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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN"
"http://www.w3.org/TR/html4/strict.dtd">
<html>
<head>
<meta http-equiv="Content-Style-Type" content="text/css">
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<title> Yorick Hardy </title>
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<tr>
<td align="center">
<h2> Yorick Hardy </h2>
<a href="https://orcid.org/0000-0003-0264-8874"><img src="https://orcid.org/sites/default/files/images/orcid_16x16.png" style="width:1em;margin-right:.5em;" alt="ORCID iD icon"></a>
<small><a href="mailto:yorick.hardy(AT)wits.ac.za">(e-mail)</a></small>
<small><a href="https://yorickhardy.github.io/">(website/</a><a href="http://yhardy.users.sourceforge.net/">alternate)</a></small>
</td>
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</table>
<hr>
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<hr>
<table width="100%" rules="none" frame="void" cellspacing="0" cellpadding="0">
<tr>
<td align="center"> <h2> Yorick Hardy </h2>
<small><a href="mailto:yorick.hardy(AT)wits.ac.za">(e-mail)</a></small>
<small><a href="https://yorickhardy.github.io/">(website/</a><a href="http://yhardy.users.sourceforge.net/">alternate)</a></small>
</td>
</tr>
</table>
<hr>
</div>
<div id="content">
<h3 id="books"> Books </h3>
<ul>
<li> <a href="https://doi.org/10.1142/9334">
Bose, Spin and Fermi Systems: Problems and Solutions
</a> </li>
<li> <a href="https://doi.org/10.1007/978-3-0348-8366-5">
Classical and Quantum Computing with C++ and Java Simulations
</a> </li>
<li> <a href="https://doi.org/10.1142/6966">
Computer Algebra with SymbolicC++
</a> </li>
<li> <a href="https://doi.org/10.1142/11338">
Matrix Calculus and Kronecker Product:
A Practical Approach to Linear and Multilinear Algebra
</a> </li>
<li> <a href="https://doi.org/10.1142/8378">
Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications
</a> </li>
<li> <a href="https://doi.org/10.1142/10135">
Problems and Solutions in Introductory and Advanced Matrix Calculus
</a> </li>
<li> <a href="https://doi.org/10.1142/10943">
Problems and Solutions in Quantum Computing and Quantum Information
</a> </li>
<li> <a href="https://doi.org/10.1142/5718">
Problems and Solutions in Scientific Computing
</a> </li>
<li> <a href="https://doi.org/10.1142/7751">
Quantum Mechanics using Computer Algebra
</a> </li>
</ul>
<h3 id="links"> Links </h3>
<ul>
<li> <a href="http://www.ahardy.za.net/">
Alexandre Hardy</a> </li>
<li> <a href="http://issc.uj.ac.za/">
International School for Scientific Computing</a> </li>
<li> <a href="http://lambda-the-ultimate.org">
Lambda the Ultimate</a> </li>
<li> <a href="http://www.netbsd.org">
NetBSD</a> operating system </li>
<li> <a href="http://issc.uj.ac.za/symbolic/symbolic.html">
SymbolicC++</a> computer algebra system
(also on <a href="http://symboliccpp.sourceforge.net">SourceForge</a>) </li>
<li> <a href="http://issc.uj.ac.za/steeb/steeb.html">
Willi-Hans Steeb</a> </li>
</ul>
<h3 id="publ"> Publications </h3>
<a href="http://arxiv.org/a/hardy_y_1">arXiv</a> also has a list of preprints.<br>
<ol>
<li> A. Fošner, Y. Hardy and B. Zinsou,<br>
<a href="https://doi.org/10.1080/03081087.2024.2303069">
Block determinants, partial determinants and the exponential map</a>,<br>
<em>Linear and Multilinear Algebra</em>,
<b>72</b>, 2879<span>—</span>2914, 2024.<br>
</li>
<li> Y. Hardy and B. Zinsou,<br>
<a href="https://doi.org/10.11948/20220073">
Canonical forms for boundary conditions of self-adjoint differential operators</a>,<br>
<em>Journal of Applied Analysis & Computation</em>,
<b>14</b>, 1854<span>—</span>1868, 2024.<br>
</li>
<li> Y. Hardy, W.-H. Steeb and G. Kemp,<br>
<a href="https://doi.org/10.1007/s10773-019-04034-6">
Matrices and Their (Un)Faithful Fermi-quadratic Forms</a>,<br>
<em>International Journal of Theoretical Physics</em>,
<b>58</b>, 1470<span>—</span>1481, 2019.
</li>
<li> Y. Hardy and A. Fošner,<br>
<a href="https://doi.org/10.1016/j.laa.2018.07.021">
Linear maps preserving Kronecker quotients</a>,<br>
<em>Linear Algebra and its Applications</em>,
<b>556</b>, 200<span>—</span>209, 2018.<br>
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1515/zna-2015-0139">
Eigenvalue Problem, Spin Systems, Lie Groups, and Parameter Dependence</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung A</em>,
<b>70</b>, 605<span>—</span>609, 2015.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1063/1.4905382">
Exponential of a matrix, a nonlinear problem, and quantum gates</a>,<br>
<em>Journal of Mathematical Physics</em>,
<b>56</b>, 012201, 2015.<br>
Preprint available on <a href="http://arxiv.org/abs/1311.2372">arXiv.org</a>.
</li>
<li> Y. Hardy,<br>
<a href="https://doi.org/10.13001/1081-3810.1611">
On Kronecker Quotients</a>,<br>
<em>Electronic Journal of Linear Algebra</em>,
<b>27</b>, 172<span>—</span>189, 2014.<br>
Also available on <a href="http://arxiv.org/abs/1403.3170">arXiv.org</a>.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1142/S0129183114500594">
Exceptional points, non-normal matrices, hierarchy of spin matrices and an eigenvalue problem</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>25</b>, 1450059, 2014.<br>
Preprint available on <a href="http://arxiv.org/abs/1301.2900">arXiv.org</a>.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.5560/ZNA.2013-0083">
Majorana Fermions on a Lattice and a Matrix Problem</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>69a</b>, 108<span>—</span>110, 2014.<br>
Preprint available on <a href="http://arxiv.org/abs/1309.2105">arXiv.org</a>.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1007/s11128-013-0650-5">
An eigenvalue problem for a Fermi system and Lie algebras</a>,<br>
<em>Quantum Information Processing</em>,
<b>13</b>, 299<span>—</span>308, 2014.<br>
Preprint available on <a href="http://arxiv.org/abs/1304.3991">arXiv.org</a>.
</li>
<li> Y. Hardy, A. Fošner and W.-H. Steeb,<br>
<a href="https://doi.org/10.1016/j.laa.2013.09.043">
Cayley transform and the Kronecker product of Hermitian matrices</a>,<br>
<em>Linear Algebra and its Applications</em>,
<b>439</b>, 4023<span>—</span>4031, 2013.<br>
Preprint available on <a href="http://arxiv.org/abs/1305.1015">arXiv.org</a>.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1142/S0129183112500957">
Hamilton Operators, Discrete Symmetries, Brute Force and SymbolicC++</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>24</b>, 1250095, 2013.<br>
Preprint available on <a href="http://arxiv.org/abs/1202.2259">arXiv.org</a>.
</li>
<li> Y. Hardy and W.-H. Steeb,<br>
<a href="https://doi.org/10.1142/S0219887812200137">
A Sequence of Quantum Gates</a>,<br>
<em>International Journal of Geometric Methods in Modern Physics</em>,
<b>9</b>, 1220013, 2012.<br>
Preprint available on <a href="http://arxiv.org/abs/1202.2259">arXiv.org</a>.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6293756">
Dynamical Systems, Vector Fields and Commutators</a>,<br>
<em>Nonlinear Dynamics of Electronic Systems, Proceedings of NDES 2012</em>,
1<span>—</span>4, 2012.
<li> W.-H. Steeb, Y. Hardy and I. Tanski,<br>
<a href="https://doi.org/10.1142/S0218127412501908">
Vector Fields, Variational Equations and Commutators</a>,<br>
<em>International Journal of Bifurcation and Chaos</em>,
<b>22</b>, 1250190, 2012.
<li> W.-H. Steeb, Y. Hardy and J. de Greef,<br>
<a href="https://doi.org/10.5560/ZNA.2012-0064">
Spin Hamilton Operators, Symmetry Breaking, Energy Level Crossing,
and Entanglement</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>67a</b>, 608<span>—</span>612, 2012.<br>
Preprint available on <a href="http://arxiv.org/abs/1110.4204">arXiv.org</a>.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1142/S1230161212500047">
Entanglement, Kronecker Product, Pauli Spin Matrices
and a Nonlinear Eigenvalue Problem</a>,<br>
<em>Open Systems & Information Dynamics</em>,
<b>19</b>, 1250004, 2012.
</li>
<li> Y. Hardy and W.-H. Steeb,<br>
<a href="https://doi.org/10.1142/S0219887811005816">
Unitary Matrices and Commutators</a>,<br>
<em>International Journal of Geometric Methods in Modern Physics</em>,
<b>8</b>, 1583<span>—</span>1592, 2011.
</li>
<li> W.-H. Steeb, Y. Hardy and I. Tanski,<br>
<a href="https://doi.org/10.1088/0253-6102/56/6/03">
Vector Product and an Integrable Dynamical System</a>,<br>
<em>Communications in Theoretical Physics</em>,
<b>56</b>, 992<span>—</span>994, 2011.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.5560/ZNA.2011-0025">
<i>N</i>-tangle, Entangled Orthonormal Basis,
and a Hierarchy of Spin Hamilton Operators</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>66a</b>, 615<span>—</span>619, 2011.
</li>
<li> Y. Hardy and W.-H. Steeb,<br>
<a href="https://doi.org/10.1142/S0129183110015890">
Genetic Algorithms and Optimization Problems
in Quantum Computing</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>21</b>, 1359<span>—</span>1375, 2010.
</li>
<li> Y. Hardy and W.-H. Steeb,<br>
<a href="https://doi.org/10.1142/S0218196710005522">
Vec-Operator, Kronecker Product and Entanglement</a>,<br>
<em>International Journal of Algebra and Computation</em>,
<b>20</b>, 71<span>—</span>76, 2010.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1515/zna-2010-6-718">
A Chaotic Map and Data Communication</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>65a</b>, 613<span>—</span>614, 2010.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1142/S0129183109014060">
Unitary Operators, Entanglement,
and Gram-Schmidt Orthogonalization</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>20</b>, 891<span>—</span>899, 2009.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.2478/s11534-009-0075-z">
A hierarchy of Hamilton operators and entanglement</a>,<br>
<em>Central European Journal of Physics</em>,
<b>7</b>, 854<span>—</span>859, 2009.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1142/S0129183108012674">
Quantum optics networks, unitary operators
and computer algebra</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>19</b>, 1069<span>—</span>1078, 2008.
</li>
<li> Y. Hardy and D. Sabatta, <br>
<a href="https://doi.org/10.1016/j.physleta.2007.02.063">
Encoding, symbolic dynamics, cryptography
and C++ implementations</a>,<br>
<em>Physics Letters A</em>,
<b>366</b>, 575<span>—</span>584, 2007.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1515/zna-2007-10-1103">
One-Dimensional Cellular Automata, Conservation Laws and Partial
Differential Equations</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>62a</b>, 569<span>—</span>572, 2007.
</li>
<li> Y. Hardy and W.-H. Steeb,<br>
<a href="https://doi.org/10.1088/0305-4470/39/6/018">
Decomposing the SWAP Quantum Gate</a>,<br>
<em>Journal of Physics A: Math. and Gen</em>,
<b>39</b>, 1463<span>—</span>1467, 2006.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1007/s10773-006-9084-5">
Quantum Gates and Hamilton Operators</a>,<br>
<em>International Journal of Theoretical Physics</em>,
<b>45</b>, 953<span>—</span>961, 2006.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1515/zna-2006-3-405">
Supersymmetric Hamilton Operator and Entanglement</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>61a</b>, 139<span>—</span>140, 2006.
</li>
<li> Y. Hardy, W.-H. Steeb and R. Stoop,<br>
<a href="https://doi.org/10.1142/S0129183105008321">
Genetic Algorithms, Floating Point Numbers and Applications</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>16</b>, 1811<span>—</span>1816, 2005.
</li>
<li> W.-H. Steeb, Y. Hardy and R. Stoop,<br>
<a href="https://doi.org/10.1515/zna-2005-11-1218">
Bell-CHSH Inequality and Genetic Algorithms</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>60a</b>, 865<span>—</span>866, 2005.
</li>
<li> Y. Hardy and W.-H. Steeb,<br>
<a href="https://doi.org/10.1023/B:IJTP.0000049019.68706.52">
Fermi-Bose Systems, Macroscopic Quantum Superposition States
and Entanglement</a>,<br>
<em>International Journal of Theoretical Physics</em>,
<b>43</b>, 2207<span>—</span>2213, 2004.
</li>
<li> Y. Hardy and W.-H. Steeb,<br>
<a href="https://doi.org/10.1023/B:IJTP.0000028867.13974.d1">
Entanglement, Hubbard Model and Symmetries</a>,<br>
<em>International Journal of Theoretical Physics</em>,
<b>43</b>, 341<span>—</span>347, 2004.
</li>
<li> Y. Hardy, W.-H. Steeb and R. Stoop,<br>
<a href="https://doi.org/10.1238/Physica.Regular.069a00166">
Entanglement, Disentanglement and Wigner Functions</a>,<br>
<em>Physica Scripta</em>,
<b>69</b>, 166<span>—</span>169, 2004.
</li>
<li> Y. Hardy, W.-H. Steeb and R. Stoop,<br>
<a href="https://doi.org/10.1023/B:IJTP.0000006011.53384.a4">
Fully Entangled Quantum States in <b>C</b><sup><b>N</b><sup>2</sup></sup>
and Bell Measurement</a>,<br>
<em>International Journal of Theoretical Physics</em>,
<b>42</b>, 2865<span>—</span>2872, 2003.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1023/A:1023335307091">
Chaotic Maps, Invariants, Bose Operators, and Coherent States</a>,<br>
<em>International Journal of Theoretical Physics</em>,
<b>42</b>, 85<span>—</span>88, 2003.
</li>
<li> W.-H. Steeb, Y. Hardy and R. Stoop,<br>
<a href="https://doi.org/10.1238/Physica.Regular.067a00464">
Lax Representation and Kronecker Product</a>,<br>
<em>Physica Scripta</em>,
<b>67</b>, 464<span>—</span>465, 2003.
</li>
<li> W.-H. Steeb, Y. Hardy and R. Stoop,<br>
<a href="https://doi.org/10.1088/0305-4470/36/24/316">
Discrete Wavelets and Perturbation Theory</a>,<br>
<em>Journal of Physics A: Mathematical and General</em>,
<b>36</b>, 6807<span>—</span>6811, 2003.
</li>
<li> Y. Hardy and W.-H. Steeb, <br>
<a href="https://doi.org/10.1142/S0129183102002912">
Gene Expression Programming and One-dimensional Chaotic Maps</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>13</b>, 13<span>—</span>24, 2002.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1515/zna-2002-0808">
Entangled Quantum States and the Kronecker Product</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>57a</b>, 689<span>—</span>691, 2002.
</li>
<li> W.-H. Steeb, Y. Hardy and R. Stoop,<br>
<a href="https://doi.org/10.1142/S0129183102003553">
Discrete Wavelets and Filtering Chaotic Signals</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>13</b>, 771<span>—</span>776, 2002.
</li>
<li> Y. Hardy and W.-H. Steeb,<br>
<a href="https://doi.org/10.1142/S0129183101001638">
Fermi Systems, Hubbard Model and a SymbolicC++ Implementation</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>12</b>, 235<span>—</span>245, 2001.
</li>
<li> W.-H. Steeb, Y. Hardy Y. and R. Stoop,<br>
<a href="https://doi.org/10.1515/zna-2001-0805">
Energy Eigenvalue Level Motion with Two Parameters</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>56a</b>, 565<span>—</span>567, 2001.
</li>
<li> W.-H. Steeb, Y. Hardy and R. Stoop,<br>
<a href="https://doi.org/10.1515/zna-2001-0919">
Bessel Functions, Recursion and a Nonlinear Field Equation</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>56a</b>, 710<span>—</span>711, 2001.
</li>
<li> Y. Hardy, W.-H. Steeb and C.M. Villet,<br>
<a href="https://doi.org/10.1515/zna-2000-11-1201">
Stabilization of Chaotic Systems with Phase Coupling</a>,<br>
<em>Zeitschrift f<span>ü</span>r Naturforschung</em>,
<b>55a</b>, 847<span>—</span>850, 2000.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1142/S0129183100000304">
Quantum Computing and SymbolicC++ Simulations</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>11</b>, 323<span>—</span>334, 2000.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1142/S0129183100000079">
Entangled Quantum States and a C++ Implementation</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>11</b>, 69<span>—</span>77, 2000.
</li>
<li> Y. Hardy, W.-H. Steeb and R. Stoop,<br>
<a href="https://doi.org/10.1142/S0129183100000031">
Jacobi Elliptic Functions, Nonlinear Evolution Equations
and Recursion</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>11</b>, 27<span>—</span>31, 2000.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1142/S012918310000119X">
Energy Eigenvalue Level Motion and a SymbolicC++ Implementation</a>,<br>
<em>International Journal of Modern Physics C</em>,
<b>11</b>, 1347<span>—</span>1356, 2000.
</li>
<li> W.-H. Steeb and Y. Hardy,<br>
<a href="https://doi.org/10.1023/A:1026457014804">
Entangled Quantum States</a>,<br>
<em>International Journal of Theoretical Physics</em>,
<b>39</b>, 2765<span>—</span>2770, 2000.
</li>
<li> Y. Hardy and W.-H. Steeb,<br>
<a href="https://doi.org/10.1023/A:1026640221874">
The Rikitake Two-Disk Dynamo System
and Domains with Periodic Orbits</a>,<br>
<em>International Journal of Theoretical Physics</em>,
<b>38</b>, 2413<span>—</span>2417, 1999.
</li>
</ol>
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