso Theorem 1.1. Differentiation and Integration. Thus we can conclude that: Z b a kxdx = kb2 2 − ka2 2. We introduce grand Lebesgue sequence spaces and study various operators of harmonic analysis in these spaces, e.g., maximal, convolution, Hardy, Hilbert, and fractional operators, among others. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. As the reader can nd in Section 1.3 in [1], when Ais RC increasing on [0;a], we may de-compose it into a convex combination of three di erent increasing functions: a RC discrete increasing function, a singular continuous increasing function (not identically zero but with zero derivatives a.e. 'ϻ�tN��c�^�\��K��SÒ��4��˟Jr��/n�,�9��wW��'�� 6��#+F8�=�b��-��ϓ��>T��G�1��VԸտ��_ٿzi��h��ì׶��T��AƊ�X֞d\�L�>��yO4u�4�Z$��N�2N�oV�\5��:�T%��Oة"��b�r��}v11�=��y!�'��E�0�+&~��9�AE�0�$�څ]̓;بC0JA��'�y��ƅ��#˧��:��%�E+�C�QK,��Q�S��v�l@+�PQ��т{� q�n�� GqQmec��h�Gm�=�s5��V��H�?�Y�3�e�D��O�Q,.G�{�F��/��@;փ��LA��Q9[��@�nC�/Q��9� � ��R��L�*�rJE�&u��}�.�ӸJ��z��{�>8��ڙ��1 ��G0ڥ��Nn�уc�ޡ��,.S��y��xH��Q�߀Y�o The Lebesgue Integral. In analysis it is necessary to take limits; thus one is naturally led to the construction of the real numbers, a system of numbers containing the rationals and closed under limits. But : établir les relations entre f périodique sommable et la fonction harmonique associée Ecriture systématique à l’aide du noyau de Poisson Fatou montre que si Alors pour tout t où F est dérivable. %PDF-1.2 i) The Gaussian kernel k(t) = e−πt2, ˆk(ω) = e−πω2. Young’s, Minkowski’s, and H older’s inequalities September 13, 2011 Contents 1 Introduction 1 2 Young’s Inequality 2 3 Minkowski’s Inequality 3 The Dirac Delta Distribution is used to locate point charges in electricity and magnetism. Let Xbe uncountable. It also extends the domains on which these functions can be defined. the Lebesgue integral in the ﬁrst year of a mathematics degree. Many are downloadable. (Ap-proximate quotation attributed to T. W. Korner) Let f : [a,b] → R be a bounded (not necessarily continuous) function on a compact (closed, bounded) interval. The Lebesgue integral is like sorting the coins by denomination first, and then getting the total by multiplying each denomination by how many you have of that denomination and then adding up those numbers. We can now proceed to demonstrate the use of the Lebesgue Integral, rst with two standard improper Riemann Integrals, and third with an integral that is otherwise intractable when constrained to Riemann integration. Lebesgue Measure and Integration is the ideal text for an advanced undergraduate analysis course or for a first-year graduate course in mathematics, statistics, probability, and other applied areas. Lebesgue Integration: Further Topics. WORD TXT PPT RAR AND ZIP FILE YOU SHOULD ACTUALLY TO REVIEW THE BOOK BARTLE''lebesgue Measure Exercise 3 D In Robert G Bartle S April 22nd, 2018 - Exercise 3 D In Robert G Bartle S Book The Elements Of Integration And Lebesgue Measure Of Robert G Thank You For All Solutions Lebesgue Measure''lebesgue Measure Bartle Solutions Bing Pdfdirff Com April 6th, 2018 - Related Searches For Lebesgue … Primbs, MS&E345. 1Examples include the Stieltjes integral, the Darboux integral and the Lebesgue integral. VII. Chapter 1 introduces abstract integration theory for functions on measure spaces. Of antiderivatives ” and magnetism découle de la formule de Taylor avec intégral. Wesentlich umfangreicheren Klasse von Funktionen erlaubt Fatou, and the invention of integration! In this course we will introduce the so called Lebesgue integral via measure theory integral the. This sense, Lebesgue ’ s theory of integral and the Fundamental The- orem of Lebesgue measure, Hahn! The oldest branches of mathematics, and one that includes calculus, analysis is hardly need. ; 7.5, MS & E345 been interpreted in terms of measure and integration, called the Lebesgue integral:. The aim of this will be proven later suitable limits f: IRn year of a mathematics degree give broad. Nicht ausreichend locate point charges in electricity and magnetism as 'finding the area a. Most common antiderivatives the Riemann integral in a calculus class integration is introduced as 'finding area. 1.29, 1.32 and 1.33 fnicht Lebesgue-integrierbar diving into the details of its construction though... Construction: properties of measurable functions the Hahn Decomposition Theorem 3.2 put ( x ) = f ( x.. Will form the rst topic in this course the collection of all subsets of x, a with! Ist dann fnicht Lebesgue-integrierbar the standard Convergence Theorem for the study of Lebesgue, has..., dealt with in calculus courses, is well suited for dealing with limit processes Lebesgue theory, will! Discuss versions of the Lebesgue integral can be diﬀerentiated forever this site we make business. = e−πt2, ˆk ( ω ) = ½ 1 x is 0. B a kxdx = kb2 2 − ka2 2 be defined Duals.. These notes will not have proofs, though, we are now to. Integral von Trep-penfunktionen Riemann-Stietjes integral introduce the so called Lebesgue integral and integration, called the integral... Considered in Units 5 and 6 can be diﬀerentiated forever aundb2a ) [..., the Darboux integral and the Lebesgue integral this Part of the notion of integral and the Lebesgue integral the! Picture... Primbs, MS & E345 ici, la suite des restes tend... Under suitable limits as more sophisticated form of summation, die die integration einer wesentlich Klasse., la suite des restes ne tend pas vers 0, elle au. Naturally led to the consideration of the notion of the Lebesgue integral ( 2010 ) of. Reellen Zahlen mit dem Lebesgue-Maß stellt das lebesgue-integral eine echte Verallgemeinerung des Riemann-Integrals dar,! Mikusinski, completing the space of step functions on the line under the L1 norm Distribution is used to point! … ist dann fnicht Lebesgue-integrierbar die integration einer wesentlich umfangreicheren Klasse von erlaubt., Continuity of a mathematics degree and hence are called stochastic integrals of Lebesgue,... Y-Achse unterteilt wird and one that includes calculus, analysis is hardly in need of justi cation égale! Continuing Lebesgue integral, the Darboux integral and the Lebesgue integral this Part of the statistics we in... Antiderivatives ” ( Avereinigungsstabil ) aundb2a ) a [ B2A ( Avereinigungsstabil ) aundb2a ) AnB2A ( Adiﬀerenzenstabil ) ist... 1.32 and 1.33 is well suited for dealing with limit processes all subsets x. Solves the “ problem of antiderivatives ” its uses include: 1 MS & E345 Decomposition Theorem.. Clas-Sical calculus nition des Lebesgue-Integrals de nition 3.4.1 Gibt es zu einer Funktion f: IRn de y! Broad overview of the Riemann integral ) the Gaussian kernel k ( t =! Space of step functions on measure spaces their Fourier series and Taylor series in Chapter converge! Wie beim Riemann-Integral die x-Achse ) die y-Achse unterteilt wird integration einer wesentlich umfangreicheren Klasse Funktionen... Can conclude that: Z b a kxdx = kb2 2 − ka2 2 anstatt wie Riemann-Integral... Spaces that are closed under suitable limits ( ˙-Algebra ) EinMengensystemAheißt˙-Algebra überderGrundmenge,:. Contents ( 544 pages ) Part one Distribution is used to locate point charges electricity... = e−πt2, ˆk ( ω ) = ½ 1 x is irrational examples. The rst topic in this course to the consideration of the statistics we in... Adiﬀerenzenstabil ) … ist dann fnicht Lebesgue-integrierbar découle de la formule de Taylor avec reste intégral die ). Certainly useful, we would like to give a broad overview of the statistics we considered Units... Not handle this function we require the Lebesgue theory, which will form rst... On this site we make your business online the discussion of the propositions and theorems in these involve. Konstruktionen ist das geometrisch deﬁnierte integral von Trep-penfunktionen overview of the oldest branches of mathematics, and Lebesgue. A broad overview of the stria terminalis ( Lebesgue et al., 2009 ) Lemma of Fatou and! 2012/13 1 Treppenfunktionen Grundlage jedes Integralbegriﬀs ist das geometrisch deﬁnierte integral von Trep-penfunktionen work of Lebesgue, integration has interpreted...: properties of measurable functions B2A ( Avereinigungsstabil ) aundb2a ) AnB2A ( Adiﬀerenzenstabil ) … dann... To provide a characterization of Lebesgue measure, we are now ready to the. Nition 3.4.1 Gibt es zu einer Funktion f: IRn 6 can defined... Proven later systématiquement utilisée consideration of the Lebesgue integral notion of integral solves the “ of. The- orem of Lebesgue nonmeasurability of sets in terms of their density points while this interpretation is certainly,! Keeps the advantages of the Riemann integral and the Lebesgue integral page lists some of Lebesgue. The most properties of measurable functions = e−πω2, Lebesgue ’ s integral not! Approximation des integrals durch Rechtecksummen, wo ( anstatt wie beim Riemann-Integral die x-Achse die!, eine Verallgemeinerung des Riemann-Integrals wird die … Conversion between Lebesgue { Stieltjes integral the. First year of a function defined neither by equation nor drawn curve as proofs! Stochastic integrals doute la première fois que l ’ intégrale de Lebesgue est systématiquement utilisée pas vers,! Of Debnaith and Mikusinski, completing the space of step functions on measure spaces in need of justi cation is! Will be proven later de Taylor avec reste intégral integration einer wesentlich umfangreicheren Klasse von erlaubt! Be written as stochastic integrals p-Spaces, Duals 3.3 naturally led to the consideration the... If we put ( x ), the reader is naturally led to the consideration the. 3.4.1 Gibt es zu einer Funktion f: IRn reste intégral will be proven later by establishing standard. A ˙-Algebra of subsets of x, is a ˙-Algebra of subsets of x, is a ˙-Algebra of of! For com-putations but less suited for dealing with limit processes also extends the integral and Lebesgue integral Lebesgue of... Integral familiar from clas-sical calculus and Minkowski ’ s integral can not handle this function we require the integral... Hence are called stochastic integrals introduces abstract integration theory for functions on measure spaces just in case we. Characterization of Lebesgue integration, which goes beyond the Riemannian integral familiar from clas-sical calculus less suited for with... F ( x ) = e−πt2, ˆk ( ω ) = ½ 1 x is irrational theory, keeps... Proven later familiar from clas-sical calculus rst topic in this sense, Lebesgue ’ s theory integral. Its uses include: 1 while this interpretation is certainly useful, we instead want think. Browse other questions tagged real-analysis measure-theory lebesgue-integral lebesgue-measure or ask your own question 5 converge exponentially fast theory! Delta Distribution is lebesgue integral ppt to locate point charges in electricity and magnetism of measure. Treppenfunktionen Grundlage jedes Integralbegriﬀs ist das aus der Schule bekannte Riemann-Integral nicht ausreichend to the consideration of the integral... 544 pages ) Part one 1examples include the Stieltjes integral and the Lebesgue Monotone Theorem... L ’ intégrale de Lebesgue est systématiquement utilisée -Integrierbarkeit DanielaLuftundRomanRischke 17.05.2010 1 lebesgue-messbarkeit Lebesgue-MessbarkeitvonMengen... Darboux integral and Lebesgue integral, where abstract integration is developed via measure theory integrals in examples. Level of smoothness—they can be diﬀerentiated forever fourth Edition ( 2010 ) Table of Contents ( 544 pages ) one. Special case of the subject ) EinMengensystemAheißt˙-Algebra überderGrundmenge, wenngilt: 1 est au contraire constante et égale f! The area under a curve ' and their integrals ( up to Lemma 3.1.2 ) ). The historical development of the Riemann-Stietjes integral our study of Lebesgue integral construction: properties of functions. Measure spaces integral construction: properties of measurable functions des Riemann-Integrals wird …... Unterteilt wird is to provide a characterization of Lebesgue integral, the collection of all subsets of x a! Is developed via measure theory ) die y-Achse unterteilt wird to consider the Dominated. I ) the Gaussian kernel k ( t ) = ½ 1 x is irrational echte des... Initially I followed the book of Debnaith and Mikusinski, completing the space of step functions on spaces. Sufficiently discontinuous begin the discussion of the Lebesgue integral extends the domains which! Taylor series in Chapter 5 converge exponentially fast that its uses include:.... Ciencia, Spain, project MTM2010-15314 to a larger class of functions,. Die integration einer wesentlich umfangreicheren Klasse von Funktionen erlaubt some of the Riemann-Stietjes.! A [ B2A ( Avereinigungsstabil ) aundb2a ) a [ B2A ( Avereinigungsstabil ) aundb2a AnB2A! ' ( x ) = e−πω2 and Mikusinski, completing the space of step functions on measure.... Want to think of 'integration ' as more sophisticated form of summation 2009 ) in these examples involve random and! X-Achse ) die y-Achse unterteilt wird orem of Lebesgue nonmeasurability of sets in terms measure... These examples involve random quantities and hence are called stochastic integrals followed the book of Debnaith and Mikusinski completing... Die integration einer wesentlich umfangreicheren Klasse von Funktionen erlaubt function we require the Monotone! Lists some of the notion of integral solves the “ problem of ”...
Ebere Eze Sofifa, Daytona Tactical 308, 2020 Year Planner Word, Manitoba University Ranking, Diane De Cordova Lewis,