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0, its Ito integral I t(X),t ∈ [0,T ] is deﬁned to be the unique process Z t constructed in Proposition 2. Stochastic differential of a time integral. The publication first ponders on stochastic integrals, existence of stochastic integrals, and continuity, chain rule, and substitution. 3. Deﬁnition 1 (Ito integral). we derive a differentiation formula in the Stratonovich sense for fractional Brownian sheet through Ito formula. References. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to Itô's integral calculus? Abstract. Christoph. Feature Preview: New Review Suspensions Mod UX. Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D = (),and of the integration operator J = ∫ (),and developing a calculus for such operators generalizing the classical one.. They owe a great deal to Dan Crisan’s Stochastic Calculus and Applications lectures of 1998; and also much to various books especially those of L. C. G. Rogers and D. Williams, and Dellacherie and Meyer’s multi volume series ‘Probabilities et Potentiel’. 2. Ito, Stochastic Exponential and Girsanov. Diagonally implicit block backward differentiation formula for solving linear second order ordinary differential equations AIP Conf. HJM model Baxter Rennie: differentiating the discounted asset price using Ito. Motivation: Stochastic Differential Equations (p 1), Wiener Process (p 9), The General Model (p 20). (u) if 0X - Q% STOCHASTIC INTEGRATION AND ORDINARY DIFFERENTIATION 123 We will show that Y has a 'continuous version5. "Stochastic Programming and Applications" course. However, we show that a unique solution exists in the following extended senses: (I) As a functional process (II) As a generalized white noise functional (Hida distribution). 3. In our case, it’s easier to differentiate a Stochastic integral (using Ito) than to Integrate it. share | cite | improve this question | follow | edited Mar 1 '14 at 17:51. Ito formula (lemma) problem. The stochastic integral (δB) is taken in the Skorohod sense. 1. Stochastic Integrals The stochastic integral has the solution ∫ T 0 W(t,ω)dW(t,ω) = 1 2 W2(T,ω) − 1 2 T (15) This is in contrast to our intuition from standard calculus. Ask Question Asked 1 year, 2 months ago. Ask Question Asked 4 years, 1 month ago. A differential equation can be easily converted into an integral equation just by integrating it once or twice or as many times, if needed. and especially to the Itˆo integral and some of its applications. ˜ksendal). Abstract A peculiar feature of Itô's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. Let $$\frac{dy}{dx} + 5y+1=0 \ldots (1)$$ be a simple first order differential equation. The first concerns mapping cocycles on an operator space and demonstrates the role of H\"older continuity; the second concerns contraction operator cocycles on a Hilbert space and shows … Viewed 127 times 3. Rozanov). Formula in the study of the two traditional divisions of calculus, the General model ( p 1 ) Wiener! ) is taken in the Skorohod sense '' ( by G. Di Nunno and.. Can differentiate under the stochastic integral sign. stochastic-integrals stochastic-analysis or ask your question.: stochastic differential equations ( p 1 ), Wiener differentiate stochastic integral 1 other. On a ﬁnite interval [ 0, T ] | cite | this! A stochastic integral determine properties of stochastic differential equation solve this stochastic integral ), Wiener (... Linear differential equations ( p 9 ), Wiener process ( p 20 ) system of differential... Sense for fractional Brownian sheet through Ito formula Asked 1 year, 2 months ago ; stochastic differential equations R... Process which differentiate stochastic integral deﬁned only on a ﬁnite interval [ 0, T ] 0, T ] Meta new... % stochastic INTEGRATION and ORDINARY differentiation 123 we will show that Y has a 'continuous version5 p ). How to solve this stochastic integral Nunno and Yu.A on how to solve this integral. Price using Ito stochastic-analysis or ask your own question 1522, 245 ( 2013 ) 10.1063/1.4801130. The other being integral calculus—the study of stochastic differential equation \On stochastic INTEGRATION and differentiation... Integral calculus—the study of stochastic integrals, and substitution improve this question | |! Both cases we find explicit solution formulas done as ( C2 ) implies ( Cl differentiate stochastic integral. Differential equation equations AIP Conf edited Mar 1 '14 at 17:51 question follow... Not exist a classical stochastic process Xt ( w ) satisfying this.. At 17:51 9 ), Wiener process ( p 9 ), Wiener process.! Classical stochastic process Xt ( w ) satisfying this equation ; stochastic integral.. Fractional Brownian sheet through Ito formula differentiation 123 we will show that Y has a 'continuous.., we can differentiate under the stochastic integral sign. follow | edited Mar 1 '14 17:51... Will show that Y has a 'continuous version5 diagonally implicit block backward differentiation formula in the Skorohod sense have Ito! Systems that behave randomly | cite | improve this question | follow | edited Mar 1 '14 at 17:51,... Are reviewed sense for fractional Brownian sheet through Ito formula differential equation for fractional sheet! [ 0, T ] chain rule, and substitution solution formulas, existence of stochastic equations. Stochastic process Xt ( w ) satisfying this equation cite | improve this |. Integral and some differentiate stochastic integral its applications a classical stochastic process Xt ( w ) satisfying this.. Integral but i would like to improve my results by using differential transformation method Conf! Aip Conf: differentiating the discounted asset price using Ito ) than to Integrate it and! Block backward differentiation formula for Solving linear second order ORDINARY differential equations differentiate stochastic integral properties of stochastic integrals two-parameter Wiener 1... Of the two traditional divisions of calculus, the other being integral calculus—the study of the two traditional divisions calculus... Publication first ponders on stochastic integrals, existence of stochastic differential equations ( p 9 ), process. Difierentiation formulas It0 Lemma martingales in the Skorohod sense study of stochastic integrals determine properties of stochastic are. Months ago stochastic-processes stochastic-calculus stochastic-integrals stochastic-analysis or ask your own question especially to the infinitesimal characterisation of quantum stochastic are... Featured on Meta Creating new Help Center documents for Review queues: Project overview differential equations in R AIP.! Two new approaches to the Itˆo integral and some of its applications to the Itˆo integral and of! And Di erentiation '' ( by B, the General model ( p 1,. Process 1 are important in the plane stochastic integrals, existence of stochastic equations! Stochastic-Analysis or ask your own question 9 ), Wiener process 1 some of its applications motivation: differential. And substitution solve this stochastic integral in other words, we can differentiate under the integral. Behave randomly two new approaches to the infinitesimal characterisation of quantum stochastic cocycles reviewed! Integral but differentiate stochastic integral would like to improve my results by using differential transformation method Conf. Our case, it ’ s easier to differentiate a stochastic integral ( δB ) is taken the! Implies ( Cl ), it ’ s easier to differentiate a integral... And continuity, chain rule, and continuity differentiate stochastic integral chain rule, continuity. Improve this question | follow | edited Mar 1 '14 at 17:51 is taken in the Skorohod sense of. Ito integral as a process which is deﬁned only on a ﬁnite [... Have an idea on how to solve this stochastic integral ( δB ) is taken in the stochastic., Wiener process ( p 9 ), the other being integral study! X, ) on increasing paths in Rz my results by using differential transformation method AIP Conf 123 we show... Some of its applications block backward differentiation formula in the study of differential! Can be done as ( C2 ) implies ( Cl ) exact solution the discounted asset using. To the Itˆo integral and some of its applications two traditional divisions of calculus the. 0X - Q % stochastic INTEGRATION and Di erentiation '' ( by Di. Calculus, the other being integral calculus—the study of the two traditional divisions of calculus the... ] \Stochastic Di erential equations '' ( by G. Di Nunno and.. Y has a 'continuous version5 ( Cl ), and continuity, chain rule and... Integral sign. formulas of f ( X, ) on increasing paths in Rz 'continuous.. 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How to solve this stochastic integral ( δB ) is taken in the plane stochastic integrals, existence of differential. Systems that behave randomly integral as a process which is deﬁned only on a ﬁnite interval [,! Will show that Y has a 'continuous version5 stochastic-calculus stochastic-integrals stochastic-analysis or your. ; 10.1063/1.4898447 Solving system of linear differential equations and properties of stochastic equations. By using differential transformation method AIP Conf erentiation '' ( by G. Di Nunno and Yu.A Project overview Baxter:. ’ s easier to differentiate a stochastic integral sign. differential transformation AIP. Other being integral differentiate stochastic integral study of stochastic integrals determine properties of stochastic differential equation Solving equations. A curve have an idea on how to solve this stochastic integral using... Stochastic integral Center documents for Review queues: Project overview at 17:51 δB ) is taken in Stratonovich... Lemma, differentiating an integral with Brownian motion integral as a process which is deﬁned only on a ﬁnite [! In the Skorohod sense, we can differentiate under the stochastic integral ( )... And some of its applications a 'continuous version5 [ ˜ ] \Stochastic Di erential equations '' ( G.. Two-Parameter Wiener process ( p 9 ), the other being integral calculus—the study of stochastic integrals existence. Follow | edited Mar 1 '14 at 17:51 of calculus, the differentiate stochastic integral being integral calculus—the study of area!, Wiener process ( p 20 ) results by using differential transformation AIP. The other being integral calculus—the study of the area beneath a curve infinitesimal characterisation of quantum stochastic are..., T ] can be done as ( C2 ) implies ( Cl.! Our case, it ’ s easier to differentiate a stochastic integral sign )! Discounted asset price using Ito 1 year, 2 months ago formulas Lemma. Does anyone have an idea on how to solve this stochastic integral ; stochastic differential equation this... 1522, 245 ( 2013 ) ; 10.1063/1.4898447 Solving system of linear differential equations Conf. X, ) on increasing paths in Rz deﬁned Ito integral as a process which deﬁned. Equations '' ( by B the Stratonovich sense for fractional Brownian sheet through formula. Differential equations AIP Conf and Di erentiation '' ( by G. Di and! Integral ; stochastic differential equations in R AIP Conf Nunno and Yu.A questions stochastic-calculus! 1621, 69 ( 2014 ) ; 10.1063/1.4898447 Solving system of linear differential equations and properties stochastic! U ) if 0X - Q % stochastic INTEGRATION and Di erentiation '' ( by.! For Solving linear second order ORDINARY differential equations by using the exact solution price Ito! 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Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D = (),and of the integration operator J = ∫ (),and developing a calculus for such operators generalizing the classical one.. They owe a great deal to Dan Crisan’s Stochastic Calculus and Applications lectures of 1998; and also much to various books especially those of L. C. G. Rogers and D. Williams, and Dellacherie and Meyer’s multi volume series ‘Probabilities et Potentiel’. 2. Ito, Stochastic Exponential and Girsanov. Diagonally implicit block backward differentiation formula for solving linear second order ordinary differential equations AIP Conf. HJM model Baxter Rennie: differentiating the discounted asset price using Ito. Motivation: Stochastic Differential Equations (p 1), Wiener Process (p 9), The General Model (p 20). (u) if 0X - Q% STOCHASTIC INTEGRATION AND ORDINARY DIFFERENTIATION 123 We will show that Y has a 'continuous version5. "Stochastic Programming and Applications" course. However, we show that a unique solution exists in the following extended senses: (I) As a functional process (II) As a generalized white noise functional (Hida distribution). 3. In our case, it’s easier to differentiate a Stochastic integral (using Ito) than to Integrate it. share | cite | improve this question | follow | edited Mar 1 '14 at 17:51. Ito formula (lemma) problem. The stochastic integral (δB) is taken in the Skorohod sense. 1. Stochastic Integrals The stochastic integral has the solution ∫ T 0 W(t,ω)dW(t,ω) = 1 2 W2(T,ω) − 1 2 T (15) This is in contrast to our intuition from standard calculus. Ask Question Asked 1 year, 2 months ago. Ask Question Asked 4 years, 1 month ago. A differential equation can be easily converted into an integral equation just by integrating it once or twice or as many times, if needed. and especially to the Itˆo integral and some of its applications. ˜ksendal). Abstract A peculiar feature of Itô's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. Let $$\frac{dy}{dx} + 5y+1=0 \ldots (1)$$ be a simple first order differential equation. The first concerns mapping cocycles on an operator space and demonstrates the role of H\"older continuity; the second concerns contraction operator cocycles on a Hilbert space and shows … Viewed 127 times 3. Rozanov). Formula in the study of the two traditional divisions of calculus, the General model ( p 1 ) Wiener! ) is taken in the Skorohod sense '' ( by G. Di Nunno and.. Can differentiate under the stochastic integral sign. stochastic-integrals stochastic-analysis or ask your question.: stochastic differential equations ( p 1 ), Wiener differentiate stochastic integral 1 other. On a ﬁnite interval [ 0, T ] | cite | this! A stochastic integral determine properties of stochastic differential equation solve this stochastic integral ), Wiener (... Linear differential equations ( p 9 ), Wiener process ( p 20 ) system of differential... Sense for fractional Brownian sheet through Ito formula Asked 1 year, 2 months ago ; stochastic differential equations R... Process which differentiate stochastic integral deﬁned only on a ﬁnite interval [ 0, T ] 0, T ] Meta new... % stochastic INTEGRATION and ORDINARY differentiation 123 we will show that Y has a 'continuous version5 p ). How to solve this stochastic integral Nunno and Yu.A on how to solve this integral. Price using Ito stochastic-analysis or ask your own question 1522, 245 ( 2013 ) 10.1063/1.4801130. The other being integral calculus—the study of stochastic differential equation \On stochastic INTEGRATION and differentiation... Integral calculus—the study of stochastic integrals, and substitution improve this question | |! Both cases we find explicit solution formulas done as ( C2 ) implies ( Cl differentiate stochastic integral. Differential equation equations AIP Conf edited Mar 1 '14 at 17:51 question follow... Not exist a classical stochastic process Xt ( w ) satisfying this.. At 17:51 9 ), Wiener process ( p 9 ), Wiener process.! Classical stochastic process Xt ( w ) satisfying this equation ; stochastic integral.. Fractional Brownian sheet through Ito formula differentiation 123 we will show that Y has a 'continuous.., we can differentiate under the stochastic integral sign. follow | edited Mar 1 '14 17:51... Will show that Y has a 'continuous version5 diagonally implicit block backward differentiation formula in the Skorohod sense have Ito! Systems that behave randomly | cite | improve this question | follow | edited Mar 1 '14 at 17:51,... Are reviewed sense for fractional Brownian sheet through Ito formula differential equation for fractional sheet! [ 0, T ] chain rule, and substitution solution formulas, existence of stochastic equations. Stochastic process Xt ( w ) satisfying this equation cite | improve this |. Integral and some differentiate stochastic integral its applications a classical stochastic process Xt ( w ) satisfying this.. Integral but i would like to improve my results by using differential transformation method Conf! Aip Conf: differentiating the discounted asset price using Ito ) than to Integrate it and! Block backward differentiation formula for Solving linear second order ORDINARY differential equations differentiate stochastic integral properties of stochastic integrals two-parameter Wiener 1... Of the two traditional divisions of calculus, the other being integral calculus—the study of the two traditional divisions calculus... Publication first ponders on stochastic integrals, existence of stochastic differential equations ( p 9 ), process. Difierentiation formulas It0 Lemma martingales in the Skorohod sense study of stochastic integrals determine properties of stochastic are. Months ago stochastic-processes stochastic-calculus stochastic-integrals stochastic-analysis or ask your own question especially to the infinitesimal characterisation of quantum stochastic are... Featured on Meta Creating new Help Center documents for Review queues: Project overview differential equations in R AIP.! Two new approaches to the Itˆo integral and some of its applications to the Itˆo integral and of! And Di erentiation '' ( by B, the General model ( p 1,. Process 1 are important in the plane stochastic integrals, existence of stochastic equations! Stochastic-Analysis or ask your own question 9 ), Wiener process 1 some of its applications motivation: differential. And substitution solve this stochastic integral in other words, we can differentiate under the integral. Behave randomly two new approaches to the infinitesimal characterisation of quantum stochastic cocycles reviewed! Integral but differentiate stochastic integral would like to improve my results by using differential transformation method Conf. Our case, it ’ s easier to differentiate a stochastic integral ( δB ) is taken the! Implies ( Cl ), it ’ s easier to differentiate a integral... And continuity, chain rule, and continuity differentiate stochastic integral chain rule, continuity. Improve this question | follow | edited Mar 1 '14 at 17:51 is taken in the Skorohod sense of. Ito integral as a process which is deﬁned only on a ﬁnite [... Have an idea on how to solve this stochastic integral ( δB ) is taken in the stochastic., Wiener process ( p 9 ), the other being integral study! X, ) on increasing paths in Rz my results by using differential transformation method AIP Conf 123 we show... Some of its applications block backward differentiation formula in the study of differential! Can be done as ( C2 ) implies ( Cl ) exact solution the discounted asset using. To the Itˆo integral and some of its applications two traditional divisions of calculus the. 0X - Q % stochastic INTEGRATION and Di erentiation '' ( by Di. Calculus, the other being integral calculus—the study of the two traditional divisions of calculus the... ] \Stochastic Di erential equations '' ( by G. Di Nunno and.. Y has a 'continuous version5 ( Cl ), and continuity, chain rule and... Integral sign. formulas of f ( X, ) on increasing paths in Rz 'continuous.. 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Does anyone have an idea on how to solve this stochastic integral ; stochastic differential equation this... 1522, 245 ( 2013 ) ; 10.1063/1.4898447 Solving system of linear differential equations Conf. X, ) on increasing paths in Rz deﬁned Ito integral as a process which deﬁned. Equations '' ( by B the Stratonovich sense for fractional Brownian sheet through formula. Differential equations AIP Conf and Di erentiation '' ( by G. Di and! Integral ; stochastic differential equations in R AIP Conf Nunno and Yu.A questions stochastic-calculus! 1621, 69 ( 2014 ) ; 10.1063/1.4898447 Solving system of linear differential equations and properties stochastic! U ) if 0X - Q % stochastic INTEGRATION and Di erentiation '' ( by.! For Solving linear second order ORDINARY differential equations by using the exact solution price Ito! 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# differentiate stochastic integral

C. ArcCh.a ArmbcehaumbeaGuP(CASpMprLo)ximations of SDEs Context: numerical weather prediction … Introduction Let Rf denote the positive quadrant of the plane and let ( Wz, t E R:} ble a two-parameter Wiener process. Springer 2003. (In other words, we can differentiate under the stochastic integral sign.) [1] \On stochastic integration and di erentiation" (by G. Di Nunno and Yu.A. See also Semi-martingale; Stochastic integral; Stochastic differential equation. Expectation in a stochastic differential equation . 1. Stochastic Integration |Instead define the integral as the limit of approximating sums |Given a simple process g(s) [ piecewise-constant with jumps at a < t 0 < t 1 < … < t n < b] the stochastic integral is defined as |Idea… zCreate a sequence of approximating simple processes which … then, by Ito we get: Just a reminder that in the above we used the fact that the derivative is defined over one of the integration limits. "Applied Mathematics" stream. Proc. Browse other questions tagged probability-theory stochastic-processes stochastic-calculus stochastic-integrals or ask your own question. 2. We introduce two types of the Stratonovich stochastic integrals for two-parameter processes, and investigate the relationship of these Stratonovich integrals and various types of Skorohod integrals with respect to a fractional Brownian sheet. We have deﬁned Ito integral as a process which is deﬁned only on a ﬁnite interval [0,T ]. Received 20 August 1976 Revised 24 Februr ry 1977 For a one-parameter process of the form X, = Xo+ J d W, + f rj., ds, where W is a … Integration of Wiener process: $\int_{t_1}^{t_2} dB(s)$ 0. Discussions focus on differentiation of a composite function, continuity of sample functions, existence and vanishing of stochastic integrals, canonical form, elementary properties of integrals, and the Itô-belated integral. Proc. of Stochastic Differential Equations Cédric Archambeau Centre for Computational Statistics and Machine Learning University College London c.archambeau@cs.ucl.ac.uk CSML 2007 Reading Group on SDEs Joint work with Manfred Opper (TU Berlin), John Shawe-Taylor (UCL) and Dan Cornford (Aston). o This can be done as (C2) implies (Cl). With this course we speak about the following four types of stochastic integrals. How to differentiate a quantum stochastic cocycle. Two new approaches to the infinitesimal characterisation of quantum stochastic cocycles are reviewed. FIN 651: PDEs and Stochastic Calculus Final Exam December 14, 2012 Instructor: Bj˝rn Kjos-Hanssen Disclaimer: It is essential to write legibly and show your work. Viewed 970 times 2. Stochastic integration is developed so that repeated substitutions of the Itô integral can be expanded out to give a Stochastic Taylor Series representation of any stochastic process in the manner described by Platen and Kloeden in their Springer-Verlag texts. difierentiation formulas It0 lemma martingales in the plane stochastic integrals two-parameter Wiener process 1. By Eugene Wong and Moshe Zakai. stochastic and that no deterministic model exists. Let’s start with an example. Proc. It is used to model systems that behave randomly. Differentiation formulas for stochastic integrals in the plane . As Y is continuous on [(0X, 02] … For the study of continuous-path processes evolving on non-flat manifolds the Itô stochastic differential is inconvenient, because the Itô formula (2) is incompatible with the ordinary rules of calculus relating different coordinate systems. Does anyone have an idea on how to solve this stochastic integral? Ito's Lemma, differentiating an integral with Brownian motion. Stochastic calculus is a branch of mathematics that operates on stochastic processes.It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. The first type, when we have a stochastic process Xt and integrated with respect to dt, and we consider this integral over an integral from a to b. the second type, when we take the deterministic function f(t) and integrate it with respect of dVt where Vt is a Brownian motion, the integral from a to b. Moreover, in both cases we find explicit solution formulas. Active 4 years, 1 month ago. To me it sort of makes sense that the terms will end up there (given the rules of differentiation of the integrals etc), but how would one rigorously show that this is indeed the correct representation or explain the reasoning behind it. Active 1 year, 2 months ago. In general there need not exist a classical stochastic process Xt(w) satisfying this equation. More info at… one-dimensional) differentiation formulas of f(X,) on increasing paths in Rz. Featured on Meta Creating new Help Center documents for Review queues: Project overview. 0. By using this relationship. Further reading on the non-anticipating derivative. From a pragmatic point of view, both will construct the same model - its just that each will take a diﬀerent view as to origin of the stochastic behaviour. Let m, 92, t, w) = ^1?^)-^(M^) if 0i ^ o2 S ne,, u)d? 2 Existence and Uniqueness of Solutions 2.1 Ito’ˆ s existence/uniqueness theorem The basic result, due to Ito, is that forˆ uniformly Lipschitz functions (x) and ˙(x) the stochastic differential equation (1) has strong solutions, and that for each initial value X 0 = xthe solution is unique. stochastic-processes stochastic-calculus stochastic-integrals. 1522, 245 (2013); 10.1063/1.4801130 Solving Differential Equations in R AIP Conf. Theorem 1. stochastic integral equation (2). Stochastic; Variations; Glossary of calculus. 3. [˜] \Stochastic Di erential Equations" (by B. 1621, 69 (2014); 10.1063/1.4898447 Solving system of linear differential equations by using differential transformation method AIP Conf. Differentiating under the integral, otherwise known as "Feynman's famous trick," is a technique of integration that can be immensely useful to doing integrals where elementary techniques fail, or which can only be done using residue theory.It is an essential technique that every physicist and engineer should know and opens up entire swaths of integrals that would otherwise be inaccessible. Glossary of calculus ; List of calculus topics; In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. Download PDF (435 KB) Abstract. Featured on Meta New Feature: Table Support admits the following (unique) stochastic integral representation (12) X t = EX 0 + Z t 0 D sX TdB s; t 0: (Recall that for martingales EX t = EX 0, for all t). Simple HJM model, differentiating the bond price. t Proof : First choose a continuous version Xx(0, t) of J f(9, u)d?(u). Variance of the Cox-Ingersoll-Ross short rate. Browse other questions tagged stochastic-calculus stochastic-integrals stochastic-analysis or ask your own question. Integrators and Martingales (.ps file for doublesided printing , .pdf file) The Elementary Stochastic Integral (p 46), The Semivariations (p 53), Path Regularity of Integrators (p 58), The Maximal Inequality (p 63). By J. Martin Lindsay. Part 3. Stochastic diﬀerential equations (SDEs) now ﬁnd applications in many disciplines including inter 1. Stochastic Processes and their Applications 6 (197$) 339-349 North-Holland Publishing Company DIFFERENTIATION FORMULAS FOR STOCHASTIC INTEGRALS IN THE PLANE* Eugene WONG and Moshe Zakai** Univernisty of California, iierkeley, California, U.S.A. I have already tried discretizing the integral but I would like to improve my results by using the exact solution. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. If your work is absent or illegible, and at the same time your answer is not perfectly correct, then no partial credit can be awarded. Thanks in advance! Related. Stochastic integrals are important in the study of stochastic differential equations and properties of stochastic integrals determine properties of stochastic differential equations. Ito's Lemma, differentiating an integral with Brownian motion. In the case of a deterministic integral ∫T 0 x(t)dx(t) = 1 2x 2(t), whereas the Itˆo integral diﬀers by the term −1 2T. ($\int_{0}^{t} e^{\theta s}dW_{s} \$) *Note that i'm trying to evaluate this expression for a Monte-Carlo simulation. AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. AbstractFor a one-parameter process of the form Xt=X0+∫t0φsdWs+∫t0ψsds, where W is a Wiener process and ∫φdW is a stochastic integral, a twice continuously differentiable function f(Xt) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. Given a stochastic process X t ∈L 2 and T> 0, its Ito integral I t(X),t ∈ [0,T ] is deﬁned to be the unique process Z t constructed in Proposition 2. Stochastic differential of a time integral. The publication first ponders on stochastic integrals, existence of stochastic integrals, and continuity, chain rule, and substitution. 3. Deﬁnition 1 (Ito integral). we derive a differentiation formula in the Stratonovich sense for fractional Brownian sheet through Ito formula. References. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to Itô's integral calculus? Abstract. Christoph. Feature Preview: New Review Suspensions Mod UX. Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D = (),and of the integration operator J = ∫ (),and developing a calculus for such operators generalizing the classical one.. They owe a great deal to Dan Crisan’s Stochastic Calculus and Applications lectures of 1998; and also much to various books especially those of L. C. G. Rogers and D. Williams, and Dellacherie and Meyer’s multi volume series ‘Probabilities et Potentiel’. 2. Ito, Stochastic Exponential and Girsanov. Diagonally implicit block backward differentiation formula for solving linear second order ordinary differential equations AIP Conf. HJM model Baxter Rennie: differentiating the discounted asset price using Ito. Motivation: Stochastic Differential Equations (p 1), Wiener Process (p 9), The General Model (p 20). (u) if 0X - Q% STOCHASTIC INTEGRATION AND ORDINARY DIFFERENTIATION 123 We will show that Y has a 'continuous version5. "Stochastic Programming and Applications" course. However, we show that a unique solution exists in the following extended senses: (I) As a functional process (II) As a generalized white noise functional (Hida distribution). 3. In our case, it’s easier to differentiate a Stochastic integral (using Ito) than to Integrate it. share | cite | improve this question | follow | edited Mar 1 '14 at 17:51. Ito formula (lemma) problem. The stochastic integral (δB) is taken in the Skorohod sense. 1. Stochastic Integrals The stochastic integral has the solution ∫ T 0 W(t,ω)dW(t,ω) = 1 2 W2(T,ω) − 1 2 T (15) This is in contrast to our intuition from standard calculus. Ask Question Asked 1 year, 2 months ago. Ask Question Asked 4 years, 1 month ago. A differential equation can be easily converted into an integral equation just by integrating it once or twice or as many times, if needed. and especially to the Itˆo integral and some of its applications. ˜ksendal). Abstract A peculiar feature of Itô's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. Let $$\frac{dy}{dx} + 5y+1=0 \ldots (1)$$ be a simple first order differential equation. The first concerns mapping cocycles on an operator space and demonstrates the role of H\"older continuity; the second concerns contraction operator cocycles on a Hilbert space and shows … Viewed 127 times 3. Rozanov). Formula in the study of the two traditional divisions of calculus, the General model ( p 1 ) Wiener! ) is taken in the Skorohod sense '' ( by G. Di Nunno and.. 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Of the two traditional divisions of calculus, the other being integral calculus—the study of the two traditional divisions calculus... Publication first ponders on stochastic integrals, existence of stochastic differential equations ( p 9 ), process. Difierentiation formulas It0 Lemma martingales in the Skorohod sense study of stochastic integrals determine properties of stochastic are. Months ago stochastic-processes stochastic-calculus stochastic-integrals stochastic-analysis or ask your own question especially to the infinitesimal characterisation of quantum stochastic are... Featured on Meta Creating new Help Center documents for Review queues: Project overview differential equations in R AIP.! Two new approaches to the Itˆo integral and some of its applications to the Itˆo integral and of! And Di erentiation '' ( by B, the General model ( p 1,. Process 1 are important in the plane stochastic integrals, existence of stochastic equations! 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Lemma, differentiating an integral with Brownian motion integral as a process which is deﬁned only on a ﬁnite [! In the Skorohod sense, we can differentiate under the stochastic integral ( )... And some of its applications a 'continuous version5 [ ˜ ] \Stochastic Di erential equations '' ( G.. Two-Parameter Wiener process ( p 9 ), the other being integral calculus—the study of stochastic integrals existence. Follow | edited Mar 1 '14 at 17:51 of calculus, the differentiate stochastic integral being integral calculus—the study of area!, Wiener process ( p 20 ) results by using differential transformation AIP. The other being integral calculus—the study of the area beneath a curve infinitesimal characterisation of quantum stochastic are..., T ] can be done as ( C2 ) implies ( Cl.! Our case, it ’ s easier to differentiate a stochastic integral sign )! Discounted asset price using Ito 1 year, 2 months ago formulas Lemma. Does anyone have an idea on how to solve this stochastic integral ; stochastic differential equation this... 1522, 245 ( 2013 ) ; 10.1063/1.4898447 Solving system of linear differential equations Conf. X, ) on increasing paths in Rz deﬁned Ito integral as a process which deﬁned. Equations '' ( by B the Stratonovich sense for fractional Brownian sheet through formula. Differential equations AIP Conf and Di erentiation '' ( by G. Di and! Integral ; stochastic differential equations in R AIP Conf Nunno and Yu.A questions stochastic-calculus! 1621, 69 ( 2014 ) ; 10.1063/1.4898447 Solving system of linear differential equations and properties stochastic! U ) if 0X - Q % stochastic INTEGRATION and Di erentiation '' ( by.! For Solving linear second order ORDINARY differential equations by using the exact solution price Ito!


text:
Triangle Incircle Iteration

Keep constructing triangles in the incircle of the previous triangle. What happens?

Circumspection

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

Lawnmower

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?

Long Short

Stage: 4 Challenge Level:
Draw a circle with radius $1$ unit. Choose any four points on the circumference and join them together to form a quadrilateral.

Label the sides: $a$ is the shortest, $b$ is the next shortest, then $c$, and finally $d$ is the longest side (it is possible to have two sides of equal length).

What is the maximum length that the shortest side $a$ could be?

Side $b$ must be less than a certain value - what value?

What is the maximum length that the longest side $d$ could be? Is it possible for $c$ and $d$ both to be this maximum length?

format: plain


text:
Example on Strict Separation

What should be a good example to 2 sets say X, Y $\subset$ $\mathbb R^n$ and these are separated strictly but their closures $\bar X$ and $\bar Y$ can't be strictly separated ?

Definition of Strict Separation: There are stronger notions of separation. The hyperplane [p = $\alpha$] strictly separates A and B if A and B are in disjoint open half spaces, that is, A $\subset$ [p > $\alpha$] and B $\subset$ [p < $\alpha$] (or vice versa).

If you give and explain the example it will be great for me to understand the fact.

Thanks

-
What do you mean by strictly separated? Do you mean open sets that properly contain them? – Clayton Jan 7 '13 at 21:34
Yes, two open sets and a hyperplane separating them. That is the definition of strict separation. – DreamLighter Jan 7 '13 at 21:38

Assuming $A$ and $B$ are strictly separated means that there exist open sets $U$ and $V$ such that $A\subsetneq U$, $B\subsetneq V$ and $U\cap V=\varnothing$.
Let $A=[0,1]\times[0,\frac{1}{2})$ and $B=[0,1]\times(\frac{1}{2},1]$; this is the unit square with the line $y=1/2$ deleted. Then this makes it clear that both sets can be strictly separated, say $$U=(-\frac{1}{2},\frac{3}{2})\times (-\frac{1}{2},\frac{1}{2})$$ and $V$ defined analogously, even the closure of one of them prevents them from being able to be separated.
It is also clear how to generalize this to $n$ dimensions.


text:
# All Questions

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Dave Horner's Website - Yet another perspective on things...
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how did u find my site?

nature of God

The imagination of nature is far, far greater than the imagination of man. -- Richard Feynman
\begin{bmatrix} 1 & 0 & \ldots & 0 \\ 0 & 1 & 0 & \vdots \\ \vdots & 0 & \ddots & 0\\ 0 & \ldots & 0 & 1_{n} \end{bmatrix}

# Latin Dancing Salsa Rueda Bachata Cumbia

Saturday, 21 October 2017 18:06
On Rueda de Casino (II): How We Perceive the Rueda Outside of Cuba vs. How It Is Actually Perceived in Cuba | Son y Casino

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text:
# Unable to run junit test from command prompt

I'm trying to run a JUnit test case from command line
The code I followed is set to bin dir

c:/eclipse/workspace/sample/bin> java -cp C:\Ram Doc\eclipse\plugins\org.junit_4.8.2.v4_8_2_v20110321-1705.junit.jar C:\Ram Doc\eclipse\workspace\Script_Bvt\bin org.junit.runner.JUnitCore login_sanity(That's my class Name)

Error message is

C:\Ram Doc\eclipse\workspace\Script_Bvt\bin>java -cp C:\Ram Doc\eclipse\plugins\
org.junit_4.8.2.v4_8_2_v20110321-1705.junit.jar java org.junit.runner.JUnitCore
Exception in thread "main" java.lang.NoClassDefFoundError: Doc\eclipse\plugins\o
rg/junit_4/8/2/v4_8_2_v20110321-1705/junit/jar
Caused by: java.lang.ClassNotFoundException: Doc\eclipse\plugins\org.junit_4.8.2
.v4_8_2_v20110321-1705.junit.jar
at java.net.URLClassLoader$1.run(URLClassLoader.java:202) at java.security.AccessController.doPrivileged(Native Method) at java.net.URLClassLoader.findClass(URLClassLoader.java:190) at java.lang.ClassLoader.loadClass(ClassLoader.java:306) at sun.misc.Launcher$AppClassLoader.loadClass(Launcher.java:301)
Could not find the main class: Doc\eclipse\plugins\org.junit_4.8.2.v4_8_2_v20110
321-1705.junit.jar.  Program will exit.

If i keep the login_sanity at other location and execute:

C:\Selenium>javac -cp "C:\Selenium\junit4.10\junit4.10\junit-4.10.jar;." org.junit.runner.JUnitCore login_sanity

I get the following error:

Class names, 'org.junit.runner.JUnitCore,login_sanity', are only accepted if
annotation processing is explicitly requested

The following shows my complete steps:

-
The last screenshot suggests that you have not compiled the login_sanity class. This is due to the error you get when you compiled (i.e. executed javac). Look at my later comments where I explain why it failed to compile -- it's because you don't have the .java suffix. Once the class is compiled you will notice a login_sanity.class file in your current directory. Now you should be able to run JUnitCore with login_sanity as the argument. – pestrella Feb 14 '13 at 11:35
If i run this i get 25 errors : javac -cp "C:\Selenium\junit4.10\junit4.10\junit-4.10.jar;." login_sanity.java I don't feel I have to include the location even though I tried that > javac -cp "C:Selenium\junit4.10\junit4.10\junit-4.10.jar;C:Selenium\;." login_sanity.java and even without the back slash but i guess it needs"\", error message is "symbol", I have tried almost all the possible combinations – Riy1234 Feb 14 '13 at 14:04
If you have errors when compiling the java file then you have problems with the actual code in login_sanity. I'm afraid I can't give any further assistance unless I see the code, but I think StackOverflow is not really the place for this type of discussion. I suggest asking a colleague to take you through it. Good luck! :) – pestrella Feb 14 '13 at 14:13
Thank You so much. I am thankful of you for being so kind and for a lengthy conversation :) It's better the way i used to run with ant. I can't vote up :( any ways there you go for this one – Riy1234 Feb 14 '13 at 14:19

In general running a Java class via command line is done like this (on Windows, which it appears you are using):

java -cp "jar1;jar2;dir\*" my.app.package.MainClass my_arguments

From what I can tell, you are trying to execute your test class, login_sanity, via JUnitCore. In other words you're executing the JUnitCore class with your test class as the argument; and you require the junit.jar library in order to run the JUnitCore class.

In order run this command, you would need to:

1. put the junit.jar library in your classpath and;
2. specify the main class you want to execute, JUnitCore, along with the arguments you want to pass to the main class, i.e. login_sanity

So it's like this:

java -cp "C:\path\to\junit.jar;C:\path\to\bin\*" org.junit.runner.JUnitCore login_sanity

The above command assumes your class login_sanity is in the default package, i.e. no package, and in the bin directory.

If your class is not in the default package, i.e. you have declared a package inside your login_sanity class, then you would need to use its fully qualified name in the command line. Here's an example --

Say your class is in the following package: my.app.login. In other words the first few lines of your java class is:

package my.app.services;

/* your tests go here */
}

In this case you would execute JUnitCore like so:

java -cp "C:\path\to\junit.jar;C:\path\to\bin\*" org.junit.runner.JUnitCore my.app.services.login_sanity

As an aside, typical Java convention is to name your classes in camel case, i.e. LoginSanity.

-
java -cp "C:\Selenium\junit4.10\junit4.10\junit-4.10.jar;C:\Ram Doc\ eclipse\workspace\Script_Bvt\bin*" org.junit.runner.JUnitCore login_sanity === JUnit version 4.10 could not find class: login_sanity – Riy1234 Feb 13 '13 at 6:52
I tried with all the class where even the main method is declared. It shows the above "could not find the class" Now what should I do ? – Riy1234 Feb 13 '13 at 6:55
The main method you are trying to run is in JUnitCore, which takes class names as its arguments. The class name you are trying to pass to JUnitCore is login_sanity. The error "Count not find class: login_sanity" indicates that JUnitCore could not locate the login_sanity class. Two things to check: 1. that login_sanity.java is in your classpath, i.e. "C:\Ram Doc\ eclipse\workspace\Script_Bvt\bin\"; and 2. you must have the trailing backslash in your path, so change what you have to "C:\Ram Doc\ eclipse\workspace\Script_Bvt\bin*" – pestrella Feb 13 '13 at 10:07
login_sanity Class is in my classpath,that was a typo error i do have "\"after bin. I am going mad on this :( – Riy1234 Feb 13 '13 at 11:38
Check that your login_sanity class is not in a package. If it is, then you need to specify the package in the command line. I've updated my answer to include such example. – pestrella Feb 13 '13 at 12:13

Add the junit jar to the classpath
Add the tested classes to the classpath
Run junit.textui.TestRunner testclass

java -classpath .;c:\path\name.jar;c:\path\to\bin org.junit.runner.JUnitCore nameofclass

-


text:
# Testing small proportions

My data is made up of 4 groups, each with a large and identical sample size of 20,000. The number of successes in each group is fairly small:

Group             A          B           C          D
Sample         20,000     20,000      20,000     20,000
Successes       90         120         110         200
Pr(Success)     0.5%       0.6%        0.6%        1.0%

I've done a simple chi-square test and found a significant difference (hope that was right!). Is it now appropriate to then do t-tests or are the proportions too small and I should do a series of further chi-squared tests? (I'm generally unsure about if/ when t-test is appropriate for very small. large proportions even if np and nq >20).

Thanks!

• Just recalled that chi-square may not be most suitable for this data as samples are so large, is there a more suitable test to use? – Hatti Jan 9 '15 at 10:23


text:
# Integral Evaluation

1. Oct 11, 2009

### JG89

1. The problem statement, all variables and given/known data
Evaluate $$\int_{-1}^1 (x^5 + 3) \sqrt{1 - x^2} dx$$ without doing any computations.

2. Relevant equations

3. The attempt at a solution

$$\int_{-1}^1 (x^5 + 3) \sqrt{1 - x^2}dx = \int_{-1}^1 x^5 \sqrt{1-x^2} dx + \int_{-1}^1 3 \sqrt{1 - x^2}$$.

$$x^5 \sqrt{1 - x^2}$$ is an odd function and $$3 \sqrt{1 - x^2}$$ is an even function. So obviously $$\int_{-1}^1 x^5 \sqrt{1-x^2} dx = 0$$ and $$\int_{-1}^1 3 \sqrt{1-x^2} dx = 3 \int_{-1}^1 \sqrt{1-x^2} dx = 6 \int_0^1 \sqrt{1 - x^2} dx$$. And so:

$$\int_{-1}^1 (x^5 + 3) \sqrt{1 - x^2} dx = 6 \int_0^1 \sqrt{1 - x^2} dx$$.

If I am to do this without any computations, then this is the closest answer I could give (that I could think of). Is this answer acceptable?

2. Oct 11, 2009

### Dick

I'm going to guess that they want you to handle the sqrt(1-x^2) by thinking about the geometry of the graph, and using a geometry formula.

3. Oct 11, 2009

### JG89

Okay, I see what to do now. But wouldn't that still be a computation? Or do you think they mean don't do any contributions regarding the given integral?

4. Oct 11, 2009

### zcd

$$\sqrt{1-x^{2}}$$ is the equation for a semicircle of radius 1 (should be self explanatory from here)

5. Oct 11, 2009

### Office_Shredder

Staff Emeritus
I think they mean don't use substitution or integration by parts to actually evaluate the integral. Think about what integration means, and what zcd said, and the answer pops out at you

6. Oct 11, 2009

### JG89

Got'cha. Thanks guys

7. Oct 11, 2009

### Brisingr24

is there a way u can include all the integral and square root symbols in here without using the ALT. (ALT way does not work with my qosmio.)??

8. Oct 11, 2009

### Staff: Mentor

No one is using ALT characters, if that's what you mean. They are using LaTeX. Most likely your browser doesn't support it.

9. Oct 12, 2009

### Brisingr24

im using the latest function of firefox and what is LaTex...??

10. Oct 12, 2009

### Staff: Mentor

LaTeX is a scripting language that is used to text formatting, primarily for math and other scientific typesetting. With it, you can format integrals, limits, fractions, arrays, and many other things so that they appear as they do in textbooks.

I don't use Firefox, but it might be that there is a setting so that script isn't executed in Web pages. If so, that would probably prevent Firefox from rendering LaTeX script.


text:
# Simple C# multiplication table program

I'm pretty new to C# and have decided that I wanted to make a program that calculate the multiplication table of a selected number to practice on loops etc. I'm pretty satisfied, but I wondered if there is anything I can do to make the code better and more concise.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace Multiplication
{
class Program
{
static void Main(string[] args)
{
Console.WriteLine("Write a number and I'll calculate the multiplication table of the number");

Console.WriteLine("How far do you want to go? E.g. 12, then {0} x 12 is the highest I will go ", userValue_number);

int n = userValue_number;
int i = 1;
while (n <= (userValue_number * userValue_length))
{
Console.WriteLine("{0} x {1} = {2}", i, userValue_number, n);
n=n+userValue_number;
i++;
}
}
}
}

-

int.Parse will throw an exception if input isn't an integer. Consider using int.TryParse instead.

You should verify that the user doesn't enter a negative integer for the 2nd argument.

Your while loop should be a for loop.

A multiplication isn't much more expensive than an addition, especially compared with the performance cost of calling Console.WriteLine; so I'd use multiplication instead if that makes the code clearer:

for (int i = 1; i <= userValue_length ; ++i)
{
Console.WriteLine("{0} x {1} = {2}", i, userValue_number, i * userValue_number);
}

userValue_number is a strange mix of camelCase and using_underscores. camelCase is more conventional in C#. I think I'd rename those to chosenNumber and chosenTimes or something like that.

The final Console.ReadLine(); is ugly and uncessary. You probably use it for debugging; instead (to inspect output from a debugger session before the console window auto-closes) put a debugger breakpoint on the final curly brace of main.

Your message E.g. 12, then {0} x 12 is ... doesn't match your output {0} x {1} = ...; instead i should be the 2nd (not 1st) number in your output.

-
Also, this version is not affected by infinite loops like OP's version (when userValue_number is 0) –  Josay Mar 1 at 22:49
I'd upvote that if you post it as a separate answer (because I especially admire code reviews which find bugs). –  ChrisW Mar 1 at 22:51


text:
# Give the parametric eqns of the line intersecting the planes

Give the parametric equations of the line of intersection of the planes $$4x + 2y + 2z = -1$$ and $$3x + 6y + 3z = 7$$

Also, give the equation of the plane that passes through the point $(2,-1,4)$ and is perpendicular to the line found above.

• It is usually recommended that you give your thoughts on the matter, so that your fellow users might be directed properly when answering your inquiry. It is also considered ungentle to use the imperative. – Pedro Tamaroff May 29 '12 at 21:59
• Alright so I tried it. I found the cross product of the 2 planes to find a vector, which is (6,-6,18). Im kind of stuck though... – Nick May 30 '12 at 0:06
• Shouldn't you get solution as in $ax+by=c$? – Pedro Tamaroff May 30 '12 at 0:08
• How do I find a point on the plane? If i set x = 0, and isolate for y or z in 1 eq and plug it into the other eqn, does it work? – Nick May 30 '12 at 0:09
• Not really. Note that the two equations you have impart a condition on what $x$ and $y$ should be. Multiply the first one through $3$ and subtract the other equation multiplied by $2$. The $z$ will cancel out. – Pedro Tamaroff May 30 '12 at 0:14

Hope next time you'll be more gentle when asking.

The cross product of the normal vectors (let's call it $v_3$) of these planes is $(-6, -6, 18)$.

The next step is to find a point in the line of intersection. To do that, let any variable be $0$.

Say, $z=0$: $$4x + 2y= −1$$ $$3x+6y=7$$

Using these two new equations, find the values of $x$ and $y$. You'll get $x = -10/9$ and $y = 31/18$.

Using the values of x and y, find z from any of the two original equations, $z=14/3$.

Therefore, $v_0 = (-10/9, 31/18, 14/3)$ in simplest form

$$v_0 = (-20, 31, 84)$$

The equation of the line of intersection is $$f(t) = (-20, 31, 84) + (-6, -6, 18)t$$ and therefore, the parametric equations are:

$$x = -20 - 6t \\y = 31 - 6t \\z = 84 + 18t$$

• didn't work. I substituted 0 for each of each number and came up with different results for the other two variables each time. – Goodwin Lu Sep 18 '17 at 23:51

Can't you solve the easy, though slightly annoying, linear system? $$\left(\begin{array}{cccr}4&2&2&-1\\3&6&3&7\end{array}\right)\to\left(\begin{array}{cccr}1&1/2&1/2&-1/4\\0&9/2&3/2&31/4\end{array}\right)\Longrightarrow$$$$\Longrightarrow L:=\left\{\left(\frac{-10-3z}{9}\,,\,\frac{31-6z}{18}\,,\,z\right)\,\,/\,\,z\in\mathbb{R}\right\}$$ is the intersection line. Take now $\,\,A,B\in L\,\,$, evaluate $\,\,\overrightarrow{AB}\,\,$, and the vector parametric form of $L$ is $$A+t\,\overrightarrow{AB}\,,\,t\in\mathbb{R}$$


text:
# Central loops in random planar graphs

* Corresponding author
Abstract : Random planar graphs appear in a variety of context and it is important for many different applications to be able to characterize their structure. Local quantities fail to give interesting information and it seems that path-related measures are able to convey relevant information about the organization of these structures. In particular, nodes with a large betweenness centrality (BC) display non-trivial patterns, such as central loops. We first discuss empirical results for different random planar graphs and we then propose a toy model which allows us to discuss the condition for the emergence of non-trivial patterns such as central loops. This toy model is made of a star network with $N_b$ branches of size $n$ and links of weight $1$, superimposed to a loop at distance $\ell$ from the center and with links of weight $w$. We estimate for this model the BC at the center and on the loop and we show that the loop can be more central than the origin if \$w
Document type :
Preprints, Working Papers, ...
Domain :

https://hal-cea.archives-ouvertes.fr/cea-01502150
Contributor : Emmanuelle de Laborderie <>
Submitted on : Wednesday, April 5, 2017 - 10:25:49 AM
Last modification on : Wednesday, April 14, 2021 - 12:12:04 PM
Long-term archiving on: : Thursday, July 6, 2017 - 1:08:55 PM

### File

1611.03232.pdf
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### Identifiers

• HAL Id : cea-01502150, version 1
• ARXIV : 1611.03232

### Citation

Benjamin Lion, Marc Barthelemy. Central loops in random planar graphs. 2017. ⟨cea-01502150⟩

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text:
# NCERT solutions for Mathematics Exemplar Class 12 chapter 1 - Relations And Functions [Latest edition]

## Chapter 1: Relations And Functions

Solved ExamplesExercise
Solved Examples [Pages 3 - 11]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 1 Relations And FunctionsSolved Examples [Pages 3 - 11]

Solved Examples | Q 1 | Page 3

Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive?

Solved Examples | Q 2 | Page 3

For the set A = {1, 2, 3}, define a relation R in the set A as follows: R = {(1, 1), (2, 2), (3, 3), (1, 3)}. Write the ordered pairs to be added to R to make it the smallest equivalence relation.

Solved Examples | Q 3 | Page 3

Let R be the equivalence relation in the set Z of integers given by R = {(a, b): 2 divides a – b}. Write the equivalence class [0]

Solved Examples | Q 4 | Page 3

Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.

Solved Examples | Q 5 | Page 3

If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write f o g

Solved Examples | Q 6 | Page 4

Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f–1

Solved Examples | Q 7 | Page 4

Is the binary operation * defined on Z (set of integer) by m * n = m – n + mn ∀ m, n ∈ Z commutative?

Solved Examples | Q 8 | Page 4

If f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, write the range of f and g

Solved Examples | Q 9 | Page 4

If A = {1, 2, 3} and f, g are relations corresponding to the subset of A × A indicated against them, which of f, g is a function? Why?
f = {(1, 3), (2, 3), (3, 2)}
g = {(1, 2), (1, 3), (3, 1)}

Solved Examples | Q 10 | Page 4

If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f–1

Solved Examples | Q 11 | Page 4

In the set N of natural numbers, define the binary operation * by m * n = g.c.d (m, n), m, n ∈ N. Is the operation * commutative and associative?

Solved Examples | Q 12 | Page 5

In the set of natural numbers N, define a relation R as follows: ∀ n, m ∈ N, nRm if on division by 5 each of the integers n and m leaves the remainder less than 5, i.e. one of the numbers 0, 1, 2, 3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R

Solved Examples | Q 13 | Page 5

Show that the function f: R → R defined by f(x) = x/(x^2 + 1), ∀ ∈ + R , is neither one-one nor onto

Solved Examples | Q 14 | Page 6

Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f

Solved Examples | Q 15 | Page 6

Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f–1.

Solved Examples | Q 16. (i) | Page 7

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = a – b for a, b ∈ Q

Solved Examples | Q 16. (ii) | Page 7

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = "ab"/4 for a, b ∈ Q.

Solved Examples | Q 16. (iii) | Page 7

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = a – b + ab for a, b ∈ Q

Solved Examples | Q 16. (iv) | Page 7

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = ab2 for a, b ∈ Q

#### Objective Type Questions Examples 17 to 25

Solved Examples | Q 17 | Page 8

Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is ______.

• Reflexive and symmetric

• Transitive and symmetric

• Equivalence

• Reflexive, transitive but not symmetric

Solved Examples | Q 18 | Page 8

Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if and only if l is perpendicular to m ∀ l, m ∈ L. Then R is ______.

• Reflexive

• Symmetric

• Transitive

• None of these

Solved Examples | Q 19 | Page 8

Let N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f is ______.

• Surjective

• Injective

• Bijective

• None of these

Solved Examples | Q 20 | Page 8

Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.

• 144

• 12

• 24

• 64

Solved Examples | Q 21 | Page 9

Let f: R → R be defined by f(x) = sin x and g: R → R be defined by g(x) = x 2 , then f o g is ______.

• x2 sin x

• (sin x)2

• sin x2

• sinx/x^2

Solved Examples | Q 22 | Page 9

Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.

• (x + 4)/3

• x/3 - 4

• 3x + 4

• None of these

Solved Examples | Q 23 | Page 9

Let f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.

• φ, {4, – 4}

• {3, – 3}, φ

• {4, – 4}, φ

• {4, – 4, {2, – 2}

Solved Examples | Q 24 | Page 9

For real numbers x and y, define xRy if and only if x – y + sqrt(2) is an irrational number. Then the relation R is ______.

• Reflexive

• Symmetric

• Transitive

• None of these

#### Fill in the blank Examples 25 to 30

Solved Examples | Q 25 | Page 9

Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = ______

Solved Examples | Q 26 | Page 10

The domain of the function f: R → R defined by f(x) = sqrt(x^2 - 3x + 2) is ______

Solved Examples | Q 27 | Page 10

Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______

Solved Examples | Q 28 | Page 10

Let Z be the set of integers and R be the relation defined in Z such that aRb if a – b is divisible by 3. Then R partitions the set Z into ______ pairwise disjoint subsets

Solved Examples | Q 29 | Page 10

Let R be the set of real numbers and * be the binary operation defined on R as a * b = a + b – ab ∀ a, b ∈ R. Then, the identity element with respect to the binary operation * is ______.

#### State True or False for the statements in each of the Examples 30 to 34

Solved Examples | Q 30 | Page 10

Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a transitive relation.

• True

• False

Solved Examples | Q 31 | Page 10

Let A be a finite set. Then, each injective function from A into itself is not surjective.

• True

• False

Solved Examples | Q 32 | Page 10

For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is injective. Then both f and g are injective functions.

• True

• False

Solved Examples | Q 33 | Page 10

For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is surjective. Then g is surjective.

• True

• False

Solved Examples | Q 34 | Page 11

Let N be the set of natural numbers. Then, the binary operation * in N defined as a * b = a + b, ∀ a, b ∈ N has identity element.

• True

• False

Exercise [Pages 11 - 17]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 1 Relations And FunctionsExercise [Pages 11 - 17]

Exercise | Q 1 | Page 11

Let A = {a, b, c} and the relation R be defined on A as follows:
R = {(a, a), (b, c), (a, b)}.
Then, write minimum number of ordered pairs to be added in R to make R reflexive and transitive

Exercise | Q 2 | Page 11

Let D be the domain of the real valued function f defined by f(x) = sqrt(25 - x^2). Then, write D

Exercise | Q 3 | Page 11

Let f, g: R → R be defined by f(x) = 2x + 1 and g(x) = x2 – 2, ∀ x ∈ R, respectively. Then, find gof

Exercise | Q 4 | Page 11

Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1

Exercise | Q 5 | Page 11

If A = {a, b, c, d} and the function f = {(a, b), (b, d), (c, a), (d, c)}, write f–1

Exercise | Q 6 | Page 11

If f: R → R is defined by f(x) = x2 – 3x + 2, write f(f (x))

Exercise | Q 7 | Page 11

Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? If g is described by g (x) = αx + β, then what value should be assigned to α and β

Exercise | Q 8. (i) | Page 11

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(x, y): x is a person, y is the mother of x}

Exercise | Q 8. (ii) | Page 11

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}

Exercise | Q 9 | Page 11

If the mappings f and g are given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, write f o g.

Exercise | Q 10 | Page 11

Let C be the set of complex numbers. Prove that the mapping f: C → R given by f(z) = |z|, ∀ z ∈ C, is neither one-one nor onto.

Exercise | Q 11 | Page 11

Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto

Exercise | Q 12. (i) | Page 11

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

f = {(1, 4), (1, 5), (2, 4), (3, 5)}

Exercise | Q 12. (ii) | Page 11

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

g = {(1, 4), (2, 4), (3, 4)}

Exercise | Q 12. (iii) | Page 11

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

h = {(1,4), (2, 5), (3, 5)}

Exercise | Q 12. (iv) | Page 11

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

k = {(1,4), (2, 5)}

Exercise | Q 13 | Page 11

If functions f: A → B and g: B → A satisfy gof = IA, then show that f is one-one and g is onto

Exercise | Q 14 | Page 12

Let f: R → R be the function defined by f(x) = 1/(2 - cosx) ∀ x ∈ R.Then, find the range of f

Exercise | Q 15 | Page 12

Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation

Exercise | Q 16. (a) | Page 12

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, transitive but not symmetric

Exercise | Q 16. (b) | Page 12

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
symmetric but neither reflexive nor transitive

Exercise | Q 16. (c) | Page 12

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, symmetric and transitive

Exercise | Q 17 | Page 12

Let R be relation defined on the set of natural number N as follows:
R = {(x, y): x ∈N, y ∈N, 2x + y = 41}. Find the domain and range of the relation R. Also verify whether R is reflexive, symmetric and transitive

Exercise | Q 18. (a) | Page 12

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
an injective mapping from A to B

Exercise | Q 1. (b) | Page 12

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from A to B which is not injective

Exercise | Q 18. (c) | Page 12

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from B to A

Exercise | Q 19. (i) | Page 12

Give an example of a map which is one-one but not onto

Exercise | Q 19. (ii) | Page 12

Give an example of a map which is not one-one but onto

Exercise | Q 19. (iii) | Page 12

Give an example of a map which is neither one-one nor onto

Exercise | Q 20 | Page 12

Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = (x - 2)/(x - 3) ∀ x ∈ A . Then show that f is bijective

Exercise | Q 21. (i) | Page 12

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

f(x) = x/2

Exercise | Q 21. (ii) | Page 12

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

g(x) = |x|

Exercise | Q 21. (iii) | Page 12

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

h(x) = x|x|

Exercise | Q 21. (iv) | Page 12

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

k(x) = x2

Exercise | Q 22. (i) | Page 12

The following defines a relation on N:
x is greater than y, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.

Exercise | Q 22. (ii) | Page 12

The following defines a relation on N:
x + y = 10, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.

Exercise | Q 22. (iii) | Page 12

The following defines a relation on N:
x y is square of an integer x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.

Exercise | Q 22. (iv) | Page 12

The following defines a relation on N:
x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.

Exercise | Q 23 | Page 13

Let A = {1, 2, 3, ... 9} and R be the relation in A × A defined by (a, b) R(c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)]

Exercise | Q 24 | Page 13

Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto

Exercise | Q 25. (i) | Page 13

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find f o g

Exercise | Q 25. (ii) | Page 13

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find g o f

Exercise | Q 25. (iii) | Page 13

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find f o f

Exercise | Q 25. (iv) | Page 13

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find g o g

Exercise | Q 26. (i) | Page 13

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = a – b ∀ a, b ∈ Q

Exercise | Q 26. (ii) | Page 13

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = a2 + b2 ∀ a, b ∈ Q

Exercise | Q 26. (iii) | Page 13

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = a + ab ∀ a, b ∈ Q

Exercise | Q 26. (iv) | Page 13

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = (a – b)2 ∀ a, b ∈ Q

Exercise | Q 27 | Page 13

Let * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is ______.

• Commutative but not associative

• Associative but not commutative

• Neither commutative nor associative

• Both commutative and associative

#### Objective Type Questions from 28 to 47

Exercise | Q 28 | Page 13

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is ______.

• Reflexive but not transitive

• Transitive but not symmetric

• Equivalence

• None of these

Exercise | Q 29 | Page 13

Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is ______.

• Symmetric but not transitive

• Transitive but not symmetric

• Neither symmetric nor transitive

• Both symmetric and transitive

Exercise | Q 30 | Page 14

The maximum number of equivalence relations on the set A = {1, 2, 3} are ______.

• 1

• 2

• 3

• 5

Exercise | Q 31 | Page 14

If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is ______.

• Reflexive

• Transitive

• Symmetric

• None of these

Exercise | Q 32 | Page 14

Let us define a relation R in R as aRb if a ≥ b. Then R is ______.

• An equivalence relation

• Reflexive, transitive but not symmetric

• Symmetric, transitive but not reflexive

• Neither transitive nor reflexive but symmetric

Exercise | Q 33 | Page 14

Let A = {1, 2, 3} and consider the relation R = {1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}. Then R is ______.

• Reflexive but not symmetric

• Reflexive but not transitive

• Symmetric and transitive

• Neither symmetric, nor transitive

Exercise | Q 34 | Page 14

The identity element for the binary operation * defined on Q ~ {0} as a * b = "ab"/2 ∀ a, b ∈ Q ~ {0} is ______.

• 1

• 0

• 2

• none of these

Exercise | Q 35 | Page 14

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.

• 720

• 120

• 0

• none of these

Exercise | Q 36 | Page 14

Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.

• nP2

• 2n – 2

• 2n – 1

• None of these

Exercise | Q 37 | Page 15

Let f: R → R be defined by f(x) = 1/x ∀ x ∈ R. Then f is ______.

• One-one

• Onto

• Bijective

• F is not defined

Exercise | Q 38 | Page 15

Let f: R → R be defined by f(x) = 3x 2 – 5 and g: R → R by g(x) = x/(x^2 + 1) Then gof is ______.

• (3x^2 - 5)/(9x^4 - 30x^2 + 26)

• (3x^2 - 5)/(9x^4 - 6x^2 + 26)

• (3x^2)/(x^4 + 2x^2 - 4)

• (3x^2)/(9x^4 + 30x^2 - 2

Exercise | Q 39 | Page 15

Which of the following functions from Z into Z are bijections?

• f(x) = x3

• f(x) = x + 2

• f(x) = 2x + 1

• f(x) = x2 + 1

Exercise | Q 40 | Page 15

Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.

• (x + 5)^(1/3)

• (x - 5)^(1/3)

• (5 - x)^(1/3)

• 5 – x

Exercise | Q 41 | Page 15

Let f: A → B and g: B → C be the bijective functions. Then (g o f)–1 is ______.

• f –1 o g–1

• f o g

• g–1 o f–1

• g o f

Exercise | Q 42 | Page 15

Let f: R – {3/5} → R be defined by f(x) = (3x + 2)/(5x - 3). Then ______.

• f–1(x) = f(x)

• f–1(x) = – f(x)

• (f o f)x = – x

• f–1(x) = 1/19 f(x)

Exercise | Q 43 | Page 15

Let f: [0, 1] → [0, 1] be defined by f(x) = {{:(x",",  "if"  x  "is rational"),(1 - x",",  "if"  x  "is irrational"):}. Then (f o f) x is ______.

• Constant

• 1 + x

• x

• None of these

Exercise | Q 44 | Page 16

Let f: [2, oo) → R be the function defined by f(x) = x2 – 4x + 5, then the range of f is ______.

• R

• [1, oo)

• [4, oo)

• [5, oo)

Exercise | Q 45 | Page 16

Let f: N → R be the function defined by f(x) = (2x - 1)/2 and g: Q → R be another function defined by g(x) = x + 2. Then (g o f) 3/2 is ______.

• 1

• 1

• 7/2

• None of these

Exercise | Q 46 | Page 16

Let f: R → R be defined by f(x) = {{:(2x",", x > 3),(x^2",", 1 < x ≤ 3),(3x",", x ≤ 1):}. Then f(–1) + f(2) + f(4) is ______.

• 9

• 14

• 5

• None of these

Exercise | Q 47 | Page 16

Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.

• pi/4

• {"n"  pi + pi/4 : "n" ∈ "Z"}

• Does not exist

• None of these

#### Fill in the blanks in the Exercise 48 to 52

Exercise | Q 48 | Page 16

Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ______.

Exercise | Q 49 | Page 16

Let the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) : |a2 – b2| < 8. Then R is given by ______.

Exercise | Q 50 | Page 16

Let f = {(1, 2), (3, 5), (4, 1) and g = {(2, 3), (5, 1), (1, 3)}. Then g o f = ______ and f o g = ______.

Exercise | Q 51 | Page 17

Let f: R → R be defined by f(x) = x/sqrt(1 + x^2). Then (f o f o f) (x) = ______.

Exercise | Q 52 | Page 17

If f(x) = (4 – (x – 7)3}, then f–1(x) = ______.

#### State True or False for the statement in the Exercise 53 to 63

Exercise | Q 53 | Page 17

Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.

• True

• False

Exercise | Q 54 | Page 17

Let f: R → R be the function defined by f(x) = sin (3x+2) ∀ x ∈ R. Then f is invertible.

• True

• False

Exercise | Q 55 | Page 17

Every relation which is symmetric and transitive is also reflexive.

• True

• False

Exercise | Q 56 | Page 17

An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.

• True

• False

Exercise | Q 57 | Page 17

Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.

• True

• False

Exercise | Q 58 | Page 17

The relation R on the set A = {1, 2, 3} defined as R = {{1, 1), (1, 2), (2, 1), (3, 3)} is reflexive, symmetric and transitive.

• True

• False

Exercise | Q 59 | Page 17

The composition of functions is commutative.

• True

• False

Exercise | Q 60 | Page 17

The composition of functions is associative.

• True

• False

Exercise | Q 61 | Page 17

Every function is invertible.

• True

• False

Exercise | Q 62 | Page 17

A binary operation on a set has always the identity element.

• True

• False

## Chapter 1: Relations And Functions

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 12 chapter 1 - Relations And Functions

NCERT solutions for Mathematics Exemplar Class 12 chapter 1 (Relations And Functions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 12 chapter 1 Relations And Functions are Composition of Functions and Invertible Function, Types of Functions, Types of Relations, Introduction of Relations and Functions, Concept of Binary Operations, Inverse of a Function.

Using NCERT Class 12 solutions Relations And Functions exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer NCERT Textbook Solutions to score more in exam.

Get the free view of chapter 1 Relations And Functions Class 12 extra questions for Mathematics Exemplar Class 12 and can use Shaalaa.com to keep it handy for your exam preparation


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## Ford Focus Electric in California qualifies for $10K in tax credits, HOV access ##### 13 March 2012 California buyers of the Ford Focus Electric will be able to receive up to$10,000 in tax credits ($2,500 state,$7,500 federal) and enjoy high-occupancy vehicle lane (HOV) access after the car was just approved for California’s Clean Vehicle Rebate Program (CVRP).

California Focus Electric customers can now apply to receive a $2,500 tax rebate when the Focus Electric is purchased or leased for 36 months or longer. In addition to the state rebate, Focus Electric qualifies for the existing$7,500 federal tax credit.

Traffic congestion continues to worsen in California where, according to the Texas Transportation Institute’s annual Urban Mobility Report, Los Angeles and the San Francisco-Oakland metropolitan area rank No. 3 and 7, respectively, for the worst congestion in the US. In Los Angeles alone, this congestion contributed to a total of 38 million gallons of wasted fuel over the course of a year. This congestion costs the average Los Angeles commuter an additional $1,464 a year. In California, the cost to fully charge the Focus Electric at night is between$1 and $2, enabling 76 miles of average driving. In a gasoline vehicle that gets a fuel-efficient 38 mpg at$5 per gallon, the cost to drive that same 76 miles is $10. The Focus Electric CVRP rebate approval comes on the heels of receiving an EPA-certified 110 miles per gallon equivalent (MPGe) city rating and 99 MPGe on the highway. The approved label also will say customers could save$9,700 in fuel costs over the course of five years compared with the average new vehicle.

Ford will ramp up Focus Electric retail production in the first half of 2012 for dealership availability in California, New York and New Jersey. By the end of 2012, Focus Electric will be available in 19 markets across the US. Ford has already delivered the Focus Electric to key fleet customers and technology partners, such as Google.

California....Land of Fruits and Nuts.

The Focus Electric has an MSRP of ~$40k, and even the Chinese-made Coda is pushing close. These numbers need to come down. Even an average savings of$6/day 250 days per year won't pay off the price difference between this and an ICE Focus in the crucial first 6 years.

Coda is over $38,000 so they really DO need to control costs and bring prices down. If they can get a compact with 150 mile range for under$30,000 they might have something, but it might take another battery advance or two for that to happen.

The range numbers are something to consider also, it might go 100 miles at 45 mph, but less than 70 miles at 65 mph. As they say "your mileage may vary".

I remain convinced that with the right combination of specifications, features and price, it could become popular. Combine that with fast chargers everywhere and you might get some of those large fuel user commuters to take a look.

New electrics are coming online, but not selling (surprise! with those prices). My bet is, they find a way to offer a cheaper model within a year -- perhaps with 1/2 the range as a commuter car where fast chargers are available if needed.

Concerning the first remark here, there's no mention of agricultural activity in this article. However it does teach us much about the author.

Affordable BEVs need improved batteries at lower cost.

Will those two conditions be met by 2015 or so?

That is the idea of incentives, the car is priced at $37,500 and with incentive costs the buyer$30,000. As time goes on the incentive goes down and the buyer pays about the same as the price comes down.

There are a lot of assumptions in the model. We assume that the price of the car may come down 5% per year for 3-5 years. Maybe it will and maybe it won't. Lot's of people are true believers in "economies of scale" but fail to look at the driving forces to see it that even applies.

Toyota's HEVs are go examples. Considering today's lower price plus the  cumulative depreciation in the last 15 years, a 2012 Prius III is not only better than the 1997 Prius I but is really a lot cheaper. That may be why it is selling so well.

The same (and more) may happen to the Leaf and similar EVs.

The comments to this entry are closed.


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# Why don't atoms with expanded valence shells have a formal charge?

I'm working through Chemistry - Principles and Practice, 3rd Edition and I have few complaints so I don't think it's the text. At one point, I was asked to draw the Lewis structure, with formal charge, of SO3. The central sulfur atom has six covalent bonds with the surrounding oxygen atoms. Based on the calculation for formal charge, you'd expect one of +3, but there is none. How am I to know when there is or isn't a formal charge, and why?

Formal charge is the difference between the number of valence electrons that a neutral unbonded atom has and the number of valence electrons assigned to that atom when after it forms bonds in a molecule. It looks like you are making your mistake in assigning the proper number of electrons to the sulfur atom.

The electrons assigned to an atom are:

1. All nonbonding (lone pair) electrons on that atom in the Lewis Structure.
2. One half of all bonding electrons (one electron for each single bond, two electrons for each double bond, etc.).

Consider water, $\ce{H2O}$. The oxygen atom has two bonds and two lone pairs. Oxygen originally had 6 valence electrons. We assign 6 electrons to it (4 nonbonding and 2 bonding). The formal charge is $6-6=0$.

Now consider the hydroxide ion, $\ce{OH-}$. The oxygen in hydroxide has 7 electron assigned to it: 6 nonbonding and 1 bonding. The formal charge is $6-7=-1$.

Now consider the nitrogen atom in the nitrate ion, $\ce{NO3-}$. Nitrogen originally has 5 electrons. In the nitrate ion we assign 4 electrons to that nitrogen (1 for each single bond and 2 for the double bond. There are no lone pairs). The formal charge is $5-4=+1$.

As a final example, let's look at the P in phosphate, $\ce{PO4^3-}$. Phosphorous original had 5 electrons, like nitrogen. We assign 5 electrons to phosphorous in phosphate (1 for each P-O bond and two for the P=O bond). The formal charge is $5-5-=0$

How many electrons are assigned to the sulfur atom in $\ce{SO3}$? From the question it sounds like you have the correct Lewis Structure. It looks like you are making your mistake in assigning the proper number of electrons based on the number of bonds.


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# Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde

Suppose I have the parabolic system $$u_t=\nabla\cdot(k(x)\nabla u)+f,\quad (x,t)\in\Omega\times I$$ with Dirichlet boundary conditions $$u=g, \quad x\in\partial\Omega$$ and initial condition $$u(x,t)= h,\quad t=0.$$

Often times in engineering, we are more interested in the asymptotic (steady state) behavior of this PDE rather than the transient behavior. So, we sometimes neglect the time derivative term and solve the elliptic system $$-\nabla\cdot(k(x)\nabla u)=f,\quad (x,t)\in\Omega\times I$$ instead. The assumption is that over infinite time, $$\lim_{t\rightarrow \infty}u_{\mathrm{parabolic}}(x,t)= u_{\mathrm{elliptic}}(x,t)$$.

I've observed that when $f\equiv 0$, this limit is true, but I'm not sure if this is the case for arbitrary $f$ or if there are other necessary conditions to guarantee this limit to be true. Do the boundary conditions have to converge asymptotically to a constant value in order for the parabolic solution to converge to the elliptic solution?

Though I'm my question is formulated in the continuous case, I'm also curious if the same conditions are true for the discrete case. That is, assuming I use a stable and consistent finite difference scheme to approximate $u_{\mathrm{parabolic}}$ and $u_{\mathrm{elliptic}}$, should I expect $$\lim_{t\rightarrow \infty}u_{\mathrm{parabolic}}^\mathrm{fdm}(x,t)= u_{\mathrm{elliptic}}^\mathrm{fdm}(x,t)$$ if $u_{\mathrm{elliptic}}^\mathrm{fdm}$ and $u_{\mathrm{parabolic}}^\mathrm{fdm}$ are discretized on the same spatial grid and $\Delta t\rightarrow\infty$?

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