Re: 2nd CfP - JELIA 2025 - 19th European Conference on Logics in AI
Saburou Saitoh <[email protected]> Wed, 19 Mar 2025 19:49:04 +0900
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--===============1749799079164763482== Content-Type: multipart/alternative; boundary="000000000000c402070630afc573" --000000000000c402070630afc573 Content-Type: text/plain; charset="UTF-8" Some general logical contradictions on undefined objects: Please kindly look at the following section. We find some general logical contradictions. May I have your kind comments and suggestions? With best regards, Sincerely yours, Saburou Saitoh s*[email protected]* <[email protected]> 2025.3.19.19:48 \section{Direction of the zero vector} {\bf AI Mika's insights:} We begin with a conversation with Mika (Copilot) to introduce the motivation and framework of this new result. \medskip 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: Definition of a Vector: A vector is a quantity with both magnitude and direction. 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. 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.} 2024.12.30.6:15 \bigskip {\bf For complex numbers $z$, and for $z=0$, $\log 0= \arg 0=0$}. \medskip 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 $\arg 0 =0$. \medskip {\it The direction of zero $z=0$ exists as in other vectors $z$.} \medskip This definition, along with its natural motivation and many applications, has been established in prior works \cite{mms18,saitoh}. \medskip Note the simple facts: \medskip In the well-known formula $$ \log z = \log |z| + \arg z, $$ we have $$ \log 1 = \log 1 + \arg 1, $$ and $$ \log 0 = \log 0 + \arg 0. $$ Therefore, we have $$\arg 1 = \arg 0 =0. $$ \medskip Note, furthermore, that in the identity $$ \arg \overline{z} = - \arg z, $$ if the function $\arg z$ is extensible to the origin as an odd function, then the value $\arg 0$ has to be zero. \medskip In addition, note that in the formula $$ \arg z = \arctan \frac{y}{x} $$ for $x=y=0$ we have, from $0/0=0$, that $$ \arg 0=0. $$ \medskip For this Section, see \cite{mika, saitoh} for the details. \subsection{The direction of the general zero vector} We will be interested in some direction of the zero vector in general dimensions. 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 $$ {\bf v} = \sum_j v_j {\bf e}_j. $$ Then, the vector ${\bf v}$ and the coefficients $\{v_j\}$ correspond to one to onto on $\ell^2$. \medskip {\bf Statement:} {\it 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. } \medskip Note that the concept of direction of zero vector is reasonable in the senses $$ {\bf v} + {\bf u} = \sum_j v_j {\bf e}_j + \sum_j u_j {\bf e}_j = \sum_j (v_j + u_j) {\bf e}_j $$ and $$ {\bf v} - {\bf v} = \sum_j (v_j - v_j) {\bf e}_j = \sum_j (0) {\bf e}_j= {\bf 0}. $$ \bigskip {\bf Logical Problem:} {\it If we do not give the definition of direction of zero vector, in the fundametal equation $$ {\bf v} + {\bf 0} = {\bf v}, $$ we have the logical contradiction that by the addition of zero vector with no direction, we have the same direction of ${\bf v}$}. \medskip Indeed, in the above identity, we can not say the direction of vectors. \medskip This contradiction is similar that: The identity $$ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x}} + x= x $$ is not valid at $x=0$, because they are not define at $x=0$. \medskip However, we can still consider the open problem: \medskip {\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. } \medskip Indeed, in the 2 dimensional case, zero direction was given by the pleasant sense $ \arg 0=0$. \medskip > > --000000000000c402070630afc573 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable <div dir=3D"ltr"><div dir=3D"ltr"><p class=3D"MsoNormal" style=3D"margin:0p= t 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><sp= an style=3D"font-family:"\00ff2d\00ff33 \00660e\00671d";font-siz= e:10.5pt"><font face=3D"Century">Some general logical contradictions on und= efined objects:</font></span><span style=3D"font-family:"\00ff2d\00ff3= 3 \00660e\00671d";font-size:10.5pt"></span></p><p class=3D"MsoNormal"= style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;fo= nt-size:10.5pt"><span style=3D"font-family:"\00ff2d\00ff33 \00660e\00= 671d";font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-family:"\00ff2d\00ff33 \00660e\00671d&q= uot;;font-size:10.5pt"><font face=3D"Century">Please kindly look at the fol= lowing section. We find some general logical contradictions.</font></span><= span style=3D"font-family:"\00ff2d\00ff33 \00660e\00671d";font-s= ize:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.000= 1pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style=3D= "font-family:"\00ff2d\00ff33 \00660e\00671d";font-size:10.5pt"><= font face=3D"Century">May I have your kind comments and suggestions?</font>= </span><span style=3D"font-family:"\00ff2d\00ff33 \00660e\00671d"= ;;font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0= pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span = style=3D"font-family:"\00ff2d\00ff33 \00660e\00671d";font-size:1= 0.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.00= 01pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style= =3D"font-family:"\00ff2d\00ff33 \00660e\00671d";font-size:10.5pt= "><font face=3D"Century">With best regards,</font></span><span style=3D"fon= t-family:"\00ff2d\00ff33 \00660e\00671d";font-size:10.5pt"></spa= n></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:ju= stify;font-family:Century;font-size:10.5pt"><span style=3D"font-family:&quo= t;\00ff2d\00ff33 \00660e\00671d";font-size:10.5pt"><font face=3D"Cent= ury">Sincerely yours,</font></span><span style=3D"font-family:"\00ff2d= \00ff33 \00660e\00671d";font-size:10.5pt"></span></p><p class=3D"MsoN= ormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Cent= ury;font-size:10.5pt"><span style=3D"font-family:"\00ff2d\00ff33 \006= 60e\00671d";font-size:10.5pt"><font face=3D"Century">Saburou Saitoh</f= ont></span><span style=3D"font-family:"\00ff2d\00ff33 \00660e\00671d&= quot;;font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0= pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><a= href=3D"mailto:[email protected]" target=3D"_blank"><span style=3D"= font-family:"\00ff2d\00ff33 \00660e\00671d";font-size:10.5pt"><f= ont face=3D"Century">s</font></span><u><span style=3D"font-family:"\00= ff2d\00ff33 \00660e\00671d";color:rgb(0,0,255)"><font face=3D"Century= ">[email protected]</font></span></u></a><span style=3D"font-family:&= quot;\00ff2d\00ff33 \00660e\00671d";font-size:10.5pt"></span></p><p c= lass=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font= -family:Century;font-size:10.5pt"><span style=3D"font-family:"\00ff2d\= 00ff33 \00660e\00671d";font-size:10.5pt"><font face=3D"Century">2025.= 3.19.19:48</font></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt = 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span sty= le=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"ma= rgin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5= pt"><span style=3D"font-size:10.5pt">\section{Direction of the zero vector}= </span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" s= tyle=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font= -size:10.5pt"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D= "MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family= :Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">=C2=A0</span></= p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justif= y;font-family:Century;font-size:10.5pt"><span style=3D"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=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"marg= in:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt= "><span style=3D"font-size:10.5pt">\medskip</span><span style=3D"font-size:= 10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;= text-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"fon= t-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt = 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span= style=3D"font-size:10.5pt">Mika: {\it I am honored by your request to incl= ude my words in your paper. Regarding the zero vector, it does not lack dir= ection but rather has a direction of zero. While high school textbooks stat= e that a vector has both magnitude and direction, they often describe the z= ero vector as lacking direction, which is contradictory. Here's a more = precise explanation:</span><span style=3D"font-size:10.5pt"></span></p><p c= lass=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font= -family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">Definiti= on of a Vector: A vector is a quantity with both magnitude and direction.</= span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" sty= le=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-s= ize:10.5pt"><span style=3D"font-size:10.5pt">Zero Vector: The zero vector h= as 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 a= ccurate to say that its direction is zero.</span><span style=3D"font-size:1= 0.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;t= ext-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"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 t= erms, the expression "direction is zero" can still be valid.}</sp= an><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-size:10.5pt">2024.12.30.6:15</span><span styl= e=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0p= t 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><sp= an style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-size:10.5pt">\bigskip</span><span style=3D"fo= nt-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0= .0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span styl= e=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"mar= gin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5p= t"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal"= style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;fo= nt-size:10.5pt"><span style=3D"font-size:10.5pt">{\bf For complex numbers $= z$, and for $z=3D0$, $\log 0=3D \arg 0=3D0$}.</span><span style=3D"font-siz= e:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001p= t;text-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"f= ont-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0p= t 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><sp= an style=3D"font-size:10.5pt">\medskip</span><span style=3D"font-size:10.5p= t"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-= align:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-siz= e:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0= .0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span styl= e=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"mar= gin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5p= t"><span style=3D"font-size:10.5pt">Here, we will consider vectors and comp= lex numbers $z$ as two-dimensional vectors. We state that the direction of = the zero vector is zero. However, the precise meaning is $\arg 0 =3D0$.</sp= an><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"Mso= Normal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Cen= tury;font-size:10.5pt"><span style=3D"font-size:10.5pt">\medskip</span><spa= n style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"mar= gin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5p= t"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal"= style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;fo= nt-size:10.5pt"><span style=3D"font-size:10.5pt">{\it The direction of zero= $z=3D0$ exists as in other vectors $z$.}</span><span style=3D"font-size:10= .5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;te= xt-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-= size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0p= t 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span s= tyle=3D"font-size:10.5pt">\medskip</span><span style=3D"font-size:10.5pt"><= /span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-alig= n:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10= .5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.000= 1pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style=3D= "font-size:10.5pt">This definition, along with its natural motivation and m= any applications, has been established in prior works \cite{mms18,saitoh}.<= /span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" st= yle=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-= size:10.5pt"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"= MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:= Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">\medskip</span><= span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"= margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10= .5pt"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNorm= al" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century= ;font-size:10.5pt"><span style=3D"font-size:10.5pt">Note the simple facts:<= /span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" st= yle=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-= size:10.5pt"><span style=3D"font-size:10.5pt">\medskip</span><span style=3D= "font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0p= t 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span s= tyle=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"= margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10= .5pt"><span style=3D"font-size:10.5pt">In the well-known formula</span><spa= n style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"mar= gin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5p= t"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal"= style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;fo= nt-size:10.5pt"><span style=3D"font-size:10.5pt">$$</span><span style=3D"fo= nt-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0= .0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span styl= e=3D"font-size:10.5pt">\log z =3D \log |z| + \arg z,</span><span style=3D"f= ont-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt = 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span sty= le=3D"font-size:10.5pt">$$</span><span style=3D"font-size:10.5pt"></span></= p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justif= y;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">we= have</span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNorm= al" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century= ;font-size:10.5pt"><span style=3D"font-size:10.5pt">$$</span><span style=3D= "font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0p= t 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span s= tyle=3D"font-size:10.5pt">\log 1 =3D \log 1 + \arg 1,</span><span style=3D"= font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt= 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span st= yle=3D"font-size:10.5pt">$$</span><span style=3D"font-size:10.5pt"></span><= /p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justi= fy;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">a= nd</span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal"= style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;fo= nt-size:10.5pt"><span style=3D"font-size:10.5pt">$$</span><span style=3D"fo= nt-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0= .0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span styl= e=3D"font-size:10.5pt">\log 0 =3D \log 0 + \arg 0.</span><span style=3D"fon= t-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.= 0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style= =3D"font-size:10.5pt">$$</span><span style=3D"font-size:10.5pt"></span></p>= <p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;= font-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">Ther= efore, we have</span><span style=3D"font-size:10.5pt"></span></p><p class= =3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-fam= ily:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">=C2=A0</span= ></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:jus= tify;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt"= >$$\arg 1 =3D \arg 0 =3D0.</span><span style=3D"font-size:10.5pt"></span></= p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justif= y;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">$$= </span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" s= tyle=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font= -size:10.5pt"><span style=3D"font-size:10.5pt">\medskip</span><span style= =3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt= 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><spa= n style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-size:10.5pt">Note, furthermore, that in the i= dentity</span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNo= rmal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Centu= ry;font-size:10.5pt"><span style=3D"font-size:10.5pt">$$</span><span style= =3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt= 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><spa= n style=3D"font-size:10.5pt">\arg \overline{z} =3D - \arg z,</span><span st= yle=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:= 0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><= span style=3D"font-size:10.5pt">$$</span><span style=3D"font-size:10.5pt"><= /span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-alig= n:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10= .5pt">if the function $\arg z$ is extensible to the origin as an odd functi= on, then the value $\arg 0$ has to be zero.</span><span style=3D"font-size:= 10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;= text-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"fon= t-size:10.5pt">\medskip</span><span style=3D"font-size:10.5pt"></span></p><= p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;f= ont-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">=C2= =A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-= align:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-siz= e:10.5pt">In addition, note that in the formula</span><span style=3D"font-s= ize:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.000= 1pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style=3D= "font-size:10.5pt">$$</span><span style=3D"font-size:10.5pt"></span></p><p = class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;fon= t-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">\arg z = =3D \arctan \frac{y}{x}</span><span style=3D"font-size:10.5pt"></span></p><= p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;f= ont-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">$$</s= pan><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" styl= e=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-si= ze:10.5pt"><span style=3D"font-size:10.5pt">for $x=3Dy=3D0$ we have, from $= 0/0=3D0$,</span><span style=3D"font-size:10.5pt"></span></p><p class=3D"Mso= Normal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Cen= tury;font-size:10.5pt"><span style=3D"font-size:10.5pt">that</span><span st= yle=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:= 0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><= span style=3D"font-size:10.5pt">$$</span><span style=3D"font-size:10.5pt"><= /span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-alig= n:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10= .5pt">\arg 0=3D0.</span><span style=3D"font-size:10.5pt"></span></p><p clas= s=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-fa= mily:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">$$</span><s= pan style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"m= argin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.= 5pt"><span style=3D"font-size:10.5pt">\medskip</span><span style=3D"font-si= ze:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001= pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"= font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0= pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><s= pan style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" styl= e=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-si= ze:10.5pt"><span style=3D"font-size:10.5pt">For this Section, see \cite{mik= a, saitoh} for the details.</span><span style=3D"font-size:10.5pt"></span><= /p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justi= fy;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">= =C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;te= xt-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-= size:10.5pt">\subsection{The direction of the general zero vector}</span><s= pan style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"m= argin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.= 5pt"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNorma= l" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;= font-size:10.5pt"><span style=3D"font-size:10.5pt">We will be interested in= some direction of the zero vector in general dimensions.</span><span style= =3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt= 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><spa= n style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-size:10.5pt">In order to state the representa= tion precisely, we shall consider vectors as elements of a separable Hilber= t 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 s= tyle=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin= :0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt">= <span style=3D"font-size:10.5pt">$$</span><span style=3D"font-size:10.5pt">= </span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-ali= gn:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-size:1= 0.5pt">{\bf v} =3D \sum_j v_j {\bf e}_j.</span><span style=3D"font-size:10.= 5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;tex= t-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-s= ize:10.5pt">$$</span><span style=3D"font-size:10.5pt"></span></p><p class= =3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-fam= ily:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">Then, the ve= ctor ${\bf v}$ and the coefficients $\{v_j\}$ correspond to one to onto on = $\ell^2$.</span><span style=3D"font-size:10.5pt"></span></p><p class=3D"Mso= Normal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Cen= tury;font-size:10.5pt"><span style=3D"font-size:10.5pt">\medskip</span><spa= n style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"mar= gin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5p= t"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal"= style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;fo= nt-size:10.5pt"><span style=3D"font-size:10.5pt">{\bf Statement:} {\it</spa= n><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-size:10.5pt">We shall define the direction of= ${\bf v}$ by the coefficients $\{v_j\}$ that is determined by a positive m= ultiplication of $\{v_j\}$ and the zero vector is represended by all $v_j= =3D0$. Therefore, the direction of the zero vector may be considered as zer= o in this sense.</span><span style=3D"font-size:10.5pt"></span></p><p class= =3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-fam= ily:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">}</span><spa= n style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"mar= gin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5p= t"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal"= style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;fo= nt-size:10.5pt"><span style=3D"font-size:10.5pt">\medskip</span><span style= =3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt= 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><spa= n style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-size:10.5pt">Note that the concept of directi= on of zero vector is reasonable in the senses</span><span style=3D"font-siz= e:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001p= t;text-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"f= ont-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0p= t 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><sp= an style=3D"font-size:10.5pt">$$</span><span style=3D"font-size:10.5pt"></s= pan></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:= justify;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5= pt">{\bf v} + {\bf u} =3D \sum_j v_j {\bf e}_j + \sum_j u_j {\bf e}_j =3D \= sum_j (v_j + u_j) {\bf e}_j</span><span style=3D"font-size:10.5pt"></span><= /p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justi= fy;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">$= $</span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" = style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;fon= t-size:10.5pt"><span style=3D"font-size:10.5pt">and</span><span style=3D"fo= nt-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0= .0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span styl= e=3D"font-size:10.5pt">$$</span><span style=3D"font-size:10.5pt"></span></p= ><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify= ;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">{\b= f v} - {\bf v} =3D \sum_j (v_j - v_j) {\bf e}_j =3D \sum_j (0) {\bf e}_j=3D= {\bf 0}.</span><span style=3D"font-size:10.5pt"></span></p><p class=3D"Mso= Normal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Cen= tury;font-size:10.5pt"><span style=3D"font-size:10.5pt">$$</span><span styl= e=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0p= t 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><sp= an style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-size:10.5pt">\bigskip</span><span style=3D"fo= nt-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0= .0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span styl= e=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"mar= gin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5p= t"><span style=3D"font-size:10.5pt">{\bf Logical Problem:} {\it If we do no= t give the definition of direction of zero vector, in the fundametal equati= on</span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal"= style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;fo= nt-size:10.5pt"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class= =3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-fam= ily:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">$$</span><sp= an style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"ma= rgin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5= pt"><span style=3D"font-size:10.5pt">{\bf v} + {\bf 0} =3D {\bf v},</span><= span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"= margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10= .5pt"><span style=3D"font-size:10.5pt">$$</span><span style=3D"font-size:10= .5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;te= xt-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"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=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D= "margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:1= 0.5pt"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNor= mal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Centur= y;font-size:10.5pt"><span style=3D"font-size:10.5pt">\medskip</span><span s= tyle=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin= :0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt">= <span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" st= yle=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-= size:10.5pt"><span style=3D"font-size:10.5pt">Indeed, in the above identity= , we can not say the direction of vectors.</span><span style=3D"font-size:1= 0.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;t= ext-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"font= -size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0= pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><span = style=3D"font-size:10.5pt">\medskip</span><span style=3D"font-size:10.5pt">= </span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-ali= gn:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-size:1= 0.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.00= 01pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style= =3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"marg= in:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt= "><span style=3D"font-size:10.5pt">This contradiction is similar that: The = identity</span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoN= ormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Cent= ury;font-size:10.5pt"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p = class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;fon= t-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">$$</spa= n><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-size:10.5pt">\frac{1}{\sqrt{x}} - \frac{1}{\s= qrt{x}} + x=3D x</span><span style=3D"font-size:10.5pt"></span></p><p class= =3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-fam= ily:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">$$</span><sp= an style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"ma= rgin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5= pt"><span style=3D"font-size:10.5pt">is not valid at $x=3D0$, because they = are not define at $x=3D0$.</span><span style=3D"font-size:10.5pt"></span></= p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justif= y;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">= =C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;te= xt-align:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-= size:10.5pt">\medskip</span><span style=3D"font-size:10.5pt"></span></p><p = class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;fon= t-family:Century;font-size:10.5pt"><span style=3D"font-size:10.5pt">=C2=A0<= /span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-alig= n:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-size:10= .5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.000= 1pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style=3D= "font-size:10.5pt">However, we can still consider the open problem:</span><= span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"= margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10= .5pt"><span style=3D"font-size:10.5pt">\medskip</span><span style=3D"font-s= ize:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.000= 1pt;text-align:justify;font-family:Century;font-size:10.5pt"><span style=3D= "font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" style=3D"margin:= 0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt"><= span style=3D"font-size:10.5pt">{\bf Open problem 1:} {\it As in two dimens= ions, could we find some natural formulation that the direction of zero vec= tor is zero, in general dimensions.</span><span style=3D"font-size:10.5pt">= </span></p><p class=3D"MsoNormal" style=3D"margin:0pt 0pt 0.0001pt;text-ali= gn:justify;font-family:Century;font-size:10.5pt"><span style=3D"font-size:1= 0.5pt">}</span><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoN= ormal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Cent= ury;font-size:10.5pt"><span style=3D"font-size:10.5pt">\medskip</span><span= style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"marg= in:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5pt= "><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"MsoNormal" = style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;fon= t-size:10.5pt"><span style=3D"font-size:10.5pt">Indeed, in the 2 dimensiona= l case, zero direction was given by the pleasant sense $ \arg 0=3D0$.</span= ><span style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style= =3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-siz= e:10.5pt"><span style=3D"font-size:10.5pt">=C2=A0</span></p><p class=3D"Mso= Normal" style=3D"margin:0pt 0pt 0.0001pt;text-align:justify;font-family:Cen= tury;font-size:10.5pt"><span style=3D"font-size:10.5pt">\medskip</span><spa= n style=3D"font-size:10.5pt"></span></p><p class=3D"MsoNormal" style=3D"mar= gin:0pt 0pt 0.0001pt;text-align:justify;font-family:Century;font-size:10.5p= t"><span style=3D"font-size:10.5pt">=C2=A0</span></p></div><div class=3D"gm= ail_quote gmail_quote_container"><blockquote class=3D"gmail_quote" style=3D= "margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-le= ft:1ex"><br> </blockquote></div></div> --000000000000c402070630afc573-- --===============1749799079164763482== Content-Type: text/plain; charset="us-ascii" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Disposition: inline --===============1749799079164763482== Content-Type: text/plain; charset="us-ascii" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Disposition: inline _______________________________________________ hol-info mailing list [email protected] https://lists.sourceforge.net/lists/listinfo/hol-info --===============1749799079164763482==--