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if, whenever $$ f , g \in K\{ x \}$$, $$ g $$ is nonzero and the order of $$ f $$ is greater then the order of $$ g $$, there is $$a \in K$$ such that $$ f (a) = 0 $$ and $$ g (a) \neq 0 $$. A differential field $$K \supset k ...
(tx + (1-t) y ) \le \max \{ u( x ), u( y ) \} \quad \text{ for every } x , y \in {\mathbb{R}^n},\quad 0 \le t \le 1. \end{ equation } (1.5) Given |$ g :{\mathbb{R}^n} \to \mathbb{R}$|, the QCE of |$ g $| is given ...
solution of semiexplicit differential -algebraic equations (DAEs) of differentiation index 2 in Hessenberg form, $$\begin{align} \dot x & = f ( x ,t) - g_x( x ,t)^{\textrm{T}}\lambda , \end{align}$$(1.1a) $$\begin ...
{-a.s.}, $$ (2.1) for all |$t\in [ 0 ,T]$|. It is well known that the above equation implies the following mild form for (1.1): $$ u(t, x )&= \int_0^1 G (t, x , y )u_0( y )\,\mathrm{d} y +\int_0^t\int_0^1 G (t-s, x , y ...
} (\forall x ^{\rho})(\exists y ^{\tau})\varphi( x , y ) \rightarrow (\exists F ^{\rho\rightarrow \tau})(\forall x ^{\rho})\varphi( x , F ( x )). \end{ equation }$$ (2.7) (3) The system |${\sf{RCA}}_{ 0 }^{\omega }$| is |$\sf{E ...
$| are constants depending on metric |$ g $| and |$k$|. It remains to compute \begin{ equation } \lim_{t \rightarrow 0 ^+}\int_{B_{r_0}( x , g )} t^{\frac{k}{4}}| \nabla_g^k b_{ g }( x , y ; t)|_{ g ( x )} \ \textrm d V_g( y ). \end ...
, Z$| which implies that it vanishes. The remaining part of |$D_XY$| in the decomposition based on (2.18) can be computed easily as follows, $$\begin{ equation *} \lambda(D_XY)=-fG(D_XN, Y )= f II( X , Y )=- g ( X ...
of a Gaussian is a Gaussian: FT[12πsexp(−x2/(2s2)]=exp(−s2k2/2) Computation of truncation selection with FT Consider a normal breeding value and environmental factor distribution p0( y )=N( 0 ,σA; y ) f ( x )=N( 0 ,σE; x ...
))$| defining the cubic |$ Y $| defines via the correspondence of the Grassmannian a section $$\begin{ equation *} s_f \in H^ 0 ( G , {\textrm{Sym}}^3 {\mathcal{K}}^{\ast} ) \end{ equation *}$$ with fiber |$ f |_{{\mathbb{P ...
)}$| be an ordinary differential operator in |$L^{2}( 0 , 1)$| of order |$m$| generated by the differential expression \begin{ equation } l( y )\equiv y ^{(m)}( x )+\sum_{k= 0 }^{m-1}p_{k}( x ) y ^{(k)}( x ), \quad 0 < x ...
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