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complex1

OpenMath Content Dictionary: complex1
Canonical URL:
http://www.openmath.org/cd/complex1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
complex1.ocd
CD as XML Encoded OpenMath:
complex1.omcd
Defines:
argument, complex_cartesian, complex_polar, conjugate, imaginary, real
Date:
2004-03-30
Version:
3 (Revision 1)
Review Date:
2006-03-30
Status:
official 
     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs

This CD is intended to be `compatible' with the MathML view of operations on and constructors for complex numbers.

complex_cartesian
Role:
application
Description:

This symbol represents a constructor function for complex numbers specified as the Cartesian coordinates of the relevant point on the complex plane. It takes two arguments, the first is a number x to denote the real part and the second a number y to denote the imaginary part of the complex number x + i y. (Where i is the square root of -1.)

Commented Mathematical property (CMP):
for all x,y | complex_cartesian(x,y) = x + iy
Formal Mathematical property (FMP): 
  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMBIND>
      <OMS cd="quant1" name="forall"/>
      <OMBVAR>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="complex1" name="complex_cartesian"/>
          <OMV name="x"/>
          <OMV name="y"/>
        </OMA>
        <OMA>
          <OMS cd="arith1" name="plus"/>
          <OMV name="x"/>
          <OMA>
            <OMS cd="arith1" name="times"/>
            <OMS cd="nums1" name="i"/>
            <OMV name="y"/>
          </OMA>
        </OMA>
      </OMA>
    </OMBIND>
  </OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
   <apply><csymbol cd="arith1">plus</csymbol>
    <ci>x</ci>
    <apply><csymbol cd="arith1">times</csymbol><csymbol cd="nums1">i</csymbol><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$x, $y -> $x | $y = $x + nums1.i * $y]

∀   x , y . x + y ⁢ i = x + i ⁢ y

Signatures:
sts
real
Role:
application
Description:

This represents the real part of a complex number

Commented Mathematical property (CMP):
for all x,y | x = real(x+iy)
Formal Mathematical property (FMP): 
  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMBIND>
      <OMS cd="quant1" name="forall"/>
      <OMBVAR>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMV name="x"/>
        <OMA>
          <OMS name="real" cd="complex1"/>
          <OMA>
            <OMS name="complex_cartesian" cd="complex1"/>
            <OMV name="x"/>
            <OMV name="y"/>
          </OMA>
        </OMA>
      </OMA>
    </OMBIND>
  </OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="complex1">real</csymbol>
    <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$x, $y -> $x = complex1.real($x | $y)]

∀   x , y . x = real ⁡ ( x + y ⁢ i )

Signatures:
sts
imaginary
Role:
application
Description:

This represents the imaginary part of a complex number

Commented Mathematical property (CMP):
for all x,y | y = imaginary(x+iy)
Formal Mathematical property (FMP): 
  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMBIND>
      <OMS cd="quant1" name="forall"/>
      <OMBVAR>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMV name="y"/>
        <OMA>
          <OMS name="imaginary" cd="complex1"/>
          <OMA>
            <OMS name="complex_cartesian" cd="complex1"/>
            <OMV name="x"/>
            <OMV name="y"/>
          </OMA>
        </OMA>
      </OMA>
    </OMBIND>
  </OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <ci>y</ci>
   <apply><csymbol cd="complex1">imaginary</csymbol>
    <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$x, $y -> $y = complex1.imaginary($x | $y)]

∀   x , y . y = imaginary ⁡ ( x + y ⁢ i )

Signatures:
sts
complex_polar
Role:
application
Description:

This symbol represents a constructor function for complex numbers specified as the polar coordinates of the relevant point on the complex plane. It takes two arguments, the first is a nonnegative number r to denote the magnitude and the second a number theta (given in radians) to denote the argument of the complex number r e^(i theta). (i and e are defined as in this CD).

Commented Mathematical property (CMP):
for all r,a | complex_polar(r,a) = r*e^(a*i)
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMBIND>
    <OMS cd="quant1" name="forall"/>
    <OMBVAR>
      <OMV name="r"/>
      <OMV name="a"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
        <OMV name="r"/>
        <OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMV name="r"/>
	<OMA>
	  <OMS cd="transc1" name="exp"/>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMV name="a"/>
	    <OMS cd="nums1" name="i"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>r</ci></bvar>
  <bvar><ci>a</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="complex1">complex_polar</csymbol><ci>r</ci><ci>a</ci></apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <ci>r</ci>
    <apply><csymbol cd="transc1">exp</csymbol>
     <apply><csymbol cd="arith1">times</csymbol><ci>a</ci><csymbol cd="nums1">i</csymbol></apply>
    </apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$r, $a -> complex1.complex_polar($r, $a) = $r * exp($a * nums1.i)]

∀   r , a . r ⁢ e a ⁢ i = r ⁢ exp ⁡ ( a ⁢ i )

Commented Mathematical property (CMP):
for all x,y,r,a | (r sin a = y and r cos a = x) implies (complex_polar(r,a) = complex_cartesian(x,y)
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
    <OMV name="y"/>
    <OMV name="r"/>
    <OMV name="a"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="arith1" name="times"/>
	  <OMV name="r"/>
	  <OMA>
	    <OMS cd="transc1" name="sin"/>
	    <OMV name="a"/>
	  </OMA>
        </OMA>
	<OMV name="y"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="arith1" name="times"/>
	  <OMV name="r"/>
	  <OMA>
	    <OMS cd="transc1" name="cos"/>
	    <OMV name="a"/>
	  </OMA>
        </OMA>
	<OMV name="x"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
        <OMV name="r"/>
        <OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="complex1" name="complex_cartesian"/>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <bvar><ci>r</ci></bvar>
  <bvar><ci>a</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="logic1">and</csymbol>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply><csymbol cd="arith1">times</csymbol>
      <ci>r</ci>
      <apply><csymbol cd="transc1">sin</csymbol><ci>a</ci></apply>
     </apply>
     <ci>y</ci>
    </apply>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply><csymbol cd="arith1">times</csymbol>
      <ci>r</ci>
      <apply><csymbol cd="transc1">cos</csymbol><ci>a</ci></apply>
     </apply>
     <ci>x</ci>
    </apply>
   </apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="complex1">complex_polar</csymbol><ci>r</ci><ci>a</ci></apply>
    <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$x, $y, $r, $a -> $r * sin($a) = $y and $r * cos($a) = $x ==> complex1.complex_polar($r, $a) = $x | $y]

∀   x , y , r , a . r ⁢ sin ⁡ ( a ) = y ∧ r ⁢ cos ⁡ ( a ) = x ⇒ r ⁢ e a ⁢ i = x + y ⁢ i

Commented Mathematical property (CMP):
for all x | if a is a real number and k is an integer then complex_polar(x,a) = complex_polar(x,a+2*pi*k)
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="set1" name="in"/>
	<OMV name="a"/>
	<OMS cd="setname1" name="R"/>
      </OMA>
      <OMA>
        <OMS cd="set1" name="in"/>
	<OMV name="k"/>
	<OMS cd="setname1" name="Z"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
	<OMV name="x"/>
	<OMV name="a"/>
      </OMA>
      <OMA>
        <OMS cd="complex1" name="complex_polar"/>
	<OMV name="x"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMV name="a"/>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMI> 2 </OMI>
	    <OMS cd="nums1" name="pi"/>
	    <OMV name="k"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="logic1">and</csymbol>
    <apply><csymbol cd="set1">in</csymbol><ci>a</ci><csymbol cd="setname1">R</csymbol></apply>
    <apply><csymbol cd="set1">in</csymbol><ci>k</ci><csymbol cd="setname1">Z</csymbol></apply>
   </apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="complex1">complex_polar</csymbol><ci>x</ci><ci>a</ci></apply>
    <apply><csymbol cd="complex1">complex_polar</csymbol>
     <ci>x</ci>
     <apply><csymbol cd="arith1">plus</csymbol>
      <ci>a</ci>
      <apply><csymbol cd="arith1">times</csymbol>
       <cn type="integer">2</cn>
       <csymbol cd="nums1">pi</csymbol>
       <ci>k</ci>
      </apply>
     </apply>
    </apply>
   </apply>
  </apply>
 </bind>
</math>
forall

[

x

] . (

implies

(

and

(

in

(

a

,

R

) ,

in

(

k

,

Z

) ) ,

eq

(

complex_polar

(

x

,

a

) ,

complex_polar

(

x

,

plus

(

a

,

times

( 2 ,

pi

,

k

) ) ) ) ) )

quant1.forall[$x -> set1.in($a, setname1.R) and set1.in($k, setname1.Z) ==> complex1.complex_polar($x, $a) = complex1.complex_polar($x, $a + 2 * nums1.pi * $k)]

∀   x . a ∈ R ∧ k ∈ Z ⇒ x ⁢ e a ⁢ i = x ⁢ e ( a + 2 ⁢ π ⁢ k ) ⁢ i

Example:
i = complex_polar(1,pi/2) 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"><OMA>
  <OMS cd="relation1" name="eq"/>
  <OMS cd="nums1" name="i"/>
  <OMA>
    <OMS cd="complex1" name="complex_polar"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMS cd="nums1" name="pi"/>
      <OMI> 2 </OMI>
    </OMA>
  </OMA>
</OMA></OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <csymbol cd="nums1">i</csymbol>
  <apply><csymbol cd="complex1">complex_polar</csymbol>
   <csymbol cd="alg1">one</csymbol>
   <apply><csymbol cd="arith1">divide</csymbol>
    <csymbol cd="nums1">pi</csymbol>
    <cn type="integer">2</cn>
   </apply>
  </apply>
 </apply>
</math>

nums1.i = complex1.complex_polar(alg1.one, nums1.pi / 2)

Signatures:
sts
argument
Role:
application
Description:

This symbol represents the unary function which returns the argument of a complex number, viz. the angle which a straight line drawn from the number to zero makes with the Real line (measured anti-clockwise). The argument to the symbol is the complex number whos argument is being taken.

Commented Mathematical property (CMP):
for all r,a | argument(complex_polar(r,a)=a)
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMBIND>
    <OMS cd="quant1" name="forall"/>
    <OMBVAR>
      <OMV name="r"/>
      <OMV name="a"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="argument"/>
        <OMA>
          <OMS cd="complex1" name="complex_polar"/>
          <OMV name="r"/>
          <OMV name="a"/>
        </OMA>
      </OMA>
      <OMV name="a"/>
    </OMA>
  </OMBIND>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>r</ci></bvar>
  <bvar><ci>a</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="complex1">argument</csymbol>
    <apply><csymbol cd="complex1">complex_polar</csymbol><ci>r</ci><ci>a</ci></apply>
   </apply>
   <ci>a</ci>
  </apply>
 </bind>
</math>

quant1.forall[$r, $a -> complex1.argument(complex1.complex_polar($r, $a)) = $a]

∀   r , a . argument ⁡ ( r ⁢ e a ⁢ i ) = a

Commented Mathematical property (CMP):
the argument of x+i*y = arctan(y/x) (if x is positive)
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="relation1" name="gt"/>
      <OMV name="x"/>
      <OMS cd="alg1" name="zero"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="complex1" name="argument"/>
        <OMA>
          <OMS cd="complex1" name="complex_cartesian"/>
          <OMV name="x"/>
          <OMV name="y"/>
        </OMA>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="arctan"/>
        <OMA>
          <OMS cd="arith1" name="divide"/>
	  <OMV name="y"/>
	  <OMV name="x"/>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="relation1">gt</csymbol><ci>x</ci><csymbol cd="alg1">zero</csymbol></apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="complex1">argument</csymbol>
    <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
   <apply><csymbol cd="transc1">arctan</csymbol>
    <apply><csymbol cd="arith1">divide</csymbol><ci>y</ci><ci>x</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

$x > alg1.zero ==> complex1.argument($x | $y) = arctan($y / $x)

x > 0 ⇒ argument ⁡ ( x + y ⁢ i ) = arctan ⁡ ( y x )

Commented Mathematical property (CMP):
the argument of x+i*y = arctan(y,x) (two-argument arctan from transc2)
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="complex1" name="argument"/>
      <OMA>
        <OMS cd="complex1" name="complex_cartesian"/>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="transc2" name="arctan"/>
      <OMV name="y"/>
      <OMV name="x"/>
    </OMA>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="complex1">argument</csymbol>
   <apply><csymbol cd="complex1">complex_cartesian</csymbol><ci>x</ci><ci>y</ci></apply>
  </apply>
  <apply><csymbol cd="transc2">arctan</csymbol><ci>y</ci><ci>x</ci></apply>
 </apply>
</math>

complex1.argument($x | $y) = transc2.arctan($y, $x)

argument ⁡ ( x + y ⁢ i ) = arctan ⁡ ( y , x )

Signatures:
sts
conjugate
Role:
application
Description:

A unary operator representing the complex conjugate of its argument.

Commented Mathematical property (CMP):
if a is a complex number then (conjugate(a) + a) is a real number
Formal Mathematical property (FMP): 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="a"/>
      <OMS cd="setname1" name="C"/>
    </OMA>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMA>
        <OMS cd="arith1" name="plus"/>
        <OMA>
          <OMS cd="complex1" name="conjugate"/>
          <OMV name="a"/>
        </OMA>
        <OMV name="a"/>
      </OMA>
      <OMS cd="setname1" name="R"/>
    </OMA>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">in</csymbol><ci>a</ci><csymbol cd="setname1">C</csymbol></apply>
  <apply><csymbol cd="set1">in</csymbol>
   <apply><csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="complex1">conjugate</csymbol><ci>a</ci></apply>
    <ci>a</ci>
   </apply>
   <csymbol cd="setname1">R</csymbol>
  </apply>
 </apply>
</math>

set1.in($a, setname1.C) ==> set1.in(complex1.conjugate($a) + $a, setname1.R)

Signatures:
sts

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