Dr. Miriam Ciavarella: New Applications
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en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 18 Sep 2014 13:46:46 GMTThu, 18 Sep 2014 13:46:46 GMTNew applications published by Dr. Miriam Ciavarellahttp://www.mapleprimes.com/images/mapleapps.gifDr. Miriam Ciavarella: New Applications
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Some Fractals with Maple
http://www.maplesoft.com/applications/view.aspx?SID=97624&ref=Feed
<p>This paper contains some procedures concerning the fractals. In particular we describe the procedures in order to draw the triadic Cantor dust, the Koch curve, the Snowflake and the procedures in order to calculate the area and the perimeter of the Snowflake curve.</p><img src="/view.aspx?si=97624/maple_icon.jpg" alt="Some Fractals with Maple" align="left"/><p>This paper contains some procedures concerning the fractals. In particular we describe the procedures in order to draw the triadic Cantor dust, the Koch curve, the Snowflake and the procedures in order to calculate the area and the perimeter of the Snowflake curve.</p>97624Fri, 08 Oct 2010 04:00:00 ZProf. Marina MarchisioProf. Marina MarchisioEichler Orders
http://www.maplesoft.com/applications/view.aspx?SID=5567&ref=Feed
<p>The aim of this worksheet is to give an explicit description in term of bases over Z, of some Eichler orders of an indefinite quaternion algebra B defined over Q. Following a work of Hashimoto, we give a procedure which returns a basis of the Eichler orders of level N. This construction provides a very useful tool for working with Eichler orders. Let q be a prime number not dividing the discriminant of B; it is well known that there are two natural inclusions of R(Nq) in R(N). We provide a new procedure in Maple which returns a basis of an Eichler order on level Nq in B and a basis of the other copy of R(Nq) in R(N).</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Eichler Orders" align="left"/><p>The aim of this worksheet is to give an explicit description in term of bases over Z, of some Eichler orders of an indefinite quaternion algebra B defined over Q. Following a work of Hashimoto, we give a procedure which returns a basis of the Eichler orders of level N. This construction provides a very useful tool for working with Eichler orders. Let q be a prime number not dividing the discriminant of B; it is well known that there are two natural inclusions of R(Nq) in R(N). We provide a new procedure in Maple which returns a basis of an Eichler order on level Nq in B and a basis of the other copy of R(Nq) in R(N).</p>5567Fri, 18 Dec 2009 05:00:00 ZDr. Miriam CiavarellaDr. Miriam CiavarellaQUADRATIC FIELDS and CLASS NUMBER FORMULA
http://www.maplesoft.com/applications/view.aspx?SID=34981&ref=Feed
<p>The aim of this document is to give same procedures in order to work explicitely with quadratic fields; in particular the idea of work was born in order to find a useful procedure to compute the class number of a quadratic filed.</p>
<div>Many problems of number theory lead to the important question in the arithmetic of algebraic number fields of decomposition of algebraic numbers into prime factors. We shall define a procedure <em>Dec </em> which returns the decomposition of algebraic numbers into prime factors in a quadratic filed. The problems of factorization are very closely connected with Fermat's (last) theorem. Historically, it was precisely the problem of Fermat's theorem which led Kummer to his fundamental work on the arithmetic of algebraic numbers. </div>
<div>It is well known the important role of the number <em>h </em>of divisor classes of algebraic number filed play in the arithmetic of the field. Thus one would like to have an explicit formula for the number <em>h</em> in terms of simpler values which depend on the filed. Although this has not been accomplished for arbitrary algebraic number fields, for certain fields of great interest, such as quadratic fields, such formulas as been found. </div>
<div>Since all divisors are products of prime divisors and the number of prime divisors is infinite, then to compute the number <em>h</em> in a finite number of steps we must use some infinite processes. This is why, in the determination of <em>h,</em> we shall have to consider infinite products, series and other analytic concepts.</div><img src="/view.aspx?si=34981/0\images\Campi_quadratici_12.gif" alt="QUADRATIC FIELDS and CLASS NUMBER FORMULA" align="left"/><p>The aim of this document is to give same procedures in order to work explicitely with quadratic fields; in particular the idea of work was born in order to find a useful procedure to compute the class number of a quadratic filed.</p>
<div>Many problems of number theory lead to the important question in the arithmetic of algebraic number fields of decomposition of algebraic numbers into prime factors. We shall define a procedure <em>Dec </em> which returns the decomposition of algebraic numbers into prime factors in a quadratic filed. The problems of factorization are very closely connected with Fermat's (last) theorem. Historically, it was precisely the problem of Fermat's theorem which led Kummer to his fundamental work on the arithmetic of algebraic numbers. </div>
<div>It is well known the important role of the number <em>h </em>of divisor classes of algebraic number filed play in the arithmetic of the field. Thus one would like to have an explicit formula for the number <em>h</em> in terms of simpler values which depend on the filed. Although this has not been accomplished for arbitrary algebraic number fields, for certain fields of great interest, such as quadratic fields, such formulas as been found. </div>
<div>Since all divisors are products of prime divisors and the number of prime divisors is infinite, then to compute the number <em>h</em> in a finite number of steps we must use some infinite processes. This is why, in the determination of <em>h,</em> we shall have to consider infinite products, series and other analytic concepts.</div>34981Thu, 17 Dec 2009 05:00:00 ZProf.ssa Marina MarchisioProf.ssa Marina MarchisioQUATERNION ALGEBRAS
http://www.maplesoft.com/applications/view.aspx?SID=5474&ref=Feed
This worksheet presents some procedures in order to make computations in a quaternion algebra B defined over the field of rational numbers Q . Quaternion algebra means something more general than the algebra of Hamilton's quaternions (for which there exists already a Maple package) . To give a quaternion algebra B is equivalent to give a pair (a,b) of non-zero rational numbers so that B is defined as the Q-algebra of basis {1,i,j,k} where the elements i,j of B verify the relations i^2=a, j^2=b, ij=-ji and k=ij. It can be very useful to define new procedures ProdQuat, InvQ, TrQ, NormQ in Maple which return the product of two elements of B, the inverse of a non-zero-element of B, the reduced trace of an element of B and the reduced norm of an element of B. Moreover, since the places of Q where B is ramified determine B up to isomorphism as an algebra, it can be very useful to give a procedure Discriminant which computes the reduced discriminant of a quaternion algebra B=(a,b).<img src="/view.aspx?si=5474//applications/images/app_image_blank_lg.jpg" alt="QUATERNION ALGEBRAS" align="left"/>This worksheet presents some procedures in order to make computations in a quaternion algebra B defined over the field of rational numbers Q . Quaternion algebra means something more general than the algebra of Hamilton's quaternions (for which there exists already a Maple package) . To give a quaternion algebra B is equivalent to give a pair (a,b) of non-zero rational numbers so that B is defined as the Q-algebra of basis {1,i,j,k} where the elements i,j of B verify the relations i^2=a, j^2=b, ij=-ji and k=ij. It can be very useful to define new procedures ProdQuat, InvQ, TrQ, NormQ in Maple which return the product of two elements of B, the inverse of a non-zero-element of B, the reduced trace of an element of B and the reduced norm of an element of B. Moreover, since the places of Q where B is ramified determine B up to isomorphism as an algebra, it can be very useful to give a procedure Discriminant which computes the reduced discriminant of a quaternion algebra B=(a,b).5474Fri, 02 Nov 2007 00:00:00 ZDr. Miriam CiavarellaDr. Miriam Ciavarella