Thus \(L^{0}=0\) as claimed. Since this has three terms, it's called a trinomial. An ideal \(I\) of \({\mathrm{Pol}}({\mathbb {R}}^{d})\) is said to be prime if it is not all of \({\mathrm{Pol}}({\mathbb {R}}^{d})\) and if the conditions \(f,g\in {\mathrm{Pol}}({\mathbb {R}}^{d})\) and \(fg\in I\) imply \(f\in I\) or \(g\in I\). However, it is good to note that generating functions are not always more suitable for such purposes than polynomials; polynomials allow more operations and convergence issues can be neglected. $$, $$ u^{\top}c(x) u = u^{\top}a(x) u \ge0. Next, the condition \({\mathcal {G}}p_{i} \ge0\) on \(M\cap\{ p_{i}=0\}\) for \(p_{i}(x)=x_{i}\) can be written as, The feasible region of this optimization problem is the convex hull of \(\{e_{j}:j\ne i\}\), and the linear objective function achieves its minimum at one of the extreme points. Indeed, \(X\) has left limits on \(\{\tau<\infty\}\) by LemmaE.4, and \(E_{0}\) is a neighborhood in \(M\) of the closed set \(E\). . \({\mathbb {E}}[\|X_{0}\|^{2k}]<\infty \), there is a constant Hajek [28, Theorem 1.3] now implies that, for any nondecreasing convex function \(\varPhi\) on , where \(V\) is a Gaussian random variable with mean \(f(0)+m T\) and variance \(\rho^{2} T\). Ackerer, D., Filipovi, D.: Linear credit risk models. What this course is about I Polynomial models provide ananalytically tractableand statistically exibleframework for nancial modeling I New factor process dynamics, beyond a ne, enter the scene I De nition of polynomial jump-di usions and basic properties I Existence and building blocks I Polynomial models in nance: option pricing, portfolio choice, risk management, economic scenario generation,.. J. R. Stat. Philos. Thus (G2) holds. tion for a data word that can be used to detect data corrup-tion. PERTURBATION { POLYNOMIALS Lecture 31 We can see how the = 0 equation (31.5) plays a role here, it is the 0 equation that starts o the process by allowing us to solve for x 0. Specifically, let \(f\in {\mathrm{Pol}}_{2k}(E)\) be given by \(f(x)=1+\|x\|^{2k}\), and note that the polynomial property implies that there exists a constant \(C\) such that \(|{\mathcal {G}}f(x)| \le Cf(x)\) for all \(x\in E\). Condition (G1) is vacuously true, and it is not hard to check that (G2) holds. \(I\) Module 1: Functions and Graphs. \(z\ge0\), and let Thus, for some coefficients \(c_{q}\). A standard argument based on the BDG inequalities and Jensens inequality (see Rogers and Williams [42, CorollaryV.11.7]) together with Gronwalls inequality yields \(\overline{\mathbb {P}}[Z'=Z]=1\). \(\rho>0\). : Markov Processes: Characterization and Convergence. This is done as in the proof of Theorem2.10 in Cuchiero etal. \(K\cap M\subseteq E_{0}\). We call them Taylor polynomials. They are therefore very common. Scand. , Note that \(E\subseteq E_{0}\) since \(\widehat{b}=b\) on \(E\). This result follows from the fact that the map \(\lambda:{\mathbb {S}}^{d}\to{\mathbb {R}}^{d}\) taking a symmetric matrix to its ordered eigenvalues is 1-Lipschitz; see Horn and Johnson [30, Theorem7.4.51]. $$, $$ 0 = \frac{{\,\mathrm{d}}^{2}}{{\,\mathrm{d}} s^{2}} (q \circ\gamma)(0) = \operatorname{Tr}\big( \nabla^{2} q(x_{0}) \gamma'(0) \gamma'(0)^{\top}\big) + \nabla q(x_{0})^{\top}\gamma''(0). with initial distribution . Many of us are familiar with this term and there would be some who are not.Some people use polynomials in their heads every day without realizing it, while others do it more consciously. and . $$, \(f,g\in {\mathrm{Pol}}({\mathbb {R}}^{d})\), https://doi.org/10.1007/s00780-016-0304-4, http://e-collection.library.ethz.ch/eserv/eth:4629/eth-4629-02.pdf. be the first time The generator polynomial will be called a CRC poly- Sending \(m\) to infinity and applying Fatous lemma gives the result. Leveraging decentralised finance derivatives to their fullest potential. J. Probab. $$, $$ \gamma_{ji}x_{i}(1-x_{i}) = a_{ji}(x) = a_{ij}(x) = h_{ij}(x)x_{j}\qquad (i\in I,\ j\in I\cup J) $$, $$ h_{ij}(x)x_{j} = a_{ij}(x) = a_{ji}(x) = h_{ji}(x)x_{i}, $$, \(a_{jj}(x)=\alpha_{jj}x_{j}^{2}+x_{j}(\phi_{j}+\psi_{(j)}^{\top}x_{I} + \pi _{(j)}^{\top}x_{J})\), \(\phi_{j}\ge(\psi_{(j)}^{-})^{\top}{\mathbf{1}}\), $$\begin{aligned} s^{-2} a_{JJ}(x_{I},s x_{J}) &= \operatorname{Diag}(x_{J})\alpha \operatorname{Diag}(x_{J}) \\ &\phantom{=:}{} + \operatorname{Diag}(x_{J})\operatorname{Diag}\big(s^{-1}(\phi+\varPsi^{\top}x_{I}) + \varPi ^{\top}x_{J}\big), \end{aligned}$$, \(\alpha+ \operatorname {Diag}(\varPi^{\top}x_{J})\operatorname{Diag}(x_{J})^{-1}\), \(\beta_{i} - (B^{-}_{i,I\setminus\{i\}}){\mathbf{1}}> 0\), \(\beta_{i} + (B^{+}_{i,I\setminus\{i\}}){\mathbf{1}}+ B_{ii}< 0\), \(\beta_{J}+B_{JI}x_{I}\in{\mathbb {R}}^{n}_{++}\), \(A(s)=(1-s)(\varLambda+{\mathrm{Id}})+sa(x)\), $$ a_{ji}(x) = x_{i} h_{ji}(x) + (1-{\mathbf{1}}^{\top}x) g_{ji}(x) $$, \({\mathrm {Pol}}_{1}({\mathbb {R}}^{d})\), $$ x_{j}h_{ij}(x) = x_{i}h_{ji}(x) + (1-{\mathbf{1}}^{\top}x) \big(g_{ji}(x) - g_{ij}(x)\big). (1) The individual summands with the coefficients (usually) included are called monomials (Becker and Weispfenning 1993, p. 191), whereas the . J.Econom. In: Yor, M., Azma, J. Wiley, Hoboken (2004), Dunkl, C.F. By the above, we have \(a_{ij}(x)=h_{ij}(x)x_{j}\) for some \(h_{ij}\in{\mathrm{Pol}}_{1}(E)\). In particular, \(\int_{0}^{t}{\boldsymbol{1}_{\{Z_{s}=0\} }}{\,\mathrm{d}} s=0\), as claimed. Finance. be a continuous semimartingale of the form. Let denote its law. This relies on(G1) and (A2), and occupies this section up to and including LemmaE.4. If a savings account with an initial Polynomials are easier to work with if you express them in their simplest form. MATH The first part of the proof applied to the stopped process \(Z^{\sigma}\) under yields \((\mu_{0}-\phi \nu_{0}){\boldsymbol{1}_{\{\sigma>0\}}}\ge0\) for all \(\phi\in {\mathbb {R}}\). Changing variables to \(s=z/(2t)\) yields \({\mathbb {P}}_{z}[\tau _{0}>\varepsilon]=\frac{1}{\varGamma(\widehat{\nu})}\int _{0}^{z/(2\varepsilon )}s^{\widehat{\nu}-1}\mathrm{e}^{-s}{\,\mathrm{d}} s\), which converges to zero as \(z\to0\) by dominated convergence. The reader is referred to Dummit and Foote [16, Chaps. The use of polynomial diffusions in financial modeling goes back at least to the early 2000s. We first prove(i). Further, by setting \(x_{i}=0\) for \(i\in J\setminus\{j\}\) and making \(x_{j}>0\) sufficiently small, we see that \(\phi_{j}+\psi_{(j)}^{\top}x_{I}\ge0\) is required for all \(x_{I}\in [0,1]^{m}\), which forces \(\phi_{j}\ge(\psi_{(j)}^{-})^{\top}{\mathbf{1}}\). (x-a)^2+\frac{f^{(3)}(a)}{3! Since \((Y^{i},W^{i})\), \(i=1,2\), are two solutions with \(Y^{1}_{0}=Y^{2}_{0}=y\), Cherny [8, Theorem3.1] shows that \((W^{1},Y^{1})\) and \((W^{2},Y^{2})\) have the same law. - 153.122.170.33. : Abstract Algebra, 3rd edn. $$, $$\begin{aligned} Y_{t} &= y_{0} + \int_{0}^{t} b_{Y}(Y_{s}){\,\mathrm{d}} s + \int_{0}^{t} \sigma_{Y}(Y_{s}){\,\mathrm{d}} W_{s}, \\ Z_{t} &= z_{0} + \int_{0}^{t} b_{Z}(Y_{s},Z_{s}){\,\mathrm{d}} s + \int_{0}^{t} \sigma _{Z}(Y_{s},Z_{s}){\,\mathrm{d}} W_{s}, \\ Z'_{t} &= z_{0} + \int_{0}^{t} b_{Z}(Y_{s},Z'_{s}){\,\mathrm{d}} s + \int_{0}^{t} \sigma _{Z}(Y_{s},Z'_{s}){\,\mathrm{d}} W_{s}. \(C\) $$, \(2 {\mathcal {G}}p({\overline{x}}) < (1-2\delta) h({\overline{x}})^{\top}\nabla p({\overline{x}})\), $$ 2 {\mathcal {G}}p \le\left(1-\delta\right) h^{\top}\nabla p \quad\text{and}\quad h^{\top}\nabla p >0 \qquad\text{on } E\cap U. . }(x-a)^3+ \cdots.\] Taylor series are extremely powerful tools for approximating functions that can be difficult to compute . Suppose p (x) = 400 - x is the model to calculate number of beds available in a hospital. The desired map \(c\) is now obtained on \(U\) by. Indeed, for any \(B\in{\mathbb {S}}^{d}_{+}\), we have, Here the first inequality uses that the projection of an ordered vector \(x\in{\mathbb {R}}^{d}\) onto the set of ordered vectors with nonnegative entries is simply \(x^{+}\). 16, 711740 (2012), Curtiss, J.H. The condition \({\mathcal {G}}q=0\) on \(M\) for \(q(x)=1-{\mathbf{1}}^{\top}x\) yields \(\beta^{\top}{\mathbf{1}}+ x^{\top}B^{\top}{\mathbf{1}}= 0\) on \(M\). Available online at http://e-collection.library.ethz.ch/eserv/eth:4629/eth-4629-02.pdf, Cuchiero, C., Keller-Ressel, M., Teichmann, J.: Polynomial processes and their applications to mathematical finance. Google Scholar, Bakry, D., mery, M.: Diffusions hypercontractives. To this end, set \(C=\sup_{x\in U} h(x)^{\top}\nabla p(x)/4\), so that \(A_{\tau(U)}\ge C\tau(U)\), and let \(\eta>0\) be a number to be determined later. \(\sigma\) for all Polynomials an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable (s). We introduce a class of Markov processes, called $m$-polynomial, for which the calculation of (mixed) moments up to order $m$ only requires the computation of matrix exponentials. $$, \(\widehat{b} :{\mathbb {R}}^{d}\to{\mathbb {R}}^{d}\), $$ \widehat{\mathcal {G}}f = \frac{1}{2}\operatorname{Tr}( \widehat{a} \nabla^{2} f) + \widehat{b} ^{\top} \nabla f $$, \(\widehat{\mathcal {G}}f={\mathcal {G}}f\), \(c:{\mathbb {R}}^{d}\to {\mathbb {R}}^{d}\), $$ c=0\mbox{ on }E \qquad \mbox{and}\qquad\nabla q^{\top}c = - \frac {1}{2}\operatorname{Tr}\big( (\widehat{a}-a) \nabla^{2} q \big) \mbox{ on } M\mbox{, for all }q\in {\mathcal {Q}}. J. Financ. \(\varepsilon>0\), By Ging-Jaeschke and Yor [26, Eq. Pick any \(\varepsilon>0\) and define \(\sigma=\inf\{t\ge0:|\nu_{t}|\le \varepsilon\}\wedge1\). Using that \(Z^{-}=0\) on \(\{\rho=\infty\}\) as well as dominated convergence, we obtain, Here \(Z_{\tau}\) is well defined on \(\{\rho<\infty\}\) since \(\tau <\infty\) on this set. Finance Assessment of present value is used in loan calculations and company valuation. In: Dellacherie, C., et al. and assume the support scalable. \(d\)-dimensional It process The zero set of the family coincides with the zero set of the ideal \(I=({\mathcal {R}})\), that is, \({\mathcal {V}}( {\mathcal {R}})={\mathcal {V}}(I)\). $$, $$ A_{t} = \int_{0}^{t} {\boldsymbol{1}_{\{X_{s}\notin U\}}} \frac{1}{p(X_{s})}\big(2 {\mathcal {G}}p(X_{s}) - h^{\top}\nabla p(X_{s})\big) {\,\mathrm{d}} s $$, \(\rho_{n}=\inf\{t\ge0: |A_{t}|+p(X_{t}) \ge n\}\), $$\begin{aligned} Z_{t} &= \log p(X_{0}) + \int_{0}^{t} {\boldsymbol{1}_{\{X_{s}\in U\}}} \frac {1}{2p(X_{s})}\big(2 {\mathcal {G}}p(X_{s}) - h^{\top}\nabla p(X_{s})\big) {\,\mathrm{d}} s \\ &\phantom{=:}{}+ \int_{0}^{t} \frac{\nabla p^{\top}\sigma(X_{s})}{p(X_{s})}{\,\mathrm{d}} W_{s}. J. Stat. It gives necessary and sufficient conditions for nonnegativity of certain It processes. For any symmetric matrix Google Scholar, Mayerhofer, E., Pfaffel, O., Stelzer, R.: On strong solutions for positive definite jump diffusions. Polynomials are also "building blocks" in other types of mathematical expressions, such as rational expressions. \end{aligned}$$, $$ { \vec{p} }^{\top}F(u) = { \vec{p} }^{\top}H(X_{t}) + { \vec{p} }^{\top}G^{\top}\int_{t}^{u} F(s) {\,\mathrm{d}} s, \qquad t\le u\le T, $$, \(F(u) = {\mathbb {E}}[H(X_{u}) \,|\,{\mathcal {F}}_{t}]\), \(F(u)=\mathrm{e}^{(u-t)G^{\top}}H(X_{t})\), $$ {\mathbb {E}}[p(X_{T}) \,|\, {\mathcal {F}}_{t} ] = F(T)^{\top}\vec{p} = H(X_{t})^{\top}\mathrm{e} ^{(T-t)G} \vec{p}, $$, $$ dX_{t} = (b+\beta X_{t})dt + \sigma(X_{t}) dW_{t}, $$, $$ \|\sigma(X_{t})\|^{2} \le C(1+\|X_{t}\|) \qquad \textit{for all }t\ge0 $$, $$ {\mathbb {E}}\big[ \mathrm{e}^{\delta\|X_{0}\|}\big]< \infty \qquad \textit{for some } \delta>0, $$, $$ {\mathbb {E}}\big[\mathrm{e}^{\varepsilon\|X_{T}\|}\big]< \infty. satisfies Math. Hence the following local existence result can be proved. As the ideal \((x_{i},1-{\mathbf{1}}^{\top}x)\) satisfies (G2) for each \(i\), the condition \(a(x)e_{i}=0\) on \(M\cap\{x_{i}=0\}\) implies that, for some polynomials \(h_{ji}\) and \(g_{ji}\) in \({\mathrm {Pol}}_{1}({\mathbb {R}}^{d})\). Synthetic Division is a method of polynomial division. By well-known arguments, see for instance Rogers and Williams [42, LemmaV.10.1 and TheoremsV.10.4 and V.17.1], it follows that, By localization, we may assume that \(b_{Z}\) and \(\sigma_{Z}\) are Lipschitz in \(z\), uniformly in \(y\). It follows from the definition that \(S\subseteq{\mathcal {I}}({\mathcal {V}}(S))\) for any set \(S\) of polynomials. Polynomial brings multiple on-chain option protocols in a single venue, encouraging arbitrage and competitive pricing. Let \(C_{0}(E_{0})\) denote the space of continuous functions on \(E_{0}\) vanishing at infinity. Example: 21 is a polynomial. The site points out that one common use of polynomials in everyday life is figuring out how much gas can be put in a car. with representation, where Appl. that only depend on be two $$, \(\tau=\inf\{t\ge0:\mu_{t}\ge0\}\wedge1\), \(0\le{\mathbb {E}}[Z_{\tau}] = {\mathbb {E}}[\int_{0}^{\tau}\mu_{s}{\,\mathrm{d}} s]<0\), \({\mathrm{d}}{\mathbb {Q}}={\mathcal {E}}(-\phi B)_{1}{\,\mathrm{d}} {\mathbb {P}}\), $$ Z_{t}=\int_{0}^{t}(\mu_{s}-\phi\nu_{s}){\,\mathrm{d}} s+\int_{0}^{t}\nu_{s}{\,\mathrm{d}} B^{\mathbb {Q}}_{s}. 1, 250271 (2003). Positive semidefiniteness requires \(a_{jj}(x)\ge0\) for all \(x\in E\). J. $$, \(t\mapsto{\mathbb {E}}[f(X_{t\wedge \tau_{m}})\,|\,{\mathcal {F}}_{0}]\), \(\int_{0}^{t\wedge\tau_{m}}\nabla f(X_{s})^{\top}\sigma(X_{s}){\,\mathrm{d}} W_{s}\), $$\begin{aligned} {\mathbb {E}}[f(X_{t\wedge\tau_{m}})\,|\,{\mathcal {F}}_{0}] &= f(X_{0}) + {\mathbb {E}}\left[\int_{0}^{t\wedge\tau_{m}}{\mathcal {G}}f(X_{s}) {\,\mathrm{d}} s\,\bigg|\, {\mathcal {F}}_{0} \right] \\ &\le f(X_{0}) + C {\mathbb {E}}\left[\int_{0}^{t\wedge\tau_{m}} f(X_{s}) {\,\mathrm{d}} s\,\bigg|\, {\mathcal {F}}_{0} \right] \\ &\le f(X_{0}) + C\int_{0}^{t}{\mathbb {E}}[ f(X_{s\wedge\tau_{m}})\,|\, {\mathcal {F}}_{0} ] {\,\mathrm{d}} s. \end{aligned}$$, \({\mathbb {E}}[f(X_{t\wedge\tau_{m}})\, |\,{\mathcal {F}} _{0}]\le f(X_{0}) \mathrm{e}^{Ct}\), $$ p(X_{u}) = p(X_{t}) + \int_{t}^{u} {\mathcal {G}}p(X_{s}) {\,\mathrm{d}} s + \int_{t}^{u} \nabla p(X_{s})^{\top}\sigma(X_{s}){\,\mathrm{d}} W_{s}. Next, it is straightforward to verify that (i) and (ii) imply (A0)(A2), so we focus on the converse direction and assume(A0)(A2) hold. 5 uses of polynomial in daily life are stated bellow:-1) Polynomials used in Finance. They are used in nearly every field of mathematics to express numbers as a result of mathematical operations. Lecture Notes in Mathematics, vol. As \(f^{2}(y)=1+\|y\|\) for \(\|y\|>1\), this implies \({\mathbb {E}}[ \mathrm{e}^{\varepsilon' \| Y_{T}\|}]<\infty\). A polynomial in one variable (i.e., a univariate polynomial) with constant coefficients is given by a_nx^n+.+a_2x^2+a_1x+a_0. Appl. be continuous functions with have the same law. Polynomial:- A polynomial is an expression consisting of indeterminate and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Then for each \(s\in[0,1)\), the matrix \(A(s)=(1-s)(\varLambda+{\mathrm{Id}})+sa(x)\) is strictly diagonally dominantFootnote 5 with positive diagonal elements. By sending \(s\) to zero, we deduce \(f=0\) and \(\alpha x=Fx\) for all \(x\) in some open set, hence \(F=\alpha\). It follows that \(a_{ij}(x)=\alpha_{ij}x_{i}x_{j}\) for some \(\alpha_{ij}\in{\mathbb {R}}\). J. We first prove that \(a(x)\) has the stated form. and \(\mu\) 7000+ polynomials are on our. Existence boils down to a stochastic invariance problem that we solve for semialgebraic state spaces. Let \(\gamma:(-1,1)\to M\) be any smooth curve in \(M\) with \(\gamma (0)=x_{0}\). For example: x 2 + 3x 2 = 4x 2, but x + x 2 cannot be written in a simpler form. We now change time via, and define \(Z_{u} = Y_{A_{u}}\). In the health field, polynomials are used by those who diagnose and treat conditions. answer key cengage advantage books introductory musicianship 8th edition 1998 chevy .. \(Y^{1}\), \(Y^{2}\) Why It Matters. 30, 605641 (2012), Stieltjes, T.J.: Recherches sur les fractions continues. Bakry and mery [4, Proposition2] then yields that \(f(X)\) and \(N^{f}\) are continuous.Footnote 3 In particular, \(X\)cannot jump to \(\Delta\) from any point in \(E_{0}\), whence \(\tau\) is a strictly positive predictable time. \(\{Z=0\}\), we have Econ. Fac. The process \(\log p(X_{t})-\alpha t/2\) is thus locally a martingale bounded from above, and hence nonexplosive by the same McKeans argument as in the proof of part(i). Taylor Polynomials. \(X\) Given a set \(V\subseteq{\mathbb {R}}^{d}\), the ideal generated by We now focus on the converse direction and assume(A0)(A2) hold. Consider the Like actuaries, statisticians are also concerned with the data collection and analysis. Its formula yields, We first claim that \(L^{0}_{t}=0\) for \(t<\tau\). Correspondence to This happens if \(X_{0}\) is sufficiently close to \({\overline{x}}\), say within a distance \(\rho'>0\). If, then for each All of them can be alternatively expressed by Rodrigues' formula, explicit form or by the recurrence law (Abramowitz and Stegun 1972 ). \(A=S\varLambda S^{\top}\), we have Then the law under \(\overline{\mathbb {P}}\) of \((W,Y,Z)\) equals the law of \((W^{1},Y^{1},Z^{1})\), and the law under \(\overline{\mathbb {P}}\) of \((W,Y,Z')\) equals the law of \((W^{2},Y^{2},Z^{2})\). 1. \(b:{\mathbb {R}}^{d}\to{\mathbb {R}}^{d}\) We first prove(i). Math. Polynomials can be used in financial planning. Defining \(\sigma_{n}=\inf\{t:\|X_{t}\|\ge n\}\), this yields, Since \(\sigma_{n}\to\infty\) due to the fact that \(X\) does not explode, we have \(V_{t}<\infty\) for all \(t\ge0\) as claimed. : The Classical Moment Problem and Some Related Questions in Analysis. Let (x-a)+ \frac{f''(a)}{2!} Ann. The extended drift coefficient is now defined by \(\widehat{b} = b + c\), and the operator \(\widehat{\mathcal {G}}\) by, In view of (E.1), it satisfies \(\widehat{\mathcal {G}}f={\mathcal {G}}f\) on \(E\) and, on \(M\) for all \(q\in{\mathcal {Q}}\), as desired. J. Econom. The first can approximate a given polynomial. Thus \(L=0\) as claimed. We first prove that there exists a continuous map \(c:{\mathbb {R}}^{d}\to {\mathbb {R}}^{d}\) such that. Exponents and polynomials are used for this analysis. Financial Planning o Polynomials can be used in financial planning. An expression of the form ax n + bx n-1 +kcx n-2 + .+kx+ l, where each variable has a constant accompanying it as its coefficient is called a polynomial of degree 'n' in variable x. If a person has a fixed amount of cash, such as $15, that person may do simple polynomial division, diving the $15 by the cost of each gallon of gas. Theorem4.4 carries over, and its proof literally goes through, to the case where \((Y,Z)\) is an arbitrary \(E\)-valued diffusion that solves (4.1), (4.2) and where uniqueness in law for \(E_{Y}\)-valued solutions to(4.1) holds, provided (4.3) is replaced by the assumption that both \(b_{Z}\) and \(\sigma_{Z}\) are locally Lipschitz in\(z\), locally in\(y\), on \(E\). Let MATH Used everywhere in engineering. Thus we may find a smooth path \(\gamma_{i}:(-1,1)\to M\) such that \(\gamma _{i}(0)=x\) and \(\gamma_{i}'(0)=S_{i}(x)\).

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