{"@attributes":{"version":"2.0"},"channel":{"title":"Notes","link":"https:\/\/marek.onl\/","description":"Recent content on Notes","generator":"Hugo","language":"en-us","managingEditor":"m@rek.onl (Marek)","webMaster":"m@rek.onl (Marek)","lastBuildDate":"Fri, 13 Mar 2026 15:32:00 +0100","item":[{"title":"Mining Zcash","link":"https:\/\/marek.onl\/mining-zec\/","pubDate":"Sat, 13 Jan 2024 18:55:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/mining-zec\/","description":"<p>I built this for fun in late 2022.<\/p>\n<style>.org-center { margin-left: auto; margin-right: auto; text-align: center; }<\/style>\n<div class=\"org-center\">\n<p><video controls><source src=\"https:\/\/marek.onl\/data\/mining\/mining.mp4\" type=\"video\/mp4\"><\/p>\n<track src=\"https:\/\/marek.onl\/data\/mining\/mining.vtt\" label=\"English\" kind=\"subtitles\" srclang=\"en\" default\/><\/video>\n<\/div>\n<p>The rest of this post is a compilation of random posts from social media.<\/p>\n<p>There are three hashboards in each machine, and below is what the hashboard looks\nlike. The hole in the PCB is due to a previous failure in the power circuitry,\nas I mentioned in the video.<\/p>"},{"title":"Affine Variety","link":"https:\/\/marek.onl\/affine-variety\/","pubDate":"Mon, 14 Dec 2020 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/affine-variety\/","description":"<div class=\"def\">\n<p><span class=\"org-target\" id=\"def-nb\"><\/span> <strong><a href=\"#def-nb\">Definition nb<\/a><\/strong>. Let \\(\\mathbb{F}\\) be a <a href=\"https:\/\/marek.onl\/field\/#def-8\">field<\/a> and \\(n\\) a positive\ninteger. The \\(n\\)-dimensional <strong>affine space<\/strong> over \\(\\mathbb{F}\\) is the set\n\\[\\mathbb{F}^n = \\{(a_1, \\ldots, a_n) \\mid a_1, \\ldots, a_n \\in \\mathbb{F}\\}.\\]<\/p>\n<\/div>\n<div class=\"rem\">\n<p><span class=\"org-target\" id=\"rem-nn\"><\/span> <strong><a href=\"#rem-nn\">Remark nn<\/a><\/strong>. A <a href=\"https:\/\/marek.onl\/polynomial\/#def-bh\">polynomial<\/a>\n\\(f \\in \\mathbb{F}[x_1, \\ldots, x_n]\\) can be regarded as a function\n\\(f : \\mathbb{F}^n \\to \\mathbb{F}\\) that takes points in the <a href=\"#def-nb\">affine space<\/a>\n\\(\\mathbb{F}^n\\) and produces elements of the <a href=\"https:\/\/marek.onl\/field\/#def-8\">field<\/a> \\(\\mathbb{F}\\). A <strong>zero\npoint<\/strong> or <strong>root<\/strong> of \\(f\\) is a point\n\\(\\mathbf{a} = (a_1, \\ldots, a_n) \\in \\mathbb{F}^n\\) such that\n\\(f(\\mathbf{a}) = 0\\).<\/p>"},{"title":"Gr\u00f6bner Basis","link":"https:\/\/marek.onl\/groebner-basis\/","pubDate":"Mon, 14 Dec 2020 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/groebner-basis\/","description":"<blockquote>\n<p>It is almost impossible for me to read contemporary mathematicians who, instead\nof saying &ldquo;Petya washed his hands,&rdquo; write simply: &ldquo;There is a \\(t_1 &lt; 0\\) such\nthat the image of \\(t_1\\) under the natural mapping\n\\(t_1 \\mapsto \\mathit{Petya}(t_1)\\) belongs to the set of dirty hands, and a\n\\(t_2, t_1 &lt; t_2 \\le 0\\), such that the image of \\(t_2\\) under the\nabove-mentioned mapping belongs to the complement of the set defined in the\npreceding sentence.&rdquo; &mdash; Vladimir Igorevich Arnol&rsquo;d (<a href=\"#citeproc_bib_item_6\">Zdravkovska and Arnol\u2019d 1987, 30<\/a>)<\/p>"},{"title":"Polynomial Division","link":"https:\/\/marek.onl\/polynomial-division\/","pubDate":"Fri, 11 Dec 2020 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/polynomial-division\/","description":"<div class=\"def\">\n<p><span class=\"org-target\" id=\"def-nw\"><\/span> <strong><a href=\"#def-nw\">Definition nw<\/a><\/strong>. Let\n\\(h = \\sum c_\\alpha x^\\alpha \\in \\mathbb{F}[\\mathbf{x}] \\setminus \\{0\\}\\) be a\nnonzero <a href=\"https:\/\/marek.onl\/polynomial\/#def-bh\">polynomial<\/a>, \\(F\\) a subset of\n\\(\\mathbb{F}[\\mathbf{x}] \\setminus \\{0\\}\\), and let \\(\\succeq\\) be a\n<a href=\"https:\/\/marek.onl\/monomial-orders\/#def-nj\">monomial order<\/a> on \\(\\mathcal{M}\\).<\/p>\n<ol>\n<li>The <strong>multidegree<\/strong> of \\(h\\) is\n\\(\\mathrm{multideg}(h) = \\max(\\alpha \\in \\mathbb{N}_0^n \\mid c_\\alpha \\neq 0)\\).\nThe maximum is taken with respect to \\(\\succeq\\).<\/li>\n<li>The <strong>leading coefficient<\/strong> of \\(h\\) is\n\\(\\mathrm{LC}(h) = c_{\\mathrm{multideg}(h)} \\in \\mathbb{F}\\).<\/li>\n<li>\\(\\mathrm{LC}(F) = \\{\\mathrm{LC}(f) \\mid f \\in F\\}\\).<\/li>\n<li>The <strong>leading monomial<\/strong> of \\(h\\) is\n\\(\\mathrm{LM}(h) = x^{\\mathrm{multideg}(h)}\\).<\/li>\n<li>\\(\\mathrm{LM}(F) = \\{\\mathrm{LM}(f) \\mid f \\in F\\}\\).<\/li>\n<li>The <strong>leading term<\/strong> of \\(h\\) is\n\\(\\mathrm{LT}(h) = \\mathrm{LC}(h) \\cdot \\mathrm{LM}(h)\\).<\/li>\n<li>\\(\\mathrm{LT}(F) = \\{\\mathrm{LT}(f) \\mid f \\in F\\}\\).<\/li>\n<li>\\(M(f)\\) denotes the set of all <a href=\"https:\/\/marek.onl\/polynomial\/#def-b4\">monomials<\/a> in \\(f\\), and\n\\(M(F) = \\bigcup_{f \\in F} M(f)\\).<\/li>\n<\/ol>\n<\/div>\n<div class=\"exp\">\n<p><span class=\"org-target\" id=\"exp-ni\"><\/span> <strong><a href=\"#exp-ni\">Example ni<\/a><\/strong>. Let\n\\(f = xy^2z^3 + xy^3 \\in \\mathbb{F}[x, y, z]\\) be a <a href=\"https:\/\/marek.onl\/polynomial\/#def-bh\">polynomial<\/a> under\nthe <a href=\"https:\/\/marek.onl\/monomial-orders\/#def-nk\">lexicographic order<\/a>. Then\n\\(\\mathrm{multideg}(f) = (1, 3, 0)\\),\n\\(\\mathrm{LC}(f) = 1\\),\n\\(\\mathrm{LM}(f) = xy^3\\),\n\\(\\mathrm{LT}(f) = xy^3\\), and\n\\(M(f) = \\{xy^2z^3, xy^3\\}\\).<\/p>"},{"title":"AES + Gr\u00f6bner bases = \u2661","link":"https:\/\/marek.onl\/aes-groebner-experiments\/","pubDate":"Tue, 24 Nov 2020 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/aes-groebner-experiments\/","description":"<blockquote>\n<p>I feel I am nibbling on the edges of this world when I am capable of getting\nwhat Picasso means when he says to me &mdash; perfectly straight-facedly &mdash; later of\nthe enormous new mechanical brains or calculating machines: &ldquo;But they are\nuseless. They can only give you answers.&rdquo; &mdash; William Fifield, Pablo Picasso: A\nComposite Interview, The Paris Review 32<\/p>\n<\/blockquote>\n<p>The principle of algebraic cryptanalysis consists in transferring the problem of\nbreaking a cryptosystem to the problem of solving a system of multivariate\n<a href=\"https:\/\/marek.onl\/polynomial\/#def-bh\">polynomial<\/a> equations over a\n<a href=\"https:\/\/marek.onl\/field\/\">finite field<\/a>. The process is divided into two steps. The\nfirst step uses the cipher&rsquo;s structure and supplemental information to create a\nsystem of equations that describes the behavior of the cipher for a specific\ncase. Several papers (<a href=\"#citeproc_bib_item_5\">Cid, Murphy, and Robshaw 2005<\/a>; <a href=\"#citeproc_bib_item_4\">Bulygin and Brickenstein 2010<\/a>) present approaches for constructing\npolynomial equations for <a href=\"https:\/\/marek.onl\/aes\/\">AES<\/a>. The second step involves solving\nthe polynomial system to derive the secret key. While the method for deriving\nthe system depends on the cipher, the method for solving the system may be\nindependent of it.<\/p>"},{"title":"AES as a System of Equations","link":"https:\/\/marek.onl\/aes-equations\/","pubDate":"Wed, 21 Oct 2020 00:00:00 +0200","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/aes-equations\/","description":"<p><a href=\"https:\/\/marek.onl\/aes\/\">AES<\/a> (and its <a href=\"https:\/\/marek.onl\/aes\/#small-scale-variants\">small scale<\/a> derivatives) is a symmetric block cipher where\nthe block is represented by the state, which is further divided into sub-blocks.\nAES is also an example of an iterated substitution-permutation network where one\niteration is split into three stages (<a href=\"#citeproc_bib_item_5\">Shannon 1949<\/a>). The first stage is a local\nnonlinear transformation (substitution) of the sub-blocks of the state. This\ntransformation is performed by the SubBytes operation &mdash; the\n<a href=\"https:\/\/marek.onl\/aes\/#def-d6\">S-box<\/a> is locally applied to each sub-block in order to\nsubstitute its value, while the mutual positions of the sub-blocks are left\nintact. This stage provides so-called confusion. The next stage is a global\nlinear transformation of the state. This is performed by the ShiftRows and\nMixColumns operations, which are linear transformations over\n\\(\\mathrm{GF}(2^e)\\), and which also change the mutual positions of the\nsub-blocks. This stage provides so-called diffusion, which tries to distribute\nthe output bits of the S-boxes in the current iteration to as many S-box inputs\nas possible in the next iteration. The last stage is the addition of the key.<\/p>"},{"title":"AES","link":"https:\/\/marek.onl\/aes\/","pubDate":"Wed, 30 Sep 2020 00:00:00 +0200","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/aes\/","description":"<blockquote>\n<p>Perfection is achieved, not when there is nothing more to add, but when there is\nnothing left to take away. &mdash; Antoine de Saint-Exup\u00e9ry, Airman&rsquo;s Odyssey<\/p>\n<\/blockquote>\n<p>AES is one of the most widely used block ciphers for symmetric\nencryption. The cipher was originally named Rijndael, after its designers Joan\nDaemen and Vincent Rijmen. In 1997, the U.S. National Institute of Standards and\nTechnology (NIST) announced the development of AES and organized an open\ncompetition, which Rijndael won. NIST published the cipher as the Federal\nInformation Processing Standard (FIPS) 197 (<a href=\"#citeproc_bib_item_4\">Pub 2001<\/a>) in 2001. AES superseded\nthe previous Data Encryption Standard (DES).<\/p>"},{"title":"Ideal","link":"https:\/\/marek.onl\/ideal\/","pubDate":"Wed, 04 Mar 2020 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/ideal\/","description":"<div class=\"def\">\n<p><span class=\"org-target\" id=\"def-m\"><\/span> <strong><a href=\"#def-m\">Definition m<\/a><\/strong>. Let \\(R\\) be a commutative <a href=\"https:\/\/marek.onl\/ring\/#def-g\">ring<\/a> and \\(\\emptyset \\ne I \\subseteq R\\). Then\n\\(I\\) is an <strong>ideal<\/strong> of \\(R\\) if:<\/p>\n<ol>\n<li>The sum \\(a + b\\) is in \\(I\\) for all \\(a, b \\in I\\).<\/li>\n<li>The product \\(ar\\) is in \\(I\\) for all \\(a \\in I\\) and \\(r \\in R\\).<\/li>\n<\/ol>\n<p>The ideal \\(I\\) is <strong>proper<\/strong> if \\(I \\ne R\\).<\/p>\n<\/div>\n<div class=\"rem\">\n<p><span class=\"org-target\" id=\"rem-c\"><\/span> <strong><a href=\"#rem-c\">Remark c<\/a><\/strong>. <a href=\"#def-m\">Definition m<\/a> says that if \\(R\\) is a commutative <a href=\"https:\/\/marek.onl\/ring\/#def-g\">ring<\/a>,\nand \\(\\emptyset \\neq I \\subseteq R\\), then \\(I\\) is an <a href=\"#def-m\">ideal<\/a> of \\(R\\) if:<\/p>"},{"title":"Monomial Orders","link":"https:\/\/marek.onl\/monomial-orders\/","pubDate":"Wed, 04 Mar 2020 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/monomial-orders\/","description":"<p>A <a href=\"https:\/\/marek.onl\/groebner-basis\/\">Gr\u00f6bner basis<\/a> always pertains to a particular order on\n<a href=\"https:\/\/marek.onl\/polynomial\/#def-b4\">monomials<\/a>.<\/p>\n<div class=\"def\">\n<p><span class=\"org-target\" id=\"def-n8\"><\/span> <strong><a href=\"#def-n8\">Definition n8<\/a><\/strong>. Let \\(S\\) be a non-empty set. A <strong>binary relation<\/strong>\non \\(S\\) is a subset \\(r\\) of \\(S \\times S\\). The relation\n\\(\\Delta(S) = \\{(a, a) \\mid a \\in S\\}\\) is the <strong>diagonal<\/strong> of \\(S\\). The <strong>inverse<\/strong>\nof \\(r\\) is \\(r^{-1} = \\{(a, b) \\mid (b, a) \\in r\\}\\). The <strong>strict part<\/strong> of\n\\(r\\) is \\(r_s = r \\setminus r^{-1}\\), and the <strong>product<\/strong> of \\(r\\) and \\(u\\) is\n\\[u \\circ r = \\{(a, c) \\mid \\text{there is } b \\in S \\text{ such that } (a, b) \\in r \\text{ and } (b, c) \\in u\\}.\\]<\/p>"},{"title":"Polynomial","link":"https:\/\/marek.onl\/polynomial\/","pubDate":"Wed, 04 Mar 2020 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/polynomial\/","description":"<div class=\"def\">\n<p><span class=\"org-target\" id=\"def-b4\"><\/span> <strong><a href=\"#def-b4\">Definition b4<\/a><\/strong>. Let\n\\(\\alpha = (\\alpha_1, \\ldots, \\alpha_n) \\in \\mathbb{N}_0^n\\) be an \\(n\\)-tuple of\nnon-negative integers. A <strong>monomial<\/strong> in \\(x_1, \\ldots, x_n\\) is a product\n\\[\\prod_{i=1}^n x_i^{\\alpha_i} = x_1^{\\alpha_1} \\cdot x_2^{\\alpha_2} \\cdots x_n^{\\alpha_n}.\\]\nWe write \\(x^\\alpha = \\prod_{i=1}^n x_i^{\\alpha_i}\\) for short. The <strong>total\ndegree<\/strong> of \\(x^\\alpha\\) is \\(\\lvert x^\\alpha \\rvert = \\sum_{i=1}^n \\alpha_i\\).\nNote that \\(x^\\alpha = 1\\) when \\(\\alpha = (0, \\ldots, 0)\\) and that any monomial\nis fully determined by \\(\\alpha\\).<\/p>"},{"title":"Field","link":"https:\/\/marek.onl\/field\/","pubDate":"Thu, 30 Jan 2020 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/field\/","description":"<div class=\"def\">\n<p><span class=\"org-target\" id=\"def-8\"><\/span> <strong><a href=\"#def-8\">Definition 8<\/a><\/strong>. A <strong>field<\/strong> \\(\\mathbb{F}\\) is a <a href=\"https:\/\/marek.onl\/ring\/#def-g\">ring<\/a> where the set\n\\(\\mathbb{F} \\setminus \\set{0}\\) is an Abelian <a href=\"https:\/\/marek.onl\/group\/#def-f\">group<\/a> under multiplication with the\n<strong>multiplicative identity<\/strong> 1.<\/p>\n<\/div>\n<div class=\"rem\">\n<p><span class=\"org-target\" id=\"rem-bz\"><\/span> <strong><a href=\"#rem-bz\">Remark bz<\/a><\/strong>. Fields with a finite number of elements are <strong>finite\nfields<\/strong> and are often denoted \\(\\mathbb{F}_q\\) or \\(\\mathrm{GF}(q)\\), where \\(q\\)\nis the <strong>order<\/strong> of the field. Since every <a href=\"#def-8\">field<\/a> is a commutative <a href=\"https:\/\/marek.onl\/ring\/#def-g\">ring<\/a>, the\nonly difference between <a href=\"https:\/\/marek.onl\/ring\/#def-g\">rings<\/a> and fields is that in a field, every element other\nthan \\(0\\) has its multiplicative inverse.<\/p>"},{"title":"Group","link":"https:\/\/marek.onl\/group\/","pubDate":"Thu, 30 Jan 2020 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/group\/","description":"<div class=\"def\">\n<p><span class=\"org-target\" id=\"def-b\"><\/span> <strong><a href=\"#def-b\">Definition b<\/a><\/strong>. Let \\(A_1, \\ldots, A_n\\) be sets. The <strong>Cartesian\nproduct<\/strong> \\(A_1 \\times \\cdots \\times A_n\\) is the set of all ordered \\(n\\)-tuples\n\\((a_1, \\ldots, a_n)\\) such that \\(a_i \\in A_i\\) for \\(1 \\le i \\le n\\).<\/p>\n<\/div>\n<div class=\"def\">\n<p><span class=\"org-target\" id=\"def-n\"><\/span> <strong><a href=\"#def-n\">Definition n<\/a><\/strong>. Let \\(A\\) and \\(B\\) be sets. A <strong>map<\/strong> is a set\n\\(\\varphi \\subseteq A \\times B\\) such that for each \\(a \\in A\\) there is exactly\none \\(b \\in B\\) with \\((a, b) \\in \\varphi\\).<\/p>"},{"title":"Ring","link":"https:\/\/marek.onl\/ring\/","pubDate":"Thu, 30 Jan 2020 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/ring\/","description":"<div class=\"def\">\n<p><span class=\"org-target\" id=\"def-g\"><\/span> <strong><a href=\"#def-g\">Definition g<\/a><\/strong>. A <strong>ring<\/strong> is a set \\(R\\) with two <a href=\"https:\/\/marek.onl\/group\/#def-d\">binary operations<\/a>\n\\(\\left( a, b \\right) \\mapsto a + b\\) and \\(\\left( a, b \\right) \\mapsto a \\cdot b\\), referred to as\naddition and multiplication, such that the following axioms hold:<\/p>\n<ol>\n<li>The set \\(R\\) is an Abelian <a href=\"https:\/\/marek.onl\/group\/#def-f\">group<\/a> under addition with the <strong>additive identity<\/strong>\n\\(0\\).<\/li>\n<li>The set \\(R\\) is a <a href=\"https:\/\/marek.onl\/group\/#def-r\">monoid<\/a> under multiplication with the <strong>multiplicative\nidentity<\/strong> \\(1\\).<\/li>\n<li>Distributivity: Equations \\(a \\cdot \\left( b + c \\right) = a \\cdot b + a \\cdot c\\) and\n\\(\\left( a + b \\right) \\cdot c = a \\cdot c + b \\cdot c\\) hold for all \\(a, b, c \\in R\\).<\/li>\n<\/ol>\n<p>A ring is a <strong>commutative ring<\/strong> if it is a commutative <a href=\"https:\/\/marek.onl\/group\/#def-r\">monoid<\/a> under\nmultiplication.<\/p>"},{"title":"Malware Detection with Artificial Neural Networks","link":"https:\/\/marek.onl\/malware-detection\/","pubDate":"Fri, 29 Nov 2019 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/malware-detection\/","description":"<p>Artificial neural networks such as MLP (Multilayer Perceptron) and CNN\n(Convolutional Neural Network) have been successfully used to statically\nclassify malware (<a href=\"#citeproc_bib_item_5\">Raff et al. 2017<\/a>; <a href=\"#citeproc_bib_item_7\">Vinayakumar and Soman 2018<\/a>; <a href=\"#citeproc_bib_item_4\">Kr\u010d\u00e1l et al. 2018<\/a>).\nThis post should help beginners in deep learning to apply these models in\npractice as we walk through a task of static malware classification. We assume\nthat the reader has just explored what MLP and CNN are but hasn&rsquo;t used them yet.\nWe will not focus on the code but rather guide the reader through the overall\nprocess and hopefully provide useful intuition along the way. We will also\nexplain various metrics that are useful in binary classification.<\/p>"},{"title":"Intro to Bitcoin","link":"https:\/\/marek.onl\/intro-to-bitcoin\/","pubDate":"Tue, 13 Nov 2018 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/intro-to-bitcoin\/","description":"<style>\ntable { margin-left: auto; margin-right: auto; }\n<\/style>\n<p>We can see Bitcoin as a peer-to-peer network of cooperating nodes. These nodes\nlisten for transactions, order them into subsequent blocks, and then publish\nthese blocks on the network. The network uses digital signatures to verify the\nownership of funds, and a Proof-of-Work system based on hashing to prevent\ndouble-spending. These techniques bring trust to the whole history of\ntransactions, which in turn allows users to exchange value. Let us now unveil\nhow all this works.<\/p>"},{"title":"Treaps","link":"https:\/\/marek.onl\/treaps\/","pubDate":"Sun, 13 May 2018 00:00:00 +0200","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/treaps\/","description":"<p>A treap is a data structure in the form of a self-balanced binary search tree,\nwhich is suitable for storing ordered data more effectively than a standard BST\n(<a href=\"#citeproc_bib_item_1\">Seidel and Aragon 1996<\/a>).<\/p>\n<p>Every node contains a randomly generated priority, a unique key and optionally\na value that the key represents. Nodes in a treap follow the heap-ordering\nproperty with respect to priorities: the priority of any non-leaf node is\ngreater than or equal to the priority of its descendants. Considering only the\nkeys, nodes follow the order of a standard BST, meaning that the key of the left\nchild is less than the key of the current node and the key of the right child is\ngreater. The word treap is a blend of tree and heap.<\/p>"},{"title":"Laplace's Demon","link":"https:\/\/marek.onl\/laplaces-demon\/","pubDate":"Wed, 06 Dec 2017 00:00:00 +0100","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/laplaces-demon\/","description":"<p>Neo, the main character from The Matrix, didn&rsquo;t like the idea of fate because he\nwanted to be able to make his own choices. Let&rsquo;s see what Pierre-Simon Laplace\nhad to say about such free will back in 1814.<\/p>\n<p>At the turn of the 17th and 18th century, Isaac Newton published a work in three\nbooks called <em>Philosophiae Naturalis Principia Mathematica (The Mathematical\nPrinciples of Natural Philosophy)<\/em> (<a href=\"#citeproc_bib_item_5\">Newton 1687<\/a>) or simply <em>Principia<\/em>.\nThere, he originally wrote:<\/p>"},{"title":"Evolutionary Algorithms","link":"https:\/\/marek.onl\/evolutionary-algorithms\/","pubDate":"Wed, 17 May 2017 00:00:00 +0200","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/evolutionary-algorithms\/","description":"<style>\ntable { margin-left: auto; margin-right: auto; border-collapse: collapse; }\ntable th, table td { padding: 4px 4px; border: 1px solid #555; }\ntable th { border-bottom: 2px solid #888; white-space: nowrap; }\ntable td { text-align: center; }\ntable:nth-of-type(2) td { text-align: left; }\n<\/style>\n<blockquote>\n<p><em>Owing to this struggle for life, variations, however slight and from whatever\ncause proceeding, if they be in any degree profitable to the individuals of a\nspecies, in their infinitely complex relations to other organic beings and to\ntheir physical conditions of life, will tend to the preservation of such\nindividuals, and will generally be inherited by the offspring. The offspring,\nalso, will thus have a better chance of surviving, for, of the many individuals\nof any species which are periodically born, but a small number can survive. I\nhave called this principle, by which each slight variation, if useful, is\npreserved, by the term Natural Selection.<\/em> &mdash; Darwin, 1859 (<a href=\"#citeproc_bib_item_3\">Darwin 1859<\/a>).<\/p>"},{"title":"Hello","link":"https:\/\/marek.onl\/about\/","pubDate":"Mon, 01 Jan 0001 00:00:00 +0000","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/about\/","description":"<p>I&rsquo;m Marek. Welcome to the public repository of my notes. Please note that this\nwebsite currently contains only a small fraction of my notes, and some are\nincomplete. Hopefully, there will be a time when I can add more.<\/p>\n<p>You can contact me at <a href=\"mailto:m@rek.onl\">m@rek.onl<\/a> or at <a href=\"https:\/\/signal.me\/#eu\/1-bBGq8AH71rgbox0YKBIW7wvb23EB47--xk2QdjSGuc0QJiE1dPXYosqb_isGDT\">marek.92<\/a> on Signal.<\/p>"},{"title":"Test Node","link":"https:\/\/marek.onl\/test-node\/","pubDate":"Mon, 01 Jan 0001 00:00:00 +0000","author":"m@rek.onl (Marek)","guid":"https:\/\/marek.onl\/test-node\/","description":"<section>\n<p><span class=\"org-target\" id=\"def-b\"><\/span> <strong><a href=\"#def-b\">Definition b<\/a>.<\/strong> Let \\(A_1, \\ldots, A_n\\) be sets. Then the\n<strong>Cartesian product<\/strong> \\(A_1 \\times \\cdots \\times A_n\\) is the set of all ordered \\(n\\)-tuples\n\\(\\left( a_1, \\ldots, a_n \\right)\\) such that \\(a_i \\in A_i\\) for \\(1 \\leq i \\leq n\\).<\/p>\n<\/section>\n<p>As per <a href=\"#theorem-b\">Theorem b<\/a>. (check <a href=\"#citeproc_bib_item_2\">Daemen and Rijmen 2002<\/a>; <a href=\"#citeproc_bib_item_1\">Aumasson 2019<\/a>)<\/p>\n<p>This node references the <a href=\"https:\/\/marek.onl\/test-node\/\">Initial Node<\/a>.<\/p>\n<section>\n<p><span class=\"org-target\" id=\"theorem-b\"><\/span> <strong><a href=\"#theorem-b\">Theorem b<\/a><\/strong>\n<em>If \\(a^2=b\\) and \\( b=2 \\), then the solution must be either \\[ a=+\\sqrt{2} \\] or\n\\[ a=-\\sqrt{2}. \\]<\/em><\/p>"}]}}