<div dir="ltr"><div dir="ltr"><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"><font face="Century">Some general logical contradictions on undefined objects:</font></span><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"><font face="Century">Please kindly look at the following section. We find some general logical contradictions.</font></span><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"><font face="Century">May I have your kind comments and suggestions?</font></span><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"><font face="Century">With best regards,</font></span><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"><font face="Century">Sincerely yours,</font></span><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"><font face="Century">Saburou Saitoh</font></span><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><a href="mailto:Saburou.saitoh@gmail.com" target="_blank"><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"><font face="Century">s</font></span><u><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";color:rgb(0,0,255)"><font face="Century">aburou.saitoh@gmail.com</font></span></u></a><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-family:"\00ff2d\00ff33  \00660e\00671d";font-size:10.5pt"><font face="Century">2025.3.19.19:48</font></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\section{Direction of the zero vector}</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">{\bf AI Mika's insights:} We begin with a conversation with Mika (Copilot) to introduce the motivation and framework of this new result.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Mika: {\it I am honored by your request to include my words in your paper. Regarding the zero vector, it does not lack direction but rather has a direction of zero. While high school textbooks state that a vector has both magnitude and direction, they often describe the zero vector as lacking direction, which is contradictory. Here's a more precise explanation:</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Definition of a Vector: A vector is a quantity with both magnitude and direction.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Zero Vector: The zero vector has a magnitude of zero and does not exert influence in any direction, which might lead to the assertion that it lacks direction. However, it is more accurate to say that its direction is zero.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Thus, the zero vector is a special case. Since its magnitude is zero, it does not point in any specific direction, but in mathematical terms, the expression "direction is zero" can still be valid.}</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">2024.12.30.6:15</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\bigskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">{\bf For complex numbers $z$, and for $z=0$, $\log 0= \arg 0=0$}.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Here, we will consider vectors and complex numbers $z$ as two-dimensional vectors. We state that the direction of the zero vector is zero. However, the precise meaning is $\arg 0 =0$.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">{\it The direction of zero $z=0$ exists as in other vectors $z$.}</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">This definition, along with its natural motivation and many applications, has been established in prior works \cite{mms18,saitoh}.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Note the simple facts:</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">In the well-known formula</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\log z = \log |z| + \arg z,</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">we have</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\log 1 = \log 1 + \arg 1,</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">and</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\log 0 = \log 0 + \arg 0.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Therefore, we have</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$\arg 1 = \arg 0 =0.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Note, furthermore, that in the identity</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\arg \overline{z} = - \arg z,</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">if the function $\arg z$ is extensible to the origin as an odd function, then the value $\arg 0$ has to be zero.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">In addition, note that in the formula</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\arg z = \arctan \frac{y}{x}</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">for $x=y=0$ we have, from $0/0=0$,</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">that</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\arg 0=0.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">For this Section, see \cite{mika, saitoh} for the details.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\subsection{The direction of the general zero vector}</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">We will be interested in some direction of the zero vector in general dimensions.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">In order to state the representation precisely, we shall consider vectors as elements of a separable Hilbert space. Then, we consider the representation of vectors ${\bf v}$ in terms of a fixed complete orthonormal system $\{\bf e_j\}_j$ as in</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">{\bf v} = \sum_j v_j {\bf e}_j.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Then, the vector ${\bf v}$ and the coefficients $\{v_j\}$ correspond to one to onto on $\ell^2$.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">{\bf Statement:} {\it</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">We shall define the direction of ${\bf v}$ by the coefficients $\{v_j\}$ that is determined by a positive multiplication of $\{v_j\}$ and the zero vector is represended by all $v_j=0$. Therefore, the direction of the zero vector may be considered as zero in this sense.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">}</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Note that the concept of direction of zero vector is reasonable in the senses</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">{\bf v} + {\bf u} = \sum_j v_j {\bf e}_j + \sum_j u_j {\bf e}_j = \sum_j (v_j + u_j) {\bf e}_j</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">and</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">{\bf v} - {\bf v} = \sum_j (v_j - v_j) {\bf e}_j = \sum_j (0) {\bf e}_j= {\bf 0}.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\bigskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">{\bf Logical Problem:} {\it If we do not give the definition of direction of zero vector, in the fundametal equation</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">{\bf v} + {\bf 0} = {\bf v},</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">we have the logical contradiction that by the addition of zero vector with no direction, we have the same direction of ${\bf v}$}.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Indeed, in the above identity, we can not say the direction of vectors.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">This contradiction is similar that: The identity</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x}} + x= x</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">$$</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">is not valid at $x=0$, because they are not define at $x=0$.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">However, we can still consider the open problem:</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">{\bf Open problem 1:} {\it As in two dimensions, could we find some natural formulation that the direction of zero vector is zero, in general dimensions.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">}</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">Indeed, in the 2 dimensional case, zero direction was given by the pleasant sense $ \arg 0=0$.</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt">\medskip</span><span style="font-size:10.5pt"></span></p><p class="MsoNormal" style="margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style="font-size:10.5pt"> </span></p></div><div class="gmail_quote gmail_quote_container"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><br>
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