OpenMath Content Dictionary: asymp1

Canonical URL:

http://www.openmath.org/cd/asymp1.ocd

CD File:

asymp1.ocd

CD as XML Encoded OpenMath:

asymp1.omcd

Defines:

asymptotic, o, O, omega, Omega, theta

Date:

1999-10-19

Version:

2

Review Date:

Status:

experimental

Uses CD:

alg1, arith1, fns1, limit1, linalg1, logic1, nums1, quant1, relation1, setname1, setname3, set1

This CD provides a representation of various asymptotic set constructors (O, \Omega, etc.) The constructors represent sets of functions : R -> R.

O

Description:

The O symbol represents a unary function which constructs a set of certain functions of type reals to reals. The condition f(n)=O(g(n)) is intended to express an upper bound condition on f.

Commented Mathematical property (CMP):

O(g) = { f:reals -> reals | exists c in positive reals and M in the naturals such that forall n geq M. |f(n)| leq c*g(n)}

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="asymp1" name="O"/>
      <OMV name="g"/>
    </OMA>
    <OMA>
      <OMS cd="set1" name="suchthat"/>
      <OMA>
        <OMS cd="setname3" name="function_set"/>
        <OMS cd="setname1" name="R"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="f"/>
        </OMBVAR>
        <OMBIND>
          <OMS cd="quant1" name="exists"/>
          <OMBVAR>
            <OMV name="c"/>
            <OMV name="M"/>
          </OMBVAR>
          <OMA>
            <OMS cd="logic1" name="and"/>
            <OMA>
              <OMS cd="set1" name="in"/>
              <OMV name="c"/>
              <OMS cd="setname1" name="R"/>
            </OMA>
            <OMA>
              <OMS cd="relation1" name="gt"/>
              <OMV name="c"/>
              <OMS cd="alg1" name="zero"/>
            </OMA>
            <OMA>
              <OMS cd="set1" name="in"/>
              <OMV name="M"/>
              <OMS cd="setname1" name="N"/>
            </OMA>
            <OMBIND>
              <OMS cd="quant1" name="forall"/>
              <OMBVAR>
                <OMV name="n"/>
              </OMBVAR>
              <OMA>
                <OMS cd="logic1" name="implies"/>
                <OMA>
                  <OMS cd="logic1" name="and"/> 
                  <OMA>
                    <OMS cd="set1" name="in"/>
                    <OMV name="n"/>
                    <OMS cd="setname1" name="N"/>
                  </OMA>
                  <OMA>
                    <OMS cd="relation1" name="geq"/>
                    <OMV name="n"/>
                    <OMV name="M"/>
                  </OMA>
                </OMA>
                <OMA>
                  <OMS cd="relation1" name="leq"/>
                  <OMA>
		    <OMS cd="arith1" name="abs"/>
		    <OMA>
                      <OMV name="f"/>
                      <OMV name="n"/>
		    </OMA>
                  </OMA>
                  <OMA>
                    <OMS cd="arith1" name="times"/>
                    <OMV name="c"/>
                    <OMA>
                      <OMV name="g"/>
                      <OMV name="n"/>
                    </OMA>
                  </OMA>
                </OMA>
              </OMA>
            </OMBIND>
          </OMA>
        </OMBIND>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

eq (O ( g) , suchthat (function_set (R, R) , lambda [ f ] . (exists [ c M ] . (and (in ( c, R) , gt ( c, zero) , in ( M, N) , forall [ n ] . (implies (and (in ( n, N) , geq ( n, M) ) , leq (abs ( f ( n) ) , times ( c, g ( n) ) ) ) ) ) ) ) ) )

$$O\left(g\right)=\left\{f\in \mathrm{function\_set}(\mathbb{R},\mathbb{R})\right|\exists \hspace{0.17em}c,M.c\in \mathbb{R}\wedge c>0\wedge M\in \mathbb{N}\wedge \forall \hspace{0.17em}n.n\in \mathbb{N}\wedge n\ge M\Rightarrow \left|f\left(n\right)\right|\le cg\left(n\right)\}$$

Signatures:

sts

o

Description:

The o symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The condition f(n) = o(g(n)) is intended to express a lower bouund condition on f. Formally we say that f(n) = o(g(n)) if and only if the limit as n tends to infinity of f(n)/g(n) exists and is equal to 0.

Commented Mathematical property (CMP):

o(g) = {f : reals -> reals | the limit as x tends to infinity of f(x)/g(x) is 0}

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="asymp1" name="o"/>
      <OMV name="g"/>
    </OMA>
    <OMA>
      <OMS cd="set1" name="suchthat"/>
      <OMA>
        <OMS cd="setname3" name="function_set"/>
        <OMS cd="setname1" name="R"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
	<OMBVAR>
	  <OMV name="f"/>
	</OMBVAR>
	<OMA>
	  <OMS cd="relation1" name="eq"/>
	  <OMA>
	    <OMS cd="limit1" name="limit"/>
	    <OMS cd="nums1" name="infinity"/>
	    <OMS cd="limit1" name="below"/>
	    <OMBIND>
	      <OMS cd="fns1" name="lambda"/>
	      <OMBVAR>
	        <OMV name="x"/>
	      </OMBVAR>
	      <OMA>
	        <OMS cd="arith1" name="divide"/>
		<OMA>
		  <OMV name="f"/>
		  <OMV name="x"/>
		</OMA>
		<OMA>
		  <OMV name="g"/>
		  <OMV name="x"/>
		</OMA>
	      </OMA>
	    </OMBIND>
	  </OMA>
	  <OMS cd="alg1" name="zero"/>
	</OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

eq (o ( g) , suchthat (function_set (R, R) , lambda [ f ] . (eq (limit (infinity, below, lambda [ x ] . (divide ( f ( x) , g ( x) ) ) ) , zero) ) ) )

$$o\left(g\right)=\left\{f\in \mathrm{function\_set}(\mathbb{R},\mathbb{R})\right|\underset{x\to {\infty }_{-}}{\mathrm{limit}}\frac{f\left(x\right)}{g\left(x\right)}=0\}$$

Signatures:

sts

theta

Description:

The theta symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The theta symbol represents a set of functions which all have the same 'rate of growth'. Formally we say that f(x) = theta(g(x)) if and only if there are constants c_1 not= 0 and c_2 not= 0 and x_0 such that for all x > x_0 it is true that c_1*g(x) < f(x) < c_2*g(x).

Commented Mathematical property (CMP):

f(x) = theta(g(x)) if and only if there are constants c_1 not= 0 and c_2 not= 0 and x_0 such that for all x > x_0 it is true that c_1*g(x) < f(x) < c_2*g(x)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="asymp1" name="theta"/>
      <OMV name="g"/>
    </OMA>
    <OMA>
      <OMS cd="set1" name="suchthat"/>
      <OMA>
        <OMS cd="setname3" name="function_set"/>
        <OMS cd="setname1" name="R"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
	<OMBVAR>
	  <OMV name="f"/>
	</OMBVAR>
	<OMBIND>
	  <OMS cd="quant1" name="exists"/>
	  <OMBVAR>
	    <OMV name="c_1"/>
	    <OMV name="c_2"/>
	    <OMV name="x_0"/>
	  </OMBVAR>
	  <OMA>
	    <OMS cd="logic1" name="and"/>
	    <OMA>
	      <OMS cd="relation1" name="neq"/>
	      <OMV name="c_1"/>
	      <OMS cd="alg1" name="zero"/>
	    </OMA>
	    <OMA>
	      <OMS cd="relation1" name="neq"/>
	      <OMV name="c_2"/>
	      <OMS cd="alg1" name="zero"/>
	    </OMA>
	    <OMBIND>
	      <OMS cd="quant1" name="forall"/>
	      <OMBVAR>
	        <OMV name="x"/>
	      </OMBVAR>
	      <OMA>
	        <OMS cd="logic1" name="and"/>
		<OMA>
		  <OMS cd="relation1" name="gt"/>
		  <OMV name="x"/>
		  <OMV name="x_0"/>
		</OMA>
		<OMA>
		  <OMS cd="relation1" name="lt"/>
		  <OMA>
		    <OMS cd="arith1" name="times"/>
		    <OMV name="c_1"/>
		    <OMA>
  		      <OMV name="g"/>
		      <OMV name="x"/>
		    </OMA>
		  </OMA>
		  <OMA>
		    <OMV name="f"/>
		    <OMV name="x"/>
		  </OMA>
		</OMA>
		<OMA>
		  <OMS cd="relation1" name="lt"/>
		  <OMA>
		    <OMV name="f"/>
		    <OMV name="x"/>
		  </OMA>
		  <OMA>
		    <OMS cd="arith1" name="times"/>
		    <OMV name="c_2"/>
		    <OMA>
  		      <OMV name="g"/>
		      <OMV name="x"/>
		    </OMA>
		  </OMA>
		</OMA>
	      </OMA>
	    </OMBIND>
	  </OMA>
	</OMBIND>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

eq (theta ( g) , suchthat (function_set (R, R) , lambda [ f ] . (exists [ c_1 c_2 x_0 ] . (and (neq ( c_1, zero) , neq ( c_2, zero) , forall [ x ] . (and (gt ( x, x_0) , lt (times ( c_1, g ( x) ) , f ( x) ) , lt ( f ( x) , times ( c_2, g ( x) ) ) ) ) ) ) ) ) )

$$\mathrm{theta}\left(g\right)=\left\{f\in \mathrm{function\_set}(\mathbb{R},\mathbb{R})\right|\exists \hspace{0.17em}{\mathrm{c\_}}_1,{\mathrm{c\_}}_2,{\mathrm{x\_}}_0.{\mathrm{c\_}}_1\ne 0\wedge {\mathrm{c\_}}_2\ne 0\wedge \forall \hspace{0.17em}x.x>{\mathrm{x\_}}_0\wedge {\mathrm{c\_}}_{1}g\left(x\right)<f\left(x\right)\wedge f\left(x\right)<{\mathrm{c\_}}_{2}g\left(x\right)\}$$

Signatures:

sts

asymptotic

Description:

The asymptotic symbol represents a binary relation between two functions of type reals to reals. The asymptotic relation between two functions returns true if the two functions have the same rate of growth and more precisely there ratio approaches 1 as the variable approaches infinity. Formally we say that f(x) is asymptotic to g(x) if and only if the limit as x tends to infinity of f(x)/g(x) = 1.

Commented Mathematical property (CMP):

f(x) is asymptotic g(x) if and only if the limit as x tends to infinity of f(x)/g(x) = 1

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="logic1" name="equivalent"/>
    <OMA>
      <OMS cd="asymp1" name="asymptotic"/>
      <OMV name="f"/>
      <OMV name="g"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="limit1" name="limit"/>
        <OMS cd="nums1" name="infinity"/>
        <OMS cd="limit1" name="below"/>
        <OMBIND>
	  <OMS cd="fns1" name="lambda"/>
	  <OMBVAR>
	    <OMV name="x"/>
	  </OMBVAR>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMA>
	      <OMV name="f"/>
	      <OMV name="x"/>
	    </OMA>
	    <OMA>
	      <OMV name="g"/>
	      <OMV name="x"/>
	    </OMA>
	  </OMA>
        </OMBIND>
      </OMA>
      <OMS cd="alg1" name="one"/>
    </OMA>
  </OMA>
</OMOBJ>

equivalent (asymptotic ( f, g) , eq (limit (infinity, below, lambda [ x ] . (divide ( f ( x) , g ( x) ) ) ) , one) )

$$\mathrm{asymptotic}(f,g)\equiv (\underset{x\to {\infty }_{-}}{\mathrm{limit}}\frac{f\left(x\right)}{g\left(x\right)}=1)$$

Signatures:

sts

omega

Description:

The omega symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The omega symbol represents a set of functions such that for any function in the set omega(g(x)), f(x); it is not true that f(x) is in o(g(x)). Formally we say that f(x) = omega(g(x)) if and only if there is an epsilon > 0 and an infinite sequence x_1, x_2, x_3, ... such that for all j then abs(f(x_j)) > epsilon * g(x_j).

Commented Mathematical property (CMP):

f(x) is omega(g(x)) if and only if it is not true that f(x) is o(g(x))

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="asymp1" name="omega"/>
      <OMV name="g"/>
    </OMA>
    <OMA>
      <OMS cd="set1" name="suchthat"/>
      <OMA>
        <OMS cd="setname3" name="function_set"/>
        <OMS cd="setname1" name="R"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
	<OMBVAR>
	  <OMV name="f"/>
	</OMBVAR>
	<OMA>
	  <OMS cd="logic1" name="not"/>
	  <OMA>
	    <OMS cd="set1" name="in"/>
	    <OMA>
	      <OMV name="f"/>
	      <OMV name="x"/>
	    </OMA>
	    <OMA>
	      <OMS cd="asymp1" name="o"/>
	      <OMV name="g"/>
	    </OMA>
	  </OMA>
	</OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

eq (omega ( g) , suchthat (function_set (R, R) , lambda [ f ] . (not (in ( f ( x) , o ( g) ) ) ) ) )

$$\mathrm{omega}\left(g\right)=\left\{f\in \mathrm{function\_set}(\mathbb{R},\mathbb{R})\right|\neg (f\left(x\right)\in o\left(g\right))\}$$

Commented Mathematical property (CMP):

f(x) = omega(g(x)) if and only if there is an epsilon > 0 and an infinite sequence x_1, x_2, x_3, ... such that for all j then abs(f(x_j)) > epsilon * g(x_j).

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="logic1" name="equivalent"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="f"/>
      <OMA>
        <OMS cd="asymp1" name="omega"/>
	<OMV name="g"/>
      </OMA>
    </OMA>
    <OMBIND>
      <OMS cd="quant1" name="exists"/>
      <OMBVAR>
	  <OMV name="epsilon"/>
	  <OMV name="seq"/>
      </OMBVAR>
      <OMA>
        <OMS cd="logic1" name="and"/>
	<OMA>
	  <OMS cd="relation1" name="gt"/>
	  <OMV name="epsilon"/>
	  <OMS cd="alg1" name="zero"/>
	</OMA>
	<OMBIND>
	  <OMS cd="quant1" name="forall"/>
	  <OMBVAR>
	    <OMV name="j"/>
	  </OMBVAR>
	  <OMA>
	    <OMS cd="relation1" name="gt"/>
	    <OMA>
	      <OMS cd="arith1" name="abs"/>
	      <OMA>
	        <OMV name="f"/>
		<OMA>
		  <OMS cd="linalg1" name="vector_selector"/>
		  <OMV name="j"/>
		  <OMV name="seq"/>
		</OMA>
	      </OMA>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="times"/>
	      <OMV name="epsilon"/>
	      <OMA>
	        <OMV name="g"/>
		<OMA>
		  <OMS cd="linalg1" name="vector_selector"/>
		  <OMV name="j"/>
		  <OMV name="seq"/>
		</OMA>
	      </OMA>
	    </OMA>
	  </OMA>
	</OMBIND>
      </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>

equivalent (in ( f, omega ( g) ) , exists [ epsilon seq ] . (and (gt ( epsilon, zero) , forall [ j ] . (gt (abs ( f (vector_selector ( j, seq) ) ) , times ( epsilon, g (vector_selector ( j, seq) ) ) ) ) ) ) )

$$f\in \mathrm{omega}\left(g\right)\equiv \exists \hspace{0.17em}\mathrm{epsilon},\mathrm{seq}.\mathrm{epsilon}>0\wedge \forall \hspace{0.17em}j.\left|f\left({\mathrm{seq}}_j\right)\right|>\mathrm{epsilon}g\left({\mathrm{seq}}_j\right)$$

Signatures:

sts

Omega

Description:

The Omega symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The Omega symbol represents a set of functions such that for any function in the set Omega(g(x)), f(x); it is not true that f(x) is in O(g(x)).

Commented Mathematical property (CMP):

f(x) is Omega(g(x)) if and only if it is not true that f(x) is O(g(x))

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="asymp1" name="Omega"/>
      <OMV name="g"/>
    </OMA>
    <OMA>
      <OMS cd="set1" name="suchthat"/>
      <OMA>
        <OMS cd="setname3" name="function_set"/>
        <OMS cd="setname1" name="R"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
	<OMBVAR>
	  <OMV name="f"/>
	</OMBVAR>
	<OMA>
	  <OMS cd="logic1" name="not"/>
	  <OMA>
	    <OMS cd="set1" name="in"/>
	    <OMA>
	      <OMV name="f"/>
	      <OMV name="x"/>
	    </OMA>
	    <OMA>
	      <OMS cd="asymp1" name="O"/>
	      <OMV name="g"/>
	    </OMA>
	  </OMA>
	</OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

eq (Omega ( g) , suchthat (function_set (R, R) , lambda [ f ] . (not (in ( f ( x) , O ( g) ) ) ) ) )

$$\mathrm{Omega}\left(g\right)=\left\{f\in \mathrm{function\_set}(\mathbb{R},\mathbb{R})\right|\neg (f\left(x\right)\in O\left(g\right))\}$$

Signatures:

sts