SD120 - Asymptotic Expansions in SageMath

Dockerfile

FROM sagemath/sagemath:9.7

ARG NB_UID=1000
ARG NB_USER=sage

USER root
RUN apt update && apt install -y python3 python3-pip

USER ${NB_UID}
ENV PATH="${PATH}:${HOME}/.local/bin"
RUN pip3 install notebook
RUN sage -pip install RISE sage_acsv
RUN ln -s $(sage -sh -c 'ls -d $SAGE_VENV/share/jupyter/kernels/sagemath') $HOME/.local/share/jupyter/kernels/sagemath-dev

# partially superfluous -- create separate directory to hold notebooks
WORKDIR ${HOME}/notebooks
COPY --chown=${NB_UID}:${NB_UID} . .
USER root
RUN chown -R ${NB_UID}:${NB_UID} .
USER ${NB_UID}

ENTRYPOINT []

SD120 - Asymptotic Expansions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "f6589899",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Asymptotic Expansions and the `AsymptoticRing`\n",
    "\n",
    "Joint work with Thomas Hagelmayer, Clemens Heuberger, Daniel Krenn, ..."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16cd6dc7",
   "metadata": {
    "slideshow": {
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    }
   },
   "source": [
    "```\n",
    "git shortlog -n -s -- src/sage/rings/asymptotic | wc -l\n",
    "      31\n",
    "```\n",
    ":-)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f59e330",
   "metadata": {
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   },
   "source": [
    "An asymptotic expansion is an expression like\n",
    "$$ 5z^3 + 4z^2 + O(z) $$\n",
    "as $z\\to\\infty$, or\n",
    "$$ 3 x^42 y^2 + 7 x^3 y^3 + O(x^2) + O(y) $$\n",
    "as $x, y\\to\\infty$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "29f973ee",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "*Reminder:* big-Oh (*Bachmann-Landau*) notation expresses a class of functions of (specified) bounded growth:\n",
    "$$f(n) = O(g(n)) :\\iff \\exists n_0 > 0, C > 0 \\forall n > n_0: |f(n)| \\leq C |g(n)|$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fac10cd1",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "*Examples:*\n",
    "$n = O(n)$, $n = O(n^2)$, $n! = O(n^n) = O(e^{n \\log n})$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "acec7400",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Want:** Computations with such expressions in SageMath.\n",
    "\n",
    "Why not just use the already existing `PowerSeriesRing`?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6608c31c",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "PSR.<n> = PowerSeriesRing(QQ)\n",
    "PSR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b2e20507",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "n + O(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5d0a0c30",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "n + O(n^2)  # in PSR, n -> 0 instead of -> oo (workaround possible ...)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "78733bb6",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "n*log(n)  # hmm. :-/"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d5739243",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Logarithmic growth is not exotic, appears naturally in e.g. $n!$ and of course also in applications like Analysis of Algorithms (e.g., Mergesort (among many others): $O(n \\log n)$ comparisons ...)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ba55894",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Want (actually):** Flexible structure able to handle a broad spectrum of expansions like\n",
    "$$ \\sqrt{\\pi} e^{n\\log n} - 42 n^2 \\log(n)^{3/2} + \\frac{1 + \\sqrt{5}}{2} n \\log(n) \\log(\\log(n)) + O(\\log(n)/n) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "720b6153",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## ... well, then we'll just implement it ourselves!\n",
    "\n",
    "- Initial implementation within GSoC 2015 (Student: BH, Mentors: Daniel Krenn, Clemens Heuberger),\n",
    "- gradual improvements over the years,\n",
    "- recently (GSoC 2022): initial implementation of new feature (Student: Thomas Hagelmayer, Mentor: BH, Clemens Heuberger)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f42ec61a",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- [Overview](https://doc.sagemath.org/html/en/reference/asymptotic/index.html)\n",
    "- [Documentation of the main structure, `AsymptoticRing`](https://doc.sagemath.org/html/en/reference/asymptotic/sage/rings/asymptotic/asymptotic_ring.html)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78b33e64",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# What can it do? Demo Time!\n",
    "\n",
    "### Creating an `AsymptoticRing`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9ead441b",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "SCR = SR.subring(no_variables=True)\n",
    "SCR  # our coefficient ring"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e12856cb",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "A = AsymptoticRing(\n",
    "    growth_group='QQ^n * n^QQ * log(n)^QQ',  # \"structure\" of the expansions\n",
    "    coefficient_ring=SCR,\n",
    "    default_prec=7,  # default precision\n",
    ")\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "21e7f2f1",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "n = A.gen()\n",
    "n  # the \"asymptotic variable\", we consider n -> oo ..."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50eca18e",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Creation of expansions *just works*™️:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "64f94bf9",
   "metadata": {},
   "outputs": [],
   "source": [
    "%display typeset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ac7e6975",
   "metadata": {},
   "outputs": [],
   "source": [
    "expansion = (\n",
    "    42 * (1/3)^n * n^(5/2) * log(n)^2 \n",
    "    + O((1/3)^n * n * log(n)^(1/2))\n",
    ")\n",
    "expansion"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3115ae7",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Basic Expansion Arithmetic\n",
    "\n",
    "Addition, subtraction, multiplication and absorption (via $O$-terms) is easy!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "dab8de01",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "(n^2 + n + 1) + (2*n + O(n^(-1)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "660e5fca",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "n^2 + O(n) - O(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "af829f44",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "O(n^2 + n + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cc4aa833",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "(1 + 1/n) * (n^3 - n^2 + n + O(n^0))  # O(n^0) = O(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "37bbac2a",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "More operations that work out of the box: division, taking $\\exp$ / $\\log$: if necessary, the expressions are expanded as series. Number of terms before cutoff depends on chosen default precision."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "11946360",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "(n - 1) / (n^3 + n^2 + O(n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1ee62ecd",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "exp(1 + 1/n^2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "65d6373d",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "log(n+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "684edd0c",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Advanced computations are possible as well, e.g. computing the power of some asymptotic expansion to another asymptotic expansion:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ab2d9272",
   "metadata": {},
   "outputs": [],
   "source": [
    "(1 + 1/n)^n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a2d7b2ff",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Asymptotic Expansion Generators\n",
    "\n",
    "Asymptotic expansions of certain well-known objects (like binomial coefficients $\\binom{kn}{n}$) can be accessed easily via *generators*: `asymptotic_expansion`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "73cca67a",
   "metadata": {},
   "outputs": [],
   "source": [
    "asymptotic_expansions.Binomial_kn_over_n(n, k=2, precision=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c97ed16",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Harmonic Numbers\n",
    "... are defined as $H_n = \\sum_{k=1}^n \\frac{1}{k}$. We have implemented their expansion directly via a generator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bdcf0094",
   "metadata": {},
   "outputs": [],
   "source": [
    "asymptotic_expansions.HarmonicNumber('n', precision=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac59dd5c",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "#### Catalan Numbers\n",
    "... are given by $C_n = \\frac{1}{n+1} \\binom{2n}{n}$, and can be obtained by dividing the asymptotic expansion of the central binomial coefficient (available via `asymptotic_expansions.Binomial_kn_over_n`) by $n+1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0223eb9e",
   "metadata": {},
   "outputs": [],
   "source": [
    "catalan_n = (\n",
    "    asymptotic_expansions.Binomial_kn_over_n(n, k=2, precision=5)\n",
    "    / (n + 1)\n",
    ")\n",
    "catalan_n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "60c04ff9",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Singularity Analysis\n",
    "\n",
    "For explicity given generating functions, our module can extract the coefficient growth by means of singularity analysis directly: this is implemented as a method of the underlying asymptotic ring.\n",
    "\n",
    "Consider the well-known\n",
    "$$ \\mathrm{Catalan}(z) = \\frac{1 - \\sqrt{1 - 4z}}{2z}. $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a3cd3042",
   "metadata": {},
   "outputs": [],
   "source": [
    "def catalan(z):\n",
    "    return (1 - sqrt(1 - 4*z))/(2*z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "08d0bfd7",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "catalan(x).series(x==0, 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a8419369",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "C_n = A.coefficients_of_generating_function(\n",
    "    function=catalan,\n",
    "    singularities=(1/4,),\n",
    "    precision=5\n",
    ")\n",
    "C_n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "45de61d4",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "catalan_n - C_n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "635b8cf9",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Harmonic numbers have the generating function\n",
    "$$ \\sum_{n\\geq 0} H_n z^n = -\\frac{\\log(1 - z)}{1 - z}. $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a92e2545",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "H_n = A.coefficients_of_generating_function(\n",
    "    function=lambda z: -log(1 - z)/(1 - z),\n",
    "    singularities=(1,),\n",
    "    precision=10\n",
    ")\n",
    "H_n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0db75840",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "asymptotic_expansions.HarmonicNumber('n', precision=10) - H_n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f52da6bd",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Inverse Functions\n",
    "There is also a generator that extracts the coefficient growth of a function $y(z)$ satisfying the implicit equation $y(z) = z \\Phi(y(z))$ (\"singular inversion\" -- Flajolet–Sedgewick, Analytic Combinatorics, Chapter VI.7); the *usual conditions* with respect to $\\Phi$ need to hold."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6e06c6b8",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "*Example:* the generating function $y(z)$ of unary-binary-ternary trees satisfies\n",
    "$$ y = z (1 + 3y + 3y^2 + y^3). $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0a28120d",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "assume(SR.an_element() > 0)  # required to make coercions work better ...\n",
    "tree_asy = asymptotic_expansions.InverseFunctionAnalysis(\n",
    "    var=n,\n",
    "    phi=lambda y: 1 + 3*y + 3*y^2 + y^3,\n",
    "    precision=7,\n",
    ")\n",
    "tree_asy"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a69fc697",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## New feature: [$B$-Terms](https://benjamin-hackl.at/downloads/talks/2023-06-29-aofa23/#/2/3)\n",
    "\n",
    "... are like $O$-terms, but without implicitly hidden constants:\n",
    "\n",
    "$$ f(n) = B_{n\\geq n_0}(C \\cdot g(n)) :\\iff \\forall n \\geq n_0: |f(n)| \\leq C |g(n)| $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c8a70b10",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "A.<n> = AsymptoticRing(\"n^QQ\", QQ, default_prec=6)\n",
    "A.B(42*n^2, valid_from=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ad18c36",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$B$-term arithmetic sometimes requires some thinking!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cf0e96c5",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "11*n + A.B(3*n, valid_from=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c1f44e30",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "A.B(42*n^2, valid_from=5) - A.B(5*n^2, valid_from=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "461d0d51",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "n^2 + A.B(19/n, valid_from=42) + O(1/n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9cd775c4",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "A.B(5*n^3, valid_from=7) + A.B(3*n, valid_from=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2ed2924e",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "... **lots** of open tasks in the context of $B$-term arithmetic!\n",
    "\n",
    "- Make arithmetic work for more complex growths than just monomials\n",
    "- Implement some form of integral estimate to make series expansions with $B$-terms work\n",
    "- $B$-term GSoC [meta issue #31922](https://github.com/sagemath/sage/issues/31922)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e7883cc",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "... and even more tasks for asymptotic expansions in general, see [meta issue #17601](https://github.com/sagemath/sage/issues/17601)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7075bb13",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Internals of the `AsymptoticRing`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "524f8bda",
   "metadata": {},
   "outputs": [],
   "source": [
    "%display plain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "46ed0f2e",
   "metadata": {},
   "outputs": [],
   "source": [
    "AR.<m, n> = AsymptoticRing(\"QQ^m * m^QQ * log(m)^QQ * n^QQ * log(n)^QQ * log(log(n))^QQ\", QQ, default_prec=5)\n",
    "expr = 42 * 3^m * n + n^2 * log(log(n))^(1/2) + O(log(m)/n) + O(n)\n",
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64c54561",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Expansion data is stored in a `MutablePoset`, order is determined from specification of growth group!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "62a9c42c",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "type(expr._summands_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f37712cf",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "print(expr._summands_.repr_full())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ca4100c",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Summands are elements from corresponding `TermMonoid` classes (e.g., `ExactTermMonoid`, `OTermMonoid`, ...)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a838f516",
   "metadata": {},
   "outputs": [],
   "source": [
    "terms = list(expr._summands_)\n",
    "[s.parent() for s in terms]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16166d91",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "... and terms primarily hold a growth, with is an element from a given growth group:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e1617b8f",
   "metadata": {},
   "outputs": [],
   "source": [
    "terms[0], terms[0].growth"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "57c451aa",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "terms[0].growth.parent()"
   ]
  }
 ],
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  "rise": {
   "overlay": "<div style='position: absolute; right: 1em; top: 1em; font-size: 1.5em;'><em>Benjamin Hackl</em></div><div style='position: absolute; left: 1em; bottom: 1em;'><img width='100' src='https://benjamin-hackl.at/downloads/logo_uni_graz.jpeg'></div><div><img style='width: 100%; height: 100%; max-width: 100%; max-height: 100%; opacity: 0.5;' src='https://benjamin-hackl.at/downloads/background_uni_graz.png'></div>"
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