From 43283469d5e539f6117b66e24d7b7c29446b6741 Mon Sep 17 00:00:00 2001 From: Petr Velycko Date: Fri, 16 May 2025 23:03:48 +0200 Subject: [PATCH] fix preklepy by D. Prochazka --- jednoznacnost-reseni.tex | 4 ++-- maximalni-reseni.tex | 2 +- skripta.pdf | Bin 227208 -> 227275 bytes zavislost-na-podmince.tex | 2 +- 4 files changed, 4 insertions(+), 4 deletions(-) diff --git a/jednoznacnost-reseni.tex b/jednoznacnost-reseni.tex index 1e32806..e1822ce 100644 --- a/jednoznacnost-reseni.tex +++ b/jednoznacnost-reseni.tex @@ -3,9 +3,9 @@ V této kapitole se budeme věnovat otázce jednoznačnosti řešení diferenciálních rovnic. V praxi to často požadujeme, například proto, aby nějaká simulace byla deterministická. \begin{definition} - Řekneme, že rovnice \eqref{eq-ode} má v $\Omega$ vlastnost \textit{globální jednoznačnosti}, jestliže pro libovolná řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0)$ pro nějaké $t_0 \in I \cup J$, potom $x(t) = y(t)$ pro všechna $t \in I \cup J$. + Řekneme, že rovnice \eqref{eq-ode} má v $\Omega$ vlastnost \textit{globální jednoznačnosti}, jestliže pro libovolná řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0)$ pro nějaké $t_0 \in I \cap J$, potom $x(t) = y(t)$ pro všechna $t \in I \cap J$. - Řekneme, že rovnice \eqref{eq-ode} má v $\Omega$ vlastnost \textit{lokální jednoznačnosti}, jestliže pro libovolná řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0)$ pro nějaké $t_0 \in I \cup J$ existuje $\delta > 0$ takové, že $x(t) = y(t)$ pro všechna $t \in (t_0 - \delta, t_0 + \delta)$. + Řekneme, že rovnice \eqref{eq-ode} má v $\Omega$ vlastnost \textit{lokální jednoznačnosti}, jestliže pro libovolná řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0)$ pro nějaké $t_0 \in I \cap J$ existuje $\delta > 0$ takové, že $x(t) = y(t)$ pro všechna $t \in (t_0 - \delta, t_0 + \delta)$. \end{definition} \begin{theorem} diff --git a/maximalni-reseni.tex b/maximalni-reseni.tex index c360e3f..b4d4421 100644 --- a/maximalni-reseni.tex +++ b/maximalni-reseni.tex @@ -27,7 +27,7 @@ V případě $f$ lipschitzovské se důkaz dá výrazně zjednodušit. Budeme uv \end{theorem} \begin{proof} - Plyne z Peanovy věty (Věta \ref{thm-peano}) a Věty \ref{thm-max-extension}. + Plyne z Peanovy věty (Věta \ref{thm-peano}) a Věty \ref{thm-max-extension}. Jednoznačnost plyne z rule of thumb v kapitole 2. \end{proof} \begin{lemma} diff --git a/skripta.pdf b/skripta.pdf index 45252b356b0e2907ab242d8318dde9ac639ad888..c9ded1ae09d205b1e7d26b074ced9f8364a41d6e 100644 GIT binary patch delta 30337 zcmY(J({?2afJD==Z95%1>DYGCv2B|twr$(CZJpTe*yh}ad6@Z#TB~Z;4!=Q|xCV`jbU60?t2^CdgU%$08e!C~5&@waALPf*)j60kq!5E# zSr2M}Tl>CI8L5+KoLJpXG%MDeV}P{kp?r%RsXVCTmn8UhQiPKFsR{}TQV|0GzPY_6 zEelI4p4{Hu#kDDKd~_8hDlbYb zDO=?;=AAwiC5Ptn%mBeF_UTj*sPOiSryGpJQ*GC*u_Jq( z_f%{pEisx`o>Exfvxzz2)}xAvK!Y|lO0-gKy>=-~7n`9UQrwvFUHT|+v*^#TYr-d= zb8gq255C#Esb+(u{pS_Q+`90G8~}I!8e3DAJn*FgMS;czSW+pu}^dJOZi;6Dymt){eHQAf+KKSsEtX3YMh_ z=#-a{-dH1KR_v{*CYpq*f@jNt7jJGYG00A7OYF@rXHsu8tM)jy(mXQ>sQ3|}5F~F- zSwN4U;K}^2H7Nn@o$1FWaB?dt9u_JaPGw|i>*3*utefIMB-U7nfF)Sip~Ya|I+7qR zxgn6|p?ssx1{S^Z4>oypM}V)n2Dmq^Aex1Zij`xDZ>Ib>+P|C~GtsH(tF_b!f7qdj zVXWm^vXAyXRl4SdocsHvc4U6I&|0@5t!g8Q$>5~fdqMN(0y_*0e?fay&5^9e(6!dZ zK_2oP>6q{53nU5Cs~Lb_daMPw2Xtrj>wX{#fR6wMBJe>y;rRt=7!a5xnb-B^CZC5f z>Hkdi=o)ni-dIv+Jxz<{uE(hQ+RGfG@tl=9!WSZhP1WFJFP(zIaT;TpHvRN(3-V|H z7F9BipPgkuvlR{$Y>~I4Mu$0l=N!>i=3q>Tw*AdaI>2c#$#p4Z`=KK~7t#mIxx>WP zy9WrQXC^E8ID;+v2be(LeC(cR&lNswS=}}mj_q3R>@?Wk&AXk3$cjSoYHpgG6MJRYyD8~5`4Ea&LiQC)KnoV(>1<-H_ zh^*^asVWN1m2f!iOw1hFzqVaE0wI{jkA6iTb@P*iD4#k1j(Ig*j$+1zVd>nv_yduLakOK^>hg?cz$ybY1z*`HnXdJPPesmd z9@GYXrGl;n0$TU-LS_QH5VIht={#!sjG7iuy;wAFao3`ec`Snw^pQCp+*-^NB%o*0 z8S%58ta%o?Xd5(N66TMAbc-Hqd+EKeUaAOAy@LC8C*-`|*5@ zI*i7_KFTLC?J|d0WL}ftM_Mg4++e3?wsov~;&-sV0P!lIJ^p)~P1B;yQDd#FY^_Nz z7qHaYXugfzCd`4Z*R;W(s3gU@BI63?nUI6ktNuNfq^Wf12?CXB^vN6I-pUU3r8ADMd6xNQjFpQgvf20I!JXynVF$x84}VphmoqyD zSJJ#SUb*JWIcT+R5hyTn{k3OuJ27&r9-{G(#RyYq6bU0hRQ}JM!n&y^sKUpK<}l3! zL>rGVuL+u>z7jdag|ZyjUs+RQnYMQuW8a)4I$O4R1ff>V9uA`+Ib6^7F` z0yQei-aj0`JcEk4oSm2nE|Lb$ymB5ZUdHD#my$bOH~FQ)Z}`5YQVoIIbJdVDHCARrl0Cmq0TIUe>w%eR7cK96UGiHljB=2+p4}||WYN+FRX+^ajfp_M zr?MwVK4sz%FC;BR5jsrf5?W~>(Xq5)h`wJ!`m-+D?5T0xH5e$?IB+?=V)G9>Ec z;ItIBF+`}buiO+`86_`mmT|axUHbIG_(Z|OuOQ;+#&6kn5DdexCg#VzoL&wKG4tdC=dh4pcevm1JLMd+ zn1*twazM5%&@k{JQ|=dEj?)+Cpo%l+n0uY;N&@ELL89%eOFXR({V@4%=nX-`e>bUa z+4*(nuT1yZ7V6-zup)V-)5PR}y_C`I-;hd21(dXYEgHNVpxuBff254G5MMB4Zhv@t zyj#bMfN!>x3OilyjK=R-ks7dRjhQe4S$yB`wKfkJ?8d6UQ7da6CK)4|han_>L$Qs@?Se*#In)pwptN8tCw zCkKk1|DCXKGd*(#%z7B=mV4e!an65AyXqrpl2iFdGh#EDsjclq;8QPkls68h z;pc|;%#nLISDC&77#G`NPiBTR7h#P9yK608S?zPvf`0 z%>T99Xr)iphm*~rE%Bq1BpG6~FR0^1p++|@-elI!*0Nc!sJG`ye}_PV#PiyJb%3N` zU<+>>q(=O`I(mv&;*|Wnkz4djlwz=g)={j&1%0Y^z_W4yoMS}EiKlUR`dWb?a8)Q* zn@A5%>*Oyj+rop z@Idi?k<-9W0OPieuY%!F5r5oYhxT7Y)Io)VQ;o9rz;Mr{(_=vLp9_4uSFzSd@yx?| zpYGPu@Hzwm;;mF2zFSG%+SmE?hWt|~z=yJl=<7+sUi~~uIGvv2OcA-~*^5oh#K+n? zul|B@tafO_=)RnB#DRQy7t-3sHGF3BHKA1cW8WFuAo(z9eF?L%Kfbk#oX1nb7TLuX zOt4M499vhPYFCDaJfRlF(>O|YDBiF9F#0U(>E~d8f7(cvbRE;r;IYxXi}`XYlFW?L z6pReb$By-MZ7eZy{Ml%%t3+r(FRhjoU`_)8t4ULHPY8)_80Bc{675C0T< zm)IVF^|`q-QXMNi2AN5e#Vtk;y_{&Vo#Hemv8@$CZp z^W{vEZ|&QVa0iU|2?@47Z5k_iVk3uTU4a@f1UXcLJo)lT&+h zaRL_9#dU$Pg}&;lk{0w^i|MNut~4PEDsXT+(9+_jV6QVub%0GslF4hc@UGz32#i*p zQwF})3EJ6E=~ACRzgk>j)L^BvMPI82;)VoFTH%+A=LGDP_kQ+@+tQO^EXUH2zI8ib z?n$uH%p}!xjciK+#Uw?ghAKibBa%N)^+djlt%6Qy78rTD9q2A^k(e7_Nah4Pf$J{S z({&gW9y+zH&^a@;J;5`q1;b1|o*R~u5JiqMo7yF_`1^QB_4qokq5s1|P*P0$p$gM0 zsUt9a_+yQ^opI6eyS)qo+YCRuj%%}6Kg?)Y{w0CX9^3!UeuuI2e0S9kTW zR%}zqDqb8vvW@l^mUymC?sJYfHhv)YqcC6(K|Oc+`8t-UgEUY z<;qoKSkK;IEgX>K8ZKgYVx|IbD@p+wOw#s_P~*13!AI2UfR8WQkXHkIc4I3B_VFJO zWU#iCe%V!rxXkuvElumF`@8p=nW^^V*^F)O&2d(!`q-)=7O{# zTYsLD6h@1Kha>opZ1IQ5x4YHFxIE00p$+6*m8uV*(Qf>JZVIIDrzJyTzj}pm`tM^I zG<$uqmxu7{D$Vg1#!h3YOlIiSjO&KABCH}Z59C>3YCH`oZz!nA-*W&7l7trNnD%H? zu602GpP3jxHn;7Lj={EOK#xh+yH(t-{o4}bfV2+il-qEVR-)sJWU6QaM*?hyLeag7 zr(*7q`i>e;Q@tz(x1Sd7ZjL5*U!0zD!iy$f@df%3fu|%R@n>31WY^e)UspQ|C%DZl zkyOhG3|j|YQzaZkQ5}HLCWVj*v9)>HDO37S3H(Jj5D%WE2|j>w4O}frk}w9sFeM=% zZqkEv3#T{2(0hxJ_e|FjcR0{I$|xJP*sUV+2vRK!$OVQ{S9i$ExkS@c>fx%ia1m`c zX~S6>Cjpn^{yd_t-JWwOr5|Z{kQ{?xj*t`;eDPwRKC1@>b_U>8;XlNQgRT>H?XcBm z$;stCG~3UYx~@^_vdm&IOPPBY>P%(M7KoG1`kl|ZD)rMechZBm4-k62q1g{}$8Y21 zI%C>eM=`1hIhGag?zLS#f;yM3k~>Kg%$QfU#EQeENrV}GKI5jX2Gif4H4UPiM+BYS ziyh%?=0PjRyI1cyL}>!Dn$?Ex|A#p6+}l_iIv1?ff3zk)z3@j?4>h{gyh;E48=;LX z#n5?B-)V)*dN1no`gbySeGk_2uEIbCm@)z92tLx~J9t&=Z1VZ>G943v4&@%Mt z8^8B%9xlTeuO@8q5=iBK35nMn9D9cTkk_CeEmSBoRMV#U{V4n(3EJ`Br!_Eup*OLK zCc^{R+5ayYI_8~S1*6qy;B@;K^

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